Daily Papers

Model families train every size from scratch. Can a pretrained large model be converted into a smaller sibling? We characterize the 1.4B->410M conversion in the Pythia family end-to-end: (i) representations align strongly across sizes (ridge R^2=0.84) while parameters align weakly; (ii) dense weight projection is functionally destructive -- provably not an assembly artifact -- because basis mixing breaks rotary, per-head, GELU, and LayerNorm structure; (iii) after the best-fit linear operator, weight residuals are statistically indistinguishable from noise under shuffle controls; (iv) conversion value therefore lives in initialization. In matched-budget continued pre-training we decompose conversion into two independent levers -- least-squares compensation (function: best zero-shot) and variance-preserving rescale (dynamics: best endpoints). Compensation is a token-efficient, low-budget win rather than a universal one: at 30M tokens it beats the strongest subcloning variant on both a width-reduced pair (84.0 +/- 1.8 vs. 89.7 +/- 3.7, 3/3 seeds) and a held-out depth-reduced pair (109.3 vs. 117.9, 3/3 seeds), reaching a given quality with fewer tokens; at a 33x larger budget the two converge to parity (40.0 vs. 40.0), both far ahead of from-scratch, which transfer initialization always beats -- by up to 18x at low budget, the margin narrowing at convergence and at the largest scale. We further map the method's boundary: at ~5x the donor scale (6.9B->1.4B) stacking both levers over-corrects, which we trace to ill-conditioning of the compensation solve at large width, pointing to dimension-aware regularization as the fix. Code, checkpoints, and the frozen evaluation corpus are released.

The Inversion-Denoising Paradigm, which is based on diffusion models, excels in diverse image editing and restoration tasks. We revisit its mechanism and reveal a critical, overlooked factor in reconstruction degradation: the approximate noise error. This error stems from approximating the noise at step t with the prediction at step t-1, resulting in severe error accumulation throughout the inversion process. We introduce Projection-Orthogonal Least Squares for Robust and Adaptive Inversion (POLARIS), which reformulates inversion from an error-compensation problem into an error-origin problem. Rather than optimizing embeddings or latent codes to offset accumulated drift, POLARIS treats the guidance scale ω as a step-wise variable and derives a mathematically grounded formula to minimize inversion error at each step. Remarkably, POLARIS improves inversion latent quality with just one line of code. With negligible performance overhead, it substantially mitigates noise approximation errors and consistently improves the accuracy of downstream tasks.

During the radiotherapy treatment of patients with lung cancer, the radiation delivered to healthy tissue around the tumor needs to be minimized, which is difficult because of respiratory motion and the latency of linear accelerator systems. In the proposed study, we first use the Lucas-Kanade pyramidal optical flow algorithm to perform deformable image registration of chest computed tomography scan images of four patients with lung cancer. We then track three internal points close to the lung tumor based on the previously computed deformation field and predict their position with a recurrent neural network (RNN) trained using real-time recurrent learning (RTRL) and gradient clipping. The breathing data is quite regular, sampled at approximately 2.5Hz, and includes artificial drift in the spine direction. The amplitude of the motion of the tracked points ranged from 12.0mm to 22.7mm. Finally, we propose a simple method for recovering and predicting 3D tumor images from the tracked points and the initial tumor image based on a linear correspondence model and Nadaraya-Watson non-linear regression. The root-mean-square error, maximum error, and jitter corresponding to the RNN prediction on the test set were smaller than the same performance measures obtained with linear prediction and least mean squares (LMS). In particular, the maximum prediction error associated with the RNN, equal to 1.51mm, is respectively 16.1% and 5.0% lower than the maximum error associated with linear prediction and LMS. The average prediction time per time step with RTRL is equal to 119ms, which is less than the 400ms marker position sampling time. The tumor position in the predicted images appears visually correct, which is confirmed by the high mean cross-correlation between the original and predicted images, equal to 0.955.

Recent advances have shown that sequential fine-tuning (SeqFT) of pre-trained vision transformers (ViTs), followed by classifier refinement using approximate distributions of class features, can be an effective strategy for class-incremental learning (CIL). However, this approach is susceptible to distribution drift, caused by the sequential optimization of shared backbone parameters. This results in a mismatch between the distributions of the previously learned classes and that of the updater model, ultimately degrading the effectiveness of classifier performance over time. To address this issue, we introduce a latent space transition operator and propose Sequential Learning with Drift Compensation (SLDC). SLDC aims to align feature distributions across tasks to mitigate the impact of drift. First, we present a linear variant of SLDC, which learns a linear operator by solving a regularized least-squares problem that maps features before and after fine-tuning. Next, we extend this with a weakly nonlinear SLDC variant, which assumes that the ideal transition operator lies between purely linear and fully nonlinear transformations. This is implemented using learnable, weakly nonlinear mappings that balance flexibility and generalization. To further reduce representation drift, we apply knowledge distillation (KD) in both algorithmic variants. Extensive experiments on standard CIL benchmarks demonstrate that SLDC significantly improves the performance of SeqFT. Notably, by combining KD to address representation drift with SLDC to compensate distribution drift, SeqFT achieves performance comparable to joint training across all evaluated datasets. Code: https://github.com/raoxuan98-hash/sldc.git.

With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. The solution of least squares (LS) problems min_x|b-Ax|_2, where A in R^{mtimes n}, arise in numerous application areas. Overdetermined standard least squares problems can be solved by using mixed precision within the iterative refinement method of Björck, which transforms the least squares problem into an (m+n)times(m+n) ''augmented'' system. It has recently been shown that mixed precision GMRES-based iterative refinement can also be used, in an approach termed GMRES-LSIR. In practice, we often encounter types of least squares problems beyond standard least squares, including weighted least squares (WLS), min_x|D^{1/2}(b-Ax)|_2, where D^{1/2} is a diagonal matrix of weights. In this paper, we discuss a mixed precision FGMRES-WLSIR algorithm for solving WLS problems using two different preconditioners.

Is it possible to perform linear regression on datasets whose labels are shuffled with respect to the inputs? We explore this question by proposing several estimators that recover the weights of a noisy linear model from labels that are shuffled by an unknown permutation. We show that the analog of the classical least-squares estimator produces inconsistent estimates in this setting, and introduce an estimator based on the self-moments of the input features and labels. We study the regimes in which each estimator excels, and generalize the estimators to the setting where partial ordering information is available in the form of experiments replicated independently. The result is a framework that enables robust inference, as we demonstrate by experiments on both synthetic and standard datasets, where we are able to recover approximate weights using only shuffled labels. Our work demonstrates that linear regression in the absence of complete ordering information is possible and can be of practical interest, particularly in experiments that characterize populations of particles, such as flow cytometry.

Lower numerical precision reduces storage and memory traffic but raises the perturbation floor. We study rank compensation: reinvesting saved memory in a larger approximation rank. For matrices, the singular-value error identity yields a directly testable sufficient condition requiring the additional singular component to offset the perturbation from storing the rank-augmented approximation in lower precision. On ten SuiteSparse matrices, all 100 truncation-dominated configurations (50 FP32 and 50 FP16) are certified non-increases and strict accuracy wins, with mean error ratio 0.963 and storage ratios 58.8% and 29.4% relative to the FP64 baseline. FP16 failures occur only in tail-rank stress tests near the perturbation floor. At the largest resident matrix-application batch, compensated FP32 and FP16 achieve geometric-mean A100 speedups of 1.28times and 2.12times; neither accelerates the smallest batch. For Tensor-Train (TT) approximation, we give a conditional a posteriori extension based on the measured truncation gain and rounded-core perturbation. Across three-way and six-way synthetic tests, FP32 and FP16 achieve combined accuracy-memory wins in 10 of 20 and 14 of 20 trials. On public hyperspectral tensors and FROSTT top-active subtensors, the corresponding counts are 44 of 60 and 54 of 60; four FP16 Salinas-A tail-stress cases fail. No certified TT case exceeds the FP64 error beyond numerical tolerance. Reconstruction of six public tensors yields geometric-mean compensated speedups of 1.38times (FP32) and 1.94times (FP16). Timings cover resident downstream kernels, not factorization, transfers, or end-to-end acceleration.

Mirror descent value iteration (MDVI), an abstraction of Kullback-Leibler (KL) and entropy-regularized reinforcement learning (RL), has served as the basis for recent high-performing practical RL algorithms. However, despite the use of function approximation in practice, the theoretical understanding of MDVI has been limited to tabular Markov decision processes (MDPs). We study MDVI with linear function approximation through its sample complexity required to identify an varepsilon-optimal policy with probability 1-delta under the settings of an infinite-horizon linear MDP, generative model, and G-optimal design. We demonstrate that least-squares regression weighted by the variance of an estimated optimal value function of the next state is crucial to achieving minimax optimality. Based on this observation, we present Variance-Weighted Least-Squares MDVI (VWLS-MDVI), the first theoretical algorithm that achieves nearly minimax optimal sample complexity for infinite-horizon linear MDPs. Furthermore, we propose a practical VWLS algorithm for value-based deep RL, Deep Variance Weighting (DVW). Our experiments demonstrate that DVW improves the performance of popular value-based deep RL algorithms on a set of MinAtar benchmarks.

We consider the problem of approximating a function from L^2 by an element of a given m-dimensional space V_m, associated with some feature map varphi, using evaluations of the function at random points x_1,dots,x_n. After recalling some results on optimal weighted least-squares using independent and identically distributed points, we consider weighted least-squares using projection determinantal point processes (DPP) or volume sampling. These distributions introduce dependence between the points that promotes diversity in the selected features varphi(x_i). We first provide a generalized version of volume-rescaled sampling yielding quasi-optimality results in expectation with a number of samples n = O(mlog(m)), that means that the expected L^2 error is bounded by a constant times the best approximation error in L^2. Also, further assuming that the function is in some normed vector space H continuously embedded in L^2, we further prove that the approximation is almost surely bounded by the best approximation error measured in the H-norm. This includes the cases of functions from L^infty or reproducing kernel Hilbert spaces. Finally, we present an alternative strategy consisting in using independent repetitions of projection DPP (or volume sampling), yielding similar error bounds as with i.i.d. or volume sampling, but in practice with a much lower number of samples. Numerical experiments illustrate the performance of the different strategies.

In this work, we re-formulate the model compression problem into the customized compensation problem: Given a compressed model, we aim to introduce residual low-rank paths to compensate for compression errors under customized requirements from users (e.g., tasks, compression ratios), resulting in greater flexibility in adjusting overall capacity without being constrained by specific compression formats. However, naively applying SVD to derive residual paths causes suboptimal utilization of the low-rank representation capacity. Instead, we propose Training-free Eigenspace Low-Rank Approximation (EoRA), a method that directly minimizes compression-induced errors without requiring gradient-based training, achieving fast optimization in minutes using a small amount of calibration data. EoRA projects compression errors into the eigenspace of input activations, leveraging eigenvalues to effectively prioritize the reconstruction of high-importance error components. Moreover, EoRA can be seamlessly integrated with fine-tuning and quantization to further improve effectiveness and efficiency. EoRA consistently outperforms previous methods in compensating errors for compressed LLaMA2/3 models on various tasks, such as language generation, commonsense reasoning, and math reasoning tasks (e.g., 31.31%/12.88% and 9.69% improvements on ARC-Easy/ARC-Challenge and MathQA when compensating LLaMA3-8B that is quantized to 4-bit and pruned to 2:4 sparsity). EoRA offers a scalable, training-free solution to compensate for compression errors, making it a powerful tool to deploy LLMs in various capacity and efficiency requirements.

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model. Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime. Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?". Our formula allow us to give sharp answers to this question, both in the positive and negative directions. More precisely, we show that the sufficient conditions for Gaussian universality (or lack of thereof) crucially depend on the alignment between the target weights and the means and covariances of the mixture clusters, which we precisely quantify. In the particular case of least-squares interpolation, we prove a strong universality property of the training error, and show it follows a simple, closed-form expression. Finally, we apply our results to real datasets, clarifying some recent discussion in the literature about Gaussian universality of the errors in this context.

Whenever we use devices to take measurements, calibration is indispensable. While the purpose of calibration is to reduce bias and uncertainty in the measurements, it can be quite difficult, expensive, and sometimes even impossible to implement. We study a challenging problem called self-calibration, i.e., the task of designing an algorithm for devices so that the algorithm is able to perform calibration automatically. More precisely, we consider the setup y = A(d) x + epsilon where only partial information about the sensing matrix A(d) is known and where A(d) linearly depends on d. The goal is to estimate the calibration parameter d (resolve the uncertainty in the sensing process) and the signal/object of interests x simultaneously. For three different models of practical relevance, we show how such a bilinear inverse problem, including blind deconvolution as an important example, can be solved via a simple linear least squares approach. As a consequence, the proposed algorithms are numerically extremely efficient, thus potentially allowing for real-time deployment. We also present a variation of the least squares approach, which leads to a~spectral method, where the solution to the bilinear inverse problem can be found by computing the singular vector associated with the smallest singular value of a certain matrix derived from the bilinear system. Explicit theoretical guarantees and stability theory are derived for both techniques; and the number of sampling complexity is nearly optimal (up to a poly-log factor). Applications in imaging sciences and signal processing are discussed and numerical simulations are presented to demonstrate the effectiveness and efficiency of our approach.

Diffusion probabilistic models (DPMs) have shown remarkable performance in visual synthesis but are computationally expensive due to the need for multiple evaluations during the sampling. Recent predictor-corrector diffusion samplers have significantly reduced the required number of function evaluations (NFE), but inherently suffer from a misalignment issue caused by the extra corrector step, especially with a large classifier-free guidance scale (CFG). In this paper, we introduce a new fast DPM sampler called DC-Solver, which leverages dynamic compensation (DC) to mitigate the misalignment of the predictor-corrector samplers. The dynamic compensation is controlled by compensation ratios that are adaptive to the sampling steps and can be optimized on only 10 datapoints by pushing the sampling trajectory toward a ground truth trajectory. We further propose a cascade polynomial regression (CPR) which can instantly predict the compensation ratios on unseen sampling configurations. Additionally, we find that the proposed dynamic compensation can also serve as a plug-and-play module to boost the performance of predictor-only samplers. Extensive experiments on both unconditional sampling and conditional sampling demonstrate that our DC-Solver can consistently improve the sampling quality over previous methods on different DPMs with a wide range of resolutions up to 1024times1024. Notably, we achieve 10.38 FID (NFE=5) on unconditional FFHQ and 0.394 MSE (NFE=5, CFG=7.5) on Stable-Diffusion-2.1. Code is available at https://github.com/wl-zhao/DC-Solver

Passive long-wave infrared (LWIR) hyperspectral imaging under a standoff geometry depends on atmospheric absorption and emission, as well as reflected radiance, thus making atmospheric compensation essential to get knowledge of a target of interest. Despite its importance, this compensation has been largely overlooked due to its practical and modeling difficulty. In this paper, we present a lightweight set-based deep learning framework that takes multiple radiance measurements, collected at different standoff ranges, as input and jointly estimates transmittance, atmospheric path radiance, and a shared downwelling spectrum. We analyze the learned representation with a sparse autoencoder and observe that several latent features do activate on geographically coherent subsets of the test data despite the absence of location supervision. Experiments on a MODTRAN generated standoff LWIR dataset demonstrate low spectral distortion across all estimated products. The dataset and code is publicly available at: https://factral.co/SAE-LWIR/

Methods based on weight compensation, which iteratively apply quantization and weight compensation to minimize the output error, have recently demonstrated remarkable success in quantizing Large Language Models (LLMs). The representative work, GPTQ, introduces several key techniques that make such iterative methods practical for LLMs with billions of parameters. GPTAQ extends this approach by introducing an asymmetric calibration process that aligns the output of each quantized layer with its full-precision counterpart, incorporating a residual error into the weight compensation framework. In this work, we revisit the formulation of the residual error. We identify a sub-optimal calibration objective in existing methods: during the intra-layer calibration process, they align the quantized output with the output from compensated weights, rather than the true output from the original full-precision model. Therefore, we redefine the objective to precisely align the quantized model's output with the original output of the full-precision model at each step. We then reveal that the residual error originates not only from the output difference of the preceding layer but also from the discrepancy between the compensated and original weights within each layer, which we name the 'compensation-aware error'. By inheriting the neuron decomposition technique from GPTAQ, we can efficiently incorporate this compensation-aware error into the weight update process. Extensive experiments on various LLMs and quantization settings demonstrate that our proposed enhancements integrate seamlessly with both GPTQ and GPTAQ, significantly improving their quantization performance. Our code is publicly available at https://github.com/list0830/ResComp.

Federated learning (FL) leverages client-server communications to train global models on decentralized data. However, communication noise or errors can impair model accuracy. To address this problem, we propose a novel FL algorithm that enhances robustness against communication noise while also reducing communication load. We derive the proposed algorithm through solving the weighted least-squares (WLS) regression problem as an illustrative example. We first frame WLS regression as a distributed convex optimization problem over a federated network employing random scheduling for improved communication efficiency. We then apply the alternating direction method of multipliers (ADMM) to iteratively solve this problem. To counteract the detrimental effects of cumulative communication noise, we introduce a key modification by eliminating the dual variable and implementing a new local model update at each participating client. This subtle yet effective change results in using a single noisy global model update at each client instead of two, improving robustness against additive communication noise. Furthermore, we incorporate another modification enabling clients to continue local updates even when not selected by the server, leading to substantial performance improvements. Our theoretical analysis confirms the convergence of our algorithm in both mean and the mean-square senses, even when the server communicates with a random subset of clients over noisy links at each iteration. Numerical results validate the effectiveness of our proposed algorithm and corroborate our theoretical findings.

We consider the problem of linear estimation, and establish an extension of the Gauss-Markov theorem, in which the bias operator is allowed to be non-zero but bounded with respect to a matrix norm of Schatten type. We derive simple and explicit formulas for the optimal estimator in the cases of Nuclear and Spectral norms (with the Frobenius case recovering ridge regression). Additionally, we analytically derive the generalization error in multiple random matrix ensembles, and compare with Ridge regression. Finally, we conduct an extensive simulation study, in which we show that the cross-validated Nuclear and Spectral regressors can outperform Ridge in several circumstances.

This paper studies the challenging problem of estimating causal effects from observational data, in the presence of unobserved confounders. The two-stage least square (TSLS) method and its variants with a standard instrumental variable (IV) are commonly used to eliminate confounding bias, including the bias caused by unobserved confounders, but they rely on the linearity assumption. Besides, the strict condition of unconfounded instruments posed on a standard IV is too strong to be practical. To address these challenging and practical problems of the standard IV method (linearity assumption and the strict condition), in this paper, we use a conditional IV (CIV) to relax the unconfounded instrument condition of standard IV and propose a non-linear CIV regression with Confounding Balancing Representation Learning, CBRL.CIV, for jointly eliminating the confounding bias from unobserved confounders and balancing the observed confounders, without the linearity assumption. We theoretically demonstrate the soundness of CBRL.CIV. Extensive experiments on synthetic and two real-world datasets show the competitive performance of CBRL.CIV against state-of-the-art IV-based estimators and superiority in dealing with the non-linear situation.

We study the class of location-scale or heteroscedastic noise models (LSNMs), in which the effect Y can be written as a function of the cause X and a noise source N independent of X, which may be scaled by a positive function g over the cause, i.e., Y = f(X) + g(X)N. Despite the generality of the model class, we show the causal direction is identifiable up to some pathological cases. To empirically validate these theoretical findings, we propose two estimators for LSNMs: an estimator based on (non-linear) feature maps, and one based on neural networks. Both model the conditional distribution of Y given X as a Gaussian parameterized by its natural parameters. When the feature maps are correctly specified, we prove that our estimator is jointly concave, and a consistent estimator for the cause-effect identification task. Although the the neural network does not inherit those guarantees, it can fit functions of arbitrary complexity, and reaches state-of-the-art performance across benchmarks.

This paper presents a novel preconditioning strategy for the classic 8-point algorithm (8-PA) for estimating an essential matrix from 360-FoV images (i.e., equirectangular images) in spherical projection. To alleviate the effect of uneven key-feature distributions and outlier correspondences, which can potentially decrease the accuracy of an essential matrix, our method optimizes a non-rigid transformation to deform a spherical camera into a new spatial domain, defining a new constraint and a more robust and accurate solution for an essential matrix. Through several experiments using random synthetic points, 360-FoV, and fish-eye images, we demonstrate that our normalization can increase the camera pose accuracy by about 20% without significantly overhead the computation time. In addition, we present further benefits of our method through both a constant weighted least-square optimization that improves further the well known Gold Standard Method (GSM) (i.e., the non-linear optimization by using epipolar errors); and a relaxation of the number of RANSAC iterations, both showing that our normalization outcomes a more reliable, robust, and accurate solution.

Overparametrization often helps improve the generalization performance. This paper proposes a dual view of overparametrization suggesting that downsampling may also help generalize. Motivated by this dual view, we characterize two out-of-sample prediction risks of the sketched ridgeless least square estimator in the proportional regime masymp n asymp p, where m is the sketching size, n the sample size, and p the feature dimensionality. Our results reveal the statistical role of downsampling. Specifically, downsampling does not always hurt the generalization performance, and may actually help improve it in some cases. We identify the optimal sketching sizes that minimize the out-of-sample prediction risks, and find that the optimally sketched estimator has stabler risk curves that eliminates the peaks of those for the full-sample estimator. We then propose a practical procedure to empirically identify the optimal sketching size. Finally, we extend our results to cover central limit theorems and misspecified models. Numerical studies strongly support our theory.

Large language models (LLMs) achieve remarkable performance across a wide range of tasks, but their deployment is constrained by substantial memory and compute requirements. Low-rank compression via singular value decomposition (SVD) is an effective remedy, but existing methods focus on how to factorize and which components to keep. We introduce SVD-Surgeon, a training-free method that brings the Optimal Brain Surgeon (OBS) framework to the singular-value basis. Treating each singular value as a parameter, it computes a closed-form update of the retained singular values that compensates, to second order in the model loss, for those removed by truncation. The same analysis yields a saliency for choosing which values to prune. As it operates directly on the singular-value factorization, SVD-Surgeon can be layered on top of existing SVD compressors. Applied to SVD-LLM, a leading SVD-based method, it improves the perplexity-compression trade-off on the OPT family and LLaMA 2-7B without any retraining.

Quantization is a promising method that reduces memory usage and computational intensity of Deep Neural Networks (DNNs), but it often leads to significant output error that hinder model deployment. In this paper, we propose Bias Compensation (BC) to minimize the output error, thus realizing ultra-low-precision quantization without model fine-tuning. Instead of optimizing the non-convex quantization process as in most previous methods, the proposed BC bypasses the step to directly minimize the quantizing output error by identifying a bias vector for compensation. We have established that the minimization of output error through BC is a convex problem and provides an efficient strategy to procure optimal solutions associated with minimal output error,without the need for training or fine-tuning. We conduct extensive experiments on Vision Transformer models and Large Language Models, and the results show that our method notably reduces quantization output error, thereby permitting ultra-low-precision post-training quantization and enhancing the task performance of models. Especially, BC improves the accuracy of ViT-B with 4-bit PTQ4ViT by 36.89% on the ImageNet-1k task, and decreases the perplexity of OPT-350M with 3-bit GPTQ by 5.97 on WikiText2.The code is in https://github.com/GongCheng1919/bias-compensation.

Factorizing a large matrix into small matrices is a popular strategy for model compression. Singular value decomposition (SVD) plays a vital role in this compression strategy, approximating a learned matrix with fewer parameters. However, SVD minimizes the squared error toward reconstructing the original matrix without gauging the importance of the parameters, potentially giving a larger reconstruction error for those who affect the task accuracy more. In other words, the optimization objective of SVD is not aligned with the trained model's task accuracy. We analyze this previously unexplored problem, make observations, and address it by introducing Fisher information to weigh the importance of parameters affecting the model prediction. This idea leads to our method: Fisher-Weighted SVD (FWSVD). Although the factorized matrices from our approach do not result in smaller reconstruction errors, we find that our resulting task accuracy is much closer to the original model's performance. We perform analysis with the transformer-based language models, showing our weighted SVD largely alleviates the mismatched optimization objectives and can maintain model performance with a higher compression rate. Our method can directly compress a task-specific model while achieving better performance than other compact model strategies requiring expensive model pre-training. Moreover, the evaluation of compressing an already compact model shows our method can further reduce 9% to 30% parameters with an insignificant impact on task accuracy.

The rapid growth in the parameter scale of large language models (LLMs) has created a high demand for efficient compression techniques. As a hardware-agnostic and highly compatible technique, low-rank compression has been widely adopted. However, existing methods typically compress each layer independently by minimizing per-layer reconstruction error, overlooking a critical limitation: the reconstruction error propagates and accumulates through the network, which leads to amplified global deviations from the full-precision baseline. To address this, we propose Self-Adaptive Error Suppression SVD (SAES-SVD), a LLMs compression framework that jointly optimizes intra-layer reconstruction and inter-layer error compensation. SAES-SVD is composed of two novel components: (1) Cumulative Error-Aware Layer Compression (CEALC), which formulates the compression objective as a combination of local reconstruction and weighted cumulative error compensation. Based on it, we derive a closed-form low-rank solution relied on second-order activation statistics, which explicitly aligns each layer's output with its full-precision counterpart to compensate for accumulated errors. (2) Adaptive Collaborative Error Suppression (ACES), which automatically adjusts the weighting coefficient to enhance the low-rank structure of the compression objective in CEALC. Specifically, the coefficient is optimized to maximize the ratio between the Frobenius norm of the compressed layer's output and that of the compression objective under a fixed rank, thus ensuring that the rank budget is utilized effectively. Extensive experiments across multiple LLM architectures and tasks show that, without fine-tuning or mixed-rank strategies, SAES-SVD consistently improves post-compression performance.

Motivated by a wide variety of applications, ranging from stochastic optimization to dimension reduction through variable selection, the problem of estimating gradients accurately is of crucial importance in statistics and learning theory. We consider here the classic regression setup, where a real valued square integrable r.v. Y is to be predicted upon observing a (possibly high dimensional) random vector X by means of a predictive function f(X) as accurately as possible in the mean-squared sense and study a nearest-neighbour-based pointwise estimate of the gradient of the optimal predictive function, the regression function m(x)=E[Ymid X=x]. Under classic smoothness conditions combined with the assumption that the tails of Y-m(X) are sub-Gaussian, we prove nonasymptotic bounds improving upon those obtained for alternative estimation methods. Beyond the novel theoretical results established, several illustrative numerical experiments have been carried out. The latter provide strong empirical evidence that the estimation method proposed works very well for various statistical problems involving gradient estimation, namely dimensionality reduction, stochastic gradient descent optimization and quantifying disentanglement.

Due to the real-time rendering performance, 3D Gaussian Splatting (3DGS) has emerged as the leading method for radiance field reconstruction. However, its reliance on spherical harmonics for color encoding inherently limits its ability to separate diffuse and specular components, making it challenging to accurately represent complex reflections. To address this, we propose a novel enhanced Gaussian kernel that explicitly models specular effects through view-dependent opacity. Meanwhile, we introduce an error-driven compensation strategy to improve rendering quality in existing 3DGS scenes. Our method begins with 2D Gaussian initialization and then adaptively inserts and optimizes enhanced Gaussian kernels, ultimately producing an augmented radiance field. Experiments demonstrate that our method not only surpasses state-of-the-art NeRF methods in rendering performance but also achieves greater parameter efficiency. Project page at: https://xiaoxinyyx.github.io/augs.

Neural scaling laws approximate a language model's loss as a power-law function of parameter count N and token count D. Following Chinchilla-style compute-optimal training, many studies fit scaling laws from runs performed under a fixed tokens-per-parameter (TPP) ratio k and set D = kN. We show that this collinear design, combined with the empirically common near-equality of the exponents governing N and D, induces an inherent ill-conditioning in the Gauss-Newton least-squares problem: the condition number of the design grows as the inverse square of the gap between the N and D-exponents. The scale coefficients become practically unidentifiable, with confidence intervals inflating by an order of magnitude or more, yielding a ``sloppy'' model whose extrapolations degrade sharply off the training ray. We prove this for four scaling-law formalisms and derive a closed-form TPP-diversity threshold that is necessary and sufficient for well-conditioned estimation. Empirically, non-collinear designs outperform collinear ones on held-out splits with a 97.3\% win rate across four laws, five corpora, multiple floating point precision modes. We further show the degeneracy is rooted in Jacobian geometry and is not an artifact of the loss function: any smooth estimation objective whose curvature involves the Jacobian inherits the same ill-conditioning.

We study the connection between multicalibration and boosting for squared error regression. First we prove a useful characterization of multicalibration in terms of a ``swap regret'' like condition on squared error. Using this characterization, we give an exceedingly simple algorithm that can be analyzed both as a boosting algorithm for regression and as a multicalibration algorithm for a class H that makes use only of a standard squared error regression oracle for H. We give a weak learning assumption on H that ensures convergence to Bayes optimality without the need to make any realizability assumptions -- giving us an agnostic boosting algorithm for regression. We then show that our weak learning assumption on H is both necessary and sufficient for multicalibration with respect to H to imply Bayes optimality. We also show that if H satisfies our weak learning condition relative to another class C then multicalibration with respect to H implies multicalibration with respect to C. Finally we investigate the empirical performance of our algorithm experimentally using an open source implementation that we make available. Our code repository can be found at https://github.com/Declancharrison/Level-Set-Boosting.

We study the problem of designing minimax procedures in linear regression under the quantile risk. We start by considering the realizable setting with independent Gaussian noise, where for any given noise level and distribution of inputs, we obtain the exact minimax quantile risk for a rich family of error functions and establish the minimaxity of OLS. This improves on the known lower bounds for the special case of square error, and provides us with a lower bound on the minimax quantile risk over larger sets of distributions. Under the square error and a fourth moment assumption on the distribution of inputs, we show that this lower bound is tight over a larger class of problems. Specifically, we prove a matching upper bound on the worst-case quantile risk of a variant of the recently proposed min-max regression procedure, thereby establishing its minimaxity, up to absolute constants. We illustrate the usefulness of our approach by extending this result to all p-th power error functions for p in (2, infty). Along the way, we develop a generic analogue to the classical Bayesian method for lower bounding the minimax risk when working with the quantile risk, as well as a tight characterization of the quantiles of the smallest eigenvalue of the sample covariance matrix.

We study the problem of online generalized linear regression in the stochastic setting, where the label is generated from a generalized linear model with possibly unbounded additive noise. We provide a sharp analysis of the classical follow-the-regularized-leader (FTRL) algorithm to cope with the label noise. More specifically, for sigma-sub-Gaussian label noise, our analysis provides a regret upper bound of O(sigma^2 d log T) + o(log T), where d is the dimension of the input vector, T is the total number of rounds. We also prove a Omega(sigma^2dlog(T/d)) lower bound for stochastic online linear regression, which indicates that our upper bound is nearly optimal. In addition, we extend our analysis to a more refined Bernstein noise condition. As an application, we study generalized linear bandits with heteroscedastic noise and propose an algorithm based on FTRL to achieve the first variance-aware regret bound.

Large-scale kernel ridge regression (KRR) is limited by the need to store a large kernel matrix K_t. To avoid storing the entire matrix K_t, Nystrom methods subsample a subset of columns of the kernel matrix, and efficiently find an approximate KRR solution on the reconstructed matrix. The chosen subsampling distribution in turn affects the statistical and computational tradeoffs. For KRR problems, recent works show that a sampling distribution proportional to the ridge leverage scores (RLSs) provides strong reconstruction guarantees for the approximation. While exact RLSs are as difficult to compute as a KRR solution, we may be able to approximate them well enough. In this paper, we study KRR problems in a sequential setting and introduce the INK-ESTIMATE algorithm, that incrementally computes the RLSs estimates. INK-ESTIMATE maintains a small sketch of K_t, that at each step is used to compute an intermediate estimate of the RLSs. First, our sketch update does not require access to previously seen columns, and therefore a single pass over the kernel matrix is sufficient. Second, the algorithm requires a fixed, small space budget to run dependent only on the effective dimension of the kernel matrix. Finally, our sketch provides strong approximation guarantees on the distance between the true kernel matrix and its approximation, and on the statistical risk of the approximate KRR solution at any time, because all our guarantees hold at any intermediate step.

Multi-view 3D reconstruction methods remain highly sensitive to photometric inconsistencies arising from camera optical characteristics and variations in image signal processing (ISP). Existing mitigation strategies such as per-frame latent variables or affine color corrections lack physical grounding and generalize poorly to novel views. We propose the Physically-Plausible ISP (PPISP) correction module, which disentangles camera-intrinsic and capture-dependent effects through physically based and interpretable transformations. A dedicated PPISP controller, trained on the input views, predicts ISP parameters for novel viewpoints, analogous to auto exposure and auto white balance in real cameras. This design enables realistic and fair evaluation on novel views without access to ground-truth images. PPISP achieves state-of-the-art performance on standard benchmarks, while providing intuitive control and supporting the integration of metadata when available. The source code is available at: https://github.com/nv-tlabs/ppisp

Neural Radiance Fields (NeRF) achieves unprecedented performance in synthesizing novel view synthesis, utilizing multi-view consistency. When capturing multiple inputs, image signal processing (ISP) in modern cameras will independently enhance them, including exposure adjustment, color correction, local tone mapping, etc. While these processings greatly improve image quality, they often break the multi-view consistency assumption, leading to "floaters" in the reconstructed radiance fields. To address this concern without compromising visual aesthetics, we aim to first disentangle the enhancement by ISP at the NeRF training stage and re-apply user-desired enhancements to the reconstructed radiance fields at the finishing stage. Furthermore, to make the re-applied enhancements consistent between novel views, we need to perform imaging signal processing in 3D space (i.e. "3D ISP"). For this goal, we adopt the bilateral grid, a locally-affine model, as a generalized representation of ISP processing. Specifically, we optimize per-view 3D bilateral grids with radiance fields to approximate the effects of camera pipelines for each input view. To achieve user-adjustable 3D finishing, we propose to learn a low-rank 4D bilateral grid from a given single view edit, lifting photo enhancements to the whole 3D scene. We demonstrate our approach can boost the visual quality of novel view synthesis by effectively removing floaters and performing enhancements from user retouching. The source code and our data are available at: https://bilarfpro.github.io.

Low-rank approximation is a fundamental technique in modern data analysis, widely utilized across various fields such as signal processing, machine learning, and natural language processing. Despite its ubiquity, the mechanics of low-rank approximation and its application in adaptation can sometimes be obscure, leaving practitioners and researchers with questions about its true capabilities and limitations. This paper seeks to clarify low-rank approximation and adaptation by offering a comprehensive guide that reveals their inner workings and explains their utility in a clear and accessible way. Our focus here is to develop a solid intuition for how low-rank approximation and adaptation operate, and why they are so effective. We begin with basic concepts and gradually build up to the mathematical underpinnings, ensuring that readers of all backgrounds can gain a deeper understanding of low-rank approximation and adaptation. We strive to strike a balance between informal explanations and rigorous mathematics, ensuring that both newcomers and experienced experts can benefit from this survey. Additionally, we introduce new low-rank decomposition and adaptation algorithms that have not yet been explored in the field, hoping that future researchers will investigate their potential applicability.

Post-training quantization is widely used for compressing large neural networks, but aggressive low-bit quantization can significantly degrade model quality. A common remedy is to augment the quantized weights with a low-rank correction, leading to approximations of the form Wapprox Q+LR. In this paper, we study this low-precision plus low-rank representation through the layer-wise reconstruction objective |XW-X(Q+LR)|_F^2, where X is a calibration matrix. We establish, to our knowledge, the first information-theoretic lower bounds for this problem under finite-alphabet and bounded low-rank compensation constraints. We then propose GPTQ-intrinsic LoRA, a training-free algorithm that incorporates the low-rank correction directly into a GPTQ-style quantization pass by appropriately augmenting the calibration Hessian. For the choice L=V_r, where V_r contains the top right singular vectors of X, we prove layer-wise reconstruction error bounds in which the usual GPTQ dependence on |X|_F^2 is replaced by the rank-r residual |X-X_r|_F^2, up to regularization terms. Under natural structural assumptions, these bounds match the information-theoretic lower bounds in their dominant scaling, up to constants and mild factors. We also introduce Bid-Up, a fixed-grid quantization refinement step that can be alternated with optimal low-rank compensation with guaranteed non-increasing layer-wise reconstruction error. Experiments on Qwen3 language models and DeiT vision transformers show that GPTQ-intrinsic LoRA improves over GPTQ and GPTQ followed by low-rank compensation, with additional gains from refinement loops.

Low-Rank Adaptation (LoRA) improves downstream performance by restricting task updates to a low-rank parameter subspace, yet how this limited capacity is allocated within a trained adapter remains unclear. Through a geometric and empirical study across multiple tasks and backbones, we find that trained LoRA updates often exhibit an inefficient spectrum: task effects concentrate in a small subset of singular directions, while many remaining components are neutral or detrimental, motivating post-hoc refinement within the learned subspace. We propose Spectral Surgery, a training-free refinement that decomposes a LoRA update with SVD, estimates per-component sensitivity using gradients on a small calibration set, and reweights singular values under a magnitude constraint while keeping the learned directions fixed. Across Llama-3.1-8B and Qwen3-8B on four benchmarks, Spectral Surgery yields consistent gains (up to +4.4 points on CommonsenseQA and +2.4 pass@1 on HumanEval) by adjusting only approx 1{,}000 scalar coefficients. These results demonstrate that SVD-structured, low-cost parameter editing can serve as a practical route to improving trained LoRA adapters in a purely post-hoc manner.

We identify the Spectral Energy Gain in extreme model compression, where low-rank binary approximations outperform tiny-rank floating-point baselines for heavy-tailed spectra. However, prior attempts fail to realize this potential, trailing state-of-the-art 1-bit methods. We attribute this degradation to Latent Geometry Misalignment: standard singular vectors exhibit high coherence (spiky distribution), the worst-case geometry for binary quantization. To realize this gain, we propose LittleBit-2, a framework employing Internal Latent Rotation and Joint Iterative Quantization (Joint-ITQ). This approach acts as a geometric preconditioner, aligning coherent latent distributions with the binary hypercube with zero inference overhead. Empirically, LittleBit-2 establishes a new state-of-the-art in the sub-1-bit regime (1sim0.1 bpp) on Llama-2 and Llama-3, matching the fidelity of leading 1-bit baselines.

The singular value decomposition (SVD) is a powerful tool in modern numerical linear algebra, which underpins computational methods such as principal component analysis (PCA), low-rank approximations, and randomized algorithms. Many practical scenarios require solving numerous small SVD problems, a regime generally referred to as "batch SVD". Existing programming models can handle this efficiently on parallel CPU architectures, but high-performance solutions for GPUs remain immature. A GPU-oriented batch SVD solver is introduced. This solver exploits the one-sided Jacobi algorithm to exploit fine-grained parallelism, and a number of algorithmic and design optimizations achieve unmatched performance. Starting from a baseline solver, a sequence of optimizations is applied to obtain incremental performance gains. Numerical experiments show that the new solver is robust across problems with different numerical properties, matrix shapes, and arithmetic precisions. Performance benchmarks on both NVIDIA and AMD systems show significant performance speedups over vendor solutions as well as existing open-source solvers.

Modern distributed training relies heavily on communication compression to reduce the communication overhead. In this work, we study algorithms employing a popular class of contractive compressors in order to reduce communication overhead. However, the naive implementation often leads to unstable convergence or even exponential divergence due to the compression bias. Error Compensation (EC) is an extremely popular mechanism to mitigate the aforementioned issues during the training of models enhanced by contractive compression operators. Compared to the effectiveness of EC in the data homogeneous regime, the understanding of the practicality and theoretical foundations of EC in the data heterogeneous regime is limited. Existing convergence analyses typically rely on strong assumptions such as bounded gradients, bounded data heterogeneity, or large batch accesses, which are often infeasible in modern machine learning applications. We resolve the majority of current issues by proposing EControl, a novel mechanism that can regulate error compensation by controlling the strength of the feedback signal. We prove fast convergence for EControl in standard strongly convex, general convex, and nonconvex settings without any additional assumptions on the problem or data heterogeneity. We conduct extensive numerical evaluations to illustrate the efficacy of our method and support our theoretical findings.

We employ random matrix theory to establish consistency of generalized cross validation (GCV) for estimating prediction risks of sketched ridge regression ensembles, enabling efficient and consistent tuning of regularization and sketching parameters. Our results hold for a broad class of asymptotically free sketches under very mild data assumptions. For squared prediction risk, we provide a decomposition into an unsketched equivalent implicit ridge bias and a sketching-based variance, and prove that the risk can be globally optimized by only tuning sketch size in infinite ensembles. For general subquadratic prediction risk functionals, we extend GCV to construct consistent risk estimators, and thereby obtain distributional convergence of the GCV-corrected predictions in Wasserstein-2 metric. This in particular allows construction of prediction intervals with asymptotically correct coverage conditional on the training data. We also propose an "ensemble trick" whereby the risk for unsketched ridge regression can be efficiently estimated via GCV using small sketched ridge ensembles. We empirically validate our theoretical results using both synthetic and real large-scale datasets with practical sketches including CountSketch and subsampled randomized discrete cosine transforms.

Anchor-based 3D Gaussian splatting (3D-GS) exploits anchor features in 3D Gaussian prediction, which has achieved impressive 3D rendering quality with reduced Gaussian redundancy. On the other hand, it often encounters the dilemma among anchor features, model size, and rendering quality - large anchor features lead to large 3D models and high-quality rendering whereas reducing anchor features degrades Gaussian attribute prediction which leads to clear artifacts in the rendered textures and geometries. We design SOGS, an anchor-based 3D-GS technique that introduces second-order anchors to achieve superior rendering quality and reduced anchor features and model size simultaneously. Specifically, SOGS incorporates covariance-based second-order statistics and correlation across feature dimensions to augment features within each anchor, compensating for the reduced feature size and improving rendering quality effectively. In addition, it introduces a selective gradient loss to enhance the optimization of scene textures and scene geometries, leading to high-quality rendering with small anchor features. Extensive experiments over multiple widely adopted benchmarks show that SOGS achieves superior rendering quality in novel view synthesis with clearly reduced model size.

Hyperparameter tuning almost always means search: fit the model at every value on a grid, score each by cross-validation, and keep the winner. For spline regression that search is unnecessary. The optimal resolution can be solved for in closed form, to the accuracy an exhaustive search reaches, at a fraction of the compute. Three ingredients make this possible: classical approximation theory pins the squared bias to a known power of the resolution G, exactly the Kolmogorov n-width of the smoothness class; the basis dimension is an explicit polynomial in G; and leave-one-out error follows from a single fit via the PRESS identity. Balancing the two known curves gives the minimizer analytically. We extend this calculus to many coordinates by replacing ambient input dimension with interaction order, the number of active low-order components in an ANOVA decomposition, yielding a scaling law in which the optimal resolution and error are power functions of the effective density (sample size per active component), with input dimension absent from the exponent. The law becomes an algorithm. KORE (Kolmogorov-optimal Order-aware Resolution Estimation) fits two pilot resolutions, solves a leverage-calibrated 2x2 system for the bias and noise scales, and evaluates the closed-form plug-in resolution with a tiny leave-one-out certificate: about a dozen fits instead of a full grid sweep, with a consistency guarantee as the sample grows. Across additive and sparse pairwise targets up to 80 input dimensions, KORE matches exhaustive 3-fold cross-validation and the full classical ladder (GCV, Mallows' Cp, AIC, BIC) while fitting roughly 8x fewer models; on 36 real tabular datasets it ranks first among 21 methods in accuracy per unit of compute, ahead of tuned boosters and kernel machines. When complexity lives in low interaction order, solving for the resolution beats searching for it.

In this paper, we examine the solvability of a functional equation in a Lipschitz space. As an application, we use our result to determine the existence and uniqueness of solutions to an equation describing a specific type of choice behavior model for the learning process of the paradise fish. Finally, we present some concrete examples where, using numerical techniques, we obtain approximations to the solution of the functional equation. As the straightforward Picard's iteration can be very expensive, we show that an analytical suboptimal least-squares approximation can be chosen in practice, resulting in very good accuracy.

We show how perceptual embeddings of the visual system can be constructed at inference-time with no training data or deep neural network features. Our perceptual embeddings are solutions to a weighted least squares (WLS) problem, defined at the pixel-level, and solved at inference-time, that can capture global and local image characteristics. The distance in embedding space is used to define a perceptual similarity metric which we call LASI: Linear Autoregressive Similarity Index. Experiments on full-reference image quality assessment datasets show LASI performs competitively with learned deep feature based methods like LPIPS (Zhang et al., 2018) and PIM (Bhardwaj et al., 2020), at a similar computational cost to hand-crafted methods such as MS-SSIM (Wang et al., 2003). We found that increasing the dimensionality of the embedding space consistently reduces the WLS loss while increasing performance on perceptual tasks, at the cost of increasing the computational complexity. LASI is fully differentiable, scales cubically with the number of embedding dimensions, and can be parallelized at the pixel-level. A Maximum Differentiation (MAD) competition (Wang & Simoncelli, 2008) between LASI and LPIPS shows that both methods are capable of finding failure points for the other, suggesting these metrics can be combined.

This paper addresses a regression problem in which output label values are the results of sensing the magnitude of a phenomenon. A low value of such labels can mean either that the actual magnitude of the phenomenon was low or that the sensor made an incomplete observation. This leads to a bias toward lower values in labels and the resultant learning because labels may have lower values due to incomplete observations, even if the actual magnitude of the phenomenon was high. Moreover, because an incomplete observation does not provide any tags indicating incompleteness, we cannot eliminate or impute them. To address this issue, we propose a learning algorithm that explicitly models incomplete observations corrupted with an asymmetric noise that always has a negative value. We show that our algorithm is unbiased as if it were learned from uncorrupted data that does not involve incomplete observations. We demonstrate the advantages of our algorithm through numerical experiments.

Supervised fine-tuning (SFT) provides the standard approach for teaching LLMs new behaviors from offline expert demonstrations. However, standard SFT uniformly fits all samples -- including those with low likelihood under the base model -- which can disproportionately drive training updates toward overfitting specific samples rather than learning the target behavior. Moreover, adapting to these unlikely samples induces substantial policy shifts that degrade prior capabilities. Existing methods mitigate this by filtering, regenerating, or down-weighting low-likelihood data. In doing so, they often suppress precisely the novel behaviors the base model has yet to learn. We propose InfoSFT, a principled weighting scheme for the SFT objective that concentrates learning signals on maximally informative, medium-confidence tokens -- those neither overly familiar to the base model nor too unlikely to cause instability. Requiring only a one-line modification to the standard token-wise loss, InfoSFT demonstrably improves generalization over vanilla SFT and likelihood-weighted baselines across math, code, and chain-of-thought tasks with diverse model families, while better preserving pre-existing capabilities.

The presence of distribution shifts poses a significant challenge for deploying modern machine learning models in real-world applications. This work focuses on the target shift problem in a regression setting (Zhang et al., 2013; Nguyen et al., 2016). More specifically, the target variable y (also known as the response variable), which is continuous, has different marginal distributions in the training source and testing domain, while the conditional distribution of features x given y remains the same. While most literature focuses on classification tasks with finite target space, the regression problem has an infinite dimensional target space, which makes many of the existing methods inapplicable. In this work, we show that the continuous target shift problem can be addressed by estimating the importance weight function from an ill-posed integral equation. We propose a nonparametric regularized approach named ReTaSA to solve the ill-posed integral equation and provide theoretical justification for the estimated importance weight function. The effectiveness of the proposed method has been demonstrated with extensive numerical studies on synthetic and real-world datasets.

Low-Rank Adaptation (LoRA) merging can efficiently combine diverse generative capabilities from multiple trained LoRAs for a diffusion model. However, existing LoRA merging techniques often suffer from severe parameter interference, causing destructive collisions in the shared parameter space. To address this, we propose Subspace Signal Routing (SSR), which resolves interference by routing internal signals instead of performing parameter-space merge. Specifically, SSR first constructs a unified subspace by concatenating candidate LoRAs along the rank dimension. Next, SSR employs an inverse correlation matrix to decorrelate mixed signals within this space. Finally, a directional guide matrix steers these purified signals into their respective task-specific subspaces. We provide a rigorous theoretical analysis proving that SSR aligns with the Ordinary Least Squares (OLS) solution, thereby ensuring mathematical optimality. We utilize the additivity of sufficient statistics to design a streaming algorithm. This enables on-the-fly updates that significantly reduce memory overhead and computation time. Extensive experiments validate that SSR significantly outperforms state-of-the-art methods while maintaining comparable efficiency. Code is available at https://github.com/nagara214/SSR-Merge.

In recent years, multiple notions of algorithmic fairness have arisen. One such notion is individual fairness (IF), which requires that individuals who are similar receive similar treatment. In parallel, matrix estimation (ME) has emerged as a natural paradigm for handling noisy data with missing values. In this work, we connect the two concepts. We show that pre-processing data using ME can improve an algorithm's IF without sacrificing performance. Specifically, we show that using a popular ME method known as singular value thresholding (SVT) to pre-process the data provides a strong IF guarantee under appropriate conditions. We then show that, under analogous conditions, SVT pre-processing also yields estimates that are consistent and approximately minimax optimal. As such, the ME pre-processing step does not, under the stated conditions, increase the prediction error of the base algorithm, i.e., does not impose a fairness-performance trade-off. We verify these results on synthetic and real data.

This research enhances linear regression models by integrating a Kalman filter and analysing curve areas to minimize loss. The goal is to develop an optimal linear regression equation using stochastic gradient descent (SGD) for weight updating. Our approach involves a stepwise process, starting with user-defined parameters. The linear regression model is trained using SGD, tracking weights and loss separately and zipping them finally. A Kalman filter is then trained based on weight and loss arrays to predict the next consolidated weights. Predictions result from multiplying input averages with weights, evaluated for loss to form a weight-versus-loss curve. The curve's equation is derived using the two-point formula, and area under the curve is calculated via integration. The linear regression equation with minimum area becomes the optimal curve for prediction. Benefits include avoiding constant weight updates via gradient descent and working with partial datasets, unlike methods needing the entire set. However, computational complexity should be considered. The Kalman filter's accuracy might diminish beyond a certain prediction range.

Block-based video codecs select coding parameters based on the input by optimizing a rate-distortion trade-off. The conventional distortion choice, the sum of squared errors (SSE), simplifies parameter selection: the SSE is the sum of block-wise SSEs, so rate-distortion optimization (RDO) can treat blocks independently. Alternatively, full-reference image quality assessment (FR-IQA) metrics such as MS-SSIM or LPIPS often align better with the human visual system than SSE, but they cannot be used in-loop: they do not decompose block-wise and typically require the fully decoded image as input. Building on existing results in metric quadratization, we approximate a broad class of FR-IQA metrics by an input-dependent quadratic distortion (IDQD), whose quadratic form matrix is derived from the Hessian of the metric evaluated at the source video. To make the distortion computable block-wise, we propose two approximations of the Hessian matrix: 1) keeping the block-diagonal, and 2) keeping only its diagonal. We propose estimators for both that require only matrix-vector products with the Hessian obtained by automatic differentiation. Across five metrics for Kodak and CLIC in VVC, IDQD-RDO achieves 14.2-36.7 % BD-rate savings under the target metric with no decoder changes and incurs 10-30 % encoding complexity overhead.

In modern recommendation systems, the standard pipeline involves training machine learning models on historical data to predict user behaviors and improve recommendations continuously. However, these data training loops can introduce interference in A/B tests, where data generated by control and treatment algorithms, potentially with different distributions, are combined. To address these challenges, we introduce a novel approach called weighted training. This approach entails training a model to predict the probability of each data point appearing in either the treatment or control data and subsequently applying weighted losses during model training. We demonstrate that this approach achieves the least variance among all estimators that do not cause shifts in the training distributions. Through simulation studies, we demonstrate the lower bias and variance of our approach compared to other methods.

Solving a linear system Ax=b is a fundamental scientific computing primitive for which numerous solvers and preconditioners have been developed. These come with parameters whose optimal values depend on the system being solved and are often impossible or too expensive to identify; thus in practice sub-optimal heuristics are used. We consider the common setting in which many related linear systems need to be solved, e.g. during a single numerical simulation. In this scenario, can we sequentially choose parameters that attain a near-optimal overall number of iterations, without extra matrix computations? We answer in the affirmative for Successive Over-Relaxation (SOR), a standard solver whose parameter omega has a strong impact on its runtime. For this method, we prove that a bandit online learning algorithm--using only the number of iterations as feedback--can select parameters for a sequence of instances such that the overall cost approaches that of the best fixed omega as the sequence length increases. Furthermore, when given additional structural information, we show that a contextual bandit method asymptotically achieves the performance of the instance-optimal policy, which selects the best omega for each instance. Our work provides the first learning-theoretic treatment of high-precision linear system solvers and the first end-to-end guarantees for data-driven scientific computing, demonstrating theoretically the potential to speed up numerical methods using well-understood learning algorithms.

Recent low-rank training methods, such as GaLore, have significantly reduced the memory required to optimize large language models (LLMs). However, these methods often suffer from time-consuming low-rank projection estimations. In particular, the singular value decomposition (SVD) in GaLore can consume more than 80\% of the total training time. To address this issue, we propose GaLore+, which uses cross-head low-rank projection to reduce the substantial time consumption in estimating low-rank projections for multi-head attention. In addition, we employ randomized subspace iteration to achieve fast SVD. To further enhance performance, we propose sparsely coded residuals to reduce the errors caused by low-rank approximation on the first- and second-order moments of the optimizers and weight updates. We evaluate GaLore+ on arithmetic reasoning and natural language generation datasets. Our experiments demonstrate that GaLore+ delivers superior performance while achieving approximately 4times fine-tuning speed compared to vanilla GaLore.

We propose a class of greedy algorithms for weighted sparse recovery by considering new loss function-based generalizations of Orthogonal Matching Pursuit (OMP). Given a (regularized) loss function, the proposed algorithms alternate the iterative construction of the signal support via greedy index selection and a signal update based on solving a local data-fitting problem restricted to the current support. We show that greedy selection rules associated with popular weighted sparsity-promoting loss functions admit explicitly computable and simple formulas. Specifically, we consider ell^0 - and ell^1 -based versions of the weighted LASSO (Least Absolute Shrinkage and Selection Operator), the Square-Root LASSO (SR-LASSO) and the Least Absolute Deviations LASSO (LAD-LASSO). Through numerical experiments on Gaussian compressive sensing and high-dimensional function approximation, we demonstrate the effectiveness of the proposed algorithms and empirically show that they inherit desirable characteristics from the corresponding loss functions, such as SR-LASSO's noise-blind optimal parameter tuning and LAD-LASSO's fault tolerance. In doing so, our study sheds new light on the connection between greedy sparse recovery and convex relaxation.

The commitment to single-precision floating-point arithmetic is widespread in the deep learning community. To evaluate whether this commitment is justified, the influence of computing precision (single and double precision) on the optimization performance of the Conjugate Gradient (CG) method (a second-order optimization algorithm) and RMSprop (a first-order algorithm) has been investigated. Tests of neural networks with one to five fully connected hidden layers and moderate or strong nonlinearity with up to 4 million network parameters have been optimized for Mean Square Error (MSE). The training tasks have been set up so that their MSE minimum was known to be zero. Computing experiments have disclosed that single-precision can keep up (with superlinear convergence) with double-precision as long as line search finds an improvement. First-order methods such as RMSprop do not benefit from double precision. However, for moderately nonlinear tasks, CG is clearly superior. For strongly nonlinear tasks, both algorithm classes find only solutions fairly poor in terms of mean square error as related to the output variance. CG with double floating-point precision is superior whenever the solutions have the potential to be useful for the application goal.

In this article, we introduce a novel normalization technique for neural network weight matrices, which we term weight conditioning. This approach aims to narrow the gap between the smallest and largest singular values of the weight matrices, resulting in better-conditioned matrices. The inspiration for this technique partially derives from numerical linear algebra, where well-conditioned matrices are known to facilitate stronger convergence results for iterative solvers. We provide a theoretical foundation demonstrating that our normalization technique smoothens the loss landscape, thereby enhancing convergence of stochastic gradient descent algorithms. Empirically, we validate our normalization across various neural network architectures, including Convolutional Neural Networks (CNNs), Vision Transformers (ViT), Neural Radiance Fields (NeRF), and 3D shape modeling. Our findings indicate that our normalization method is not only competitive but also outperforms existing weight normalization techniques from the literature.

Scene coordinate regression achieves impressive results in outdoor LiDAR localization but requires days of training. Since training needs to be repeated for each new scene, long training times make these methods impractical for time-sensitive applications, such as autonomous driving, drones, and robotics. We identify large coverage areas and vast data in large-scale outdoor scenes as key challenges that limit fast training. In this paper, we propose LightLoc, the first method capable of efficiently learning localization in a new scene at light speed. LightLoc introduces two novel techniques to address these challenges. First, we introduce sample classification guidance to assist regression learning, reducing ambiguity from similar samples and improving training efficiency. Second, we propose redundant sample downsampling to remove well-learned frames during training, reducing training time without compromising accuracy. Additionally, the fast training and confidence estimation capabilities of sample classification enable its integration into SLAM, effectively eliminating error accumulation. Extensive experiments on large-scale outdoor datasets demonstrate that LightLoc achieves state-of-the-art performance with a 50x reduction in training time than existing methods. Our code is available at https://github.com/liw95/LightLoc.

This paper models the availability of bikes at San Francisco Bay Area Bike Share stations using machine learning algorithms. Random Forest (RF) and Least-Squares Boosting (LSBoost) were used as univariate regression algorithms, and Partial Least-Squares Regression (PLSR) was applied as a multivariate regression algorithm. The univariate models were used to model the number of available bikes at each station. PLSR was applied to reduce the number of required prediction models and reflect the spatial correlation between stations in the network. Results clearly show that univariate models have lower error predictions than the multivariate model. However, the multivariate model results are reasonable for networks with a relatively large number of spatially correlated stations. Results also show that station neighbors and the prediction horizon time are significant predictors. The most effective prediction horizon time that produced the least prediction error was 15 minutes.

Large pre-trained models (LPMs), such as large language models, have become ubiquitous and are employed in many applications. These models are often adapted to a desired domain or downstream task through a fine-tuning stage. This paper proposes SQFT, an end-to-end solution for low-precision sparse parameter-efficient fine-tuning of LPMs, allowing for effective model manipulation in resource-constrained environments. Additionally, an innovative strategy enables the merging of sparse weights with low-rank adapters without losing sparsity and accuracy, overcoming the limitations of previous approaches. SQFT also addresses the challenge of having quantized weights and adapters with different numerical precisions, enabling merging in the desired numerical format without sacrificing accuracy. Multiple adaptation scenarios, models, and comprehensive sparsity levels demonstrate the effectiveness of SQFT. Models and code are available at https://github.com/IntelLabs/Hardware-Aware-Automated-Machine-Learning.

Lately, post-training quantization methods have gained considerable attention, as they are simple to use, and require only a small unlabeled calibration set. This small dataset cannot be used to fine-tune the model without significant over-fitting. Instead, these methods only use the calibration set to set the activations' dynamic ranges. However, such methods always resulted in significant accuracy degradation, when used below 8-bits (except on small datasets). Here we aim to break the 8-bit barrier. To this end, we minimize the quantization errors of each layer separately by optimizing its parameters over the calibration set. We empirically demonstrate that this approach is: (1) much less susceptible to over-fitting than the standard fine-tuning approaches, and can be used even on a very small calibration set; and (2) more powerful than previous methods, which only set the activations' dynamic ranges. Furthermore, we demonstrate how to optimally allocate the bit-widths for each layer, while constraining accuracy degradation or model compression by proposing a novel integer programming formulation. Finally, we suggest model global statistics tuning, to correct biases introduced during quantization. Together, these methods yield state-of-the-art results for both vision and text models. For instance, on ResNet50, we obtain less than 1\% accuracy degradation --- with 4-bit weights and activations in all layers, but the smallest two. We open-sourced our code.

The growing number of parameters and computational demands of large language models (LLMs) present significant challenges for their efficient deployment. Recently, there is an increasing interest in quantizing weights to extremely low precision while offsetting the resulting error with low-rank, high-precision error reconstruction terms. The combination of quantization and low-rank approximation is now popular in both adapter-based, parameter-efficient fine-tuning methods such as LoftQ and low-precision inference techniques including ZeroQuant-V2. Usually, the low-rank terms are calculated via the singular value decomposition (SVD) of the weight quantization error, minimizing the Frobenius and spectral norms of the weight approximation error. Recent methods like LQ-LoRA and LQER introduced hand-crafted heuristics to minimize errors in layer outputs (activations) rather than weights, resulting improved quantization results. However, these heuristic methods lack an analytical solution to guide the design of quantization error reconstruction terms. In this paper, we revisit this problem and formulate an analytical framework, named Quantization Error Reconstruction Analysis (QERA), and offer a closed-form solution to the problem. We show QERA benefits both existing low-precision fine-tuning and inference methods -- QERA achieves a fine-tuned accuracy gain of Δ_{acc} = 6.05% of 2-bit RoBERTa-base on GLUE compared to LoftQ; and obtains Δ_{acc} = 2.97% higher post-training quantization accuracy of 4-bit Llama-3.1-70B on average than ZeroQuant-V2 and Δ_{ppl} = - 0.28 lower perplexity on WikiText2 than LQER.

In this paper, we present a simple optimization-based preprocessing technique called Weight Magnitude Reduction (MagR) to improve the performance of post-training quantization. For each linear layer, we adjust the pre-trained floating-point weights by solving an ell_infty-regularized optimization problem. This process greatly diminishes the maximum magnitude of the weights and smooths out outliers, while preserving the layer's output. The preprocessed weights are centered more towards zero, which facilitates the subsequent quantization process. To implement MagR, we address the ell_infty-regularization by employing an efficient proximal gradient descent algorithm. Unlike existing preprocessing methods that involve linear transformations and subsequent post-processing steps, which can introduce significant overhead at inference time, MagR functions as a non-linear transformation, eliminating the need for any additional post-processing. This ensures that MagR introduces no overhead whatsoever during inference. Our experiments demonstrate that MagR achieves state-of-the-art performance on the Llama family of models. For example, we achieve a Wikitext2 perplexity of 5.95 on the LLaMA2-70B model for per-channel INT2 weight quantization without incurring any inference overhead.

The recent advancements in 3D Gaussian splatting (3D-GS) have not only facilitated real-time rendering through modern GPU rasterization pipelines but have also attained state-of-the-art rendering quality. Nevertheless, despite its exceptional rendering quality and performance on standard datasets, 3D-GS frequently encounters difficulties in accurately modeling specular and anisotropic components. This issue stems from the limited ability of spherical harmonics (SH) to represent high-frequency information. To overcome this challenge, we introduce Spec-Gaussian, an approach that utilizes an anisotropic spherical Gaussian (ASG) appearance field instead of SH for modeling the view-dependent appearance of each 3D Gaussian. Additionally, we have developed a coarse-to-fine training strategy to improve learning efficiency and eliminate floaters caused by overfitting in real-world scenes. Our experimental results demonstrate that our method surpasses existing approaches in terms of rendering quality. Thanks to ASG, we have significantly improved the ability of 3D-GS to model scenes with specular and anisotropic components without increasing the number of 3D Gaussians. This improvement extends the applicability of 3D GS to handle intricate scenarios with specular and anisotropic surfaces.

Recent 3D Gaussian Splatting (3DGS) Dropout methods address overfitting under sparse-view conditions by randomly nullifying Gaussian opacities. However, we identify a neighbor compensation effect in these approaches: dropped Gaussians are often compensated by their neighbors, weakening the intended regularization. Moreover, these methods overlook the contribution of high-degree spherical harmonic coefficients (SH) to overfitting. To address these issues, we propose DropAnSH-GS, a novel anchor-based Dropout strategy. Rather than dropping Gaussians independently, our method randomly selects certain Gaussians as anchors and simultaneously removes their spatial neighbors. This effectively disrupts local redundancies near anchors and encourages the model to learn more robust, globally informed representations. Furthermore, we extend the Dropout to color attributes by randomly dropping higher-degree SH to concentrate appearance information in lower-degree SH. This strategy further mitigates overfitting and enables flexible post-training model compression via SH truncation. Experimental results demonstrate that DropAnSH-GS substantially outperforms existing Dropout methods with negligible computational overhead, and can be readily integrated into various 3DGS variants to enhance their performances. Project Website: https://sk-fun.fun/DropAnSH-GS

Recently, a range of neural network-based methods for image rendering have been introduced. One such widely-researched neural radiance field (NeRF) relies on a neural network to represent 3D scenes, allowing for realistic view synthesis from a small number of 2D images. However, most NeRF models are constrained by long training and inference times. In comparison, Gaussian Splatting (GS) is a novel, state-of-the-art technique for rendering points in a 3D scene by approximating their contribution to image pixels through Gaussian distributions, warranting fast training and swift, real-time rendering. A drawback of GS is the absence of a well-defined approach for its conditioning due to the necessity to condition several hundred thousand Gaussian components. To solve this, we introduce the Gaussian Mesh Splatting (GaMeS) model, which allows modification of Gaussian components in a similar way as meshes. We parameterize each Gaussian component by the vertices of the mesh face. Furthermore, our model needs mesh initialization on input or estimated mesh during training. We also define Gaussian splats solely based on their location on the mesh, allowing for automatic adjustments in position, scale, and rotation during animation. As a result, we obtain a real-time rendering of editable GS.

Activation outliers in large-scale transformer models pose a fundamental challenge to model quantization, creating excessively large ranges that cause severe accuracy drops during quantization. We empirically observe that outlier severity intensifies with pre-training scale (e.g., progressing from CLIP to the more extensively trained SigLIP and SigLIP2). Through theoretical analysis as well as empirical correlation studies, we establish the direct link between these activation outliers and dominant singular values of the weights. Building on this insight, we propose Selective Spectral Decay (S^2D), a geometrically-principled conditioning method that surgically regularizes only the weight components corresponding to the largest singular values during fine-tuning. Through extensive experiments, we demonstrate that S^2D significantly reduces activation outliers and produces well-conditioned representations that are inherently quantization-friendly. Models trained with S^2D achieve up to 7% improved PTQ accuracy on ImageNet under W4A4 quantization and 4% gains when combined with QAT. These improvements also generalize across downstream tasks and vision-language models, enabling the scaling of increasingly large and rigorously trained models without sacrificing deployment efficiency.

LLM-as-a-judge has emerged as a cornerstone technique for evaluating large language models by leveraging LLM reasoning to score prompt-response pairs. Since LLM judgments are stochastic, practitioners commonly query each pair multiple times to estimate mean scores accurately. This raises a critical challenge: given a fixed computational budget B, how to optimally allocate queries across K prompt-response pairs to minimize estimation error? % We present a principled variance-adaptive approach leveraging multi-armed bandit theory and concentration inequalities. Our method dynamically allocates queries based on estimated score variances, concentrating resources where uncertainty is highest. Further, our algorithm is shown to achieve a worst-case score-estimation error of Oleft(frac{sum_{i=1^K σ_i^2}{B}}right), σ_i^2 being the unknown score variance for pair i in [K] with near-optimal budget allocation. % Experiments on Summarize-From-Feedback and HelpSteer2 demonstrate that our method significantly outperforms uniform allocation, reducing worst-case estimation error while maintaining identical budgets. Our work establishes a theoretical foundation for efficient LLM evaluation with practical implications for AI safety, model alignment, and automated assessment at scale.

Post-training quantization (PTQ) offers an efficient approach to compressing large language models (LLMs), significantly reducing memory access and computational costs. Existing compensation-based weight calibration methods often rely on a second-order Taylor expansion to model quantization error, under the assumption that the first-order term is negligible in well-trained full-precision models. However, we reveal that the progressive compensation process introduces accumulated first-order deviations between latent weights and their full-precision counterparts, making this assumption fundamentally flawed. To address this, we propose FOEM, a novel PTQ method that explicitly incorporates first-order gradient terms to improve quantization error compensation. FOEM approximates gradients by performing a first-order Taylor expansion around the pre-quantization weights. This yields an approximation based on the difference between latent and full-precision weights as well as the Hessian matrix. When substituted into the theoretical solution, the formulation eliminates the need to explicitly compute the Hessian, thereby avoiding the high computational cost and limited generalization of backpropagation-based gradient methods. This design introduces only minimal additional computational overhead. Extensive experiments across a wide range of models and benchmarks demonstrate that FOEM consistently outperforms the classical GPTQ method. In 3-bit weight-only quantization, FOEM reduces the perplexity of Llama3-8B by 17.3% and increases the 5-shot MMLU accuracy from 53.8% achieved by GPTAQ to 56.1%. Moreover, FOEM can be seamlessly combined with advanced techniques such as SpinQuant, delivering additional gains under the challenging W4A4KV4 setting and further narrowing the performance gap with full-precision baselines, surpassing existing state-of-the-art methods.

Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector θinR^d of coefficients is constrained in either ell_1-norm (for Lasso) or in ell_2-norm (for Ridge). We study the complexity of quantum algorithms for finding varepsilon-minimizers for these minimization problems. We show that for Lasso we can get a quadratic quantum speedup in terms of d by speeding up the cost-per-iteration of the Frank-Wolfe algorithm, while for Ridge the best quantum algorithms are linear in d, as are the best classical algorithms. As a byproduct of our quantum lower bound for Lasso, we also prove the first classical lower bound for Lasso that is tight up to polylog-factors.

We consider the problem of estimating how well a model class is capable of fitting a distribution of labeled data. We show that it is often possible to accurately estimate this "learnability" even when given an amount of data that is too small to reliably learn any accurate model. Our first result applies to the setting where the data is drawn from a d-dimensional distribution with isotropic covariance (or known covariance), and the label of each datapoint is an arbitrary noisy function of the datapoint. In this setting, we show that with O(d) samples, one can accurately estimate the fraction of the variance of the label that can be explained via the best linear function of the data. In contrast to this sublinear sample size, finding an approximation of the best-fit linear function requires on the order of d samples. Our sublinear sample results and approach also extend to the non-isotropic setting, where the data distribution has an (unknown) arbitrary covariance matrix: we show that, if the label y of point x is a linear function with independent noise, y = langle x , βrangle + noise with |β| bounded, the variance of the noise can be estimated to error ε with O(d^{1-1/1/ε}) if the covariance matrix has bounded condition number, or O(d^{1-sqrtε}) if there are no bounds on the condition number. We also establish that these sample complexities are optimal, to constant factors. Finally, we extend these techniques to the setting of binary classification, where we obtain analogous sample complexities for the problem of estimating the prediction error of the best linear classifier, in a natural model of binary labeled data. We demonstrate the practical viability of our approaches on several real and synthetic datasets.

Spatial audio quality is a highly multifaceted concept, with many interactions between environmental, geometrical, anatomical, psychological, and contextual considerations. Methods for characterization or evaluation of the geometrical components of spatial audio quality, however, remain scarce, despite being perhaps the least subjective aspect of spatial audio quality to quantify. By considering interchannel time and level differences relative to a reference signal, it is possible to construct a signal model to isolate some of the spatial distortion. By using a combination of least-square optimization and heuristics, we propose a signal decomposition method to isolate the spatial error from a processed signal, in terms of interchannel gain leakages and changes in relative delays. This allows the computation of simple energy-ratio metrics, providing objective measures of spatial and non-spatial signal qualities, with minimal assumptions and no dataset dependency. Experiments demonstrate the robustness of the method against common spatial signal degradation introduced by, e.g., audio compression and music source separation. Implementation is available at https://github.com/karnwatcharasupat/spauq.

Data-driven modeling in real-world regression tasks often suffers from limited training samples, high collection costs, and noisy observations. Inspired by the impact of data augmentation in vision and language, we propose a novel Counterfactual Residual Data Augmentation (CRDA) technique for tabular regression. Our key insight is that once a regressor has modeled the systematic component of the data, the remaining noise can be viewed as an invariant residual that remains stable under small perturbations of carefully selected features. We exploit this residual invariance to generate new, yet realistic, training samples, effectively expanding the dataset without requiring additional real data. Our method is model-agnostic and readily applicable to various types of regressors. In experiments across datasets from a variety of benchmark repositories, on average, CRDA reduces an MLP Regressor's MSE by 22.9% and an XGBoost Regressor's MSE by 6.4%. When compared to existing state-of-the-art data generators and augmentation techniques, CRDA consistently outperforms in MSE reduction. By adding principled counterfactual variations to the training data, our method offers a simple and efficient remedy for noise-prone, small-sample regression settings.

Noise plagues many numerical datasets, where the recorded values in the data may fail to match the true underlying values due to reasons including: erroneous sensors, data entry/processing mistakes, or imperfect human estimates. We consider general regression settings with covariates and a potentially corrupted response whose observed values may contain errors. By accounting for various uncertainties, we introduced veracity scores that distinguish between genuine errors and natural data fluctuations, conditioned on the available covariate information in the dataset. We propose a simple yet efficient filtering procedure for eliminating potential errors, and establish theoretical guarantees for our method. We also contribute a new error detection benchmark involving 5 regression datasets with real-world numerical errors (for which the true values are also known). In this benchmark and additional simulation studies, our method identifies incorrect values with better precision/recall than other approaches.

In this paper, we study the problem of principal component analysis with generative modeling assumptions, adopting a general model for the observed matrix that encompasses notable special cases, including spiked matrix recovery and phase retrieval. The key assumption is that the underlying signal lies near the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. We propose a quadratic estimator, and show that it enjoys a statistical rate of order frac{klog L{m}}, where m is the number of samples. We also provide a near-matching algorithm-independent lower bound. Moreover, we provide a variant of the classic power method, which projects the calculated data onto the range of the generative model during each iteration. We show that under suitable conditions, this method converges exponentially fast to a point achieving the above-mentioned statistical rate. We perform experiments on various image datasets for spiked matrix and phase retrieval models, and illustrate performance gains of our method to the classic power method and the truncated power method devised for sparse principal component analysis.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

Large Language Models (LLMs) have transformed both everyday life and scientific research. However, adapting LLMs from general-purpose models to specialized tasks remains challenging, particularly in resource-constrained environments. Low-Rank Adaptation (LoRA), a prominent method within Parameter-Efficient Fine-Tuning (PEFT), has emerged as a promising approach to LLMs by approximating model weight updates using low-rank decomposition. However, LoRA is limited by its uniform rank ( r ) allocation to each incremental matrix, and existing rank allocation techniques aimed at addressing this issue remain computationally inefficient, complex, and unstable, hindering practical applications. To address these limitations, we propose Sensitivity-LoRA, an efficient fine-tuning method that dynamically allocates ranks to weight matrices based on both their global and local sensitivities. It leverages the second-order derivatives (Hessian Matrix) of the loss function to effectively capture weight sensitivity, enabling optimal rank allocation with minimal computational overhead. Our experimental results have demonstrated robust effectiveness, efficiency and stability of Sensitivity-LoRA across diverse tasks and benchmarks.

The Shapley value is a ubiquitous framework for attribution in machine learning, encompassing feature importance, data valuation, and causal inference. However, its exact computation is generally intractable, necessitating efficient approximation methods. While the most effective and popular estimators leverage the paired sampling heuristic to reduce estimation error, the theoretical mechanism driving this improvement has remained opaque. In this work, we provide an elegant and fundamental justification for paired sampling: we prove that the Shapley value depends exclusively on the odd component of the set function, and that paired sampling orthogonalizes the regression objective to filter out the irrelevant even component. Leveraging this insight, we propose OddSHAP, a novel consistent estimator that performs polynomial regression solely on the odd subspace. By utilizing the Fourier basis to isolate this subspace and employing a proxy model to identify high-impact interactions, OddSHAP overcomes the combinatorial explosion of higher-order approximations. Through an extensive benchmark, we find that OddSHAP achieves state-of-the-art estimation accuracy at larger sampling budgets.

The trade-off between regret and computational cost is a fundamental problem for online kernel regression, and previous algorithms worked on the trade-off can not keep optimal regret bounds at a sublinear computational complexity. In this paper, we propose two new algorithms, AOGD-ALD and NONS-ALD, which can keep nearly optimal regret bounds at a sublinear computational complexity, and give sufficient conditions under which our algorithms work. Both algorithms dynamically maintain a group of nearly orthogonal basis used to approximate the kernel mapping, and keep nearly optimal regret bounds by controlling the approximate error. The number of basis depends on the approximate error and the decay rate of eigenvalues of the kernel matrix. If the eigenvalues decay exponentially, then AOGD-ALD and NONS-ALD separately achieves a regret of O(L(f)) and O(d_{eff}(mu)T) at a computational complexity in O(ln^2{T}). If the eigenvalues decay polynomially with degree pgeq 1, then our algorithms keep the same regret bounds at a computational complexity in o(T) in the case of p>4 and pgeq 10, respectively. L(f) is the cumulative losses of f and d_{eff}(mu) is the effective dimension of the problem. The two regret bounds are nearly optimal and are not comparable.

Previous research has shown that the principal singular vectors of a pre-trained model's weight matrices capture critical knowledge. In contrast, those associated with small singular values may contain noise or less reliable information. As a result, the LoRA-based parameter-efficient fine-tuning (PEFT) approach, which does not constrain the use of the spectral space, may not be effective for tasks that demand high representation capacity. In this study, we enhance existing PEFT techniques by incorporating the spectral information of pre-trained weight matrices into the fine-tuning process. We investigate spectral adaptation strategies with a particular focus on the additive adjustment of top singular vectors. This is accomplished by applying singular value decomposition (SVD) to the pre-trained weight matrices and restricting the fine-tuning within the top spectral space. Extensive speaker verification experiments on VoxCeleb1 and CN-Celeb1 demonstrate enhanced tuning performance with the proposed approach. Code is released at https://github.com/lizhepolyu/SpectralFT.

One of the major limits of kernel ridge regression (KRR) is that storing and manipulating the kernel matrix K_n for n samples requires O(n^2) space, which rapidly becomes unfeasible for large n. Nystrom approximations reduce the space complexity to O(nm) by sampling m columns from K_n. Uniform sampling preserves KRR accuracy (up to epsilon) only when m is proportional to the maximum degree of freedom of K_n, which may require O(n) columns for datasets with high coherence. Sampling columns according to their ridge leverage scores (RLS) gives accurate Nystrom approximations with m proportional to the effective dimension, but computing exact RLS also requires O(n^2) space. (Calandriello et al. 2016) propose INK-Estimate, an algorithm that processes the dataset incrementally and updates RLS, effective dimension, and Nystrom approximations on-the-fly. Its space complexity scales with the effective dimension but introduces a dependency on the largest eigenvalue of K_n, which in the worst case is O(n). In this paper we introduce SQUEAK, a new algorithm that builds on INK-Estimate but uses unnormalized RLS. As a consequence, the algorithm is simpler, does not need to estimate the effective dimension for normalization, and achieves a space complexity that is only a constant factor worse than exact RLS sampling.

Offline reinforcement learning (RL), where the agent aims to learn the optimal policy based on the data collected by a behavior policy, has attracted increasing attention in recent years. While offline RL with linear function approximation has been extensively studied with optimal results achieved under certain assumptions, many works shift their interest to offline RL with non-linear function approximation. However, limited works on offline RL with non-linear function approximation have instance-dependent regret guarantees. In this paper, we propose an oracle-efficient algorithm, dubbed Pessimistic Nonlinear Least-Square Value Iteration (PNLSVI), for offline RL with non-linear function approximation. Our algorithmic design comprises three innovative components: (1) a variance-based weighted regression scheme that can be applied to a wide range of function classes, (2) a subroutine for variance estimation, and (3) a planning phase that utilizes a pessimistic value iteration approach. Our algorithm enjoys a regret bound that has a tight dependency on the function class complexity and achieves minimax optimal instance-dependent regret when specialized to linear function approximation. Our work extends the previous instance-dependent results within simpler function classes, such as linear and differentiable function to a more general framework.

Bundle adjustment is the common way to solve localization and mapping. It is an iterative process in which a system of non-linear equations is solved using two optimization methods, weighted by a damping factor. In the classic approach, the latter is chosen heuristically by the Levenberg-Marquardt algorithm on each iteration. This might take many iterations, making the process computationally expensive, which might be harmful to real-time applications. We propose to replace this heuristic by viewing the problem in a holistic manner, as a game, and formulating it as a reinforcement-learning task. We set an environment which solves the non-linear equations and train an agent to choose the damping factor in a learned manner. We demonstrate that our approach considerably reduces the number of iterations required to reach the bundle adjustment's convergence, on both synthetic and real-life scenarios. We show that this reduction benefits the classic approach and can be integrated with other bundle adjustment acceleration methods.

We deal with the problem of gradient estimation for stochastic differentiable relaxations of algorithms, operators, simulators, and other non-differentiable functions. Stochastic smoothing conventionally perturbs the input of a non-differentiable function with a differentiable density distribution with full support, smoothing it and enabling gradient estimation. Our theory starts at first principles to derive stochastic smoothing with reduced assumptions, without requiring a differentiable density nor full support, and we present a general framework for relaxation and gradient estimation of non-differentiable black-box functions f:R^ntoR^m. We develop variance reduction for gradient estimation from 3 orthogonal perspectives. Empirically, we benchmark 6 distributions and up to 24 variance reduction strategies for differentiable sorting and ranking, differentiable shortest-paths on graphs, differentiable rendering for pose estimation, as well as differentiable cryo-ET simulations.

We present a method for Sparse view reconstruction with surface element splatting that runs within 3 minutes on a consumer grade GPU. While few methods address sparse radiance field learning from noisy or unposed sparse cameras, shape recovery remains relatively underexplored in this setting. Several radiance and shape learning test-time optimization methods address the sparse posed setting by learning data priors or using combinations of external monocular geometry priors. Differently, we propose an efficient and simple pipeline harnessing a single recent 3D foundation model. We leverage its various task heads, notably point maps and camera initializations to instantiate a bundle adjusting 2D Gaussian Splatting (2DGS) model, and image correspondences to guide camera optimization midst 2DGS training. Key to our contribution is a novel formulation of splatted color variance along rays, which can be computed efficiently. Reducing this moment in training leads to more accurate shape reconstructions. We demonstrate state-of-the-art performances in the sparse uncalibrated setting in reconstruction and novel view benchmarks based on established multi-view datasets.

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

Data fields sampled on irregularly spaced points arise in many applications in the sciences and engineering. For regular grids, Convolutional Neural Networks (CNNs) have been successfully used to gaining benefits from weight sharing and invariances. We generalize CNNs by introducing methods for data on unstructured point clouds based on Generalized Moving Least Squares (GMLS). GMLS is a non-parametric technique for estimating linear bounded functionals from scattered data, and has recently been used in the literature for solving partial differential equations. By parameterizing the GMLS estimator, we obtain learning methods for operators with unstructured stencils. In GMLS-Nets the necessary calculations are local, readily parallelizable, and the estimator is supported by a rigorous approximation theory. We show how the framework may be used for unstructured physical data sets to perform functional regression to identify associated differential operators and to regress quantities of interest. The results suggest the architectures to be an attractive foundation for data-driven model development in scientific machine learning applications.

Parameter-efficient fine-tuning (PEFT) aims to adapt pre-trained vision models to downstream tasks. Among PEFT paradigms, sparse tuning achieves remarkable performance by adjusting only the weights most relevant to downstream tasks, rather than densely tuning the entire weight matrix. Current methods follow a two-stage paradigm. First, it locates task-relevant weights by gradient information, which overlooks the parameter adjustments during fine-tuning and limits the performance. Second, it updates only the located weights by applying a sparse mask to the gradient of the weight matrix, which results in high memory usage due to the storage of all weight matrices in the optimizer. In this paper, we propose a one-stage method named SNELLA to overcome the above limitations. For memory usage, SNELLA selectively updates the weight matrix by adding it to another sparse matrix that is merged by two low-rank learnable matrices. We extend the low-rank decomposition by introducing nonlinear kernel functions, thereby increasing the rank of the resulting merged matrix to prevent the interdependency among weight updates, enabling better adaptation to downstream tasks. For locating task-relevant weights, we propose an adaptive bi-level sparsity allocation mechanism that encourages weights to compete across and inside layers based on their importance scores in an end-to-end manner. Extensive experiments are conducted on classification, segmentation, and generation tasks using different pre-trained vision models. The results show that SNELLA achieves SOTA performance with low memory usage. Notably, SNELLA obtains 1.8% (91.9% v.s. 90.1%) higher Top-1 accuracy on the FGVC benchmark compared to SPT-LoRA. Compared to previous methods, SNELLA achieves a memory reduction of 31.1%-39.9% across models with parameter scales from 86M to 632M. Our source codes are available at https://github.com/ssfgunner/SNELL.

The two-stage fine-tuning (FT) method, linear probing (LP) then fine-tuning (LP-FT), outperforms linear probing and FT alone. This holds true for both in-distribution (ID) and out-of-distribution (OOD) data. One key reason for its success is the preservation of pre-trained features, achieved by obtaining a near-optimal linear head during LP. However, despite the widespread use of large language models, there has been limited exploration of more complex architectures such as Transformers. In this paper, we analyze the training dynamics of LP-FT for classification tasks on the basis of the neural tangent kernel (NTK) theory. Our analysis decomposes the NTK matrix into two components. This decomposition highlights the importance of the linear head norm alongside the prediction accuracy at the start of the FT stage. We also observe a significant increase in the linear head norm during LP, which stems from training with the cross-entropy (CE) loss. This increase in the linear head norm effectively reduces changes in learned features. Furthermore, we find that this increased norm can adversely affect model calibration, which can be corrected using temperature scaling. Additionally, we extend our analysis with the NTK to the low-rank adaptation (LoRA) method and validate its effectiveness. Our experiments using a Transformer-based model on multiple natural language processing datasets confirm our theoretical analysis. Our study demonstrates the effectiveness of LP-FT for fine-tuning language models. Code is available at https://github.com/tom4649/lp-ft_ntk.

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or unitarily invariant, i.e., they depend only on the eigenvalues or singular values of the matrix. This paper investigates how spectral matrix cones -- cones defined from epigraphs and perspectives of spectral or unitarily invariant functions -- can be used to enhance first-order conic solvers for semidefinite programs. Our main result shows that projecting a matrix can be reduced to projecting its eigenvalues or singular values, which we demonstrate can be done at a negligible cost compared to the eigenvalue or singular value decomposition itself. We have integrated support for spectral matrix cone projections into the Splitting Conic Solver (SCS). Numerical experiments show that SCS with this enhancement can achieve speedups of up to an order of magnitude for solving semidefinite programs arising in experimental design, robust principal component analysis, and graph partitioning.

Low-rank adaptation (LoRA) has become a standard approach for fine-tuning large foundation models. However, our theoretical understanding of LoRA remains limited as prior analyses of LoRA's training dynamics either rely on linearization arguments or consider highly simplified setups. In this work, we analyze the LoRA loss landscape without such restrictive assumptions. We define two regimes: a "special regime", which includes idealized setups where linearization arguments hold, and a "generic regime" representing more realistic setups where linearization arguments do not hold. In the generic regime, we show that LoRA training converges to a global minimizer with low rank and small magnitude, or a qualitatively distinct solution with high rank and large magnitude. Finally, we argue that the zero-initialization and weight decay in LoRA training induce an implicit bias toward the low-rank, small-magnitude region of the parameter space -- where global minima lie -- thus shedding light on why LoRA training usually succeeds in finding global minima.

Supervised low-level vision models rely on pixel-wise losses against paired references, yet paired training sets exhibit per-pair photometric inconsistency, say, different image pairs demand different global brightness, color, or white-balance mappings. This inconsistency enters through task-intrinsic photometric transfer (e.g., low-light enhancement) or unintended acquisition shifts (e.g., de-raining), and in either case causes an optimization pathology. Standard reconstruction losses allocate disproportionate gradient budget to conflicting per-pair photometric targets, crowding out content restoration. In this paper, we investigate this issue and prove that, under least-squares decomposition, the photometric and structural components of the prediction-target residual are orthogonal, and that the spatially dense photometric component dominates the gradient energy. Motivated by this analysis, we propose Photometric Alignment Loss (PAL). This flexible supervision objective discounts nuisance photometric discrepancy via closed-form affine color alignment while preserving restoration-relevant supervision, requiring only covariance statistics and tiny matrix inversion with negligible overhead. Across 6 tasks, 16 datasets, and 16 architectures, PAL consistently improves metrics and generalization. The implementation is in the appendix.

We tackle the problem of estimating a Manhattan frame, i.e. three orthogonal vanishing points, and the unknown focal length of the camera, leveraging a prior vertical direction. The direction can come from an Inertial Measurement Unit that is a standard component of recent consumer devices, e.g., smartphones. We provide an exhaustive analysis of minimal line configurations and derive two new 2-line solvers, one of which does not suffer from singularities affecting existing solvers. Additionally, we design a new non-minimal method, running on an arbitrary number of lines, to boost the performance in local optimization. Combining all solvers in a hybrid robust estimator, our method achieves increased accuracy even with a rough prior. Experiments on synthetic and real-world datasets demonstrate the superior accuracy of our method compared to the state of the art, while having comparable runtimes. We further demonstrate the applicability of our solvers for relative rotation estimation. The code is available at https://github.com/cvg/VP-Estimation-with-Prior-Gravity.

Despite 3D Gaussian Splatting (3DGS) excelling in most configurations, it lacks generalization across novel viewpoints in a few-shot scenario because it overfits to the sparse observations. We revisit 3DGS optimization from a machine learning perspective, framing novel view synthesis as a generalization problem to unseen viewpoints-an underexplored direction. We propose Frequency-Adaptive Sharpness Regularization (FASR), which reformulates the 3DGS training objective, thereby guiding 3DGS to converge toward a better generalization solution. Although Sharpness-Aware Minimization (SAM) similarly reduces the sharpness of the loss landscape to improve generalization of classification models, directly employing it to 3DGS is suboptimal due to the discrepancy between the tasks. Specifically, it hinders reconstructing high-frequency details due to excessive regularization, while reducing its strength leads to under-penalizing sharpness. To address this, we reflect the local frequency of images to set the regularization weight and the neighborhood radius when estimating the local sharpness. It prevents floater artifacts in novel viewpoints and reconstructs fine details that SAM tends to oversmooth. Across datasets with various configurations, our method consistently improves a wide range of baselines. Code will be available at https://bbangsik13.github.io/FASR.

Singular Value Decomposition (SVD) has become an important technique for reducing the computational burden of Vision Language Models (VLMs), which play a central role in tasks such as image captioning and visual question answering. Although multiple prior works have proposed efficient SVD variants to enable low-rank operations, we find that in practice it remains difficult to achieve substantial latency reduction during model execution. To address this limitation, we introduce a new computational pattern and apply SVD at a finer granularity, enabling real and measurable improvements in execution latency. Furthermore, recognizing that weight elements differ in their relative importance, we adaptively allocate relative importance to each element during SVD process to better preserve accuracy, then extend this framework with quantization applied to both weights and activations, resulting in a highly efficient VLM. Collectively, we introduce~Weighted SVD (WSVD), which outperforms other approaches by achieving over 1.8times decoding speedup while preserving accuracy. We open source our code at: https://github.com/SAI-Lab-NYU/WSVD{https://github.com/SAI-Lab-NYU/WSVD

Recent works have empirically analyzed in-context learning and shown that transformers trained on synthetic linear regression tasks can learn to implement ridge regression, which is the Bayes-optimal predictor, given sufficient capacity [Aky\"urek et al., 2023], while one-layer transformers with linear self-attention and no MLP layer will learn to implement one step of gradient descent (GD) on a least-squares linear regression objective [von Oswald et al., 2022]. However, the theory behind these observations remains poorly understood. We theoretically study transformers with a single layer of linear self-attention, trained on synthetic noisy linear regression data. First, we mathematically show that when the covariates are drawn from a standard Gaussian distribution, the one-layer transformer which minimizes the pre-training loss will implement a single step of GD on the least-squares linear regression objective. Then, we find that changing the distribution of the covariates and weight vector to a non-isotropic Gaussian distribution has a strong impact on the learned algorithm: the global minimizer of the pre-training loss now implements a single step of pre-conditioned GD. However, if only the distribution of the responses is changed, then this does not have a large effect on the learned algorithm: even when the response comes from a more general family of nonlinear functions, the global minimizer of the pre-training loss still implements a single step of GD on a least-squares linear regression objective.

We consider the problem of learning a target function corresponding to a single hidden layer neural network, with a quadratic activation function after the first layer, and random weights. We consider the asymptotic limit where the input dimension and the network width are proportionally large. Recent work [Cui & al '23] established that linear regression provides Bayes-optimal test error to learn such a function when the number of available samples is only linear in the dimension. That work stressed the open challenge of theoretically analyzing the optimal test error in the more interesting regime where the number of samples is quadratic in the dimension. In this paper, we solve this challenge for quadratic activations and derive a closed-form expression for the Bayes-optimal test error. We also provide an algorithm, that we call GAMP-RIE, which combines approximate message passing with rotationally invariant matrix denoising, and that asymptotically achieves the optimal performance. Technically, our result is enabled by establishing a link with recent works on optimal denoising of extensive-rank matrices and on the ellipsoid fitting problem. We further show empirically that, in the absence of noise, randomly-initialized gradient descent seems to sample the space of weights, leading to zero training loss, and averaging over initialization leads to a test error equal to the Bayes-optimal one.

Parameter-Efficient Fine-Tuning (PEFT) has emerged as a practical paradigm for adapting large language models (LLMs) without updating all parameters. Most existing approaches, such as LoRA and PiSSA, rely on low-rank decompositions of weight updates. However, the low-rank assumption may restrict expressivity, particularly in task-specific adaptation scenarios where singular values are distributed relatively uniformly. To address this limitation, we propose CoSA (Compressed Sensing-Based Adaptation), a new PEFT method extended from compressed sensing theory. Instead of constraining weight updates to a low-rank subspace, CoSA expresses them through fixed random projection matrices and a compact learnable core. We provide a formal theoretical analysis of CoSA as a synthesis process, proving that weight updates can be compactly encoded into a low-dimensional space and mapped back through random projections. Extensive experimental results show that CoSA provides a principled perspective for efficient and expressive multi-scale model adaptation. Specifically, we evaluate CoSA on 10 diverse tasks, including natural language understanding and generation, employing 5 models of different scales from RoBERTa, Llama, and Qwen families. Across these settings, CoSA consistently matches or outperforms state-of-the-art PEFT methods.

Large pre-trained models (LPMs) have demonstrated exceptional performance in diverse natural language processing and computer vision tasks. However, fully fine-tuning these models poses substantial memory challenges, particularly in resource-constrained environments. Parameter-efficient fine-tuning (PEFT) methods, such as LoRA, mitigate this issue by adjusting only a small subset of parameters. Nevertheless, these methods typically employ random initialization for low-rank matrices, which can lead to inefficiencies in gradient descent and diminished generalizability due to suboptimal starting points. To address these limitations, we propose SVFit, a novel PEFT approach that leverages singular value decomposition (SVD) to initialize low-rank matrices using critical singular values as trainable parameters. Specifically, SVFit performs SVD on the pre-trained weight matrix to obtain the best rank-r approximation matrix, emphasizing the most critical singular values that capture over 99% of the matrix's information. These top-r singular values are then used as trainable parameters to scale the fundamental subspaces of the matrix, facilitating rapid domain adaptation. Extensive experiments across various pre-trained models in natural language understanding, text-to-image generation, and image classification tasks reveal that SVFit outperforms LoRA while requiring 16 times fewer trainable parameters.

In our era of enormous neural networks, empirical progress has been driven by the philosophy that more is better. Recent deep learning practice has found repeatedly that larger model size, more data, and more computation (resulting in lower training loss) improves performance. In this paper, we give theoretical backing to these empirical observations by showing that these three properties hold in random feature (RF) regression, a class of models equivalent to shallow networks with only the last layer trained. Concretely, we first show that the test risk of RF regression decreases monotonically with both the number of features and the number of samples, provided the ridge penalty is tuned optimally. In particular, this implies that infinite width RF architectures are preferable to those of any finite width. We then proceed to demonstrate that, for a large class of tasks characterized by powerlaw eigenstructure, training to near-zero training loss is obligatory: near-optimal performance can only be achieved when the training error is much smaller than the test error. Grounding our theory in real-world data, we find empirically that standard computer vision tasks with convolutional neural tangent kernels clearly fall into this class. Taken together, our results tell a simple, testable story of the benefits of overparameterization, overfitting, and more data in random feature models.

Adaptive rounding methods such as GPTQ, or equivalently Babai's nearest plane algorithm, round a real matrix to integers under a quadratic metric. They process the entries in a fixed order, one at a time, propagating each rounding error to the entries not yet processed through a triangular feedback matrix. We study the two-sided version of this task, in which fixed nonsingular basis matrices act on both the left and the right of the residual; the familiar one-sided case is the special case of an identity right basis. Vectorizing the matrix turns the two-sided objective into a quadratic metric whose Gram matrix is a Kronecker product, so the one-dimensional algorithm applies verbatim, but takes quartic time in the matrix dimension. We present GPTQ-2D, which produces the identical rounded matrix in cubic time. It rounds the entries anti-diagonal by anti-diagonal; entries on the same anti-diagonal are independent and are rounded in parallel.

Current exposure correction methods have three challenges, labor-intensive paired data annotation, limited generalizability, and performance degradation in low-level computer vision tasks. In this work, we introduce an innovative Unsupervised Exposure Correction (UEC) method that eliminates the need for manual annotations, offers improved generalizability, and enhances performance in low-level downstream tasks. Our model is trained using freely available paired data from an emulated Image Signal Processing (ISP) pipeline. This approach does not need expensive manual annotations, thereby minimizing individual style biases from the annotation and consequently improving its generalizability. Furthermore, we present a large-scale Radiometry Correction Dataset, specifically designed to emphasize exposure variations, to facilitate unsupervised learning. In addition, we develop a transformation function that preserves image details and outperforms state-of-the-art supervised methods [12], while utilizing only 0.01% of their parameters. Our work further investigates the broader impact of exposure correction on downstream tasks, including edge detection, demonstrating its effectiveness in mitigating the adverse effects of poor exposure on low-level features. The source code and dataset are publicly available at https://github.com/BeyondHeaven/uec_code.

This paper presents the Adaptive Personalized Control System (APECS) architecture, a novel framework for human-in-the-loop control. An architecture is developed which defines appropriate constraints for the system objectives. A method for enacting Lipschitz and sector bounds on the resulting controller is derived to ensure desirable control properties. An analysis of worst-case loss functions and the optimal loss function weighting is made to implement an effective training scheme. Finally, simulations are carried out to demonstrate the effectiveness of the proposed architecture. This architecture resulted in a 4.5% performance increase compared to the human operator and 9% to an unconstrained feedforward neural network trained in the same way.

In this paper, we introduce a novel theoretical framework for multi-task regression, applying random matrix theory to provide precise performance estimations, under high-dimensional, non-Gaussian data distributions. We formulate a multi-task optimization problem as a regularization technique to enable single-task models to leverage multi-task learning information. We derive a closed-form solution for multi-task optimization in the context of linear models. Our analysis provides valuable insights by linking the multi-task learning performance to various model statistics such as raw data covariances, signal-generating hyperplanes, noise levels, as well as the size and number of datasets. We finally propose a consistent estimation of training and testing errors, thereby offering a robust foundation for hyperparameter optimization in multi-task regression scenarios. Experimental validations on both synthetic and real-world datasets in regression and multivariate time series forecasting demonstrate improvements on univariate models, incorporating our method into the training loss and thus leveraging multivariate information.

Many political science theories relate to latent variables, but such quantities cannot be observed directly and must instead be estimated from data with inherent uncertainty. In regression models, when a variable is measured with error, its slope coefficient is known to be biased toward zero. We show how measurement error interacts with unique aspects of latent variable estimation, identification restrictions in particular, and demonstrate how common error adjustment strategies can worsen bias. We introduce a method for adjusting coefficients on latent predictors, which reduces bias and typically increases the magnitude of estimated coefficients, often dramatically. We illustrate these dynamics using several different estimation strategies for the latent predictors. Corrected estimates using our proposed method show stronger relationships -- sometimes up to 50% larger -- than those from naive regression. Our findings highlight the importance of considering measurement error in latent predictors and the inadequacy of many commonly used approaches for dealing with this issue.

The complex nature of medical image segmentation calls for models that are specifically designed to capture detailed, domain-specific features. Large foundation models offer considerable flexibility, yet the cost of fine-tuning these models remains a significant barrier. Parameter-Efficient Fine-Tuning (PEFT) methods, such as Low-Rank Adaptation (LoRA), efficiently update model weights with low-rank matrices but may suffer from underfitting when the chosen rank is insufficient to capture domain-specific nuances. Conversely, full-rank Singular Value Decomposition (SVD) based methods provide comprehensive updates by modifying all singular values, yet they often lack flexibility and exhibit variable performance across datasets. We propose SALT (Singular Value Adaptation with Low-Rank Transformation), a method that selectively adapts the most influential singular values using trainable scale and shift parameters while complementing this with a low-rank update for the remaining subspace. This hybrid approach harnesses the advantages of both LoRA and SVD, enabling effective adaptation without relying on increasing model size or depth. Evaluated on 5 challenging medical datasets, ranging from as few as 20 samples to 1000, SALT outperforms state-of-the-art PEFT (LoRA and SVD) by 2% to 5% in Dice with only 3.9% trainable parameters, demonstrating robust adaptation even in low-resource settings. The code for SALT is available at: https://github.com/BioMedIA-MBZUAI/SALT

Data-driven algorithm design automates hyperparameter tuning, but its statistical foundations remain limited because model performance can depend on hyperparameters in implicit and highly non-smooth ways. Existing guarantees focus on the simple case of a one-dimensional (scalar) hyperparameter. This leaves the practically important, multi-dimensional hyperparameter tuning setting unresolved. We address this open question by establishing the first general framework for establishing generalization guarantees for tuning multi-dimensional hyperparameters in data-driven settings. Our approach strengthens the generalization guarantee framework for semi-algebraic function classes by exploiting tools from real algebraic geometry, yielding sharper, more broadly applicable guarantees. For completeness, we also instantiate the first lower bound for this general setting. We further extend the analysis to hyperparameter tuning using the validation loss under minimal assumptions, and derive improved bounds when additional structure is available. Finally, we demonstrate the scope of the framework with new learnability results, including data-driven weighted group lasso and weighted fused lasso.

The optimal stopping problem is a category of decision problems with a specific constrained configuration. It is relevant to various real-world applications such as finance and management. To solve the optimal stopping problem, state-of-the-art algorithms in dynamic programming, such as the least-squares Monte Carlo (LSMC), are employed. This type of algorithm relies on path simulations using only the last price of the underlying asset as a state representation. Also, the LSMC was thinking for option valuation where risk-neutral probabilities can be employed to account for uncertainty. However, the general optimal stopping problem goals may not fit the requirements of the LSMC showing auto-correlated prices. We employ a data-driven method that uses Monte Carlo simulation to train and test artificial neural networks (ANN) to solve the optimal stopping problem. Using ANN to solve decision problems is not entirely new. We propose a different architecture that uses convolutional neural networks (CNN) to deal with the dimensionality problem that arises when we transform the whole history of prices into a Markovian state. We present experiments that indicate that our proposed architecture improves results over the previous implementations under specific simulated time series function sets. Lastly, we employ our proposed method to compare the optimal exercise of the financial options problem with the LSMC algorithm. Our experiments show that our method can capture more accurate exercise opportunities when compared to the LSMC. We have outstandingly higher (above 974\% improvement) expected payoff from these exercise policies under the many Monte Carlo simulations that used the real-world return database on the out-of-sample (test) data.

The rapid advancement in large language models (LLMs) comes with a significant increase in their parameter size, presenting challenges for adaptation and fine-tuning. Parameter-efficient fine-tuning (PEFT) methods are widely used to adapt LLMs for downstream tasks efficiently. In this paper, we propose Singular Values and Orthonormal Regularized Singular Vectors Adaptation, or SORSA, a novel PEFT method. We introduce a method to analyze the variation of the parameters by performing singular value decomposition (SVD) and discuss and analyze SORSA's superiority in minimizing the alteration in the SVD aspect. Each SORSA adapter consists of two main parts: trainable principal singular weights W_p = U_p Sigma_p V^top_p, and frozen residual weights W_r = U_r Sigma_r V^top_r. These parts are initialized by performing SVD on pre-trained weights. Moreover, we implement and analyze an orthonormal regularizer, which could effectively transfer the scaling information into Sigma_p and ultimately allows the training process to be more efficient. SORSA adapters could be merged during inference, thus eliminating any inference latency. After all, SORSA shows a faster convergence than PiSSA and LoRA in our experiments. On the MATH benchmark, Llama 2 7B adapted using SORSA achieved 10.36% accuracy, outperforming LoRA (5.50%), Full FT (7.22%), and PiSSA (7.44%). On the GSM-8K benchmark, SORSA achieved 56.03% accuracy, surpassing LoRA (42.30%), Full FT (49.05%), and PiSSA (53.07%). We conclude that SORSA offers a new perspective on parameter-efficient fine-tuning, demonstrating remarkable performance. The code is available at https://github.com/Gunale0926/SORSA.

In recent years, Parameter-Efficient Fine-Tuning (PEFT) methods like Low-Rank Adaptation (LoRA) have significantly enhanced the adaptability of large-scale pre-trained models. Weight-Decomposed Low-Rank Adaptation (DoRA) improves upon LoRA by separating the magnitude and direction components of the weight matrix, leading to superior performance. However, DoRA's improvements are limited to the vertical dimension, resulting in an asymmetrical pattern between horizontal and vertical dimensions. This paper introduces BoRA, an innovative extension of LoRA and DoRA, characterized by symmetrical properties across horizontal and vertical dimensions. Our approach optimizes the weight matrix symmetrically by adjusting both column-wise and row-wise magnitudes. Extensive experiments demonstrate that BoRA surpasses state-of-the-art PEFT methods, including LoRA and DoRA, achieving superior results across various benchmarks.

During lung radiotherapy, the position of infrared reflective objects on the chest can be recorded to estimate the tumor location. However, radiotherapy systems have a latency inherent to robot control limitations that impedes the radiation delivery precision. Prediction with online learning of recurrent neural networks (RNN) allows for adaptation to non-stationary respiratory signals, but classical methods such as RTRL and truncated BPTT are respectively slow and biased. This study investigates the capabilities of unbiased online recurrent optimization (UORO) to forecast respiratory motion and enhance safety in lung radiotherapy. We used 9 observation records of the 3D position of 3 external markers on the chest and abdomen of healthy individuals breathing during intervals from 73s to 222s. The sampling frequency was 10Hz, and the amplitudes of the recorded trajectories range from 6mm to 40mm in the superior-inferior direction. We forecast the 3D location of each marker simultaneously with a horizon value between 0.1s and 2.0s, using an RNN trained with UORO. We compare its performance with an RNN trained with RTRL, LMS, and offline linear regression. We provide closed-form expressions for quantities involved in the loss gradient calculation in UORO, thereby making its implementation efficient. Training and cross-validation were performed during the first minute of each sequence. On average over the horizon values considered and the 9 sequences, UORO achieves the lowest root-mean-square (RMS) error and maximum error among the compared algorithms. These errors are respectively equal to 1.3mm and 8.8mm, and the prediction time per time step was lower than 2.8ms (Dell Intel core i9-9900K 3.60 GHz). Linear regression has the lowest RMS error for the horizon values 0.1s and 0.2s, followed by LMS for horizon values between 0.3s and 0.5s, and UORO for horizon values greater than 0.6s.

Self-distillation (SD) is the process of retraining a student on a mixture of ground-truth labels and the teacher's own predictions using the same architecture and training data. Although SD has been empirically shown to often improve generalization, its formal guarantees remain limited. We study SD for ridge regression in unconstrained setting in which the mixing weight ξ may be outside the unit interval. Conditioned on the training data and without any distributional assumptions, we prove that for any squared prediction risk (including out-of-distribution), the optimally mixed student strictly improves upon the ridge teacher for every regularization level λ> 0 at which the teacher ridge risk R(λ) is nonstationary (i.e., R'(λ) neq 0). We obtain a closed-form expression for the optimal mixing weight ξ^star(λ) for any value of λ and show that it obeys the sign rule: sign(ξ^star(λ))=-sign(R'(λ)). In particular, ξ^star(λ) can be negative, which is the case in over-regularized regimes. To quantify the risk improvement due to SD, we derive exact deterministic equivalents for the optimal SD risk in the proportional asymptotics regime (where the sample and feature sizes n and p both diverge but their aspect ratio p/n converges) under general anisotropic covariance and deterministic signals. Our asymptotic analysis extends standard second-order ridge deterministic equivalents to their fourth-order analogs using block linearization, which may be of independent interest. From a practical standpoint, we propose a consistent one-shot tuning method to estimate ξ^star without grid search, sample splitting, or refitting. Experiments on real-world datasets and pretrained neural network features support our theory and the one-shot tuning method.

Post-Training Quantization (PTQ) has emerged as an effective technique for alleviating the substantial computational and memory overheads of Vision-Language Models (VLMs) by compressing both weights and activations without retraining the full model. Existing PTQ methods primarily rely on static identification and global compensation of sensitive or outlier channels, yet they often overlook the distributional differences of these important channels across inputs, leading to unsatisfactory quantization. In this work, we observe that the distributions and occurrence frequencies of important channels vary significantly both across modalities and among tokens, even within the same modality. Accordingly, we propose Quant Experts (QE), a token-aware adaptive error compensation with mixture-of-experts for VLMs quantization. QE divides the important channels into token-independent and token-dependent groups. For the former, a shared expert is designed for most tokens to compensate for global quantization error using a low-rank adapter. For the latter, routed experts including multiple routed low-rank adapters are elaborated to compensate for local quantization error related to specific tokens. Extensive experiments demonstrate that QE consistently enhances task accuracy across various quantization settings and model scales, ranging from 2B to 70B parameters, while maintaining performance comparable to full-precision models.

Regression trees are among the most interpretable yet expressive model classes in machine learning. Historically, greedy induction has been the dominant approach for constructing well-performing regression trees. While optimal methods based on dynamic programming and branch-and-bound exist, they are computationally prohibitive for general linear regression trees, despite often achieving substantially better performance than greedy approaches. Recent work has shown that specialized lookahead strategies can dramatically improve runtime while maintaining near-optimal performance, primarily in classification settings. In this work, we develop a novel algorithm for near-optimal, sparse, piecewise linear regression trees that combines a lookahead-style search strategy with efficient rank-one Cholesky updates of the Gram matrix. We demonstrate, both theoretically and empirically, that our method achieves a favorable trade-off between computational efficiency, predictive accuracy, and sparsity, and scales significantly better than the current state of the art.

Non-line-of-sight (NLOS) imaging is conducted to infer invisible scenes from indirect light on visible objects. The neural transient field (NeTF) was proposed for representing scenes as neural radiance fields in NLOS scenes. We propose NLOS neural implicit surface (NLOS-NeuS), which extends the NeTF to neural implicit surfaces with a signed distance function (SDF) for reconstructing three-dimensional surfaces in NLOS scenes. We introduce two constraints as loss functions for correctly learning an SDF to avoid non-zero level-set surfaces. We also introduce a lower bound constraint of an SDF based on the geometry of the first-returning photons. The experimental results indicate that these constraints are essential for learning a correct SDF in NLOS scenes. Compared with previous methods with discretized representation, NLOS-NeuS with the neural continuous representation enables us to reconstruct smooth surfaces while preserving fine details in NLOS scenes. To the best of our knowledge, this is the first study on neural implicit surfaces with volume rendering in NLOS scenes.

Zeroth-order optimization addresses problems where gradient information is inaccessible or impractical to compute. While most existing methods rely on first-order approximations, incorporating second-order (curvature) information can, in principle, significantly accelerate convergence. However, the high cost of function evaluations required to estimate Hessian matrices often limits practical applicability. We present the subspace-based approximate Hessian (ZO-SAH) method, a zeroth-order optimization algorithm that mitigates these costs by focusing on randomly selected two-dimensional subspaces. Within each subspace, ZO-SAH estimates the Hessian by fitting a quadratic polynomial to the objective function and extracting its second-order coefficients. To further reduce function-query costs, ZO-SAH employs a periodic subspace-switching strategy that reuses function evaluations across optimization steps. Experiments on eight benchmark datasets, including logistic regression and deep neural network training tasks, demonstrate that ZO-SAH achieves significantly faster convergence than existing zeroth-order methods.

This paper presents the results of the fourth edition of the Monocular Depth Estimation Challenge (MDEC), which focuses on zero-shot generalization to the SYNS-Patches benchmark, a dataset featuring challenging environments in both natural and indoor settings. In this edition, we revised the evaluation protocol to use least-squares alignment with two degrees of freedom to support disparity and affine-invariant predictions. We also revised the baselines and included popular off-the-shelf methods: Depth Anything v2 and Marigold. The challenge received a total of 24 submissions that outperformed the baselines on the test set; 10 of these included a report describing their approach, with most leading methods relying on affine-invariant predictions. The challenge winners improved the 3D F-Score over the previous edition's best result, raising it from 22.58% to 23.05%.

Language Models (LLMs) are often quantized to lower precision to reduce the memory cost and latency in inference. However, quantization often degrades model performance, thus fine-tuning is required for various down-stream tasks. Traditional fine-tuning methods such as stochastic gradient descent and Adam optimization require backpropagation, which are error-prone in the low-precision settings. To overcome these limitations, we propose the Quantized Zeroth-Order (QuZO) framework, specifically designed for fine-tuning LLMs through low-precision (e.g., 4- or 8-bit) forward passes. Our method can avoid the error-prone low-precision straight-through estimator, and utilizes optimized stochastic rounding to mitigate the increased bias. QuZO simplifies the training process, while achieving results comparable to first-order methods in {rm FP}8 and superior accuracy in {rm INT}8 and {rm INT}4 training. Experiments demonstrate that low-bit training QuZO achieves performance comparable to MeZO optimization on GLUE, Multi-Choice, and Generation tasks, while reducing memory cost by 2.94 times in LLaMA2-7B fine-tuning compared to quantized first-order methods.