Daily Papers

Optimal transport is an important tool in machine learning, allowing to capture geometric properties of the data through a linear program on transport polytopes. We present a single-loop optimization algorithm for minimizing general convex objectives on these domains, utilizing the principles of Sinkhorn matrix scaling and mirror descent. The proposed algorithm is robust to noise, and can be used in an online setting. We provide theoretical guarantees for convex objectives and experimental results showcasing it effectiveness on both synthetic and real-world data.

Computing the optimal transport distance between statistical distributions is a fundamental task in machine learning. One remarkable recent advancement is entropic regularization and the Sinkhorn algorithm, which utilizes only matrix scaling and guarantees an approximated solution with near-linear runtime. Despite the success of the Sinkhorn algorithm, its runtime may still be slow due to the potentially large number of iterations needed for convergence. To achieve possibly super-exponential convergence, we present Sinkhorn-Newton-Sparse (SNS), an extension to the Sinkhorn algorithm, by introducing early stopping for the matrix scaling steps and a second stage featuring a Newton-type subroutine. Adopting the variational viewpoint that the Sinkhorn algorithm maximizes a concave Lyapunov potential, we offer the insight that the Hessian matrix of the potential function is approximately sparse. Sparsification of the Hessian results in a fast O(n^2) per-iteration complexity, the same as the Sinkhorn algorithm. In terms of total iteration count, we observe that the SNS algorithm converges orders of magnitude faster across a wide range of practical cases, including optimal transportation between empirical distributions and calculating the Wasserstein W_1, W_2 distance of discretized densities. The empirical performance is corroborated by a rigorous bound on the approximate sparsity of the Hessian matrix.

The recent emergence of new algorithms for permuting models into functionally equivalent regions of the solution space has shed some light on the complexity of error surfaces, and some promising properties like mode connectivity. However, finding the right permutation is challenging, and current optimization techniques are not differentiable, which makes it difficult to integrate into a gradient-based optimization, and often leads to sub-optimal solutions. In this paper, we propose a Sinkhorn re-basin network with the ability to obtain the transportation plan that better suits a given objective. Unlike the current state-of-art, our method is differentiable and, therefore, easy to adapt to any task within the deep learning domain. Furthermore, we show the advantage of our re-basin method by proposing a new cost function that allows performing incremental learning by exploiting the linear mode connectivity property. The benefit of our method is compared against similar approaches from the literature, under several conditions for both optimal transport finding and linear mode connectivity. The effectiveness of our continual learning method based on re-basin is also shown for several common benchmark datasets, providing experimental results that are competitive with state-of-art results from the literature.

A new class of affine scaling matrices for the interior point Newton-type methods is considered to solve the nonlinear systems with simple bounds. We review the essential properties of a scaling matrix and consider several well-known scaling matrices proposed in the literature. We define a new scaling matrix that is the convex combination of these matrices. The proposed scaling matrix inherits those interesting properties of the individual matrices and satisfies additional desired requirements. The numerical experiments demonstrate the superiority of the new scaling matrix in solving several important test problems.

Optimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another. Recent works have drawn inspiration from Brenier's theorem, which states that when the ground cost is the squared-Euclidean distance, the ``best'' map to morph a continuous measure in P(Rd) into another must be the gradient of a convex function. To exploit that result, [Makkuva+ 2020, Korotin+2020] consider maps T=nabla f_theta, where f_theta is an input convex neural network (ICNN), as defined by Amos+2017, and fit theta with SGD using samples. Despite their mathematical elegance, fitting OT maps with ICNNs raises many challenges, due notably to the many constraints imposed on theta; the need to approximate the conjugate of f_theta; or the limitation that they only work for the squared-Euclidean cost. More generally, we question the relevance of using Brenier's result, which only applies to densities, to constrain the architecture of candidate maps fitted on samples. Motivated by these limitations, we propose a radically different approach to estimating OT maps: Given a cost c and a reference measure rho, we introduce a regularizer, the Monge gap M^c_{rho}(T) of a map T. That gap quantifies how far a map T deviates from the ideal properties we expect from a c-OT map. In practice, we drop all architecture requirements for T and simply minimize a distance (e.g., the Sinkhorn divergence) between Tsharpmu and nu, regularized by M^c_rho(T). We study M^c_{rho}, and show how our simple pipeline outperforms significantly other baselines in practice.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

Dirichlet Process mixture models (DPMM) in combination with Gaussian kernels have been an important modeling tool for numerous data domains arising from biological, physical, and social sciences. However, this versatility in applications does not extend to strong theoretical guarantees for the underlying parameter estimates, for which only a logarithmic rate is achieved. In this work, we (re)introduce and investigate a metric, named Orlicz-Wasserstein distance, in the study of the Bayesian contraction behavior for the parameters. We show that despite the overall slow convergence guarantees for all the parameters, posterior contraction for parameters happens at almost polynomial rates in outlier regions of the parameter space. Our theoretical results provide new insight in understanding the convergence behavior of parameters arising from various settings of hierarchical Bayesian nonparametric models. In addition, we provide an algorithm to compute the metric by leveraging Sinkhorn divergences and validate our findings through a simulation study.

We study the matrix models pi:C(S_N^+)to M_N(C(X)) which are flat, in the sense that the standard generators of C(S_N^+) are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at N=4, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of C(S_N^+), inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.

3D Gaussian Splatting (3DGS) has demonstrated its potential in reconstructing scenes from unposed images. However, optimization-based 3DGS methods struggle with sparse views due to limited prior knowledge. Meanwhile, feed-forward Gaussian approaches are constrained by input formats, making it challenging to incorporate more input views. To address these challenges, we propose RegGS, a 3D Gaussian registration-based framework for reconstructing unposed sparse views. RegGS aligns local 3D Gaussians generated by a feed-forward network into a globally consistent 3D Gaussian representation. Technically, we implement an entropy-regularized Sinkhorn algorithm to efficiently solve the optimal transport Mixture 2-Wasserstein (MW_2) distance, which serves as an alignment metric for Gaussian mixture models (GMMs) in Sim(3) space. Furthermore, we design a joint 3DGS registration module that integrates the MW_2 distance, photometric consistency, and depth geometry. This enables a coarse-to-fine registration process while accurately estimating camera poses and aligning the scene. Experiments on the RE10K and ACID datasets demonstrate that RegGS effectively registers local Gaussians with high fidelity, achieving precise pose estimation and high-quality novel-view synthesis. Project page: https://3dagentworld.github.io/reggs/.

Projection maintenance is one of the core data structure tasks. Efficient data structures for projection maintenance have led to recent breakthroughs in many convex programming algorithms. In this work, we further extend this framework to the Kronecker product structure. Given a constraint matrix {sf A} and a positive semi-definite matrix Win R^{ntimes n} with a sparse eigenbasis, we consider the task of maintaining the projection in the form of {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}, where {sf B}={sf A}(Wotimes I) or {sf B}={sf A}(W^{1/2}otimes W^{1/2}). At each iteration, the weight matrix W receives a low rank change and we receive a new vector h. The goal is to maintain the projection matrix and answer the query {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}h with good approximation guarantees. We design a fast dynamic data structure for this task and it is robust against an adaptive adversary. Following the beautiful and pioneering work of [Beimel, Kaplan, Mansour, Nissim, Saranurak and Stemmer, STOC'22], we use tools from differential privacy to reduce the randomness required by the data structure and further improve the running time.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

Empirical scaling laws prescribe how to allocate parameters, data, and compute, while maximal-update parameterization (muP) enables learning-rate transfer across widths by equalizing early-time update magnitudes. However, in modern scale-invariant architectures, training quickly enters an optimizer-governed steady state where normalization layers create backward scale sensitivity and the effective learning rate becomes width dependent, degrading muP transfer. We address this by introducing a weight-decay scaling rule for AdamW that preserves sublayer gain across widths. Empirically, the singular-value spectrum of each matrix parameter scales in norm as eta/lambda with an approximately invariant shape; under width scaling d, we observe that the top singular value scales approximately as eta/lambdacdot d^{0.75}. Combining this observation with the muP learning-rate rule eta_2propto d^{-1} for matrix-like parameters implies an empirical weight-decay scaling rule lambda_2propto d that approximately keeps sublayer gains width invariant. Together with vector-like parameters trained at eta_1=Theta_d(1) and lambda_1=0, this yields zero-shot transfer of both learning rate and weight decay from proxy to target widths, removing per-width sweeps. We validate the rule on LLaMA-style Transformers and in a minimal synthetic setting, and we provide a simple diagnostic, matching top singular values, to check sublayer-gain invariance. Our results extend muP beyond the near-init regime by explicitly controlling steady-state scales set by the optimizer, offering a practical recipe for width-robust hyperparameter transfer under AdamW.

Popular parameter-efficient fine-tuning (PEFT) methods, such as LoRA and its variants, freeze pre-trained model weights \(W\) and inject learnable matrices \(\Delta W\). These \(\Delta W\) matrices are structured for efficient parameterization, often using techniques like low-rank approximations or scaling vectors. However, these methods typically show a performance gap compared to full fine-tuning. Although recent PEFT methods have narrowed this gap, they do so at the cost of additional learnable parameters. We propose SVFT, a simple approach that fundamentally differs from existing methods: the structure imposed on \(\Delta W\) depends on the specific weight matrix \(W\). Specifically, SVFT updates \(W\) as a sparse combination of outer products of its singular vectors, training only the coefficients (scales) of these sparse combinations. This approach allows fine-grained control over expressivity through the number of coefficients. Extensive experiments on language and vision benchmarks show that SVFT recovers up to 96% of full fine-tuning performance while training only 0.006 to 0.25% of parameters, outperforming existing methods that only recover up to 85% performance using 0.03 to 0.8% of the trainable parameter budget.

Diagnosing deep neural networks (DNNs) through the eigenspectrum of weight matrices has been an active area of research in recent years. At a high level, eigenspectrum analysis of DNNs involves measuring the heavytailness of the empirical spectral densities (ESD) of weight matrices. It provides insight into how well a model is trained and can guide decisions on assigning better layer-wise training hyperparameters. In this paper, we address a challenge associated with such eigenspectrum methods: the impact of the aspect ratio of weight matrices on estimated heavytailness metrics. We demonstrate that matrices of varying sizes (and aspect ratios) introduce a non-negligible bias in estimating heavytailness metrics, leading to inaccurate model diagnosis and layer-wise hyperparameter assignment. To overcome this challenge, we propose FARMS (Fixed-Aspect-Ratio Matrix Subsampling), a method that normalizes the weight matrices by subsampling submatrices with a fixed aspect ratio. Instead of measuring the heavytailness of the original ESD, we measure the average ESD of these subsampled submatrices. We show that measuring the heavytailness of these submatrices with the fixed aspect ratio can effectively mitigate the aspect ratio bias. We validate our approach across various optimization techniques and application domains that involve eigenspectrum analysis of weights, including image classification in computer vision (CV) models, scientific machine learning (SciML) model training, and large language model (LLM) pruning. Our results show that despite its simplicity, FARMS uniformly improves the accuracy of eigenspectrum analysis while enabling more effective layer-wise hyperparameter assignment in these application domains. In one of the LLM pruning experiments, FARMS reduces the perplexity of the LLaMA-7B model by 17.3% when compared with the state-of-the-art method.

Our knowledge of supermassive black holes (SMBHs) and their relation to their host galaxies is still limited, and there are only around 150 SMBHs that have their masses directly measured and confirmed. Better black hole mass scaling relations will help us reveal the physics of black holes, as well as predict black hole masses that are not yet measured. Here, we apply symbolic regression, combined with random forest to those directly-measured black hole masses and host galaxy properties, and find a collection of higher-dimensional (N-D) black hole mass scaling relations. These N-D black hole mass scaling relations have scatter smaller than any of the existing black hole mass scaling relations. One of the best among them involves the parameters of central stellar velocity dispersion, bulge-to-total ratio, and density at the black hole's sphere-of-influence with an intrinsic scatter of epsilon=0.083, dex, significantly lower than epsilon sim 0.3, dex for the M-sigma relation. These relations will inspire black hole physics, test black hole models implemented in simulations, and estimate unknown black hole masses on an unprecedented precision.

Vector-quantized networks (VQNs) have exhibited remarkable performance across various tasks, yet they are prone to training instability, which complicates the training process due to the necessity for techniques such as subtle initialization and model distillation. In this study, we identify the local minima issue as the primary cause of this instability. To address this, we integrate an optimal transport method in place of the nearest neighbor search to achieve a more globally informed assignment. We introduce OptVQ, a novel vector quantization method that employs the Sinkhorn algorithm to optimize the optimal transport problem, thereby enhancing the stability and efficiency of the training process. To mitigate the influence of diverse data distributions on the Sinkhorn algorithm, we implement a straightforward yet effective normalization strategy. Our comprehensive experiments on image reconstruction tasks demonstrate that OptVQ achieves 100% codebook utilization and surpasses current state-of-the-art VQNs in reconstruction quality.

Dense linear layers are the dominant computational bottleneck in foundation models. Identifying more efficient alternatives to dense matrices has enormous potential for building more compute-efficient models, as exemplified by the success of convolutional networks in the image domain. In this work, we systematically explore structured matrices as replacements for dense matrices. We show that different structures often require drastically different initialization scales and learning rates, which are crucial to performance, especially as models scale. Using insights from the Maximal Update Parameterization, we determine the optimal scaling for initialization and learning rates of these unconventional layers. Finally, we measure the scaling laws of different structures to compare how quickly their performance improves with compute. We propose a novel matrix family containing Monarch matrices, the Block Tensor-Train (BTT), which we show performs better than dense matrices for the same compute on multiple tasks. On CIFAR-10/100 with augmentation, BTT achieves exponentially lower training loss than dense when training MLPs and ViTs. BTT matches dense ViT-S/32 performance on ImageNet-1k with 3.8 times less compute and is more efficient than dense for training small GPT-2 language models.

In this study, we propose Feature-aligned N-BEATS as a domain generalization model for univariate time series forecasting problems. The proposed model is an extension of the doubly residual stacking architecture of N-BEATS (Oreshkin et al. [34]) into a representation learning framework. The model is a new structure that involves marginal feature probability measures (i.e., pushforward measures of multiple source domains) induced by the intricate composition of residual operators of N-BEATS in each stack and aligns them stack-wise via an entropic regularized Wasserstein distance referred to as the Sinkhorn divergence (Genevay et al. [14]). The loss function consists of a typical forecasting loss for multiple source domains and an alignment loss calculated with the Sinkhorn divergence, which allows the model to learn invariant features stack-wise across multiple source data sequences while retaining N-BEATS's interpretable design. We conduct a comprehensive experimental evaluation of the proposed approach and the results demonstrate the model's forecasting and generalization capabilities in comparison with methods based on the original N-BEATS.

We propose ScaledGD(\lambda), a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor representations, ScaledGD(\lambda) starts from a small random initialization, and proceeds by gradient descent with a specific form of damped preconditioning to combat bad curvatures induced by overparameterization and ill-conditioning. At the expense of light computational overhead incurred by preconditioners, ScaledGD(\lambda) is remarkably robust to ill-conditioning compared to vanilla gradient descent (GD) even with overprameterization. Specifically, we show that, under the Gaussian design, ScaledGD(\lambda) converges to the true low-rank matrix at a constant linear rate after a small number of iterations that scales only logarithmically with respect to the condition number and the problem dimension. This significantly improves over the convergence rate of vanilla GD which suffers from a polynomial dependency on the condition number. Our work provides evidence on the power of preconditioning in accelerating the convergence without hurting generalization in overparameterized learning.

The rapid growth of model scale has necessitated substantial computational resources for fine-tuning. Existing approach such as Low-Rank Adaptation (LoRA) has sought to address the problem of handling the large updated parameters in full fine-tuning. However, LoRA utilize random initialization and optimization of low-rank matrices to approximate updated weights, which can result in suboptimal convergence and an accuracy gap compared to full fine-tuning. To address these issues, we propose LoLDU, a Parameter-Efficient Fine-Tuning (PEFT) approach that significantly reduces trainable parameters by 2600 times compared to regular PEFT methods while maintaining comparable performance. LoLDU leverages Lower-Diag-Upper Decomposition (LDU) to initialize low-rank matrices for faster convergence and orthogonality. We focus on optimizing the diagonal matrix for scaling transformations. To the best of our knowledge, LoLDU has the fewest parameters among all PEFT approaches. We conducted extensive experiments across 4 instruction-following datasets, 6 natural language understanding (NLU) datasets, 8 image classification datasets, and image generation datasets with multiple model types (LLaMA2, RoBERTa, ViT, and Stable Diffusion), providing a comprehensive and detailed analysis. Our open-source code can be accessed at https://github.com/SKDDJ/LoLDU{https://github.com/SKDDJ/LoLDU}.

Efficient LLM pre-training requires well-tuned hyperparameters (HPs), including learning rate {\eta} and weight decay {\lambda}. We study scaling laws for HPs: formulas for how to scale HPs as we scale model size N, dataset size D, and batch size B. Recent work suggests the AdamW timescale, B/({\eta}{\lambda}D), should remain constant across training settings, and we verify the implication that optimal {\lambda} scales linearly with B, for a fixed N,D. However, as N,D scale, we show the optimal timescale obeys a precise power law in the tokens-per-parameter ratio, D/N. This law thus provides a method to accurately predict {\lambda}opt in advance of large-scale training. We also study scaling laws for optimal batch size Bopt (the B enabling lowest loss at a given N,D) and critical batch size Bcrit (the B beyond which further data parallelism becomes ineffective). In contrast with prior work, we find both Bopt and Bcrit scale as power laws in D, independent of model size, N. Finally, we analyze how these findings inform the real-world selection of Pareto-optimal N and D under dual training time and compute objectives.

3D Gaussian Splatting (3DGS) is increasingly popular for 3D reconstruction due to its superior visual quality and rendering speed. However, 3DGS training currently occurs on a single GPU, limiting its ability to handle high-resolution and large-scale 3D reconstruction tasks due to memory constraints. We introduce Grendel, a distributed system designed to partition 3DGS parameters and parallelize computation across multiple GPUs. As each Gaussian affects a small, dynamic subset of rendered pixels, Grendel employs sparse all-to-all communication to transfer the necessary Gaussians to pixel partitions and performs dynamic load balancing. Unlike existing 3DGS systems that train using one camera view image at a time, Grendel supports batched training with multiple views. We explore various optimization hyperparameter scaling strategies and find that a simple sqrt(batch size) scaling rule is highly effective. Evaluations using large-scale, high-resolution scenes show that Grendel enhances rendering quality by scaling up 3DGS parameters across multiple GPUs. On the Rubble dataset, we achieve a test PSNR of 27.28 by distributing 40.4 million Gaussians across 16 GPUs, compared to a PSNR of 26.28 using 11.2 million Gaussians on a single GPU. Grendel is an open-source project available at: https://github.com/nyu-systems/Grendel-GS

Clinically deployed deep learning-based segmentation models are known to fail on data outside of their training distributions. While clinicians review the segmentations, these models tend to perform well in most instances, which could exacerbate automation bias. Therefore, detecting out-of-distribution images at inference is critical to warn the clinicians that the model likely failed. This work applied the Mahalanobis distance (MD) post hoc to the bottleneck features of four Swin UNETR and nnU-net models that segmented the liver on T1-weighted magnetic resonance imaging and computed tomography. By reducing the dimensions of the bottleneck features with either principal component analysis or uniform manifold approximation and projection, images the models failed on were detected with high performance and minimal computational load. In addition, this work explored a non-parametric alternative to the MD, a k-th nearest neighbors distance (KNN). KNN drastically improved scalability and performance over MD when both were applied to raw and average-pooled bottleneck features.

We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made important advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another interesting type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank-bounded matrices. For low-rank matrices the Hessian of this loss can theoretically blow up, which creates challenges to analyze convergence of optimizaton methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss and convergence results for finite step size gradient descent under certain assumptions on the initial weights.

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.

Analytical framework for predicting General Matrix Multiplication (GEMM) performance on modern GPUs, focusing on runtime, power consumption, and energy efficiency. Our study employs two approaches: a custom-implemented tiled matrix multiplication kernel for fundamental analysis, and NVIDIA's CUTLASS library for comprehensive performance data collection across advanced configurations. Using the NVIDIA RTX 4070 as our experimental platform, we developed a Random Forest-based prediction model with multi-output regression capability. Through analysis of both naive tiled matrix multiplication with varying tile sizes (1 to 32) and 16,128 CUTLASS GEMM operations across diverse configurations, we identified critical performance patterns related to matrix dimensions, thread block configurations, and memory access patterns. Our framework achieved exceptional accuracy with an R^2 score of 0.98 for runtime prediction (mean error 15.57%) and 0.78 for power prediction (median error 5.42%). The system successfully predicts performance across matrix sizes, demonstrating robust scaling behavior. Our results show that optimal tile size selection can improve performance by up to 3.2x while reducing power consumption by 22% compared to baseline configurations. Analysis of shared memory utilization and SM occupancy reveals that tile sizes of 16x16 achieve the best balance between parallelism and resource usage. The implementation of our framework, including prediction models and analysis tools, is available as an open-source project at GPPerf [https://github.com/pavlyhalim/GPPerf].

Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or in the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root. For the forward propagation, one method is to use Matrix Taylor Polynomial (MTP), and the other method is to use Matrix Pad\'e Approximants (MPA). The backward gradient is computed by iteratively solving the continuous-time Lyapunov equation using the matrix sign function. Both methods yield considerable speed-up compared with the SVD or the Newton-Schulz iteration. Experimental results on the de-correlated batch normalization and second-order vision transformer demonstrate that our methods can also achieve competitive and even slightly better performances. The code is available at https://github.com/KingJamesSong/FastDifferentiableMatSqrt{https://github.com/KingJamesSong/FastDifferentiableMatSqrt}.

High-performance sparse matrix-matrix (SpMM) multiplication is paramount for science and industry, as the ever-increasing sizes of data prohibit using dense data structures. Yet, existing hardware, such as Tensor Cores (TC), is ill-suited for SpMM, as it imposes strict constraints on data structures that cannot be met by unstructured sparsity found in many applications. To address this, we introduce (S)parse (Ma)trix Matrix (T)ensor Core-accelerated (SMaT): a novel SpMM library that utilizes TCs for unstructured sparse matrices. Our block-sparse library leverages the low-level CUDA MMA (matrix-matrix-accumulate) API, maximizing the performance offered by modern GPUs. Algorithmic optimizations such as sparse matrix permutation further improve performance by minimizing the number of non-zero blocks. The evaluation on NVIDIA A100 shows that SMaT outperforms SotA libraries (DASP, cuSPARSE, and Magicube) by up to 125x (on average 2.6x). SMaT can be used to accelerate many workloads in scientific computing, large-model training, inference, and others.

We study the estimation of a planted signal hidden in a recently introduced nested matrix-tensor model, which is an extension of the classical spiked rank-one tensor model, motivated by multi-view clustering. Prior work has theoretically examined the performance of a tensor-based approach, which relies on finding a best rank-one approximation, a problem known to be computationally hard. A tractable alternative approach consists in computing instead the best rank-one (matrix) approximation of an unfolding of the observed tensor data, but its performance was hitherto unknown. We quantify here the performance gap between these two approaches, in particular by deriving the precise algorithmic threshold of the unfolding approach and demonstrating that it exhibits a BBP-type transition behavior. This work is therefore in line with recent contributions which deepen our understanding of why tensor-based methods surpass matrix-based methods in handling structured tensor data.

We present a principled approach to uncover the structure of visual data by solving a novel deep learning task coined visual permutation learning. The goal of this task is to find the permutation that recovers the structure of data from shuffled versions of it. In the case of natural images, this task boils down to recovering the original image from patches shuffled by an unknown permutation matrix. Unfortunately, permutation matrices are discrete, thereby posing difficulties for gradient-based methods. To this end, we resort to a continuous approximation of these matrices using doubly-stochastic matrices which we generate from standard CNN predictions using Sinkhorn iterations. Unrolling these iterations in a Sinkhorn network layer, we propose DeepPermNet, an end-to-end CNN model for this task. The utility of DeepPermNet is demonstrated on two challenging computer vision problems, namely, (i) relative attributes learning and (ii) self-supervised representation learning. Our results show state-of-the-art performance on the Public Figures and OSR benchmarks for (i) and on the classification and segmentation tasks on the PASCAL VOC dataset for (ii).

Traditionally, data selection has been studied in settings where all samples from prospective sources are fully revealed to a machine learning developer. However, in practical data exchange scenarios, data providers often reveal only a limited subset of samples before an acquisition decision is made. Recently, there have been efforts to fit scaling laws that predict model performance at any size and data source composition using the limited available samples. However, these scaling functions are black-box, computationally expensive to fit, highly susceptible to overfitting, or/and difficult to optimize for data selection. This paper proposes a framework called <projektor>, which predicts model performance and supports data selection decisions based on partial samples of prospective data sources. Our approach distinguishes itself from existing work by introducing a novel *two-stage* performance inference process. In the first stage, we leverage the Optimal Transport distance to predict the model's performance for any data mixture ratio within the range of disclosed data sizes. In the second stage, we extrapolate the performance to larger undisclosed data sizes based on a novel parameter-free mapping technique inspired by neural scaling laws. We further derive an efficient gradient-based method to select data sources based on the projected model performance. Evaluation over a diverse range of applications demonstrates that <projektor> significantly improves existing performance scaling approaches in terms of both the accuracy of performance inference and the computation costs associated with constructing the performance predictor. Also, <projektor> outperforms by a wide margin in data selection effectiveness compared to a range of other off-the-shelf solutions.

This paper introduces a novel theoretical simplification of the Diffusion Schr\"odinger Bridge (DSB) that facilitates its unification with Score-based Generative Models (SGMs), addressing the limitations of DSB in complex data generation and enabling faster convergence and enhanced performance. By employing SGMs as an initial solution for DSB, our approach capitalizes on the strengths of both frameworks, ensuring a more efficient training process and improving the performance of SGM. We also propose a reparameterization technique that, despite theoretical approximations, practically improves the network's fitting capabilities. Our extensive experimental evaluations confirm the effectiveness of the simplified DSB, demonstrating its significant improvements. We believe the contributions of this work pave the way for advanced generative modeling. The code is available at https://github.com/checkcrab/SDSB.

Clustering non-Euclidean data is difficult, and one of the most used algorithms besides hierarchical clustering is the popular algorithm Partitioning Around Medoids (PAM), also simply referred to as k-medoids. In Euclidean geometry the mean-as used in k-means-is a good estimator for the cluster center, but this does not hold for arbitrary dissimilarities. PAM uses the medoid instead, the object with the smallest dissimilarity to all others in the cluster. This notion of centrality can be used with any (dis-)similarity, and thus is of high relevance to many domains such as biology that require the use of Jaccard, Gower, or more complex distances. A key issue with PAM is its high run time cost. We propose modifications to the PAM algorithm to achieve an O(k)-fold speedup in the second SWAP phase of the algorithm, but will still find the same results as the original PAM algorithm. If we slightly relax the choice of swaps performed (at comparable quality), we can further accelerate the algorithm by performing up to k swaps in each iteration. With the substantially faster SWAP, we can now also explore alternative strategies for choosing the initial medoids. We also show how the CLARA and CLARANS algorithms benefit from these modifications. It can easily be combined with earlier approaches to use PAM and CLARA on big data (some of which use PAM as a subroutine, hence can immediately benefit from these improvements), where the performance with high k becomes increasingly important. In experiments on real data with k=100, we observed a 200-fold speedup compared to the original PAM SWAP algorithm, making PAM applicable to larger data sets as long as we can afford to compute a distance matrix, and in particular to higher k (at k=2, the new SWAP was only 1.5 times faster, as the speedup is expected to increase with k).

Recently, Visual Autoregressive (VAR) Models introduced a groundbreaking advancement in the field of image generation, offering a scalable approach through a coarse-to-fine "next-scale prediction" paradigm. However, the state-of-the-art algorithm of VAR models in [Tian, Jiang, Yuan, Peng and Wang, NeurIPS 2024] takes O(n^4) time, which is computationally inefficient. In this work, we analyze the computational limits and efficiency criteria of VAR Models through a fine-grained complexity lens. Our key contribution is identifying the conditions under which VAR computations can achieve sub-quadratic time complexity. Specifically, we establish a critical threshold for the norm of input matrices used in VAR attention mechanisms. Above this threshold, assuming the Strong Exponential Time Hypothesis (SETH) from fine-grained complexity theory, a sub-quartic time algorithm for VAR models is impossible. To substantiate our theoretical findings, we present efficient constructions leveraging low-rank approximations that align with the derived criteria. This work initiates the study of the computational efficiency of the VAR model from a theoretical perspective. Our technique will shed light on advancing scalable and efficient image generation in VAR frameworks.

Linear time-invariant state space models (SSM) are a classical model from engineering and statistics, that have recently been shown to be very promising in machine learning through the Structured State Space sequence model (S4). A core component of S4 involves initializing the SSM state matrix to a particular matrix called a HiPPO matrix, which was empirically important for S4's ability to handle long sequences. However, the specific matrix that S4 uses was actually derived in previous work for a particular time-varying dynamical system, and the use of this matrix as a time-invariant SSM had no known mathematical interpretation. Consequently, the theoretical mechanism by which S4 models long-range dependencies actually remains unexplained. We derive a more general and intuitive formulation of the HiPPO framework, which provides a simple mathematical interpretation of S4 as a decomposition onto exponentially-warped Legendre polynomials, explaining its ability to capture long dependencies. Our generalization introduces a theoretically rich class of SSMs that also lets us derive more intuitive S4 variants for other bases such as the Fourier basis, and explains other aspects of training S4, such as how to initialize the important timescale parameter. These insights improve S4's performance to 86% on the Long Range Arena benchmark, with 96% on the most difficult Path-X task.

We analyze the layerwise effective dimension (rank of the feature matrix) in fully-connected ReLU networks of finite width. Specifically, for a fixed batch of m inputs and random Gaussian weights, we derive closed-form expressions for the expected rank of the \mtimes n hidden activation matrices. Our main result shows that E[EDim(ell)]=m[1-(1-2/pi)^ell]+O(e^{-c m}) so that the rank deficit decays geometrically with ratio 1-2 / pi approx 0.3634. We also prove a sub-Gaussian concentration bound, and identify the "revival" depths at which the expected rank attains local maxima. In particular, these peaks occur at depths ell_k^*approx(k+1/2)pi/log(1/rho) with height approx (1-e^{-pi/2}) m approx 0.79m. We further show that this oscillatory rank behavior is a finite-width phenomenon: under orthogonal weight initialization or strong negative-slope leaky-ReLU, the rank remains (nearly) full. These results provide a precise characterization of how random ReLU layers alternately collapse and partially revive the subspace of input variations, adding nuance to prior work on expressivity of deep networks.

Clustering non-Euclidean data is difficult, and one of the most used algorithms besides hierarchical clustering is the popular algorithm Partitioning Around Medoids (PAM), also simply referred to as k-medoids clustering. In Euclidean geometry the mean-as used in k-means-is a good estimator for the cluster center, but this does not exist for arbitrary dissimilarities. PAM uses the medoid instead, the object with the smallest dissimilarity to all others in the cluster. This notion of centrality can be used with any (dis-)similarity, and thus is of high relevance to many domains and applications. A key issue with PAM is its high run time cost. We propose modifications to the PAM algorithm that achieve an O(k)-fold speedup in the second ("SWAP") phase of the algorithm, but will still find the same results as the original PAM algorithm. If we relax the choice of swaps performed (while retaining comparable quality), we can further accelerate the algorithm by eagerly performing additional swaps in each iteration. With the substantially faster SWAP, we can now explore faster initialization strategies, because (i) the classic ("BUILD") initialization now becomes the bottleneck, and (ii) our swap is fast enough to compensate for worse starting conditions. We also show how the CLARA and CLARANS algorithms benefit from the proposed modifications. While we do not study the parallelization of our approach in this work, it can easily be combined with earlier approaches to use PAM and CLARA on big data (some of which use PAM as a subroutine, hence can immediately benefit from these improvements), where the performance with high k becomes increasingly important. In experiments on real data with k=100,200, we observed a 458x respectively 1191x speedup compared to the original PAM SWAP algorithm, making PAM applicable to larger data sets, and in particular to higher k.

This paper develops a hierarchical clustering algorithm for weighted projective spaces P_{q}, utilizing a Finsler metric d_F([z], [w]) and its rational analogue d_{F,Q}([z], [w]) to define distances that preserve the non-Euclidean geometry of these quotient manifolds. Defined via geodesic integrals of a scaling invariant Finsler norm weighted by the grades q = (q_0, q_1, dots, q_n), these metrics satisfy true metric properties including the triangle inequality, overcoming the limitations of the non-metric dissimilarity measure from prior work.

Regularized optimal transport (OT) is now increasingly used as a loss or as a matching layer in neural networks. Entropy-regularized OT can be computed using the Sinkhorn algorithm but it leads to fully-dense transportation plans, meaning that all sources are (fractionally) matched with all targets. To address this issue, several works have investigated quadratic regularization instead. This regularization preserves sparsity and leads to unconstrained and smooth (semi) dual objectives, that can be solved with off-the-shelf gradient methods. Unfortunately, quadratic regularization does not give direct control over the cardinality (number of nonzeros) of the transportation plan. We propose in this paper a new approach for OT with explicit cardinality constraints on the transportation plan. Our work is motivated by an application to sparse mixture of experts, where OT can be used to match input tokens such as image patches with expert models such as neural networks. Cardinality constraints ensure that at most k tokens are matched with an expert, which is crucial for computational performance reasons. Despite the nonconvexity of cardinality constraints, we show that the corresponding (semi) dual problems are tractable and can be solved with first-order gradient methods. Our method can be thought as a middle ground between unregularized OT (recovered in the limit case k=1) and quadratically-regularized OT (recovered when k is large enough). The smoothness of the objectives increases as k increases, giving rise to a trade-off between convergence speed and sparsity of the optimal plan.

Strategies for the generation of periodic discrete structures with identical two-point correlation are developed. Starting from a pair of root structures, which are not related by translation, phase inversion or axis reflections, child structures of arbitrary resolution (i.e., pixel or voxel numbers) and number of phases (i.e., material phases/species) can be generated by means of trivial embedding based phase extension, application of kernels and/or phase coalescence, such that the generated structures inherit the two-point-correlation equivalence. Proofs of the inheritance property are provided by means of the Discrete Fourier Transform theory. A Python 3 implementation of the results is offered by the authors through the Github repository https://github.com/DataAnalyticsEngineering/EQ2PC in order to make the provided results reproducible and useful for all interested readers. Examples for the generation of structures are demonstrated, together with applications in the homogenization theory of periodic media.

Optimal transport (OT) aims to find a map T that transports mass from one probability measure to another while minimizing a cost function. Recently, neural OT solvers have gained popularity in high dimensional biological applications such as drug perturbation, due to their superior computational and memory efficiency compared to traditional exact Sinkhorn solvers. However, the overly complex high dimensional maps learned by neural OT solvers often suffer from poor interpretability. Prior work addressed this issue in the context of exact OT solvers by introducing displacement-sparse maps via designed elastic cost, but such method failed to be applied to neural OT settings. In this work, we propose an intuitive and theoretically grounded approach to learning displacement-sparse maps within neural OT solvers. Building on our new formulation, we introduce a novel smoothed ell_0 regularizer that outperforms the ell_1 based alternative from prior work. Leveraging Input Convex Neural Network's flexibility, we further develop a heuristic framework for adaptively controlling sparsity intensity, an approach uniquely enabled by the neural OT paradigm. We demonstrate the necessity of this adaptive framework in large-scale, high-dimensional training, showing not only improved accuracy but also practical ease of use for downstream applications.

Orthogonal gradient updates have emerged as a promising direction in optimization for machine learning. However, traditional approaches such as SVD/QR decomposition incur prohibitive computational costs of O(n^3) and underperform compared to well-tuned SGD with momentum, since momentum is applied only after strict orthogonalization. Recent advances, such as Muon, improve efficiency by applying momentum before orthogonalization and producing semi-orthogonal matrices via Newton-Schulz iterations, reducing complexity to O(n^2). Nevertheless, quadratic costs remain a bottleneck. In this work, we study the semi-orthogonal properties of momentum-based updates and develop a method to bound momentum updates under a spectral-norm trust region, preserving directional information without requiring explicit semi-orthogonalization. We propose AuON (Alternative Unit-norm momentum updates by Normalized nonlinear scaling), a linear-time optimizer that achieves strong performance without constructing semi-orthogonal matrices, while preserving structural alignment and reconditioning ill-posed updates. Our approach combines hyperbolic-cosine RMS scaling transformations with normalization, demonstrating both effectiveness and computational efficiency compared to Newton-Schulz methods. We further introduce a hybrid variant (Hybrid-AuON) that applies a single Newton-Schulz iteration. Experiments across vision and language benchmarks show that AuON and its hybrid variant achieve performance comparable to strong baselines such as AdamW and Muon. Code is available at: https://github.com/ryyzn9/AuON

Low-Rank Adaptation (LoRA) has significantly advanced parameter-efficient fine-tuning of large pretrained models. LoRA augments the pre-trained weights of a model by adding the product of two smaller matrices that together form a low-rank matrix update. Recent research has shown that scale disparities between these two matrices often cause unstable training dynamics, leading to suboptimal performance. In this paper, we propose SingLoRA, which reformulates low-rank adaptation by learning the weights update as a decomposition of a single low-rank matrix multiplied by its transpose. This simple design inherently removes inter-matrix scale conflicts, ensuring stable optimization, and roughly halves the parameter count. We analyze SingLoRA within the infinite-width neural network framework, showing that it guarantees stable feature learning by construction. Extensive experiments on multiple tasks validate these benefits. In common sense reasoning, fine-tuning LLama 7B on MNLI with SingLoRA achieves 91.3% accuracy - surpassing LoRA (89.1%) and LoRA+ (90.2%) - while using only 60% of their parameter budget. In image generation, fine-tuning Stable Diffusion with SingLoRA significantly improves image fidelity on DreamBooth, achieving a DINO similarity score of 0.151, compared to scores of 0.148 and 0.143 for DoRA and LoRA, respectively.

Large neural networks excel in many domains, but they are expensive to train and fine-tune. A popular approach to reduce their compute or memory requirements is to replace dense weight matrices with structured ones (e.g., sparse, low-rank, Fourier transform). These methods have not seen widespread adoption (1) in end-to-end training due to unfavorable efficiency--quality tradeoffs, and (2) in dense-to-sparse fine-tuning due to lack of tractable algorithms to approximate a given dense weight matrix. To address these issues, we propose a class of matrices (Monarch) that is hardware-efficient (they are parameterized as products of two block-diagonal matrices for better hardware utilization) and expressive (they can represent many commonly used transforms). Surprisingly, the problem of approximating a dense weight matrix with a Monarch matrix, though nonconvex, has an analytical optimal solution. These properties of Monarch matrices unlock new ways to train and fine-tune sparse and dense models. We empirically validate that Monarch can achieve favorable accuracy-efficiency tradeoffs in several end-to-end sparse training applications: speeding up ViT and GPT-2 training on ImageNet classification and Wikitext-103 language modeling by 2x with comparable model quality, and reducing the error on PDE solving and MRI reconstruction tasks by 40%. In sparse-to-dense training, with a simple technique called "reverse sparsification," Monarch matrices serve as a useful intermediate representation to speed up GPT-2 pretraining on OpenWebText by 2x without quality drop. The same technique brings 23% faster BERT pretraining than even the very optimized implementation from Nvidia that set the MLPerf 1.1 record. In dense-to-sparse fine-tuning, as a proof-of-concept, our Monarch approximation algorithm speeds up BERT fine-tuning on GLUE by 1.7x with comparable accuracy.

We study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian. We also derive similar results for the Generalised Gauss-Newton matrix approximation of the Hessian. As a consequence of our theorems we derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. %For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. Based on our investigations of the sub-sampled Hessian we develop a stochastic Lanczos quadrature based on the fly learning rate and momentum learner, which avoids the need for expensive multiple evaluations for these key hyper-parameters and shows good preliminary results on the Pre-Residual Architecure for CIFAR-100.

In this study, we propose a simple method for fault-tolerant Strassen-like matrix multiplications. The proposed method is based on using two distinct Strassen-like algorithms instead of replicating a given one. We have realized that using two different algorithms, new check relations arise resulting in more local computations. These local computations are found using computer aided search. To improve performance, special parity (extra) sub-matrix multiplications (PSMMs) are generated (two of them) at the expense of increasing communication/computation cost of the system. Our preliminary results demonstrate that the proposed method outperforms a Strassen-like algorithm with two copies and secures a very close performance to three copy version using only 2 PSMMs, reducing the total number of compute nodes by around 24\% i.e., from 21 to 16.

Motivated by the problem of fast processing of attention matrices, we study fast algorithms for computing matrix-vector products for asymmetric Gaussian Kernel matrices Kin R^{ntimes n}. K's columns are indexed by a set of n keys k_1,k_2ldots, k_nin R^d, rows by a set of n queries q_1,q_2,ldots,q_nin R^d , and its i,j entry is K_{ij} = e^{-|q_i-k_j|_2^2/2sigma^2} for some bandwidth parameter sigma>0. Given a vector xin R^n and error parameter epsilon>0, our task is to output a yin R^n such that |Kx-y|_2leq epsilon |x|_2 in time subquadratic in n and linear in d. Our algorithms rely on the following modelling assumption about the matrices K: the sum of the entries of K scales linearly in n, as opposed to worst case quadratic growth. We validate this assumption experimentally, for Gaussian kernel matrices encountered in various settings such as fast attention computation in LLMs. We obtain the first subquadratic-time algorithm that works under this assumption, for unrestricted vectors.

The superfluid phase transition dynamics and associated spontaneous vortex formation with the crossing of the critical temperature in a disk geometry is studied in the framework of the AdS/CFT correspondence by solving the Einstein-Abelian-Higgs model in an AdS_4 black hole. For a slow quench, the vortex density admits a universal scaling law with the cooling rate as predicted by the Kibble-Zurek mechanism (KZM), while for fast quenches, the density shows a universal scaling behavior as a function of the final temperature, that lies beyond the KZM prediction. The vortex number distribution in both the power-law and saturation regimes can be approximated by a normal distribution. However, the study of the universal scaling of the cumulants reveals non-normal features and indicates that vortex statistics in the newborn superfluid is best described by the Poisson binomial distribution, previously predicted in the KZM regime [Phys. Rev. Lett. 124, 240602 (2020)]. This is confirmed by studying the cumulant scalings as a function of the quench time and the quench depth. Our work supports the existence of a universal defect number distribution that accommodates the KZM scaling, its breakdown at fast quenches, and the additional universal scaling laws as a function of the final value of the control parameter.

Scale has become a main ingredient in obtaining strong machine learning models. As a result, understanding a model's scaling properties is key to effectively designing both the right training setup as well as future generations of architectures. In this work, we argue that scale and training research has been needlessly complex due to reliance on the cosine schedule, which prevents training across different lengths for the same model size. We investigate the training behavior of a direct alternative - constant learning rate and cooldowns - and find that it scales predictably and reliably similar to cosine. Additionally, we show that stochastic weight averaging yields improved performance along the training trajectory, without additional training costs, across different scales. Importantly, with these findings we demonstrate that scaling experiments can be performed with significantly reduced compute and GPU hours by utilizing fewer but reusable training runs.

Recently, the statistical effect of fermionic superradiance is approved by series of experiments both in free space and in a cavity. The Pauli blocking effect can be visualized by a 1/2 scaling of Dicke transition critical pumping strength against particle number Nat for fermions in a trap. However, the Fermi surface nesting effect, which manifests the enhancement of superradiance by Fermi statistics is still very hard to be identified. Here we studied the influence of localized fermions on the trap edge when both pumping optical lattice and the trap are presented. We find due to localization, the statistical effect in superradiant transition is enhanced. Two new scalings of critical pumping strength are observed as 4/3, and 2/3 for mediate particle number, and the Pauli blocking scaling 1/3 (2d case) in large particle number limit is unaffected. Further, we find the 4/3 scaling is subject to a power law increasing with rising ratio between recoil energy and trap frequency in pumping laser direction. The divergence of this scaling of critical pumping strength against N_{rm at} in E_R/omega_xrightarrow+infty limit can be identified as the Fermi surface nesting effect. Thus we find a practical experimental scheme for visualizing the long-desired Fermi surface nesting effect with the help of trap induced localization in a two-dimensional Fermi gas in a cavity.

In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve mgg 1 lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling I blocks and sampling B samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of O(mepsilon^{-3I(I<m)}{II} + mepsilon^{-3}{IB}) for finding an epsilon-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.

This paper studies the role of over-parametrization in solving non-convex optimization problems. The focus is on the important class of low-rank matrix sensing, where we propose an infinite hierarchy of non-convex problems via the lifting technique and the Burer-Monteiro factorization. This contrasts with the existing over-parametrization technique where the search rank is limited by the dimension of the matrix and it does not allow a rich over-parametrization of an arbitrary degree. We show that although the spurious solutions of the problem remain stationary points through the hierarchy, they will be transformed into strict saddle points (under some technical conditions) and can be escaped via local search methods. This is the first result in the literature showing that over-parametrization creates a negative curvature for escaping spurious solutions. We also derive a bound on how much over-parametrization is requited to enable the elimination of spurious solutions.

In this paper, we propose a new class of metric for table structure recognition (TSR) evaluation, called grid table similarity (GriTS). Unlike prior metrics, GriTS evaluates the correctness of a predicted table directly in its natural form as a matrix. To create a similarity measure between matrices, we generalize the two-dimensional largest common substructure (2D-LCS) problem, which is NP-hard, to the 2D most similar substructures (2D-MSS) problem and propose a polynomial-time heuristic for solving it. This algorithm produces both an upper and a lower bound on the true similarity between matrices. We show using evaluation on a large real-world dataset that in practice there is almost no difference between these bounds. We compare GriTS to other metrics and empirically validate that matrix similarity exhibits more desirable behavior than alternatives for TSR performance evaluation. Finally, GriTS unifies all three subtasks of cell topology recognition, cell location recognition, and cell content recognition within the same framework, which simplifies the evaluation and enables more meaningful comparisons across different types of TSR approaches. Code will be released at https://github.com/microsoft/table-transformer.

The Milky Way is known to contain a stellar bar, as are a significant fraction of disc galaxies across the universe. Our understanding of bar evolution, both theoretically and through analysis of simulations indicates that bars both grow in amplitude and slow down over time through interaction and angular momentum exchange with the galaxy's dark matter halo. Understanding the physical mechanisms underlying this coupling requires modelling of the structural deformations to the potential that are mutually induced between components. In this work we use Basis Function Expansion (BFE) in combination with multichannel Singular Spectral Analysis (mSSA) as a non-parametric analysis tool to illustrate the coupling between the bar and the dark halo in a single high-resolution isolated barred disc galaxy simulation. We demonstrate the power of mSSA to extract and quantify explicitly coupled dynamical modes, determining growth rates, pattern speeds and phase lags for different stages of evolution of the stellar bar and the dark matter response. BFE & mSSA together grant us the ability to explore the importance and physical mechanisms of bar-halo coupling, and other dynamically coupled structures across a wide range of dynamical environments.

With the rapid deployment of SCADA systems, how to effectively analyze industrial signals and detect abnormal states is an urgent need for the industry. Due to the significant heterogeneity of these signals, which we summarize as the M5 problem, previous works only focus on small sub-problems and employ specialized models, failing to utilize the synergies between modalities and the powerful scaling law. However, we argue that the M5 signals can be modeled in a unified manner due to the intrinsic similarity. As a result, we propose FISHER, a Foundation model for multi-modal Industrial Signal compreHEnsive Representation. To support arbitrary sampling rates, FISHER considers the increment of sampling rate as the concatenation of sub-band information. Specifically, FISHER takes the STFT sub-band as the modeling unit and adopts a teacher student SSL framework for pre-training. We also develop the RMIS benchmark, which evaluates the representations of M5 industrial signals on multiple health management tasks. Compared with top SSL models, FISHER showcases versatile and outstanding capabilities with a general performance gain up to 5.03%, along with much more efficient scaling curves. We also investigate the scaling law on downstream tasks and derive potential avenues for future works. FISHER is now open-sourced on https://github.com/jianganbai/FISHER

Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this approach. In particular, we show that the orthogonalized gradient method can be seen as a first-order trust-region optimization method, where the trust-region is defined in terms of the matrix spectral norm. Motivated by this observation, we develop the stochastic non-Euclidean trust-region gradient method with momentum, which recovers the Muon optimizer (Jordan et al., 2024) as a special case, along with normalized SGD and signSGD with momentum (Cutkosky and Mehta, 2020; Sun et al., 2023). In addition, we prove state-of-the-art convergence results for the proposed algorithm in a range of scenarios, which involve arbitrary non-Euclidean norms, constrained and composite problems, and non-convex, star-convex, first- and second-order smooth functions. Finally, our theoretical findings provide an explanation for several practical observations, including the practical superiority of Muon compared to the Orthogonal-SGDM algorithm of Tuddenham et al. (2022) and the importance of weight decay in the training of large-scale language models.

In this work we analyze strategies for convolutional neural network scaling; that is, the process of scaling a base convolutional network to endow it with greater computational complexity and consequently representational power. Example scaling strategies may include increasing model width, depth, resolution, etc. While various scaling strategies exist, their tradeoffs are not fully understood. Existing analysis typically focuses on the interplay of accuracy and flops (floating point operations). Yet, as we demonstrate, various scaling strategies affect model parameters, activations, and consequently actual runtime quite differently. In our experiments we show the surprising result that numerous scaling strategies yield networks with similar accuracy but with widely varying properties. This leads us to propose a simple fast compound scaling strategy that encourages primarily scaling model width, while scaling depth and resolution to a lesser extent. Unlike currently popular scaling strategies, which result in about O(s) increase in model activation w.r.t. scaling flops by a factor of s, the proposed fast compound scaling results in close to O(s) increase in activations, while achieving excellent accuracy. This leads to comparable speedups on modern memory-limited hardware (e.g., GPU, TPU). More generally, we hope this work provides a framework for analyzing and selecting scaling strategies under various computational constraints.

Automatic differentiation through digital signal processing algorithms for virtual analogue modelling has recently gained popularity. These algorithms are typically more computationally efficient than black-box neural networks that rely on dense matrix multiplications. Due to their differentiable nature, they can be integrated with neural networks and jointly trained using gradient descent algorithms, resulting in more efficient systems. Furthermore, signal processing algorithms have significantly fewer parameters than neural networks, allowing the application of the Newton-Raphson method. This method offers faster and more robust convergence than gradient descent at the cost of quadratic storage. This paper presents a method to emulate analogue levelling amplifiers using a feed-forward digital compressor with parameters optimised via the Newton-Raphson method. We demonstrate that a digital compressor can successfully approximate the behaviour of our target unit, the Teletronix LA-2A. Different strategies for computing the Hessian matrix are benchmarked. We leverage parallel algorithms for recursive filters to achieve efficient training on modern GPUs. The resulting model is made into a VST plugin and is open-sourced at https://github.com/aim-qmul/4a2a.

This paper rigorously shows how over-parameterization changes the convergence behaviors of gradient descent (GD) for the matrix sensing problem, where the goal is to recover an unknown low-rank ground-truth matrix from near-isotropic linear measurements. First, we consider the symmetric setting with the symmetric parameterization where M^* in R^{n times n} is a positive semi-definite unknown matrix of rank r ll n, and one uses a symmetric parameterization XX^top to learn M^*. Here X in R^{n times k} with k > r is the factor matrix. We give a novel Omega (1/T^2) lower bound of randomly initialized GD for the over-parameterized case (k >r) where T is the number of iterations. This is in stark contrast to the exact-parameterization scenario (k=r) where the convergence rate is exp (-Omega (T)). Next, we study asymmetric setting where M^* in R^{n_1 times n_2} is the unknown matrix of rank r ll min{n_1,n_2}, and one uses an asymmetric parameterization FG^top to learn M^* where F in R^{n_1 times k} and G in R^{n_2 times k}. Building on prior work, we give a global exact convergence result of randomly initialized GD for the exact-parameterization case (k=r) with an exp (-Omega(T)) rate. Furthermore, we give the first global exact convergence result for the over-parameterization case (k>r) with an exp(-Omega(alpha^2 T)) rate where alpha is the initialization scale. This linear convergence result in the over-parameterization case is especially significant because one can apply the asymmetric parameterization to the symmetric setting to speed up from Omega (1/T^2) to linear convergence. On the other hand, we propose a novel method that only modifies one step of GD and obtains a convergence rate independent of alpha, recovering the rate in the exact-parameterization case.

We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank real symmetric tensors. This algorithm calculates one new CP component at a time, alternating between applying the shifted symmetric higher-order power method (SS-HOPM) to a certain modified tensor, constructed from a matrix flattening of the original tensor; and using appropriate deflation steps. We obtain rigorous guarantees for SPM regarding convergence and global optima for input tensors of dimension d and order m of CP rank up to O(d^{lfloor m/2rfloor}), via results in classical algebraic geometry and optimization theory. As a by-product of our analysis we prove that SS-HOPM converges unconditionally, settling a conjecture in [Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM Journal on Matrix Analysis and Applications 32(4), 1095-1124 (2011)]. We present numerical experiments which demonstrate that SPM is efficient and robust to noise, being up to one order of magnitude faster than state-of-the-art CP decomposition algorithms in certain experiments. Furthermore, prior knowledge of the CP rank is not required by SPM.

We study the scaling properties of avalanche activity in the two-dimensional Abelian sandpile model. Instead of the conventional avalanche size distribution, we analyze the site activity distribution, which measures how often a site participates in avalanches when grains are added across the lattice. Using numerical simulations for system sizes up to \(L = 160\), averaged over \(10^4\) configurations, we determine the probability distribution \(P(A, L)\) of site activities. The results show that \(P(A, L)\) follows a finite-size scaling form \[ P(A, L) \sim L^{-2} F\Big(A{L^2}\Big). \] For small values \(A \ll L^2\) the scaling function behaves as \[ F(u) \sim u^{-1/2}, \quad corresponding to \quad P(A) \sim 1{L}, \] while for large activities \(A \sim O(L^2)\) the distribution decays as \[ F(u) \sim \exp\big(-c_3 u - c_4 u^2\big). \] The crossover between these two regimes occurs at \[ A^* \sim 0.1 \, L^2, \] marking the threshold between typical and highly excitable sites. This characterization of local avalanche activity provides complementary information to the usual avalanche size statistics, highlighting how local regions serve as frequent conduits for critical dynamics. These results may help connect sandpile models to real-world self-organized critical systems where only partial local activity can be observed.

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices lauritzen2019maximum,slawski2015estimation and even one observation for diagonally dominant M-matrices truell2021maximum. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted ell_1-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

The Fisher information is a fundamental concept for characterizing the sensitivity of parameters in neural networks. However, leveraging the full observed Fisher information is too expensive for large models, so most methods rely on simple diagonal approximations. While efficient, this approach ignores parameter correlations, often resulting in reduced performance on downstream tasks. In this work, we mitigate these limitations and propose Generalized Fisher-Weighted SVD (GFWSVD), a post-training LLM compression technique that accounts for both diagonal and off-diagonal elements of the Fisher information matrix, providing a more accurate reflection of parameter importance. To make the method tractable, we introduce a scalable adaptation of the Kronecker-factored approximation algorithm for the observed Fisher information. We demonstrate the effectiveness of our method on LLM compression, showing improvements over existing compression baselines. For example, at a 20 compression rate on the MMLU benchmark, our method outperforms FWSVD, which is based on a diagonal approximation of the Fisher information, by 5 percent, SVD-LLM by 3 percent, and ASVD by 6 percent compression rate.

Despite recent progress in optimal hyperparameter transfer under model and dataset scaling, no unifying explanatory principle has been established. Using the Scion optimizer, we discover that joint optimal scaling across model and dataset sizes is governed by a single invariant: the operator norm of the output layer. Across models with up to 1.3B parameters trained on up to 138B tokens, the optimal learning rate/batch size pair (eta^{ast}, B^{ast}) consistently has the same operator norm value - a phenomenon we term norm transfer. This constant norm condition is necessary but not sufficient: while for each dataset size, multiple (eta, B) reach the optimal norm, only a unique (eta^{ast}, B^{ast}) achieves the best loss. As a sufficient condition, we provide the first measurement of (eta^{ast}, B^{ast}) scaling with dataset size for Scion, and find that the scaling rules are consistent with those of the Adam optimizer. Tuning per-layer-group learning rates also improves model performance, with the output layer being the most sensitive and hidden layers benefiting from lower learning rates. We provide practical insights on norm-guided optimal scaling and release our Distributed Scion (Disco) implementation with logs from over two thousand runs to support research on LLM training dynamics at scale.

This paper studies the scale and redshift variation of density and velocity distributions in self-gravitating collisionless dark matter flow by a halo-based non-projection approach. All particles are divided into halo and out-of-halo particles for redshift variation of distributions. Without projecting particle fields onto a structured grid, the scale variation is analyzed by identifying all particle pairs on different scales r. We demonstrate that: i) Delaunay tessellation can be used to reconstruct the density field. The density correlation, spectrum, and dispersion functions were obtained, modeled, and compared with the N-body simulation; ii) the velocity distributions are symmetric on both small and large scales and are non-symmetric with a negative skewness on intermediate scales due to the inverse energy cascade at a constant rate varepsilon_u; iii) On small scales, the even order moments of pairwise velocity Delta u_L follow a two-thirds law (-varepsilon_ur)^{2/3}, while the odd order moments follow a linear scaling langle(Delta u_L)^{2n+1}rangle=(2n+1)langle(Delta u_L)^{2n}ranglelangleDelta u_Lrangler; iv) The scale variation of the velocity distributions was studied for longitudinal velocities u_L or u_L^{'}, pairwise velocity (velocity difference) Delta u_L=u_L^{'}-u_L and velocity sum Sigma u_L=u^{'}_L+u_L. Fully developed velocity fields are never Gaussian on any scale, despite that they can initially be Gaussian; v) On small scales, u_L and Sigma u_L can be modeled by a X distribution to maximize the system entropy; vi) On large scales, Delta u_L and Sigma u_L can be modeled by a logistic or a X distribution; vii) the redshift variation of the velocity distributions follows the evolution of the X distribution involving a shape parameter alpha(z) decreasing with time.

Multi-layer perceptron (MLP) is a fundamental component of deep learning that has been extensively employed for various problems. However, recent empirical successes in MLP-based architectures, particularly the progress of the MLP-Mixer, have revealed that there is still hidden potential in improving MLPs to achieve better performance. In this study, we reveal that the MLP-Mixer works effectively as a wide MLP with certain sparse weights. Initially, we clarify that the mixing layer of the Mixer has an effective expression as a wider MLP whose weights are sparse and represented by the Kronecker product. This expression naturally defines a permuted-Kronecker (PK) family, which can be regarded as a general class of mixing layers and is also regarded as an approximation of Monarch matrices. Subsequently, because the PK family effectively constitutes a wide MLP with sparse weights, one can apply the hypothesis proposed by Golubeva, Neyshabur and Gur-Ari (2021) that the prediction performance improves as the width (sparsity) increases when the number of weights is fixed. We empirically verify this hypothesis by maximizing the effective width of the MLP-Mixer, which enables us to determine the appropriate size of the mixing layers quantitatively.

Adapting large-scale pretrained language models to downstream tasks via fine-tuning is the standard method for achieving state-of-the-art performance on NLP benchmarks. However, fine-tuning all weights of models with millions or billions of parameters is sample-inefficient, unstable in low-resource settings, and wasteful as it requires storing a separate copy of the model for each task. Recent work has developed parameter-efficient fine-tuning methods, but these approaches either still require a relatively large number of parameters or underperform standard fine-tuning. In this work, we propose Compacter, a method for fine-tuning large-scale language models with a better trade-off between task performance and the number of trainable parameters than prior work. Compacter accomplishes this by building on top of ideas from adapters, low-rank optimization, and parameterized hypercomplex multiplication layers. Specifically, Compacter inserts task-specific weight matrices into a pretrained model's weights, which are computed efficiently as a sum of Kronecker products between shared "slow" weights and "fast" rank-one matrices defined per Compacter layer. By only training 0.047% of a pretrained model's parameters, Compacter performs on par with standard fine-tuning on GLUE and outperforms standard fine-tuning on SuperGLUE and low-resource settings. Our code is publicly available at~https://github.com/rabeehk/compacter.

Hoffmann et al. (2022) propose three methods for estimating a compute-optimal scaling law. We attempt to replicate their third estimation procedure, which involves fitting a parametric loss function to a reconstruction of data from their plots. We find that the reported estimates are inconsistent with their first two estimation methods, fail at fitting the extracted data, and report implausibly narrow confidence intervals--intervals this narrow would require over 600,000 experiments, while they likely only ran fewer than 500. In contrast, our rederivation of the scaling law using the third approach yields results that are compatible with the findings from the first two estimation procedures described by Hoffmann et al.

Learned Iterative Shrinkage-Thresholding Algorithm (LISTA) introduces the concept of unrolling an iterative algorithm and training it like a neural network. It has had great success on sparse recovery. In this paper, we show that adding momentum to intermediate variables in the LISTA network achieves a better convergence rate and, in particular, the network with instance-optimal parameters is superlinearly convergent. Moreover, our new theoretical results lead to a practical approach of automatically and adaptively calculating the parameters of a LISTA network layer based on its previous layers. Perhaps most surprisingly, such an adaptive-parameter procedure reduces the training of LISTA to tuning only three hyperparameters from data: a new record set in the context of the recent advances on trimming down LISTA complexity. We call this new ultra-light weight network HyperLISTA. Compared to state-of-the-art LISTA models, HyperLISTA achieves almost the same performance on seen data distributions and performs better when tested on unseen distributions (specifically, those with different sparsity levels and nonzero magnitudes). Code is available: https://github.com/VITA-Group/HyperLISTA.

We study the properties of common loss surfaces through their Hessian matrix. In particular, in the context of deep learning, we empirically show that the spectrum of the Hessian is composed of two parts: (1) the bulk centered near zero, (2) and outliers away from the bulk. We present numerical evidence and mathematical justifications to the following conjectures laid out by Sagun et al. (2016): Fixing data, increasing the number of parameters merely scales the bulk of the spectrum; fixing the dimension and changing the data (for instance adding more clusters or making the data less separable) only affects the outliers. We believe that our observations have striking implications for non-convex optimization in high dimensions. First, the flatness of such landscapes (which can be measured by the singularity of the Hessian) implies that classical notions of basins of attraction may be quite misleading. And that the discussion of wide/narrow basins may be in need of a new perspective around over-parametrization and redundancy that are able to create large connected components at the bottom of the landscape. Second, the dependence of small number of large eigenvalues to the data distribution can be linked to the spectrum of the covariance matrix of gradients of model outputs. With this in mind, we may reevaluate the connections within the data-architecture-algorithm framework of a model, hoping that it would shed light into the geometry of high-dimensional and non-convex spaces in modern applications. In particular, we present a case that links the two observations: small and large batch gradient descent appear to converge to different basins of attraction but we show that they are in fact connected through their flat region and so belong to the same basin.

Low-Rank Adaptation (LoRA), which introduces a product of two trainable low-rank matrices into frozen pre-trained weights, is widely used for efficient fine-tuning of language models in federated learning (FL). However, when combined with differentially private stochastic gradient descent (DP-SGD), LoRA faces substantial noise amplification: DP-SGD perturbs per-sample gradients, and the matrix multiplication of the LoRA update (BA) intensifies this effect. Freezing one matrix (e.g., A) reduces the noise but restricts model expressiveness, often resulting in suboptimal adaptation. To address this, we propose FedSVD, a simple yet effective method that introduces a global reparameterization based on singular value decomposition (SVD). In our approach, each client optimizes only the B matrix and transmits it to the server. The server aggregates the B matrices, computes the product BA using the previous A, and refactorizes the result via SVD. This yields a new adaptive A composed of the orthonormal right singular vectors of BA, and an updated B containing the remaining SVD components. This reparameterization avoids quadratic noise amplification, while allowing A to better capture the principal directions of the aggregate updates. Moreover, the orthonormal structure of A bounds the gradient norms of B and preserves more signal under DP-SGD, as confirmed by our theoretical analysis. As a result, FedSVD consistently improves stability and performance across a variety of privacy settings and benchmarks, outperforming relevant baselines under both private and non-private regimes.

Shampoo is an online and stochastic optimization algorithm belonging to the AdaGrad family of methods for training neural networks. It constructs a block-diagonal preconditioner where each block consists of a coarse Kronecker product approximation to full-matrix AdaGrad for each parameter of the neural network. In this work, we provide a complete description of the algorithm as well as the performance optimizations that our implementation leverages to train deep networks at-scale in PyTorch. Our implementation enables fast multi-GPU distributed data-parallel training by distributing the memory and computation associated with blocks of each parameter via PyTorch's DTensor data structure and performing an AllGather primitive on the computed search directions at each iteration. This major performance enhancement enables us to achieve at most a 10% performance reduction in per-step wall-clock time compared against standard diagonal-scaling-based adaptive gradient methods. We validate our implementation by performing an ablation study on training ImageNet ResNet50, demonstrating Shampoo's superiority over standard training recipes with minimal hyperparameter tuning.

Low-precision formats such as float8 have been introduced in machine learning accelerated hardware to improve computational efficiency for large language models training and inference. Nevertheless, adoption by the ML community has been slowed down by the complex, and sometimes brittle, techniques required to match higher precision training accuracy. In this work, we present Scalify, a end-to-end scale propagation paradigm for computational graphs, generalizing and formalizing existing tensor scaling methods. Experiment results show that Scalify supports out-of-the-box float8 matrix multiplication and gradients representation, as well as float16 optimizer state storage. Our JAX implementation of Scalify is open-sourced at https://github.com/graphcore-research/jax-scalify

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.

Variance reduction (VR) techniques have contributed significantly to accelerating learning with massive datasets in the smooth and strongly convex setting (Schmidt et al., 2017; Johnson & Zhang, 2013; Roux et al., 2012). However, such techniques have not yet met the same success in the realm of large-scale deep learning due to various factors such as the use of data augmentation or regularization methods like dropout (Defazio & Bottou, 2019). This challenge has recently motivated the design of novel variance reduction techniques tailored explicitly for deep learning (Arnold et al., 2019; Ma & Yarats, 2018). This work is an additional step in this direction. In particular, we exploit the ubiquitous clustering structure of rich datasets used in deep learning to design a family of scalable variance reduced optimization procedures by combining existing optimizers (e.g., SGD+Momentum, Quasi Hyperbolic Momentum, Implicit Gradient Transport) with a multi-momentum strategy (Yuan et al., 2019). Our proposal leads to faster convergence than vanilla methods on standard benchmark datasets (e.g., CIFAR and ImageNet). It is robust to label noise and amenable to distributed optimization. We provide a parallel implementation in JAX.

This paper is the second in the series Commutative Scaling of Width and Depth (WD) about commutativity of infinite width and depth limits in deep neural networks. Our aim is to understand the behaviour of neural functions (functions that depend on a neural network model) as width and depth go to infinity (in some sense), and eventually identify settings under which commutativity holds, i.e. the neural function tends to the same limit no matter how width and depth limits are taken. In this paper, we formally introduce and define the commutativity framework, and discuss its implications on neural network design and scaling. We study commutativity for the neural covariance kernel which reflects how network layers separate data. Our findings extend previous results established in [55] by showing that taking the width and depth to infinity in a deep neural network with skip connections, when branches are suitably scaled to avoid exploding behaviour, result in the same covariance structure no matter how that limit is taken. This has a number of theoretical and practical implications that we discuss in the paper. The proof techniques in this paper are novel and rely on tools that are more accessible to readers who are not familiar with stochastic calculus (used in the proofs of WD(I))).

In studies of transferable learning, scaling laws are obtained for various important foundation models to predict their properties and performance at larger scales. We show here how scaling law derivation can also be used for model and dataset comparison, allowing to decide which procedure is to be preferred for pre-training. For the first time, full scaling laws based on dense measurements across a wide span of model and samples seen scales are derived for two important language-vision learning procedures, CLIP and MaMMUT, that use either contrastive only or contrastive and captioning text generative loss. Ensuring sufficient prediction accuracy for held out points, we use derived scaling laws to compare both models, obtaining evidence for MaMMUT's stronger improvement with scale and better sample efficiency than standard CLIP. To strengthen validity of the comparison, we show scaling laws for various downstream tasks, classification, retrieval, and segmentation, and for different open datasets, DataComp, DFN and Re-LAION, observing consistently the same trends. We show that comparison can also be performed when deriving scaling laws with a constant learning rate schedule, reducing compute cost. Accurate derivation of scaling laws provides thus means to perform model and dataset comparison across scale spans, avoiding misleading conclusions based on measurements from single reference scales only, paving the road for systematic comparison and improvement of open foundation models and datasets for their creation. We release all the pre-trained models with their intermediate checkpoints, including openMaMMUT-L/14, which achieves 80.3% zero-shot ImageNet-1k accuracy, trained on 12.8B samples from DataComp-1.4B. Code for reproducing experiments in the paper and raw experiments data can be found at https://github.com/LAION-AI/scaling-laws-for-comparison.

The success of deep learning is due in large part to our ability to solve certain massive non-convex optimization problems with relative ease. Though non-convex optimization is NP-hard, simple algorithms -- often variants of stochastic gradient descent -- exhibit surprising effectiveness in fitting large neural networks in practice. We argue that neural network loss landscapes often contain (nearly) a single basin after accounting for all possible permutation symmetries of hidden units a la Entezari et al. 2021. We introduce three algorithms to permute the units of one model to bring them into alignment with a reference model in order to merge the two models in weight space. This transformation produces a functionally equivalent set of weights that lie in an approximately convex basin near the reference model. Experimentally, we demonstrate the single basin phenomenon across a variety of model architectures and datasets, including the first (to our knowledge) demonstration of zero-barrier linear mode connectivity between independently trained ResNet models on CIFAR-10. Additionally, we identify intriguing phenomena relating model width and training time to mode connectivity. Finally, we discuss shortcomings of the linear mode connectivity hypothesis, including a counterexample to the single basin theory.

State space models (SSM) have recently been shown to be very effective as a deep learning layer as a promising alternative to sequence models such as RNNs, CNNs, or Transformers. The first version to show this potential was the S4 model, which is particularly effective on tasks involving long-range dependencies by using a prescribed state matrix called the HiPPO matrix. While this has an interpretable mathematical mechanism for modeling long dependencies, it introduces a custom representation and algorithm that can be difficult to implement. On the other hand, a recent variant of S4 called DSS showed that restricting the state matrix to be fully diagonal can still preserve the performance of the original model when using a specific initialization based on approximating S4's matrix. This work seeks to systematically understand how to parameterize and initialize such diagonal state space models. While it follows from classical results that almost all SSMs have an equivalent diagonal form, we show that the initialization is critical for performance. We explain why DSS works mathematically, by showing that the diagonal restriction of S4's matrix surprisingly recovers the same kernel in the limit of infinite state dimension. We also systematically describe various design choices in parameterizing and computing diagonal SSMs, and perform a controlled empirical study ablating the effects of these choices. Our final model S4D is a simple diagonal version of S4 whose kernel computation requires just 2 lines of code and performs comparably to S4 in almost all settings, with state-of-the-art results for image, audio, and medical time-series domains, and averaging 85\% on the Long Range Arena benchmark.

We consider the solvable neural scaling model with three parameters: data complexity, target complexity, and model-parameter-count. We use this neural scaling model to derive new predictions about the compute-limited, infinite-data scaling law regime. To train the neural scaling model, we run one-pass stochastic gradient descent on a mean-squared loss. We derive a representation of the loss curves which holds over all iteration counts and improves in accuracy as the model parameter count grows. We then analyze the compute-optimal model-parameter-count, and identify 4 phases (+3 subphases) in the data-complexity/target-complexity phase-plane. The phase boundaries are determined by the relative importance of model capacity, optimizer noise, and embedding of the features. We furthermore derive, with mathematical proof and extensive numerical evidence, the scaling-law exponents in all of these phases, in particular computing the optimal model-parameter-count as a function of floating point operation budget.

In this paper, we introduce a new class of score-based generative models (SGMs) designed to handle high-cardinality data distributions by leveraging concepts from mean-field theory. We present mean-field chaos diffusion models (MF-CDMs), which address the curse of dimensionality inherent in high-cardinality data by utilizing the propagation of chaos property of interacting particles. By treating high-cardinality data as a large stochastic system of interacting particles, we develop a novel score-matching method for infinite-dimensional chaotic particle systems and propose an approximation scheme that employs a subdivision strategy for efficient training. Our theoretical and empirical results demonstrate the scalability and effectiveness of MF-CDMs for managing large high-cardinality data structures, such as 3D point clouds.

Scaling laws offer valuable insights into the design of time series foundation models (TSFMs). However, previous research has largely focused on the scaling laws of TSFMs for in-distribution (ID) data, leaving their out-of-distribution (OOD) scaling behavior and the influence of model architectures less explored. In this work, we examine two common TSFM architectures, encoder-only and decoder-only Transformers, and investigate their scaling behavior on both ID and OOD data. These models are trained and evaluated across varying parameter counts, compute budgets, and dataset sizes. Our experiments reveal that the log-likelihood loss of TSFMs exhibits similar scaling behavior in both OOD and ID settings. We further compare the scaling properties across different architectures, incorporating two state-of-the-art TSFMs as case studies, showing that model architecture plays a significant role in scaling. The encoder-only Transformers demonstrate better scalability than the decoder-only Transformers, while the architectural enhancements in the two advanced TSFMs primarily improve ID performance but reduce OOD scalability. While scaling up TSFMs is expected to drive performance breakthroughs, the lack of a comprehensive understanding of TSFM scaling laws has hindered the development of a robust framework to guide model scaling. We fill this gap in this work by synthesizing our findings and providing practical guidelines for designing and scaling larger TSFMs with enhanced model capabilities.

Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In this work, we propose a novel algorithm to find efficient low-rank subnetworks. Remarkably, these subnetworks are determined and adapted already during the training phase and the overall time and memory resources required by both training and evaluating them are significantly reduced. The main idea is to restrict the weight matrices to a low-rank manifold and to update the low-rank factors rather than the full matrix during training. To derive training updates that are restricted to the prescribed manifold, we employ techniques from dynamic model order reduction for matrix differential equations. This allows us to provide approximation, stability, and descent guarantees. Moreover, our method automatically and dynamically adapts the ranks during training to achieve the desired approximation accuracy. The efficiency of the proposed method is demonstrated through a variety of numerical experiments on fully-connected and convolutional networks.