arXiv subjects

Michał Dereziński

Publications and source records attributed to Michał Dereziński.

At least 37 records · Page 2Linked to original sources

arXiv2025

Optimal Oblivious Subspace Embeddings with Near-optimal Sparsity

An oblivious subspace embedding is a random $m\times n$ matrix $Π$ such that, for any $d$-dimensional subspace, with high probability $Π$ preserves the norms of all vectors in that subspace within a $1\pmε$ factor. In this work, we give an oblivious subspace embedding with the optimal dimension $m=Θ(d/ε^2)$ that has a near-optimal sparsity of $\tilde O(1/ε)$ non-zero entries per column of $Π$. This is the first result to nearly match the conjecture of Nelson and Nguyen [FOCS 2013] in terms of the best sparsity attainable by an optimal oblivious subspace embedding, improving on a prior bound of $\tilde O(1/ε^6)$ non-zeros per column [Chenakkod et al., STOC 2024]. We further extend our approach to the non-oblivious setting, proposing a new family of Leverage Score Sparsified embeddings with Independent Columns, which yield faster runtimes for matrix approximation and regression tasks. In our analysis, we develop a new method which uses a decoupling argument together with the cumulant method for bounding the edge universality error of isotropic random matrices. To achieve near-optimal sparsity, we combine this general-purpose approach with new traces inequalities that leverage the specific structure of our subspace embedding construction.

cs.DS↗

arXiv2025

Faster Linear Systems and Matrix Norm Approximation via Multi-level Sketched Preconditioning

We present a new class of preconditioned iterative methods for solving linear systems of the form $Ax = b$. Our methods are based on constructing a low-rank Nyström approximation to $A$ using sparse random matrix sketching. This approximation is used to construct a preconditioner, which itself is inverted quickly using additional levels of random sketching and preconditioning. We prove that the convergence of our methods depends on a natural average condition number of $A$, which improves as the rank of the Nyström approximation increases. Concretely, this allows us to obtain faster runtimes for a number of fundamental linear algebraic problems: 1. We show how to solve any $n\times n$ linear system that is well-conditioned except for $k$ outlying large singular values in $\tilde{O}(n^{2.065} + k^ω)$ time, improving on a recent result of [Dereziński, Yang, STOC 2024] for all $k \gtrsim n^{0.78}$. 2. We give the first $\tilde{O}(n^2 + {d_λ}^ω$) time algorithm for solving a regularized linear system $(A + λI)x = b$, where $A$ is positive semidefinite with effective dimension $d_λ=\mathrm{tr}(A(A+λI)^{-1})$. This problem arises in applications like Gaussian process regression. 3. We give faster algorithms for approximating Schatten $p$-norms and other matrix norms. For example, for the Schatten 1-norm (nuclear norm), we give an algorithm that runs in $\tilde{O}(n^{2.11})$ time, improving on an $\tilde{O}(n^{2.18})$ method of [Musco et al., ITCS 2018]. All results are proven in the real RAM model of computation. Interestingly, previous state-of-the-art algorithms for most of the problems above relied on stochastic iterative methods, like stochastic coordinate and gradient descent. Our work takes a completely different approach, instead leveraging tools from matrix sketching.

cs.DS↗

arXiv2025

Surrogate-based Autotuning for Randomized Sketching Algorithms in Regression Problems

Algorithms from Randomized Numerical Linear Algebra (RandNLA) are known to be effective in handling high-dimensional computational problems, providing high-quality empirical performance as well as strong probabilistic guarantees. However, their practical application is complicated by the fact that the user needs to set various algorithm-specific tuning parameters which are different than those used in traditional NLA. This paper demonstrates how a surrogate-based autotuning approach can be used to address fundamental problems of parameter selection in RandNLA algorithms. In particular, we provide a detailed investigation of surrogate-based autotuning for sketch-and-precondition (SAP) based randomized least squares methods, which have been one of the great success stories in modern RandNLA. Empirical results show that our surrogate-based autotuning approach can achieve near-optimal performance with much less tuning cost than a random search (up to about 4x fewer trials of different parameter configurations). Moreover, while our experiments focus on least squares, our results demonstrate a general-purpose autotuning pipeline applicable to any kind of RandNLA algorithm.

cs.LG↗

arXiv2024

Stochastic Newton Proximal Extragradient Method

Stochastic second-order methods achieve fast local convergence in strongly convex optimization by using noisy Hessian estimates to precondition the gradient. However, these methods typically reach superlinear convergence only when the stochastic Hessian noise diminishes, increasing per-iteration costs over time. Recent work in [arXiv:2204.09266] addressed this with a Hessian averaging scheme that achieves superlinear convergence without higher per-iteration costs. Nonetheless, the method has slow global convergence, requiring up to $\tilde{O}(κ^2)$ iterations to reach the superlinear rate of $\tilde{O}((1/t)^{t/2})$, where $κ$ is the problem's condition number. In this paper, we propose a novel stochastic Newton proximal extragradient method that improves these bounds, achieving a faster global linear rate and reaching the same fast superlinear rate in $\tilde{O}(κ)$ iterations. We accomplish this by extending the Hybrid Proximal Extragradient (HPE) framework, achieving fast global and local convergence rates for strongly convex functions with access to a noisy Hessian oracle.

math.OC↗

arXiv2024

Recent and Upcoming Developments in Randomized Numerical Linear Algebra for Machine Learning

Large matrices arise in many machine learning and data analysis applications, including as representations of datasets, graphs, model weights, and first and second-order derivatives. Randomized Numerical Linear Algebra (RandNLA) is an area which uses randomness to develop improved algorithms for ubiquitous matrix problems. The area has reached a certain level of maturity; but recent hardware trends, efforts to incorporate RandNLA algorithms into core numerical libraries, and advances in machine learning, statistics, and random matrix theory, have lead to new theoretical and practical challenges. This article provides a self-contained overview of RandNLA, in light of these developments.

cs.LG↗

arXiv2024

Solving Dense Linear Systems Faster Than via Preconditioning

We give a stochastic optimization algorithm that solves a dense $n\times n$ real-valued linear system $Ax=b$, returning $\tilde x$ such that $\|A\tilde x-b\|\leq ε\|b\|$ in time: $$\tilde O((n^2+nk^{ω-1})\log1/ε),$$ where $k$ is the number of singular values of $A$ larger than $O(1)$ times its smallest positive singular value, $ω< 2.372$ is the matrix multiplication exponent, and $\tilde O$ hides a poly-logarithmic in $n$ factor. When $k=O(n^{1-θ})$ (namely, $A$ has a flat-tailed spectrum, e.g., due to noisy data or regularization), this improves on both the cost of solving the system directly, as well as on the cost of preconditioning an iterative method such as conjugate gradient. In particular, our algorithm has an $\tilde O(n^2)$ runtime when $k=O(n^{0.729})$. We further adapt this result to sparse positive semidefinite matrices and least squares regression. Our main algorithm can be viewed as a randomized block coordinate descent method, where the key challenge is simultaneously ensuring good convergence and fast per-iteration time. In our analysis, we use theory of majorization for elementary symmetric polynomials to establish a sharp convergence guarantee when coordinate blocks are sampled using a determinantal point process. We then use a Markov chain coupling argument to show that similar convergence can be attained with a cheaper sampling scheme, and accelerate the block coordinate descent update via matrix sketching.

cs.DS↗

arXiv2024

Optimal Embedding Dimension for Sparse Subspace Embeddings

A random $m\times n$ matrix $S$ is an oblivious subspace embedding (OSE) with parameters $ε>0$, $δ\in(0,1/3)$ and $d\leq m\leq n$, if for any $d$-dimensional subspace $W\subseteq R^n$, $P\big(\,\forall_{x\in W}\ (1+ε)^{-1}\|x\|\leq\|Sx\|\leq (1+ε)\|x\|\,\big)\geq 1-δ.$ It is known that the embedding dimension of an OSE must satisfy $m\geq d$, and for any $θ> 0$, a Gaussian embedding matrix with $m\geq (1+θ) d$ is an OSE with $ε= O_θ(1)$. However, such optimal embedding dimension is not known for other embeddings. Of particular interest are sparse OSEs, having $s\ll m$ non-zeros per column, with applications to problems such as least squares regression and low-rank approximation. We show that, given any $θ> 0$, an $m\times n$ random matrix $S$ with $m\geq (1+θ)d$ consisting of randomly sparsified $\pm1/\sqrt s$ entries and having $s= O(\log^4(d))$ non-zeros per column, is an oblivious subspace embedding with $ε= O_θ(1)$. Our result addresses the main open question posed by Nelson and Nguyen (FOCS 2013), who conjectured that sparse OSEs can achieve $m=O(d)$ embedding dimension, and it improves on $m=O(d\log(d))$ shown by Cohen (SODA 2016). We use this to construct the first oblivious subspace embedding with $O(d)$ embedding dimension that can be applied faster than current matrix multiplication time, and to obtain an optimal single-pass algorithm for least squares regression. We further extend our results to Leverage Score Sparsification (LESS), which is a recently introduced non-oblivious embedding technique. We use LESS to construct the first subspace embedding with low distortion $ε=o(1)$ and optimal embedding dimension $m=O(d/ε^2)$ that can be applied in current matrix multiplication time.

cs.DS↗

arXiv2024

Distributed Least Squares in Small Space via Sketching and Bias Reduction

Matrix sketching is a powerful tool for reducing the size of large data matrices. Yet there are fundamental limitations to this size reduction when we want to recover an accurate estimator for a task such as least square regression. We show that these limitations can be circumvented in the distributed setting by designing sketching methods that minimize the bias of the estimator, rather than its error. In particular, we give a sparse sketching method running in optimal space and current matrix multiplication time, which recovers a nearly-unbiased least squares estimator using two passes over the data. This leads to new communication-efficient distributed averaging algorithms for least squares and related tasks, which directly improve on several prior approaches. Our key novelty is a new bias analysis for sketched least squares, giving a sharp characterization of its dependence on the sketch sparsity. The techniques include new higher-moment restricted Bai-Silverstein inequalities, which are of independent interest to the non-asymptotic analysis of deterministic equivalents for random matrices that arise from sketching.

cs.DS↗

arXiv2024

Second-order Information Promotes Mini-Batch Robustness in Variance-Reduced Gradients

We show that, for finite-sum minimization problems, incorporating partial second-order information of the objective function can dramatically improve the robustness to mini-batch size of variance-reduced stochastic gradient methods, making them more scalable while retaining their benefits over traditional Newton-type approaches. We demonstrate this phenomenon on a prototypical stochastic second-order algorithm, called Mini-Batch Stochastic Variance-Reduced Newton ($\texttt{Mb-SVRN}$), which combines variance-reduced gradient estimates with access to an approximate Hessian oracle. In particular, we show that when the data size $n$ is sufficiently large, i.e., $n\gg α^2κ$, where $κ$ is the condition number and $α$ is the Hessian approximation factor, then $\texttt{Mb-SVRN}$ achieves a fast linear convergence rate that is independent of the gradient mini-batch size $b$, as long $b$ is in the range between $1$ and $b_{\max}=O(n/(α\log n))$. Only after increasing the mini-batch size past this critical point $b_{\max}$, the method begins to transition into a standard Newton-type algorithm which is much more sensitive to the Hessian approximation quality. We demonstrate this phenomenon empirically on benchmark optimization tasks showing that, after tuning the step size, the convergence rate of $\texttt{Mb-SVRN}$ remains fast for a wide range of mini-batch sizes, and the dependence of the phase transition point $b_{\max}$ on the Hessian approximation factor $α$ aligns with our theoretical predictions.

math.OC↗

arXiv2024

HERTA: A High-Efficiency and Rigorous Training Algorithm for Unfolded Graph Neural Networks

As a variant of Graph Neural Networks (GNNs), Unfolded GNNs offer enhanced interpretability and flexibility over traditional designs. Nevertheless, they still suffer from scalability challenges when it comes to the training cost. Although many methods have been proposed to address the scalability issues, they mostly focus on per-iteration efficiency, without worst-case convergence guarantees. Moreover, those methods typically add components to or modify the original model, thus possibly breaking the interpretability of Unfolded GNNs. In this paper, we propose HERTA: a High-Efficiency and Rigorous Training Algorithm for Unfolded GNNs that accelerates the whole training process, achieving a nearly-linear time worst-case training guarantee. Crucially, HERTA converges to the optimum of the original model, thus preserving the interpretability of Unfolded GNNs. Additionally, as a byproduct of HERTA, we propose a new spectral sparsification method applicable to normalized and regularized graph Laplacians that ensures tighter bounds for our algorithm than existing spectral sparsifiers do. Experiments on real-world datasets verify the superiority of HERTA as well as its adaptability to various loss functions and optimizers.

cs.LG↗

arXiv2023

Sharp Analysis of Sketch-and-Project Methods via a Connection to Randomized Singular Value Decomposition

Sketch-and-project is a framework which unifies many known iterative methods for solving linear systems and their variants, as well as further extensions to non-linear optimization problems. It includes popular methods such as randomized Kaczmarz, coordinate descent, variants of the Newton method in convex optimization, and others. In this paper, we develop a theoretical framework for obtaining sharp guarantees on the convergence rate of sketch-and-project methods. Our approach is the first to: (1) show that the convergence rate improves at least linearly with the sketch size, and even faster when the data matrix exhibits certain spectral decays; and (2) allow for sparse sketching matrices, which are more efficient than dense sketches and more robust than sub-sampling methods. In particular, our results explain an observed phenomenon that a radical sparsification of the sketching matrix does not affect the per iteration convergence rate of sketch-and-project. To obtain our results, we develop new non-asymptotic spectral bounds for the expected sketched projection matrix, which are of independent interest; and we establish a connection between the convergence rates of iterative sketch-and-project solvers and the approximation error of randomized singular value decomposition, which is a widely used one-shot sketching algorithm for low-rank approximation. Our experiments support the theory and demonstrate that even extremely sparse sketches exhibit the convergence properties predicted by our framework.

math.OC↗

arXiv2023

Algorithmic Gaussianization through Sketching: Converting Data into Sub-gaussian Random Designs

Algorithmic Gaussianization is a phenomenon that can arise when using randomized sketching or sampling methods to produce smaller representations of large datasets: For certain tasks, these sketched representations have been observed to exhibit many robust performance characteristics that are known to occur when a data sample comes from a sub-gaussian random design, which is a powerful statistical model of data distributions. However, this phenomenon has only been studied for specific tasks and metrics, or by relying on computationally expensive methods. We address this by providing an algorithmic framework for gaussianizing data distributions via averaging, proving that it is possible to efficiently construct data sketches that are nearly indistinguishable (in terms of total variation distance) from sub-gaussian random designs. In particular, relying on a recently introduced sketching technique called Leverage Score Sparsified (LESS) embeddings, we show that one can construct an $n\times d$ sketch of an $N\times d$ matrix $A$, where $n\ll N$, that is nearly indistinguishable from a sub-gaussian design, in time $O(\text{nnz}(A)\log N + nd^2)$, where $\text{nnz}(A)$ is the number of non-zero entries in $A$. As a consequence, strong statistical guarantees and precise asymptotics available for the estimators produced from sub-gaussian designs (e.g., for least squares and Lasso regression, covariance estimation, low-rank approximation, etc.) can be straightforwardly adapted to our sketching framework. We illustrate this with a new approximation guarantee for sketched least squares, among other examples.

cs.LG↗

arXiv2023

Randomized Numerical Linear Algebra : A Perspective on the Field With an Eye to Software

Randomized numerical linear algebra - RandNLA, for short - concerns the use of randomization as a resource to develop improved algorithms for large-scale linear algebra computations. The origins of contemporary RandNLA lay in theoretical computer science, where it blossomed from a simple idea: randomization provides an avenue for computing approximate solutions to linear algebra problems more efficiently than deterministic algorithms. This idea proved fruitful in the development of scalable algorithms for machine learning and statistical data analysis applications. However, RandNLA's true potential only came into focus upon integration with the fields of numerical analysis and "classical" numerical linear algebra. Through the efforts of many individuals, randomized algorithms have been developed that provide full control over the accuracy of their solutions and that can be every bit as reliable as algorithms that might be found in libraries such as LAPACK. Recent years have even seen the incorporation of certain RandNLA methods into MATLAB, the NAG Library, NVIDIA's cuSOLVER, and SciKit-Learn. For all its success, we believe that RandNLA has yet to realize its full potential. In particular, we believe the scientific community stands to benefit significantly from suitably defined "RandBLAS" and "RandLAPACK" libraries, to serve as standards conceptually analogous to BLAS and LAPACK. This 200-page monograph represents a step toward defining such standards. In it, we cover topics spanning basic sketching, least squares and optimization, low-rank approximation, full matrix decompositions, leverage score sampling, and sketching data with tensor product structures (among others). Much of the provided pseudo-code has been tested via publicly available MATLAB and Python implementations.

math.NA↗

arXiv2022

Hessian Averaging in Stochastic Newton Methods Achieves Superlinear Convergence

We consider minimizing a smooth and strongly convex objective function using a stochastic Newton method. At each iteration, the algorithm is given an oracle access to a stochastic estimate of the Hessian matrix. The oracle model includes popular algorithms such as Subsampled Newton and Newton Sketch. Despite using second-order information, these existing methods do not exhibit superlinear convergence, unless the stochastic noise is gradually reduced to zero during the iteration, which would lead to a computational blow-up in the per-iteration cost. We propose to address this limitation with Hessian averaging: instead of using the most recent Hessian estimate, our algorithm maintains an average of all the past estimates. This reduces the stochastic noise while avoiding the computational blow-up. We show that this scheme exhibits local $Q$-superlinear convergence with a non-asymptotic rate of $(Υ\sqrt{\log (t)/t}\,)^{t}$, where $Υ$ is proportional to the level of stochastic noise in the Hessian oracle. A potential drawback of this (uniform averaging) approach is that the averaged estimates contain Hessian information from the global phase of the method, i.e., before the iterates converge to a local neighborhood. This leads to a distortion that may substantially delay the superlinear convergence until long after the local neighborhood is reached. To address this drawback, we study a number of weighted averaging schemes that assign larger weights to recent Hessians, so that the superlinear convergence arises sooner, albeit with a slightly slower rate. Remarkably, we show that there exists a universal weighted averaging scheme that transitions to local convergence at an optimal stage, and still exhibits a superlinear convergence rate nearly (up to a logarithmic factor) matching that of uniform Hessian averaging.

math.OC↗

arXiv2022

Precise expressions for random projections: Low-rank approximation and randomized Newton

It is often desirable to reduce the dimensionality of a large dataset by projecting it onto a low-dimensional subspace. Matrix sketching has emerged as a powerful technique for performing such dimensionality reduction very efficiently. Even though there is an extensive literature on the worst-case performance of sketching, existing guarantees are typically very different from what is observed in practice. We exploit recent developments in the spectral analysis of random matrices to develop novel techniques that provide provably accurate expressions for the expected value of random projection matrices obtained via sketching. These expressions can be used to characterize the performance of dimensionality reduction in a variety of common machine learning tasks, ranging from low-rank approximation to iterative stochastic optimization. Our results apply to several popular sketching methods, including Gaussian and Rademacher sketches, and they enable precise analysis of these methods in terms of spectral properties of the data. Empirical results show that the expressions we derive reflect the practical performance of these sketching methods, down to lower-order effects and even constant factors.

cs.LG↗

arXiv2022

Unbiased estimators for random design regression

In linear regression we wish to estimate the optimum linear least squares predictor for a distribution over $d$-dimensional input points and real-valued responses, based on a small sample. Under standard random design analysis, where the sample is drawn i.i.d. from the input distribution, the least squares solution for that sample can be viewed as the natural estimator of the optimum. Unfortunately, this estimator almost always incurs an undesirable bias coming from the randomness of the input points, which is a significant bottleneck in model averaging. In this paper we show that it is possible to draw a non-i.i.d. sample of input points such that, regardless of the response model, the least squares solution is an unbiased estimator of the optimum. Moreover, this sample can be produced efficiently by augmenting a previously drawn i.i.d. sample with an additional set of $d$ points, drawn jointly according to a certain determinantal point process constructed from the input distribution rescaled by the squared volume spanned by the points. Motivated by this, we develop a theoretical framework for studying volume-rescaled sampling, and in the process prove a number of new matrix expectation identities. We use them to show that for any input distribution and $ε>0$ there is a random design consisting of $O(d\log d+ d/ε)$ points from which an unbiased estimator can be constructed whose expected square loss over the entire distribution is bounded by $1+ε$ times the loss of the optimum. We provide efficient algorithms for generating such unbiased estimators in a number of practical settings and support our claims experimentally.

stat.ML↗

arXiv2021

Domain Sparsification of Discrete Distributions using Entropic Independence

We present a framework for speeding up the time it takes to sample from discrete distributions $μ$ defined over subsets of size $k$ of a ground set of $n$ elements, in the regime $k\ll n$. We show that having estimates of marginals $\mathbb{P}_{S\sim μ}[i\in S]$, the task of sampling from $μ$ can be reduced to sampling from distributions $ν$ supported on size $k$ subsets of a ground set of only $n^{1-α}\cdot \operatorname{poly}(k)$ elements. Here, $1/α\in [1, k]$ is the parameter of entropic independence for $μ$. Further, the sparsified distributions $ν$ are obtained by applying a sparse (mostly $0$) external field to $μ$, an operation that often retains algorithmic tractability of sampling from $ν$. This phenomenon, which we dub domain sparsification, allows us to pay a one-time cost of estimating the marginals of $μ$, and in return reduce the amortized cost needed to produce many samples from the distribution $μ$, as is often needed in upstream tasks such as counting and inference. For a wide range of distributions where $α=Ω(1)$, our result reduces the domain size, and as a corollary, the cost-per-sample, by a $\operatorname{poly}(n)$ factor. Examples include monomers in a monomer-dimer system, non-symmetric determinantal point processes, and partition-constrained Strongly Rayleigh measures. Our work significantly extends the reach of prior work of Anari and Dereziński who obtained domain sparsification for distributions with a log-concave generating polynomial (corresponding to $α=1$). As a corollary of our new analysis techniques, we also obtain a less stringent requirement on the accuracy of marginal estimates even for the case of log-concave polynomials; roughly speaking, we show that constant-factor approximation is enough for domain sparsification, improving over $O(1/k)$ relative error established in prior work.

cs.DS↗

arXiv2021

Newton-LESS: Sparsification without Trade-offs for the Sketched Newton Update

In second-order optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration. Randomized sketching has emerged as a powerful technique for constructing estimates of the Hessian which can be used to perform approximate Newton steps. This involves multiplication by a random sketching matrix, which introduces a trade-off between the computational cost of sketching and the convergence rate of the optimization algorithm. A theoretically desirable but practically much too expensive choice is to use a dense Gaussian sketching matrix, which produces unbiased estimates of the exact Newton step and which offers strong problem-independent convergence guarantees. We show that the Gaussian sketching matrix can be drastically sparsified, significantly reducing the computational cost of sketching, without substantially affecting its convergence properties. This approach, called Newton-LESS, is based on a recently introduced sketching technique: LEverage Score Sparsified (LESS) embeddings. We prove that Newton-LESS enjoys nearly the same problem-independent local convergence rate as Gaussian embeddings, not just up to constant factors but even down to lower order terms, for a large class of optimization tasks. In particular, this leads to a new state-of-the-art convergence result for an iterative least squares solver. Finally, we extend LESS embeddings to include uniformly sparsified random sign matrices which can be implemented efficiently and which perform well in numerical experiments.

math.OC↗