arXiv subjects

Michał Dereziński

Publications and source records attributed to Michał Dereziński.

At least 19 recordsLinked to original sources

arXiv2026

Inference for Newton Methods with Accelerated Sketch-and-Project via Random Scaling

We study an online sketched Newton method that approximates the Newton direction at each step via a state-of-the-art sketching solver, called the generalized accelerated sketch-and-project solver (GAS), thereby mitigating the computational bottleneck of classical second-order methods. The GAS solver improves upon vanilla, unaccelerated sketch-and-project solvers by achieving accelerated convergence through Nesterov momentum updates, and accommodates a flexible projection metric whose proper choice further reduces computational cost. Building on this design, we establish asymptotic normality of the averaged sketched Newton iterates and characterize their limiting covariance matrix. The resulting covariance recovers that of the unaccelerated sketched Newton method under a specific choice of acceleration parameters, converges more rapidly (in the number of sketching steps) to the minimax-optimal covariance in general, and is smaller than that of the last iterate produced by the accelerated method. Finally, we strengthen these results by establishing a functional central limit theorem for the Newton iterates, which allows us to bypass explicit covariance estimation and develop an online inference procedure based on random scaling. Specifically, we construct a pivotal test statistic by appropriately rescaling the averaged iterates, so that its limiting distribution is free of any unknown parameters, enabling asymptotically valid online inference. Numerical experiments demonstrate superior performance of the proposed inference procedure.

stat.ML↗

arXiv2026

The matrix-vector complexity of $Ax=b$

Matrix--vector algorithms, particularly Krylov subspace methods, are widely viewed as the most effective algorithms for solving large systems of linear equations. This paper establishes lower bounds on the worst-case number of matrix--vector products needed by such an algorithm to approximately solve a general linear system. The first main result is that, for any matrix--vector algorithm which is allowed the use of randomization and can perform products with both a matrix and its transpose, $Ω(κ\log(1/\varepsilon))$ matrix--vector products are necessary to solve a linear system with condition number $κ$ to accuracy $\varepsilon$, matching an upper bound for conjugate gradient on the normal equations. The second main result is that one-sided algorithms, which lack access to the transpose, must use $n$ matrix--vector products to solve an $n \times n$ linear system, even when the problem is perfectly conditioned. Both main results include explicit constants that match known upper bounds up to a factor of four. These results rigorously demonstrate the limitations of matrix--vector algorithms and confirm the optimality of widely used Krylov subspace algorithms.

cs.DS↗

arXiv2026

How AI settled the complexity of the oldest SGD algorithm

In 1937, Stefan Kaczmarz proposed a simple algorithm for solving systems of linear equations. This algorithm turned out to be the earliest known example of stochastic gradient descent, a ubiquitous computing paradigm that drives the training of modern AI models such as ChatGPT and Gemini. Now, those AI models have joined forces to discover the worst-case complexity of the Kaczmarz algorithm. This paper tells the story of how it happened.

cs.LG↗

arXiv2026

Debiasing Random Oblique Projections for Subsampled OLS and Fast CUR in High Dimensions

Random sampling is a fundamental tool in modern machine learning and numerical linear algebra for reducing the computational cost of large-scale matrix problems. Existing analyses, however, rely primarily on subspace embedding guarantees, which do not precisely characterize the statistical bias of nonlinear random oblique projections induced by sampling, which arises ubiquitously in subsampled least squares and fast low-rank approximation methods. Because (pseudo)inversion is nonlinear, these random oblique projections can be systematically biased even when the underlying sketch is unbiased, thereby introducing hidden bias into downstream least squares and low-rank approximation solutions. In this work, we develop a unified non-asymptotic theory for random oblique projections in high dimensions. We show that standard random sampling schemes generally induce a systematic statistical bias overlooked by classical subspace embedding-style analyses, and we propose a principled debiasing framework to correct it. We illustrate the power of the theory through two canonical applications. For subsampled least squares, we obtain sharp bias--variance characterizations, reveal previously unrecognized statistical suboptimality in widely used sampling schemes, and identify when debiasing yields provable improvements. For fast CUR decomposition, we develop a debiased approach with improved approximation accuracy. Numerical experiments further validate our theoretical findings.

math.NA↗

arXiv2026

Towards Universal Convergence of Backward Error in Linear System Solvers

The quest for an algorithm that solves an $n\times n$ linear system in $O(n^2)$ time complexity, or $O(n^2 \text{poly}(1/ε))$ when solving up to $ε$ relative error, is a long-standing open problem in numerical linear algebra and theoretical computer science. There are two predominant paradigms for measuring relative error: forward error (i.e., distance from the output to the optimum solution) and backward error (i.e., distance to the nearest problem solved by the output). In most prior studies, convergence of iterative linear system solvers is measured via various notions of forward error, and as a result, depends heavily on the conditioning of the input. Yet, the numerical analysis literature has long advocated for backward error as the more practically relevant notion of approximation. In this work, we show that -- surprisingly -- the classical and simple Richardson iteration incurs at most $1/k$ (relative) backward error after $k$ iterations on any positive semidefinite (PSD) linear system, irrespective of its condition number. This universal convergence rate implies an $O(n^2/ε)$ complexity algorithm for solving a PSD linear system to $ε$ backward error, and we establish similar or better complexity when using a variety of Krylov solvers beyond Richardson. Then, by directly minimizing backward error over a Krylov subspace, we attain an even faster $O(1/k^2)$ universal rate, and we turn this into an efficient algorithm, MINBERR, with complexity $O(n^2/\sqrtε)$. Finally, we extend this approach via normal equations to solving general linear systems in $O(n^2\log(n)/ε)$ time complexity. We report strong numerical performance of our algorithms on benchmark problems.

math.NA↗

arXiv2026

Perfect Parallelization in Mini-Batch SGD with Classical Momentum Acceleration

Accelerating stochastic gradient methods with classical momentum schemes, such as Polyak's heavy ball, has proven highly successful in training large-scale machine learning models, particularly when combined with the hardware acceleration of large mini-batch computations. Yet, the effect of classical momentum on stochastic mini-batch optimization has been poorly understood theoretically, with prior works requiring strong noise assumptions and extremely large mini-batches. In this work, we develop a general theory of stochastic momentum acceleration for optimizing over quadratics in the interpolation regime, a popular abstraction for studying deep learning dynamics which also includes classical methods such as randomized Kaczmarz and coordinate descent. Our framework encompasses both heavy ball and Nesterov-style momentum, allows for arbitrary mini-batch sizes, and makes minimal assumptions on the stochastic noise. In particular, we show that acceleration from classical momentum is directly proportional to the gradient mini-batch size (up to a natural saturation point), thereby enabling perfect parallelization of mini-batch computations. Our theory also provides a simple choice for the momentum parameter, which is shown to be effective empirically.

cs.LG↗

arXiv2026

Numerical Instabilities in the Kaczmarz Method and Stabilization by Iterative Refinement

The randomized Kaczmarz method and its accelerated variants are a powerful class of iterative methods for solving large-scale linear systems, offering guaranteed convergence with low per-iteration cost. However, their numerical stability remains poorly understood. In this work, we investigate the stability properties of both classical and accelerated randomized Kaczmarz methods, with an emphasis on how error propagates across iterations and interacts with acceleration. We show that both classical and accelerated randomized Kaczmarz fail to be forward stable. To address this issue, we propose the integration of iterative refinement into randomized Kaczmarz frameworks. We demonstrate that refinement can effectively control error accumulation and recover high-accuracy solutions, even when the system is ill-conditioned. Numerical experiments corroborate our theoretical findings and illustrate the practical benefits of combining refinement with both classical and accelerated Kaczmarz methods.

math.NA↗

arXiv2026

Accelerating Power Method with Fast Sketching for Stronger Low-Rank Approximation

The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this procedure becomes a major bottleneck. We develop an algorithmic and theoretical framework for accelerating the power method using fast sketching, which is a popular paradigm in randomized linear algebra. Our framework leads to simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation, which attain strong numerical performance on benchmark problems. The key novelty in our analysis is the use of regularized spectral approximation, a property of fast sketching methods which proves more flexible in generalizing power method guarantees than traditional arguments.

math.NA↗

arXiv2026

Well-Conditioned Oblivious Perturbations in Linear Space

Perturbing a deterministic $n$-dimensional matrix with small Gaussian noise is a cornerstone of smoothed analysis of algorithms [Spielman and Teng, JACM 2004], as it reduces the condition number of the input to $O(n)$, and with it the complexity of many matrix algorithms. However, when deployed algorithmically, these perturbations are expensive due to the cost of generating and storing $n^2$ Gaussian random variables. We propose a perturbation that requires generating and storing $O(n)$ random numbers in $O(\log n)$ bits of precision, and reduces the condition number of any deterministic matrix to $O(n)$, matching Gaussian perturbations. Our result in particular implies a better complexity for the perturbed conjugate gradient algorithm, showing that we can solve an $n\times n$ linear system in linear space to within an arbitrarily small constant backward error using $O(n)$ matrix-vector products. In our construction, we introduce the concept of a pattern matrix, which is a dense deterministic matrix that maps all sparse vectors into dense vectors, and we combine it with a sparse perturbation whose entries are dependent and located in a non-uniform fashion. In order to analyze this construction, we develop new techniques for lower bounding the smallest singular value of a random matrix with dependent entries.

cs.DS↗

arXiv2026

Last-Iterate Convergence of Randomized Kaczmarz and SGD with Greedy Step Size

We study last-iterate convergence of SGD with greedy step size over smooth quadratics in the interpolation regime, a setting which captures the classical Randomized Kaczmarz algorithm as well as other popular iterative linear system solvers. For these methods, we show that the $t$-th iterate attains an $O(1/t^{3/4})$ convergence rate, addressing a question posed by Attia, Schliserman, Sherman, and Koren, who gave an $O(1/t^{1/2})$ guarantee for this setting. In the proof, we introduce the family of stochastic contraction processes, whose behavior can be described by the evolution of a certain deterministic eigenvalue equation, which we analyze via a careful discrete-to-continuous reduction.

cs.LG↗

arXiv2026

Have ASkotch: A Neat Solution for Large-scale Kernel Ridge Regression

Kernel ridge regression (KRR) is a fundamental computational tool, appearing in problems that range from computational chemistry to health analytics, with a particular interest due to its starring role in Gaussian process regression. However, full KRR solvers are challenging to scale to large datasets: both direct (i.e., Cholesky decomposition) and iterative methods (i.e., PCG) incur prohibitive computational and storage costs. The standard approach to scale KRR to large datasets chooses a set of inducing points and solves an approximate version of the problem, inducing points KRR. However, the resulting solution tends to have worse predictive performance than the full KRR solution. In this work, we introduce a new solver, ASkotch, for full KRR that provides better solutions faster than state-of-the-art solvers for full and inducing points KRR. ASkotch is a scalable, accelerated, iterative method for full KRR that provably obtains linear convergence. Under appropriate conditions, we show that ASkotch obtains condition-number-free linear convergence. This convergence analysis rests on the theory of ridge leverage scores and determinantal point processes. ASkotch outperforms state-of-the-art KRR solvers on a testbed of 23 large-scale KRR regression and classification tasks derived from a wide range of application domains, demonstrating the superiority of full KRR over inducing points KRR. Our work opens up the possibility of as-yet-unimagined applications of full KRR across a number of disciplines.

cs.LG↗

arXiv2025

Optimal Subspace Embeddings: Resolving Nelson-Nguyen Conjecture Up to Sub-Polylogarithmic Factors

We give a proof of the conjecture of Nelson and Nguyen [FOCS 2013] on the optimal dimension and sparsity of oblivious subspace embeddings, up to sub-polylogarithmic factors: For any $n\geq d$ and $ε\geq d^{-O(1)}$, there is a random $\tilde O(d/ε^2)\times n$ matrix $Π$ with $\tilde O(\log(d)/ε)$ non-zeros per column such that for any $A\in\mathbb{R}^{n\times d}$, with high probability, $(1-ε)\|Ax\|\leq\|ΠAx\|\leq(1+ε)\|Ax\|$ for all $x\in\mathbb{R}^d$, where $\tilde O(\cdot)$ hides only sub-polylogarithmic factors in $d$. Our result in particular implies a new fastest sub-current matrix multiplication time reduction of size $\tilde O(d/ε^2)$ for a broad class of $n\times d$ linear regression tasks. A key novelty in our analysis is a matrix concentration technique we call iterative decoupling, which we use to fine-tune the higher-order trace moment bounds attainable via existing random matrix universality tools [Brailovskaya and van Handel, GAFA 2024].

cs.DS↗

arXiv2025

Turbocharging Gaussian Process Inference with Approximate Sketch-and-Project

Gaussian processes (GPs) play an essential role in biostatistics, scientific machine learning, and Bayesian optimization for their ability to provide probabilistic predictions and model uncertainty. However, GP inference struggles to scale to large datasets (which are common in modern applications), since it requires the solution of a linear system whose size scales quadratically with the number of samples in the dataset. We propose an approximate, distributed, accelerated sketch-and-project algorithm ($\texttt{ADASAP}$) for solving these linear systems, which improves scalability. We use the theory of determinantal point processes to show that the posterior mean induced by sketch-and-project rapidly converges to the true posterior mean. In particular, this yields the first efficient, condition number-free algorithm for estimating the posterior mean along the top spectral basis functions, showing that our approach is principled for GP inference. $\texttt{ADASAP}$ outperforms state-of-the-art solvers based on conjugate gradient and coordinate descent across several benchmark datasets and a large-scale Bayesian optimization task. Moreover, $\texttt{ADASAP}$ scales to a dataset with $> 3 \cdot 10^8$ samples, a feat which has not been accomplished in the literature.

cs.LG↗

arXiv2025

Randomized Kaczmarz Methods with Beyond-Krylov Convergence

Randomized Kaczmarz methods form a family of linear system solvers which converge by repeatedly projecting their iterates onto randomly sampled equations. While effective in some contexts, such as highly over-determined least squares, Kaczmarz methods are traditionally deemed secondary to Krylov subspace methods, since this latter family of solvers can exploit outliers in the input's singular value distribution to attain fast convergence on ill-conditioned systems. In this paper, we introduce Kaczmarz++, an accelerated randomized block Kaczmarz algorithm that exploits outlying singular values in the input to attain a fast Krylov-style convergence. Moreover, we show that Kaczmarz++ captures large outlying singular values provably faster than popular Krylov methods, for both over- and under-determined systems. We also develop an optimized variant for positive semidefinite systems, called CD++, demonstrating empirically that it is competitive in arithmetic operations with both CG and GMRES on a collection of benchmark problems. To attain these results, we introduce several novel algorithmic improvements to the Kaczmarz framework, including adaptive momentum acceleration, Tikhonov-regularized projections, and a memoization scheme for reusing information from previously sampled equation blocks.

math.NA↗

arXiv2025

Faster Low-Rank Approximation and Kernel Ridge Regression via the Block-Nyström Method

The Nyström method is a popular low-rank approximation technique for large matrices that arise in kernel methods and convex optimization. Yet, when the data exhibits heavy-tailed spectral decay, the effective dimension of the problem often becomes so large that even the Nyström method may be outside of our computational budget. To address this, we propose Block-Nyström, an algorithm that injects a block-diagonal structure into the Nyström method, thereby significantly reducing its computational cost while recovering strong approximation guarantees. We show that Block-Nyström can be used to construct improved preconditioners for second-order optimization, as well as to efficiently solve kernel ridge regression for statistical learning over Hilbert spaces. Our key technical insight is that, within the same computational budget, combining several smaller Nyström approximations leads to stronger tail estimates of the input spectrum than using one larger approximation. Along the way, we provide a novel recursive preconditioning scheme for efficiently inverting the Block-Nyström matrix, and provide new statistical learning bounds for a broad class of approximate kernel ridge regression solvers.

cs.DS↗

arXiv2025

Approaching Optimality for Solving Dense Linear Systems with Low-Rank Structure

We provide new high-accuracy randomized algorithms for solving linear systems and regression problems that are well-conditioned except for $k$ large singular values. For solving such $d \times d$ positive definite system our algorithms succeed whp. and run in time $\tilde O(d^2 + k^ω)$. For solving such regression problems in a matrix $\mathbf{A} \in \mathbb{R}^{n \times d}$ our methods succeed whp. and run in time $\tilde O(\mathrm{nnz}(\mathbf{A}) + d^2 + k^ω)$ where $ω$ is the matrix multiplication exponent and $\mathrm{nnz}(\mathbf{A})$ is the number of non-zeros in $\mathbf{A}$. Our methods nearly-match a natural complexity limit under dense inputs for these problems and improve upon a trade-off in prior approaches that obtain running times of either $\tilde O(d^{2.065}+k^ω)$ or $\tilde O(d^2 + dk^{ω-1})$ for $d\times d$ systems. Moreover, we show how to obtain these running times even under the weaker assumption that all but $k$ of the singular values have a suitably bounded generalized mean. Consequently, we give the first nearly-linear time algorithm for computing a multiplicative approximation to the nuclear norm of an arbitrary dense matrix. Our algorithms are built on three general recursive preconditioning frameworks, where matrix sketching and low-rank update formulas are carefully tailored to the problems' structure.

cs.DS↗

arXiv2025

Fine-grained Analysis and Faster Algorithms for Iteratively Solving Linear Systems

Despite being a key bottleneck in many machine learning tasks, the cost of solving large linear systems has proven challenging to quantify due to problem-dependent quantities such as condition numbers. To tackle this, we consider a fine-grained notion of complexity for solving linear systems, which is motivated by applications where the data exhibits low-dimensional structure, including spiked covariance models and kernel machines, and when the linear system is explicitly regularized, such as ridge regression. Concretely, let $κ_\ell$ be the ratio between the $\ell$th largest and the smallest singular value of $n\times n$ matrix $A$. We give a stochastic algorithm based on the Sketch-and-Project paradigm, that solves the linear system $Ax = b$, that is, finds $\bar{x}$ such that $\|A\bar{x} - b\| \le ε\|b\|$, in time $\bar O(κ_\ell\cdot n^2\log 1/ε)$, for any $\ell = O(n^{0.729})$. This is a direct improvement over preconditioned conjugate gradient, and it provides a stronger separation between stochastic linear solvers and algorithms accessing $A$ only through matrix-vector products. Our main technical contribution is the new analysis of the first and second moments of the random projection matrix that arises in Sketch-and-Project.

cs.DS↗

arXiv2025

Stochastic Variance-Reduced Newton: Accelerating Finite-Sum Minimization with Large Batches

Stochastic variance reduction has proven effective at accelerating first-order algorithms for solving convex finite-sum optimization tasks such as empirical risk minimization. Incorporating second-order information has proven helpful in further improving the performance of these first-order methods. Yet, comparatively little is known about the benefits of using variance reduction to accelerate popular stochastic second-order methods such as Subsampled Newton. To address this, we propose Stochastic Variance-Reduced Newton (SVRN), a finite-sum minimization algorithm that provably accelerates existing stochastic Newton methods from $O(α\log(1/ε))$ to $O\big(\frac{\log(1/ε)}{\log(n)}\big)$ passes over the data, i.e., by a factor of $O(α\log(n))$, where $n$ is the number of sum components and $α$ is the approximation factor in the Hessian estimate. Surprisingly, this acceleration gets more significant the larger the data size $n$, which is a unique property of SVRN. Our algorithm retains the key advantages of Newton-type methods, such as easily parallelizable large-batch operations and a simple unit step size. We use SVRN to accelerate Subsampled Newton and Iterative Hessian Sketch algorithms, and show that it compares favorably to popular first-order methods with variance~reduction.

math.OC↗