arXiv subjects

Aaron Sidford

Publications and source records attributed to Aaron Sidford.

At least 19 recordsLinked to original sources

arXiv2026

Matching Composition and Efficient Weight Reduction in Dynamic Matching

We consider the foundational problem of maintaining a $(1-\varepsilon)$-approximate maximum weight matching (MWM) in an $n$-node dynamic graph undergoing edge insertions and deletions. We provide a general reduction that reduces the problem on graphs with a weight range of $\mathrm{poly}(n)$ to $\mathrm{poly}(1/\varepsilon)$ at the cost of just an additive $\mathrm{poly}(1/\varepsilon)$ in update time. This improves upon the prior reduction of Gupta-Peng (FOCS 2013) which reduces the problem to a weight range of $\varepsilon^{-O(1/\varepsilon)}$ with a multiplicative cost of $O(\log n)$. When combined with a reduction of Bernstein-Dudeja-Langley (STOC 2021) this yields a reduction from dynamic $(1-\varepsilon)$-approximate MWM in bipartite graphs with a weight range of $\mathrm{poly}(n)$ to dynamic $(1-\varepsilon)$-approximate maximum cardinality matching in bipartite graphs at the cost of a multiplicative $\mathrm{poly}(1/\varepsilon)$ in update time, thereby resolving an open problem in [GP'13; BDL'21]. Additionally, we show that our approach is amenable to MWM problems in streaming, shared-memory work-depth, and massively parallel computation models. We also apply our techniques to obtain an efficient dynamic algorithm for rounding weighted fractional matchings in general graphs. Underlying our framework is a new structural result about MWM that we call the "matching composition lemma" and new dynamic matching subroutines that may be of independent interest.

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arXiv2026

Fast, Parallel, Query-Efficient Binary Classification

We study the fundamental classification problem of computing a separating hyperplane for a binary-labeled dataset of size $n$ with normalized $d$-dimensional features. Letting $Φ\in \mathbb{R}^{n \times d}$ denote the feature matrix and $γ$ the margin of the maximum-margin separating hyperplane, we present a randomized algorithm that solves this problem in $\tilde{O}(γ^{-2/3}\, \operatorname{nnz}(Φ) + γ^{-2(ω+1)/3})$-sequential running time (work), $\tilde{O}(γ^{-2/3})$-parallel (computational) depth, and accesses $Φ$ only through $\tilde{O}(γ^{-2/3})$-matrix-vector queries (matvecs). We also present a second, faster randomized algorithm with a $\tilde{O}(γ^{-2/3}\, \operatorname{nnz}(Φ) + γ^{-2})$-sequential running time that uses $\tilde{O}(γ^{-2/3})$-matvecs to $Φ$, but achieves only $\tilde{O}(γ^{-4/3})$-parallel depth. Both algorithms match the near-optimal deterministic matvec complexity recently established by Kornowski and Shamir [2025], Karmarkar et al. [2026] and achieve improved sequential runtime and parallel depth, albeit at the expense of using randomness.

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arXiv2026

Computing Lewis weights to high precision using local relative smoothness

We provide algorithms that compute $ε$-estimates of the $\ell_p$-Lewis weights of a matrix $A \in \mathbb{R}^{m \times n}$ for $p \geq 4$ using $O(p^2 \log(m/ε))$ rounds of leverage score computation, where $\ell_p$-Lewis weights and leverage scores are both standard measures of row importance. This improves upon the state-of-the-art round complexity of $O(p^3 \log(m/ε))$ due to Fazel, Lee, Padmanabha, and Sidford (2022). We obtain our results by carefully applying a local variant of relatively smooth gradient descent to primal and dual forms of the $\ell_p$-Lewis weight optimization problem and providing tools to convert between different notions of approximate $\ell_p$-Lewis weights.

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arXiv2026

Solving Zero-Sum Games with Fewer Matrix-Vector Products

In this paper we consider the problem of computing an $ε$-approximate Nash Equilibrium of a zero-sum game in a payoff matrix $A \in \mathbb{R}^{m \times n}$ with $O(1)$-bounded entries given access to a matrix-vector product oracle for $A$ and its transpose $A^\top$. We provide a deterministic algorithm that solves the problem using $\tilde{O}(ε^{-8/9})$-oracle queries, where $\tilde{O}(\cdot)$ hides factors polylogarithmic in $m$, $n$, and $ε^{-1}$. Our result improves upon the state-of-the-art query complexity of $\tilde{O}(ε^{-1})$ established by [Nemirovski, 2004] and [Nesterov, 2005]. We obtain this result through a general framework that yields improved deterministic query complexities for solving a broader class of minimax optimization problems which includes computing a linear classifier (hard-margin support vector machine) as well as linear regression.

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arXiv2026

Solving Matrix Games with Near-Optimal Matvec Complexity

We study the problem of computing an $ε$-approximate Nash equilibrium of a two-player, bilinear game with a bounded payoff matrix $A \in \mathbb{R}^{m \times n}$, when the players' strategies are constrained to lie in simple sets. We provide algorithms which solve this problem in $\tilde{O}(ε^{-2/3})$ matrix-vector multiplies (matvecs) in two well-studied cases: $\ell_1$-$\ell_1$ (or zero-sum) games, where the players' strategies are both in the probability simplex, and $\ell_2$-$\ell_1$ games (encompassing hard-margin SVMs), where the players' strategies are in the unit Euclidean ball and probability simplex respectively. These results improve upon the previous state-of-the-art complexities of $\tilde{O}(ε^{-8/9})$ for $\ell_1$-$\ell_1$ and $\tilde{O}(ε^{-7/9})$ for $\ell_2$-$\ell_1$ due to [KOS '25]. In both settings our results are nearly-optimal as they match lower bounds of [KS '25] up to polylogarithmic factors.

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arXiv2026

Randomization for Faster Exact Optimization of Discounted Markov Decision Processes

We provide faster deterministic and randomized algorithms for exactly solving discounted Markov Decision Processes (DMDPs). We obtain our results by efficiently reducing computing optimal values and policies in DMDPs to the easier tasks of policy evaluation and computing approximately optimal values in DMDPs. We provide both a straightforward deterministic reduction and a more efficient randomized variant that, together with advances in approximately solving DMDPs, yield our results.

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arXiv2026

Convex optimization with $p$-norm oracles

In recent years, there have been significant advances in efficiently solving $\ell_s$-regression using linear system solvers and $\ell_2$-regression [Adil-Kyng-Peng-Sachdeva, J. ACM'24]. Would efficient smoothed $\ell_p$-norm solvers lead to even faster rates for solving $\ell_s$-regression when $2 \leq p < s$? In this paper, we give an affirmative answer to this question and show how to solve $\ell_s$-regression using $\tilde{O}(n^{\fracν{1+ν}})$ iterations of solving smoothed $\ell_p$ regression problems, where $ν:= \frac{1}{p} - \frac{1}{s}$. To obtain this result, we provide improved accelerated rates for convex optimization problems when given access to an $\ell_p^s(λ)$-proximal oracle, which, for a point $c$, returns the solution of the regularized problem $\min_{x} f(x) + λ||x-c||_p^s$. Additionally, we show that these rates for the $\ell_p^s(λ)$-proximal oracle are optimal for algorithms that query in the span of the outputs of the oracle, and we further apply our techniques to settings of high-order and quasi-self-concordant optimization.

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arXiv2025

From Incremental Transitive Cover to Strongly Polynomial Maximum Flow

We provide faster strongly polynomial time algorithms solving maximum flow in structured $n$-node $m$-arc networks. Our results imply an $n^{ω+ o(1)}$-time strongly polynomial time algorithms for computing a maximum bipartite $b$-matching where $ω$ is the matrix multiplication constant. Additionally, they imply an $m^{1 + o(1)} W$-time algorithm for solving the problem on graphs with a given tree decomposition of width $W$. We obtain these results by strengthening and efficiently implementing an approach in Orlin's (STOC 2013) state-of-the-art $O(mn)$ time maximum flow algorithm. We develop a general framework that reduces solving maximum flow with arbitrary capacities to (1) solving a sequence of maximum flow problems with polynomial bounded capacities and (2) dynamically maintaining a size-bounded supersets of the transitive closure under arc additions; we call this problem \emph{incremental transitive cover}. Our applications follow by leveraging recent weakly polynomial, almost linear time algorithms for maximum flow due to Chen, Kyng, Liu, Peng, Gutenberg, Sachdeva (FOCS 2022) and Brand, Chen, Kyng, Liu, Peng, Gutenberg, Sachdeva, Sidford (FOCS 2023), and by developing incremental transitive cover data structures.

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arXiv2025

Isotropic Noise in Stochastic and Quantum Convex Optimization

We consider the problem of minimizing a $d$-dimensional Lipschitz convex function using a stochastic gradient oracle. We introduce and motivate a setting where the noise of the stochastic gradient is isotropic in that it is bounded in every direction with high probability. We then develop an algorithm for this setting which improves upon prior results by a factor of $d$ in certain regimes, and as a corollary, achieves a new state-of-the-art complexity for sub-exponential noise. We give matching lower bounds (up to polylogarithmic factors) for both results. Additionally, we develop an efficient quantum isotropifier, a quantum algorithm which converts a variance-bounded quantum sampling oracle into one that outputs an unbiased estimate with isotropic error. Combining our results, we obtain improved dimension-dependent rates for quantum stochastic convex optimization.

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arXiv2025

Balancing Gradient and Hessian Queries in Non-Convex Optimization

We develop optimization methods which offer new trade-offs between the number of gradient and Hessian computations needed to compute the critical point of a non-convex function. We provide a method that for any twice-differentiable $f\colon \mathbb R^d \rightarrow \mathbb R$ with $L_2$-Lipschitz Hessian, input initial point with $Δ$-bounded sub-optimality, and sufficiently small $ε> 0$, outputs an $ε$-critical point, i.e., a point $x$ such that $\|\nabla f(x)\| \leq ε$, using $\tilde{O}(L_2^{1/4} n_H^{-1/2}Δε^{-9/4})$ queries to a gradient oracle and $n_H$ queries to a Hessian oracle for any positive integer $n_H$. As a consequence, we obtain an improved gradient query complexity of $\tilde{O}(d^{1/3}L_2^{1/2}Δε^{-3/2})$ in the case of bounded dimension and of $\tilde{O}(L_2^{3/4}Δ^{3/2}ε^{-9/4})$ in the case where we are allowed only a \emph{single} Hessian query. We obtain these results through a more general algorithm which can handle approximate Hessian computations and recovers the state-of-the-art bound of computing an $ε$-critical point with $O(L_1^{1/2}L_2^{1/4}Δε^{-7/4})$ gradient queries provided that $f$ also has an $L_1$-Lipschitz gradient.

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arXiv2025

Generalized Flow in Nearly-linear Time on Moderately Dense Graphs

In this paper we consider generalized flow problems where there is an $m$-edge $n$-node directed graph $G = (V,E)$ and each edge $e \in E$ has a loss factor $γ_e >0$ governing whether the flow is increased or decreased as it crosses edge $e$. We provide a randomized $\tilde{O}( (m + n^{1.5}) \cdot \mathrm{polylog}(\frac{W}δ))$ time algorithm for solving the generalized maximum flow and generalized minimum cost flow problems in this setting where $δ$ is the target accuracy and $W$ is the maximum of all costs, capacities, and loss factors and their inverses. This improves upon the previous state-of-the-art $\tilde{O}(m \sqrt{n} \cdot \log^2(\frac{W}δ) )$ time algorithm, obtained by combining the algorithm of [Daitch-Spielman, 2008] with techniques from [Lee-Sidford, 2014]. To obtain this result we provide new dynamic data structures and spectral results regarding the matrices associated to generalized flows and apply them through the interior point method framework of [Brand-Lee-Liu-Saranurak-Sidford-Song-Wang, 2021].

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arXiv2025

Reusing Samples in Variance Reduction

We provide a general framework to improve trade-offs between the number of full batch and sample queries used to solve structured optimization problems. Our results apply to a broad class of randomized optimization algorithms that iteratively solve sub-problems to high accuracy. We show that such algorithms can be modified to reuse the randomness used to query the input across sub-problems. Consequently, we improve the trade-off between the number of gradient (full batch) and individual function (sample) queries for finite sum minimization, the number of matrix-vector multiplies (full batch) and random row (sample) queries for top-eigenvector computation, and the number of matrix-vector multiplies with the transition matrix (full batch) and generative model (sample) queries for optimizing Markov Decision Processes. To facilitate our analysis we introduce the notion of pseudo-independent algorithms, a generalization of pseudo-deterministic algorithms [Gat and Goldwasser 2011], that quantifies how independent the output of a randomized algorithm is from a randomness source.

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arXiv2025

Improved girth approximation in weighted undirected graphs

Let $G = (V,E,\ell)$ be a $n$-node $m$-edge weighted undirected graph, where $\ell: E \rightarrow (0,\infty)$ is a real \emph{length} function defined on its edges, and let $g$ denote the girth of $G$, i.e., the length of its shortest cycle. We present an algorithm that, for any input, integer $k \geq 1$, in $O(kn^{1+1/k}\log{n} + m(k+\log{n}))$ expected time finds a cycle of length at most $\frac{4k}{3}g$. This algorithm nearly matches a $O(n^{1+1/k}\log{n})$-time algorithm of \cite{KadriaRSWZ22} which applied to unweighted graphs of girth $3$. For weighted graphs, this result also improves upon the previous state-of-the-art algorithm that in $O((n^{1+1/k}\log n+m)\log (nM))$ time, where $\ell: E \rightarrow [1, M]$ is an integral length function, finds a cycle of length at most $2kg$~\cite{KadriaRSWZ22}. For $k=1$ this result improves upon the result of Roditty and Tov~\cite{RodittyT13}.

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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.

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arXiv2025

Accelerated Approximate Optimization of Multi-Commodity Flows on Directed Graphs

We provide $m^{1+o(1)}kε^{-1}$-time algorithms for computing multiplicative $(1 - ε)$-approximate solutions to multi-commodity flow problems with $k$-commodities on $m$-edge directed graphs, including concurrent multi-commodity flow and maximum multi-commodity flow. To obtain our results, we provide new optimization tools of potential independent interest. First, we provide an improved optimization method for solving $\ell_{q, p}$-regression problems to high accuracy. This method makes $\tilde{O}_{q, p}(k)$ queries to a high accuracy convex minimization oracle for an individual block, where $\tilde{O}_{q, p}(\cdot)$ hides factors depending only on $q$, $p$, or $\mathrm{poly}(\log m)$, improving upon the $\tilde{O}_{q, p}(k^2)$ bound of [Chen-Ye, ICALP 2024]. As a result, we obtain the first almost-linear time algorithm that solves $\ell_{q, p}$ flows on directed graphs to high accuracy. Second, we present optimization tools to reduce approximately solving composite $\ell_{1, \infty}$-regression problems to solving $m^{o(1)}ε^{-1}$ instances of composite $\ell_{q, p}$-regression problem. The method builds upon recent advances in solving box-simplex games [Jambulapati-Tian, NeurIPS 2023] and the area convex regularizer introduced in [Sherman, STOC 2017] to obtain faster rates for constrained versions of the problem. Carefully combining these techniques yields our directed multi-commodity flow algorithm.

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arXiv2024

Extracting Dual Solutions via Primal Optimizers

We provide a general method to convert a "primal" black-box algorithm for solving regularized convex-concave minimax optimization problems into an algorithm for solving the associated dual maximin optimization problem. Our method adds recursive regularization over a logarithmic number of rounds where each round consists of an approximate regularized primal optimization followed by the computation of a dual best response. We apply this result to obtain new state-of-the-art runtimes for solving matrix games in specific parameter regimes, obtain improved query complexity for solving the dual of the CVaR distributionally robust optimization (DRO) problem, and recover the optimal query complexity for finding a stationary point of a convex function.

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arXiv2024

Entropy Regularization and Faster Decremental Matching in General Graphs

We provide an algorithm that maintains, against an adaptive adversary, a $(1-\varepsilon)$-approximate maximum matching in $n$-node $m$-edge general (not necessarily bipartite) undirected graph undergoing edge deletions with high probability with (amortized) $O(\mathrm{poly}(\varepsilon^{-1}, \log n))$ time per update. We also obtain the same update time for maintaining a fractional approximate weighted matching (and hence an approximation to the value of the maximum weight matching) and an integral approximate weighted matching in dense graphs. Our unweighted result improves upon the prior state-of-the-art which includes a $\mathrm{poly}(\log{n}) \cdot 2^{O(1/\varepsilon^2)}$ update time [Assadi-Bernstein-Dudeja 2022] and an $O(\sqrt{m} \varepsilon^{-2})$ update time [Gupta-Peng 2013], and our weighted result improves upon the $O(\sqrt{m}\varepsilon^{-O(1/\varepsilon)}\log{n})$ update time due to [Gupta-Peng 2013]. To obtain our results, we generalize a recent optimization approach to dynamic algorithms from [Jambulapati-Jin-Sidford-Tian 2022]. We show that repeatedly solving entropy-regularized optimization problems yields a lazy updating scheme for fractional decremental problems with a near-optimal number of updates. To apply this framework we develop optimization methods compatible with it and new dynamic rounding algorithms for the matching polytope.

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arXiv2024

Stability of the Lanczos Method for Matrix Function Approximation

The ubiquitous Lanczos method can approximate $f(A)x$ for any symmetric $n \times n$ matrix $A$, vector $x$, and function $f$. In exact arithmetic, the method's error after $k$ iterations is bounded by the error of the best degree-$k$ polynomial uniformly approximating $f(x)$ on the range $[λ_{min}(A), λ_{max}(A)]$. However, despite decades of work, it has been unclear if this powerful guarantee holds in finite precision. We resolve this problem, proving that when $\max_{x \in [λ_{min}, λ_{max}]}|f(x)| \le C$, Lanczos essentially matches the exact arithmetic guarantee if computations use roughly $\log(nC\|A\|)$ bits of precision. Our proof extends work of Druskin and Knizhnerman [DK91], leveraging the stability of the classic Chebyshev recurrence to bound the stability of any polynomial approximating $f(x)$. We also study the special case of $f(A) = A^{-1}$, where stronger guarantees hold. In exact arithmetic Lanczos performs as well as the best polynomial approximating $1/x$ at each of $A$'s eigenvalues, rather than on the full eigenvalue range. In seminal work, Greenbaum gives an approach to extending this bound to finite precision: she proves that finite precision Lanczos and the related CG method match any polynomial approximating $1/x$ in a tiny range around each eigenvalue [Gre89]. For $A^{-1}$, this bound appears stronger than ours. However, we exhibit matrices with condition number $κ$ where exact arithmetic Lanczos converges in $polylog(κ)$ iterations, but Greenbaum's bound predicts $Ω(κ^{1/5})$ iterations. It thus cannot offer significant improvement over the $O(κ^{1/2})$ bound achievable via our result. Our analysis raises the question of if convergence in less than $poly(κ)$ iterations can be expected in finite precision, even for matrices with clustered, skewed, or otherwise favorable eigenvalue distributions.

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