arXiv subjects

Kevin Tian

Publications and source records attributed to Kevin Tian.

At least 37 records · Page 2Linked to original sources

arXiv2025

Spike-and-Slab Posterior Sampling in High Dimensions

Posterior sampling with the spike-and-slab prior [MB88], a popular multimodal distribution used to model uncertainty in variable selection, is considered the theoretical gold standard method for Bayesian sparse linear regression [CPS09, Roc18]. However, designing provable algorithms for performing this sampling task is notoriously challenging. Existing posterior samplers for Bayesian sparse variable selection tasks either require strong assumptions about the signal-to-noise ratio (SNR) [YWJ16], only work when the measurement count grows at least linearly in the dimension [MW24], or rely on heuristic approximations to the posterior. We give the first provable algorithms for spike-and-slab posterior sampling that apply for any SNR, and use a measurement count sublinear in the problem dimension. Concretely, assume we are given a measurement matrix $\mathbf{X} \in \mathbb{R}^{n\times d}$ and noisy observations $\mathbf{y} = \mathbf{X}\mathbfθ^\star + \mathbfξ$ of a signal $\mathbfθ^\star$ drawn from a spike-and-slab prior $π$ with a Gaussian diffuse density and expected sparsity k, where $\mathbfξ \sim \mathcal{N}(\mathbb{0}_n, σ^2\mathbf{I}_n)$. We give a polynomial-time high-accuracy sampler for the posterior $π(\cdot \mid \mathbf{X}, \mathbf{y})$, for any SNR $σ^{-1}$ > 0, as long as $n \geq k^3 \cdot \text{polylog}(d)$ and $X$ is drawn from a matrix ensemble satisfying the restricted isometry property. We further give a sampler that runs in near-linear time $\approx nd$ in the same setting, as long as $n \geq k^5 \cdot \text{polylog}(d)$. To demonstrate the flexibility of our framework, we extend our result to spike-and-slab posterior sampling with Laplace diffuse densities, achieving similar guarantees when $σ= O(\frac{1}{k})$ is bounded.

stat.ML↗

arXiv2025

Omnipredicting Single-Index Models with Multi-Index Models

Recent work on supervised learning [GKR+22] defined the notion of omnipredictors, i.e., predictor functions $p$ over features that are simultaneously competitive for minimizing a family of loss functions $\mathcal{L}$ against a comparator class $\mathcal{C}$. Omniprediction requires approximating the Bayes-optimal predictor beyond the loss minimization paradigm, and has generated significant interest in the learning theory community. However, even for basic settings such as agnostically learning single-index models (SIMs), existing omnipredictor constructions require impractically-large sample complexities and runtimes, and output complex, highly-improper hypotheses. Our main contribution is a new, simple construction of omnipredictors for SIMs. We give a learner outputting an omnipredictor that is $\varepsilon$-competitive on any matching loss induced by a monotone, Lipschitz link function, when the comparator class is bounded linear predictors. Our algorithm requires $\approx \varepsilon^{-4}$ samples and runs in nearly-linear time, and its sample complexity improves to $\approx \varepsilon^{-2}$ if link functions are bi-Lipschitz. This significantly improves upon the only prior known construction, due to [HJKRR18, GHK+23], which used $\gtrsim \varepsilon^{-10}$ samples. We achieve our construction via a new, sharp analysis of the classical Isotron algorithm [KS09, KKKS11] in the challenging agnostic learning setting, of potential independent interest. Previously, Isotron was known to properly learn SIMs in the realizable setting, as well as constant-factor competitive hypotheses under the squared loss [ZWDD24]. As they are based on Isotron, our omnipredictors are multi-index models with $\approx \varepsilon^{-2}$ prediction heads, bringing us closer to the tantalizing goal of proper omniprediction for general loss families and comparators.

cs.LG↗

arXiv2025

Learning Noisy Halfspaces with a Margin: Massart is No Harder than Random

We study the problem of PAC learning $γ$-margin halfspaces with Massart noise. We propose a simple proper learning algorithm, the Perspectron, that has sample complexity $\widetilde{O}((εγ)^{-2})$ and achieves classification error at most $η+ε$ where $η$ is the Massart noise rate. Prior works [DGT19,CKMY20] came with worse sample complexity guarantees (in both $ε$ and $γ$) or could only handle random classification noise [DDK+23,KIT+23] -- a much milder noise assumption. We also show that our results extend to the more challenging setting of learning generalized linear models with a known link function under Massart noise, achieving a similar sample complexity to the halfspace case. This significantly improves upon the prior state-of-the-art in this setting due to [CKMY20], who introduced this model.

cs.LG↗

arXiv2024

Eulerian Graph Sparsification by Effective Resistance Decomposition

We provide an algorithm that, given an $n$-vertex $m$-edge Eulerian graph with polynomially bounded weights, computes an $\breve{O}(n\log^{2} n \cdot \varepsilon^{-2})$-edge $\varepsilon$-approximate Eulerian sparsifier with high probability in $\breve{O}(m\log^3 n)$ time (where $\breve{O}(\cdot)$ hides $\text{polyloglog}(n)$ factors). Due to a reduction from [Peng-Song, STOC '22], this yields an $\breve{O}(m\log^3 n + n\log^6 n)$-time algorithm for solving $n$-vertex $m$-edge Eulerian Laplacian systems with polynomially-bounded weights with high probability, improving upon the previous state-of-the-art runtime of $Ω(m\log^8 n + n\log^{23} n)$. We also give a polynomial-time algorithm that computes $O(\min(n\log n \cdot \varepsilon^{-2} + n\log^{5/3} n \cdot \varepsilon^{-4/3}, n\log^{3/2} n \cdot \varepsilon^{-2}))$-edge sparsifiers, improving the best such sparsity bound of $O(n\log^2 n \cdot \varepsilon^{-2} + n\log^{8/3} n \cdot \varepsilon^{-4/3})$ [Sachdeva-Thudi-Zhao, ICALP '24]. Finally, we show that our techniques extend to yield the first $O(m\cdot\text{polylog}(n))$ time algorithm for computing $O(n\varepsilon^{-1}\cdot\text{polylog}(n))$-edge graphical spectral sketches, as well as a natural Eulerian generalization we introduce. In contrast to prior Eulerian graph sparsification algorithms which used either short cycle or expander decompositions, our algorithms use a simple efficient effective resistance decomposition scheme we introduce. Our algorithms apply a natural sampling scheme and electrical routing (to achieve degree balance) to such decompositions. Our analysis leverages new asymmetric variance bounds specialized to Eulerian Laplacians and tools from discrepancy theory.

cs.DS↗

arXiv2024

Testing Calibration in Nearly-Linear Time

In the recent literature on machine learning and decision making, calibration has emerged as a desirable and widely-studied statistical property of the outputs of binary prediction models. However, the algorithmic aspects of measuring model calibration have remained relatively less well-explored. Motivated by [BGHN23], which proposed a rigorous framework for measuring distances to calibration, we initiate the algorithmic study of calibration through the lens of property testing. We define the problem of calibration testing from samples where given $n$ draws from a distribution $\mathcal{D}$ on $(predictions, binary outcomes)$, our goal is to distinguish between the case where $\mathcal{D}$ is perfectly calibrated, and the case where $\mathcal{D}$ is $\varepsilon$-far from calibration. We make the simple observation that the empirical smooth calibration linear program can be reformulated as an instance of minimum-cost flow on a highly-structured graph, and design an exact dynamic programming-based solver for it which runs in time $O(n\log^2(n))$, and solves the calibration testing problem information-theoretically optimally in the same time. This improves upon state-of-the-art black-box linear program solvers requiring $Ω(n^ω)$ time, where $ω> 2$ is the exponent of matrix multiplication. We also develop algorithms for tolerant variants of our testing problem improving upon black-box linear program solvers, and give sample complexity lower bounds for alternative calibration measures to the one considered in this work. Finally, we present experiments showing the testing problem we define faithfully captures standard notions of calibration, and that our algorithms scale efficiently to accommodate large sample sizes.

cs.LG↗

arXiv2024

Black-Box $k$-to-$1$-PCA Reductions: Theory and Applications

The $k$-principal component analysis ($k$-PCA) problem is a fundamental algorithmic primitive that is widely-used in data analysis and dimensionality reduction applications. In statistical settings, the goal of $k$-PCA is to identify a top eigenspace of the covariance matrix of a distribution, which we only have black-box access to via samples. Motivated by these settings, we analyze black-box deflation methods as a framework for designing $k$-PCA algorithms, where we model access to the unknown target matrix via a black-box $1$-PCA oracle which returns an approximate top eigenvector, under two popular notions of approximation. Despite being arguably the most natural reduction-based approach to $k$-PCA algorithm design, such black-box methods, which recursively call a $1$-PCA oracle $k$ times, were previously poorly-understood. Our main contribution is significantly sharper bounds on the approximation parameter degradation of deflation methods for $k$-PCA. For a quadratic form notion of approximation we term ePCA (energy PCA), we show deflation methods suffer no parameter loss. For an alternative well-studied approximation notion we term cPCA (correlation PCA), we tightly characterize the parameter regimes where deflation methods are feasible. Moreover, we show that in all feasible regimes, $k$-cPCA deflation algorithms suffer no asymptotic parameter loss for any constant $k$. We apply our framework to obtain state-of-the-art $k$-PCA algorithms robust to dataset contamination, improving prior work in sample complexity by a $\mathsf{poly}(k)$ factor.

math.NA↗

arXiv2024

Closing the Computational-Query Depth Gap in Parallel Stochastic Convex Optimization

We develop a new parallel algorithm for minimizing Lipschitz, convex functions with a stochastic subgradient oracle. The total number of queries made and the query depth, i.e., the number of parallel rounds of queries, match the prior state-of-the-art, [CJJLLST23], while improving upon the computational depth by a polynomial factor for sufficiently small accuracy. When combined with previous state-of-the-art methods our result closes a gap between the best-known query depth and the best-known computational depth of parallel algorithms. Our method starts with a ball acceleration framework of previous parallel methods, i.e., [CJJJLST20, ACJJS21], which reduce the problem to minimizing a regularized Gaussian convolution of the function constrained to Euclidean balls. By developing and leveraging new stability properties of the Hessian of this induced function, we depart from prior parallel algorithms and reduce these ball-constrained optimization problems to stochastic unconstrained quadratic minimization problems. Although we are unable to prove concentration of the asymmetric matrices that we use to approximate this Hessian, we nevertheless develop an efficient parallel method for solving these quadratics. Interestingly, our algorithms can be improved using fast matrix multiplication and use nearly-linear work if the matrix multiplication exponent is 2.

math.OC↗

arXiv2024

Private Stochastic Convex Optimization with Heavy Tails: Near-Optimality from Simple Reductions

We study the problem of differentially private stochastic convex optimization (DP-SCO) with heavy-tailed gradients, where we assume a $k^{\text{th}}$-moment bound on the Lipschitz constants of sample functions rather than a uniform bound. We propose a new reduction-based approach that enables us to obtain the first optimal rates (up to logarithmic factors) in the heavy-tailed setting, achieving error $G_2 \cdot \frac 1 {\sqrt n} + G_k \cdot (\frac{\sqrt d}{nε})^{1 - \frac 1 k}$ under $(ε, δ)$-approximate differential privacy, up to a mild $\textup{polylog}(\frac{1}δ)$ factor, where $G_2^2$ and $G_k^k$ are the $2^{\text{nd}}$ and $k^{\text{th}}$ moment bounds on sample Lipschitz constants, nearly-matching a lower bound of [Lowy and Razaviyayn 2023]. We further give a suite of private algorithms in the heavy-tailed setting which improve upon our basic result under additional assumptions, including an optimal algorithm under a known-Lipschitz constant assumption, a near-linear time algorithm for smooth functions, and an optimal linear time algorithm for smooth generalized linear models.

cs.DS↗

arXiv2023

A CS guide to the quantum singular value transformation

We present a simplified exposition of some pieces of [Gilyén, Su, Low, and Wiebe, STOC'19, arXiv:1806.01838], which introduced a quantum singular value transformation (QSVT) framework for applying polynomial functions to block-encoded matrices. The QSVT framework has garnered substantial recent interest from the quantum algorithms community, as it was demonstrated by [GSLW19] to encapsulate many existing algorithms naturally phrased as an application of a matrix function. First, we posit that the lifting of quantum singular processing (QSP) to QSVT is better viewed not through Jordan's lemma (as was suggested by [GSLW19]) but as an application of the cosine-sine decomposition, which can be thought of as a more explicit and stronger version of Jordan's lemma. Second, we demonstrate that the constructions of bounded polynomial approximations given in [GSLW19], which use a variety of ad hoc approaches drawing from Fourier analysis, Chebyshev series, and Taylor series, can be unified under the framework of truncation of Chebyshev series, and indeed, can in large part be matched via a bounded variant of a standard meta-theorem from [Trefethen, 2013]. We hope this work finds use to the community as a companion guide for understanding and applying the powerful framework of [GSLW19].

quant-ph↗

arXiv2023

Revisiting Area Convexity: Faster Box-Simplex Games and Spectrahedral Generalizations

We investigate different aspects of area convexity [Sherman '17], a mysterious tool introduced to tackle optimization problems under the challenging $\ell_\infty$ geometry. We develop a deeper understanding of its relationship with more conventional analyses of extragradient methods [Nemirovski '04, Nesterov '07]. We also give improved solvers for the subproblems required by variants of the [Sherman '17] algorithm, designed through the lens of relative smoothness [Bauschke-Bolte-Teboulle '17, Lu-Freund-Nesterov '18]. Leveraging these new tools, we give a state-of-the-art first-order algorithm for solving box-simplex games (a primal-dual formulation of $\ell_\infty$ regression) in a $d \times n$ matrix with bounded rows, using $O(\log d \cdot ε^{-1})$ matrix-vector queries. As a consequence, we obtain improved complexities for approximate maximum flow, optimal transport, min-mean-cycle, and other basic combinatorial optimization problems. We also develop a near-linear time algorithm for a matrix generalization of box-simplex games, capturing a family of problems closely related to semidefinite programs recently used as subroutines in robust statistics and numerical linear algebra.

math.OC↗

arXiv2023

ReSQueing Parallel and Private Stochastic Convex Optimization

We introduce a new tool for stochastic convex optimization (SCO): a Reweighted Stochastic Query (ReSQue) estimator for the gradient of a function convolved with a (Gaussian) probability density. Combining ReSQue with recent advances in ball oracle acceleration [CJJJLST20, ACJJS21], we develop algorithms achieving state-of-the-art complexities for SCO in parallel and private settings. For a SCO objective constrained to the unit ball in $\mathbb{R}^d$, we obtain the following results (up to polylogarithmic factors). We give a parallel algorithm obtaining optimization error $ε_{\text{opt}}$ with $d^{1/3}ε_{\text{opt}}^{-2/3}$ gradient oracle query depth and $d^{1/3}ε_{\text{opt}}^{-2/3} + ε_{\text{opt}}^{-2}$ gradient queries in total, assuming access to a bounded-variance stochastic gradient estimator. For $ε_{\text{opt}} \in [d^{-1}, d^{-1/4}]$, our algorithm matches the state-of-the-art oracle depth of [BJLLS19] while maintaining the optimal total work of stochastic gradient descent. Given $n$ samples of Lipschitz loss functions, prior works [BFTT19, BFGT20, AFKT21, KLL21] established that if $n \gtrsim d ε_{\text{dp}}^{-2}$, $(ε_{\text{dp}}, δ)$-differential privacy is attained at no asymptotic cost to the SCO utility. However, these prior works all required a superlinear number of gradient queries. We close this gap for sufficiently large $n \gtrsim d^2 ε_{\text{dp}}^{-3}$, by using ReSQue to design an algorithm with near-linear gradient query complexity in this regime.

math.OC↗

arXiv2023

Structured Semidefinite Programming for Recovering Structured Preconditioners

We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including the following. We give an algorithm which, given positive definite $\mathbf{K} \in \mathbb{R}^{d \times d}$ with $\mathrm{nnz}(\mathbf{K})$ nonzero entries, computes an $ε$-optimal diagonal preconditioner in time $\widetilde{O}(\mathrm{nnz}(\mathbf{K}) \cdot \mathrm{poly}(κ^\star,ε^{-1}))$, where $κ^\star$ is the optimal condition number of the rescaled matrix. We give an algorithm which, given $\mathbf{M} \in \mathbb{R}^{d \times d}$ that is either the pseudoinverse of a graph Laplacian matrix or a constant spectral approximation of one, solves linear systems in $\mathbf{M}$ in $\widetilde{O}(d^2)$ time. Our diagonal preconditioning results improve state-of-the-art runtimes of $Ω(d^{3.5})$ attained by general-purpose semidefinite programming, and our solvers improve state-of-the-art runtimes of $Ω(d^ω)$ where $ω> 2.3$ is the current matrix multiplication constant. We attain our results via new algorithms for a class of semidefinite programs (SDPs) we call matrix-dictionary approximation SDPs, which we leverage to solve an associated problem we call matrix-dictionary recovery.

cs.DS↗

arXiv2023

Matrix Completion in Almost-Verification Time

We give a new framework for solving the fundamental problem of low-rank matrix completion, i.e., approximating a rank-$r$ matrix $\mathbf{M} \in \mathbb{R}^{m \times n}$ (where $m \ge n$) from random observations. First, we provide an algorithm which completes $\mathbf{M}$ on $99\%$ of rows and columns under no further assumptions on $\mathbf{M}$ from $\approx mr$ samples and using $\approx mr^2$ time. Then, assuming the row and column spans of $\mathbf{M}$ satisfy additional regularity properties, we show how to boost this partial completion guarantee to a full matrix completion algorithm by aggregating solutions to regression problems involving the observations. In the well-studied setting where $\mathbf{M}$ has incoherent row and column spans, our algorithms complete $\mathbf{M}$ to high precision from $mr^{2+o(1)}$ observations in $mr^{3 + o(1)}$ time (omitting logarithmic factors in problem parameters), improving upon the prior state-of-the-art [JN15] which used $\approx mr^5$ samples and $\approx mr^7$ time. Under an assumption on the row and column spans of $\mathbf{M}$ we introduce (which is satisfied by random subspaces with high probability), our sample complexity improves to an almost information-theoretically optimal $mr^{1 + o(1)}$, and our runtime improves to $mr^{2 + o(1)}$. Our runtimes have the appealing property of matching the best known runtime to verify that a rank-$r$ decomposition $\mathbf{U}\mathbf{V}^\top$ agrees with the sampled observations. We also provide robust variants of our algorithms that, given random observations from $\mathbf{M} + \mathbf{N}$ with $\|\mathbf{N}\|_{F} \le Δ$, complete $\mathbf{M}$ to Frobenius norm distance $\approx r^{1.5}Δ$ in the same runtimes as the noiseless setting. Prior noisy matrix completion algorithms [CP10] only guaranteed a distance of $\approx \sqrt{n}Δ$.

cs.LG↗

arXiv2023

Linear-Sized Sparsifiers via Near-Linear Time Discrepancy Theory

Discrepancy theory provides powerful tools for producing higher-quality objects which "beat the union bound" in fundamental settings throughout combinatorics and computer science. However, this quality has often come at the price of more expensive algorithms. We introduce a new framework for bridging this gap, by allowing for the efficient implementation of discrepancy-theoretic primitives. Our framework repeatedly solves regularized optimization problems to low accuracy to approximate the partial coloring method of [Rot17], and simplifies and generalizes recent work of [JSS23] on fast algorithms for Spencer's theorem. In particular, our framework only requires that the discrepancy body of interest has exponentially large Gaussian measure and is expressible as a sublevel set of a symmetric, convex function. We combine this framework with new tools for proving Gaussian measure lower bounds to give improved algorithms for a variety of sparsification and coloring problems. As a first application, we use our framework to obtain an $\widetilde{O}(m \cdot ε^{-3.5})$ time algorithm for constructing an $ε$-approximate spectral sparsifier of an $m$-edge graph, matching the sparsity of [BSS14] up to constant factors and improving upon the $\widetilde{O}(m \cdot ε^{-6.5})$ runtime of [LeeS17]. We further give a state-of-the-art algorithm for constructing graph ultrasparsifiers and an almost-linear time algorithm for constructing linear-sized degree-preserving sparsifiers via discrepancy theory; in the latter case, such sparsifiers were not known to exist previously. We generalize these results to their analogs in sparsifying isotropic sums of positive semidefinite matrices. Finally, to demonstrate the versatility of our technique, we obtain a nearly-input-sparsity time constructive algorithm for Spencer's theorem (where we recover a recent result of [JSS23]).

cs.DS↗

arXiv2023

Quadratic Speedups in Parallel Sampling from Determinantal Distributions

We study the problem of parallelizing sampling from distributions related to determinants: symmetric, nonsymmetric, and partition-constrained determinantal point processes, as well as planar perfect matchings. For these distributions, the partition function, a.k.a. the count, can be obtained via matrix determinants, a highly parallelizable computation; Csanky proved it is in NC. However, parallel counting does not automatically translate to parallel sampling, as classic reductions between the two are inherently sequential. We show that a nearly quadratic parallel speedup over sequential sampling can be achieved for all the aforementioned distributions. If the distribution is supported on subsets of size $k$ of a ground set, we show how to approximately produce a sample in $\widetilde{O}(k^{\frac{1}{2} + c})$ time with polynomially many processors for any constant $c>0$. In the two special cases of symmetric determinantal point processes and planar perfect matchings, our bound improves to $\widetilde{O}(\sqrt k)$ and we show how to sample exactly in these cases. As our main technical contribution, we fully characterize the limits of batching for the steps of sampling-to-counting reductions. We observe that only $O(1)$ steps can be batched together if we strive for exact sampling, even in the case of nonsymmetric determinantal point processes. However, we show that for approximate sampling, $\widetildeΩ(k^{\frac{1}{2}-c})$ steps can be batched together, for any entropically independent distribution, which includes all mentioned classes of determinantal point processes. Entropic independence and related notions have been the source of breakthroughs in Markov chain analysis in recent years, so we expect our framework to prove useful for distributions beyond those studied in this work.

cs.DS↗

arXiv2023

Quantum Speedups for Zero-Sum Games via Improved Dynamic Gibbs Sampling

We give a quantum algorithm for computing an $ε$-approximate Nash equilibrium of a zero-sum game in a $m \times n$ payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time $\widetilde{O}(\sqrt{m + n}\cdot ε^{-2.5} + ε^{-3})$ and outputs a classical representation of the $ε$-approximate Nash equilibrium. This improves upon the best prior quantum runtime of $\widetilde{O}(\sqrt{m + n} \cdot ε^{-3})$ obtained by [vAG19] and the classic $\widetilde{O}((m + n) \cdot ε^{-2})$ runtime due to [GK95] whenever $ε= Ω((m +n)^{-1})$. We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.

quant-ph↗

arXiv2022

Private Convex Optimization in General Norms

We propose a new framework for differentially private optimization of convex functions which are Lipschitz in an arbitrary norm $\|\cdot\|$. Our algorithms are based on a regularized exponential mechanism which samples from the density $\propto \exp(-k(F+μr))$ where $F$ is the empirical loss and $r$ is a regularizer which is strongly convex with respect to $\|\cdot\|$, generalizing a recent work of [Gopi, Lee, Liu '22] to non-Euclidean settings. We show that this mechanism satisfies Gaussian differential privacy and solves both DP-ERM (empirical risk minimization) and DP-SCO (stochastic convex optimization) by using localization tools from convex geometry. Our framework is the first to apply to private convex optimization in general normed spaces and directly recovers non-private SCO rates achieved by mirror descent as the privacy parameter $ε\to \infty$. As applications, for Lipschitz optimization in $\ell_p$ norms for all $p \in (1, 2)$, we obtain the first optimal privacy-utility tradeoffs; for $p = 1$, we improve tradeoffs obtained by the recent works [Asi, Feldman, Koren, Talwar '21, Bassily, Guzman, Nandi '21] by at least a logarithmic factor. Our $\ell_p$ norm and Schatten-$p$ norm optimization frameworks are complemented with polynomial-time samplers whose query complexity we explicitly bound.

cs.LG↗

arXiv2022

Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching

Box-simplex games are a family of bilinear minimax objectives which encapsulate graph-structured problems such as maximum flow [She17], optimal transport [JST19], and bipartite matching [AJJ+22]. We develop efficient near-linear time, high-accuracy solvers for regularized variants of these games. Beyond the immediate applications of such solvers for computing Sinkhorn distances, a prominent tool in machine learning, we show that these solvers can be used to obtain improved running times for maintaining a (fractional) $ε$-approximate maximum matching in a dynamic decremental bipartite graph against an adaptive adversary. We give a generic framework which reduces this dynamic matching problem to solving regularized graph-structured optimization problems to high accuracy. Through our reduction framework, our regularized box-simplex game solver implies a new algorithm for dynamic decremental bipartite matching in total time $\tilde{O}(m \cdot ε^{-3})$, from an initial graph with $m$ edges and $n$ nodes. We further show how to use recent advances in flow optimization [CKL+22] to improve our runtime to $m^{1 + o(1)} \cdot ε^{-2}$, thereby demonstrating the versatility of our reduction-based approach. These results improve upon the previous best runtime of $\tilde{O}(m \cdot ε^{-4})$ [BGS20] and illustrate the utility of using regularized optimization problem solvers for designing dynamic algorithms.

cs.DS↗