arXiv subjects

Darina Dvinskikh

Publications and source records attributed to Darina Dvinskikh.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Rates for Federated Variational Inequalities under different relaxations of bounded variance assumption

We study how interpolation affects local steps and parallel sampling in monotone variational inequalities (VIs), with $m$ workers, $K$ oracle calls per worker per round, and $N$ rounds. Under the strong growth condition (SGC), we determine the minimax rate for affine VIs, attained by local extragradient with worker averaging. For nonlinear monotone VIs and convex-concave saddle problems with Lipschitz operators satisfying SGC, we improve the stochastic terms of bounded-variance expected-gap guarantees from $(KN)^{-1/2}$ to $(KN)^{-1}$ with local steps and from $(mKN)^{-1/2}$ to $(mKN)^{-1}$ with minibatches. Under strong monotonicity, we improve the noise-dependent condition-number scaling of existing extragradient bounds from quadratic to linear. The resulting monotone-gap and strongly monotone accuracy bounds match lower bounds up to logarithmic factors in $m$. Under generalized smoothness and SGC, adaptive steps give polynomial dependence on the initial radius. For affine VIs with globally bounded oracle variance, we also attain the minimax rate: the deterministic rate over $KN$ sequential calls plus the statistical rate over all $mKN$ samples. For composite VIs with smooth convex sample interpolation and an exactly evaluated monotone Lipschitz component, accelerated mirror-prox sliding achieves an $N^{-2}$ gradient term, a $(KN)^{-1}$ monotone term, and a $(mKN)^{-1}$ sampling term, without assuming SGC.

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arXiv2026

Minimax Call Complexity, Rounds and Admissible Noise in Zeroth-Order Optimization of Highly Smooth Functions

We study zeroth-order optimization of convex and strongly convex functions with Hölder smoothness of order $β>2$ from values corrupted by random noise with bias independent of the method's random directions, a Lipschitz systematic error and a bounded adversarial error. We measure the number of function evaluations (calls), of sequential rounds of parallel calls, and the largest tolerable systematic and adversarial errors. For strongly convex functions with a bounded Hessian, we prove a lower bound of order $d^2\varepsilon^{-β/(β-1)}$ on the number of calls, where $d$ is the dimension and $\varepsilon$ the accuracy, and find the sharp power of $d$ under the stronger Hilbert--Schmidt smoothness for all $β>2$. An accelerated finite-difference method attains them in $O(\sqrtκ\log(1/\varepsilon))$ rounds for condition number $κ$, like Nesterov's method with exact gradients, and tolerates the optimal systematic and, up to logarithms, adversarial errors; rate-optimal methods need $Ω(\log d)$ rounds. The same number of calls is attained when the Hessian grows at most linearly with the gradient and, for centered noise, without an upper bound on the Hessian, also in logarithmically many rounds. For convex functions, our lower bound improves the known one polynomially in $d$ and is tight at the curvature of the hard instances; the systematic level is again optimal. These bounds extend to general norms and constraints; on the $\ell_1$ ball, mirror descent needs calls nearly linear in $d$. Experiments confirm the theory.

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arXiv2026

Asking the Crowd the Right Question: Bias-Cancelling Weights for Federated Learning

A federated objective is a weighted sum of client risks, and the weights are almost always fixed in advance. We treat them instead as the only instrument of a wisdom-of-crowds mechanism: clients are noisy views of one truth, each seeing it through an independent distortion that is unbiased across the crowd. That the optimal weights are inversely proportional to the clients' error energies is classical; we begin at the question that answer presupposes, which energies belong there and whether a crowd can recover them from itself. Excess risk on the truth is of the exact order of the aggregate bias energy, so no optimizer can repair a bad weight vector; the truth itself is identifiable only up to a linear tilt, so subtracting estimated client biases provably reproduces uniform weighting. The expected per-client second moments, however, are exactly identified from the law of the crowd's disagreement by a well-conditioned linear inversion, a step random-effects meta-analysis cannot take because a source reports once; their realized counterparts are estimable up to an incoherence floor the algorithm can measure. This yields CROWD, which reads the disagreement off the optimization trajectory at no extra cost and matches a Bayesian minimax lower bound in the same constant: per instance as the horizon grows, and unconditionally as the prior becomes diffuse. For arbitrary distortions it stays competitive with the optimal weights, at a ratio governed by a geometric incoherence the algorithm can measure. On real scans split into sites with their own miscalibrated detectors it attains the oracle excess risk; on a companion federation that pulls bias and noise apart, weighting by noise variance is worse than not weighting at all, and CROWD is not.

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arXiv2026

Gradient-Free Methods for Stochastic Convex Optimization with Stochastic Functional Constraints

We develop accelerated gradient-free methods for stochastic convex optimization with constraints defined by expectations. Our batched primal-dual sliding method uses two-point evaluations sharing a random sample and guarantees expected objective error and expected maximum constraint violation at most $\varepsilon$. It achieves $O(\varepsilon^{-1/2})$ sequential oracle rounds for smooth data and $O(d^{1/4}/\varepsilon)$ for nonsmooth data in dimension $d$, recovering the accuracy and dimension dependence of the corresponding unconstrained accelerated methods. The smooth rate is optimal in accuracy. Each round collects all samples needed for its inner primal-dual updates. Total evaluations retain quadratic dependence on inverse accuracy, with only polylogarithmic dependence on the number of constraints under vector feedback. Complementary lower bounds distinguish the dimension cost of gradient estimation from the unavoidable logarithmic cost of estimating noisy constraint levels. For smooth strongly convex objectives, restarts give logarithmic round complexity and objective evaluation cost linear in inverse accuracy, while constraint-level estimation necessarily remains quadratic. Known affine constraints require no constraint-oracle calls.

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arXiv2026

Lower Bounds For Gradient-Free Convex Optimization And Convex-Concave Saddle-Point Problems

We study the query complexity of optimization with exact scalar-value information. For globally $L$-smooth convex functions on $\mathbb R^d$ with a minimizer in a Euclidean ball of radius $R$, we prove the lower bound $Ω(d\min\{d,\sqrt{LR^2/\varepsilon}\})$ for adaptive randomized algorithms in the stated accuracy range. Queries are unrestricted, and the bound contains no logarithmic factor. It holds for quadratics and matches the minimum of two elementary upper bounds on that class. Prescribed strong convexity introduces the additional cap $\sqrt{L/μ}$. The proof compares the mean of a matrix with the mean of its inverse on arbitrary affine slices and bounds posterior moments after adaptive measurements. We extend these information bounds to saddle-point problems with independent block dimensions. For compact bilinear games, the coupling alone yields $Ω(r\min\{r,\sqrt{CR_xR_y/\varepsilon}\})$ queries, where $r=\min\lbrace d_x,d_y\rbrace$. A strongly convex--strongly concave construction yields a condition-dependent lower bound even with known isotropic diagonal Hessians. Separate reductions establish scalar query lower bounds under mixed value and partial-gradient access, including for outer objective accuracy. A Euclidean-image radius yields smooth and nonsmooth extensions to independently specified norms and known penalties.

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arXiv2026

Accelerated Stochastic Method under $(H_0,H_1)$-Smoothness and Heavy-Tailed Noise

We develop an accelerated stochastic method for convex $(H_0,H_1)$-smooth optimization under heavy-tailed noise. The unbiased oracle has a finite $p$-th noise moment, $1<p\le2$, with constant, gradient-dependent, and gap-dependent terms. Our method combines accelerated updates with clipping, projection, and phase restarts. Each iteration uses a single stochastic-gradient sample, without minibatching. We prove high-probability convergence to any desired accuracy despite clipping bias. The deterministic terms retain square-root dependence on both $H_0$ and $H_1$. In our three-component noise model, strong convexity leaves a polynomial accuracy cost only for the constant component; without it, the dependence is logarithmic even for $p<2$. More generally, for noise moments scaling as $(f(x)-f^\star)^α$, the stochastic cost becomes logarithmic at $α=p$ for convex objectives and $α=p/2$ under strong convexity.

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arXiv2026

Accelerated Consensus and Decentralized Optimization over Slowly Varying Networks

We study average consensus over slowly time-varying networks and its application to decentralized optimization. We introduce WAVE, a windowed Chebyshev method that limits the accumulated effect of network variation by restarting the recurrence after finite windows. If $χ$ bounds the network condition number and successive communication operators satisfy $\|L_{k+1}-L_k\|\leqβ$, WAVE reaches $ε$-consensus in $O((\sqrtχ+\min\lbraceβχ^2,χ\rbrace)\ln(e/ε))$ communication rounds. This rate matches the fixed-network $\sqrtχ$ dependence when $β=O(χ^{-3/2})$ and smoothly interpolates across the full range of network variation. For piecewise-constant networks where each change is detected when it occurs and consecutive changes are at least $τ$ rounds apart, WAVE reaches $ε$-consensus in $O((\sqrtχ+χ/τ)\ln(e/ε))$ communication rounds. Finally, we use WAVE as the consensus step in an accelerated decentralized optimization method for $α$-smooth convex local objectives with a $μ$-strongly convex average. For sufficiently slow network variation, every agent obtains an $ε$-solution to the global optimization problem after $\widetilde O(\sqrt{α/μ}\,\sqrtχ)$ communication rounds, matching the fixed-network dependence.

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arXiv2026

The Complexity of Convex Optimization with Mismatched Geometry

Optimal first-order methods on non-Euclidean domains such as the $\ell_1$ ball $B_1^n(R)=\{x\in\mathbb R^n:\|x\|_1\le R\}$ pair the prox-function with the norm in which smoothness is measured. When the gradient is $L$-Lipschitz in the Euclidean norm only, the accelerated method with a Euclidean prox-setup reduces the functional gap $f(x_N)-\min_{B_1^n(R)}f$ to $O(LR^2/N^2)$ after $N$ first-order queries (an entropic $\ell_1$ prox-setup replaces $L$ by the $\ell_1\to\ell_\infty$ constant $L_1\le L$, at the cost of a factor $\log n$ and with the same exponent). The lower bound of Guzmán and Nemirovski is of order $LR^2/N^3$, and whether the upper bound can be brought down to that order is a question of A. S. Nemirovski. We show that for this mismatched problem the minimax value of the gap is of order $LR^2/N^3$ up to logarithmic factors, for deterministic and for randomized methods alike, and already in dimension proportional to the number of queries. The upper bound is attained by a Steiner-point level method: every supporting hyperplane seen so far is kept as a level cut, and the next query is made near the Steiner point of the resulting localization polytope. The analysis rests on a single geometric fact: along nested subsets of the ball the Steiner points travel a distance that is polylogarithmic in the dimension, in contrast to $\sqrt n$ for the Euclidean ball. All queries stay feasible, and a randomized selector keeps the internal work polynomial. The same geometry yields optimal rates for nonsmooth objectives, Hölder gradients, higher-order oracles and Lipschitz monotone operators, the last with a matching deterministic lower bound. For convex quadratics, a curvature-learning method attains the optimal rate on every $\ell_p$ ball with $1\le p<2$ and without logarithmic loss. Experiments confirm the predicted $N^{-3}$ behaviour on objectives that are hard for Euclidean methods.

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arXiv2026

Clipped or Unclipped? Finite-Sample Trade-offs for Averaged SGD under Heavy-Tailed Noise

Gradient clipping is widely used to stabilize training, but it need not improve the statistical accuracy of averaged SGD, even under heavy-tailed noise. We derive a finite-sample comparison of clipped and unclipped Polyak-Ruppert averaged SGD under finite conditional $p$-th moments, $p\ge2$. Our main result gives explicit accuracy and confidence conditions under which, for $p>2$, the Gaussian term dominates the unclipped deviation bound, so clipping need not improve its leading order. By balancing clipping bias and concentration, we obtain a bound in which the heavy-tail correction depends logarithmically rather than polynomially on the inverse failure probability. At $p=2$, this improves the confidence dependence of the leading bound. We establish sharpness of the unclipped heavy-tail term through an exact one-dimensional quadratic recursion and extend the comparison to projected convex SGD. We also prove concrete costs of clipping: every fixed finite threshold increases asymptotic variance on a scalar Gaussian quadratic, while whole-gradient clipping can shift the limiting point under asymmetric noise.

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arXiv2026

Localize, Restart, Accelerate: Stochastic Optimization under Generalized Smoothness

We study stochastic convex optimization under asymmetric $(L_0,L_1)$-generalized smoothness, a model motivated by machine-learning objectives whose local curvature may grow with the gradient norm. We assume an unbiased first-order oracle with additive norm-sub-Gaussian noise. Acceleration is difficult in this setting because momentum may enter regions of much larger curvature, while stochastic gradients cannot reliably certify an unrestricted trajectory. We propose \textsf{ARC-SG}, a two-phase accelerated method: Phase~I reduces excessively large gradients using a generalized-smoothness-aware stochastic step, then Phase~II solves strongly convex proximal subproblems by a restarted accelerated solver confined to certified smoothness balls. Exact proximal points do not increase the gradient norm, allowing these certificates to propagate through the outer loop. The contribution is a query-by-query certified-localization construction with explicit generalized-smoothness factors and a strongly convex restart extension. \textsf{ARC-SG} achieves, with high probability, an accelerated optimization contribution and smooth-subclass-optimal statistical dependence on accuracy, up to logarithmic and generalized-smoothness factors. Its convex accuracy exponents agree with a contemporaneous public stochastic-acceleration result under a broader smoothness and affine-variance model; our distinction is the certified geometry, explicit parameter accounting, and strongly convex guarantee. The results recover classical accelerated stochastic rates when $L_1=0$. Experiments on objectives with unbounded gradients illustrate the two-phase mechanism and its finite-budget advantage.

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arXiv2026

Stochastic Optimization and Data Science

This paper aims to motivate stochastic optimization problems from a statistical perspective and a statistical learning perspective, where the goal is to maximize the log-likelihood or minimize the population risk. We briefly describe the two main approaches: offline (Monte Carlo / Sample Average Approximation) and online (Stochastic Approximation) approaches -- to solve the expectation minimization problems.

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arXiv2026

Wall-Clock Complexity for Zeroth-Order Optimization with Tunable Oracle Fidelity

Zeroth-order (black-box) optimization is applied when gradients are unavailable and objective evaluations rely on expensive simulations. In many such applications, the oracle fidelity is tunable: higher-accuracy queries reduce noise but incur higher computational costs. To capture this trade-off, we study an accuracy-aware wall-clock model where each query with fidelity $δ$ has a cost $c(δ)$, and we minimize the total time $T_{\mathrm{total}} = \sum_{k=1}^{N} c(δ_k)$, subject to a target accuracy constraint. We show how the choice of oracle type, noise model, and optimization scheme induces explicit wall-clock-optimal choices for the algorithmic parameters. For instance, we demonstrate that accelerated methods can be wall-clock inferior to non-accelerated schemes. Furthermore, we characterize the conditions under which a constant fidelity strategy is optimal in the Big-O sense. Our framework provides a unified methodology to translate convergence guarantees into practical fidelity and batching recommendations.

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arXiv2024

About some works of Boris Polyak on convergence of gradient methods and their development

The paper presents a review of the state-of-the-art of subgradient and accelerated methods of convex optimization, including in the presence of disturbances and access to various information about the objective function (function value, gradient, stochastic gradient, higher derivatives). For nonconvex problems, the Polak-Lojasiewicz condition is considered and a review of the main results is given. The behavior of numerical methods in the presence of sharp minima is considered. The purpose of this survey is to show the influence of the works of B.T. Polyak (1935 -- 2023) on gradient optimization methods and their neighborhoods on the modern development of numerical optimization methods.

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arXiv2024

Accelerated zero-order SGD under high-order smoothness and overparameterized regime

We present a novel gradient-free algorithm to solve a convex stochastic optimization problem, such as those encountered in medicine, physics, and machine learning (e.g., adversarial multi-armed bandit problem), where the objective function can only be computed through numerical simulation, either as the result of a real experiment or as feedback given by the function evaluations from an adversary. Thus we suppose that only a black-box access to the function values of the objective is available, possibly corrupted by adversarial noise: deterministic or stochastic. The noisy setup can arise naturally from modeling randomness within a simulation or by computer discretization, or when exact values of function are forbidden due to privacy issues, or when solving non-convex problems as convex ones with an inexact function oracle. By exploiting higher-order smoothness, fulfilled, e.g., in logistic regression, we improve the performance of zero-order methods developed under the assumption of classical smoothness (or having a Lipschitz gradient). The proposed algorithm enjoys optimal oracle complexity and is designed under an overparameterization setup, i.e., when the number of model parameters is much larger than the size of the training dataset. Overparametrized models fit to the training data perfectly while also having good generalization and outperforming underparameterized models on unseen data. We provide convergence guarantees for the proposed algorithm under both types of noise. Moreover, we estimate the maximum permissible adversarial noise level that maintains the desired accuracy in the Euclidean setup, and then we extend our results to a non-Euclidean setup. Our theoretical results are verified on the logistic regression problem.

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arXiv2024

Bregman Proximal Method for Efficient Communications under Similarity

We propose a novel stochastic distributed method for both monotone and strongly monotone variational inequalities with Lipschitz operator and proper convex regularizers arising in various applications from game theory to adversarial training. By exploiting similarity, our algorithm overcomes the communication bottleneck that is a major issue in distributed optimization. The proposed method enjoys optimal communication complexity. All the existing distributed algorithms achieving the lower bounds under similarity condition essentially utilize the Euclidean setup. In contrast to them, our method is built upon the Bregman proximal maps and it is compatible with an arbitrary problem geometry. Thereby the proposed method fills an existing gap in this area of research. Our theoretical results are confirmed by numerical experiments on a stochastic matrix game.

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arXiv2024

Gradient-free algorithm for saddle point problems under overparametrization

This paper focuses on solving a stochastic saddle point problem (SPP) under an overparameterized regime for the case, when the gradient computation is impractical. As an intermediate step, we generalize Same-sample Stochastic Extra-gradient algorithm (Gorbunov et al., 2022) to a biased oracle and estimate novel convergence rates. As the result of the paper we introduce an algorithm, which uses gradient approximation instead of a gradient oracle. We also conduct an analysis to find the maximum admissible level of adversarial noise and the optimal number of iterations at which our algorithm can guarantee achieving the desired accuracy.

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arXiv2024

Decentralized Distributed Optimization for Saddle Point Problems

We consider distributed convex-concave saddle point problems over arbitrary connected undirected networks and propose a decentralized distributed algorithm for their solution. The local functions distributed across the nodes are assumed to have global and local groups of variables. For the proposed algorithm we prove non-asymptotic convergence rate estimates with explicit dependence on the network characteristics. To supplement the convergence rate analysis, we propose lower bounds for strongly-convex-strongly-concave and convex-concave saddle-point problems over arbitrary connected undirected networks. We illustrate the considered problem setting by a particular application to distributed calculation of non-regularized Wasserstein barycenters.

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arXiv2024

Randomized gradient-free methods in convex optimization

This review presents modern gradient-free methods to solve convex optimization problems. By gradient-free methods, we mean those that use only (noisy) realizations of the objective value. We are motivated by various applications where gradient information is prohibitively expensive or even unavailable. We mainly focus on three criteria: oracle complexity, iteration complexity, and the maximum permissible noise level.

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