arXiv subjects

Alexander Gasnikov

Publications and source records attributed to Alexander Gasnikov.

At least 19 recordsLinked to original sources

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

Convex-Concave Interpolation and Application of PEP to Bilinear-Coupled Saddle-Point Problem

The Performance estimation problem (PEP) approach reformulates finding the exact worst-case performance of an algorithm as the solution to an optimization problem. Tractable formulation of the problem requires necessary and sufficient interpolation conditions. We present the interpolation conditions for convex--concave functions, bilinear functions, and composite functions with bilinear coupling. We also construct PEP for first-order methods for composite saddle-point problem.

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arXiv2026

Sliding Methods for H\"older-Smooth Convex--Concave Minimax Optimization with Bilinear Coupling

We study convex-concave minimax optimization problems with bilinear coupling of the form $\min_{x\in \mathcal X}\max_{y\in \mathcal Y} \; f(x)+\langle y,\mathbf{B}x\rangle-g(y),$ where the functions $f$ and $g$ have H\"older continuous (sub)gradients. This setting covers a broad range of regimes, from nonsmooth problems with bounded subgradient variation to smooth problems with Lipschitz continuous gradients; for a smooth component used in the coupling-induced regularizer, its Lipschitz-gradient constant is assumed to hold in the ambient space. We propose a sliding method that exploits the composite structure of the problem by querying the oracles associated with $f$, $g$, and the bilinear coupling operator at prescribed frequencies determined by their individual properties. The method is based on a recursive sliding scheme for monotone variational inequalities. We establish convergence guarantees under H\"older continuity and show how the resulting complexity bounds depend explicitly on the H\"older exponents, H\"older constants, strong convexity parameters, and spectral properties of the coupling matrix. Our analysis covers nonstrongly convex and partially strongly convex regimes. For stochastic problems, we prove a uniform expected-gap bound in the degenerate regime and, under ambient smoothness and positive effective curvature, convergence up to an explicit noise floor. Numerical experiments reproduce the predicted H\"older exponents and confirm that the number of gradient evaluations required for each function separates according to its own smoothness level rather than the worse of the two. A tomographic benchmark shows runtime gains when gradient evaluations are more expensive than the additional matrix-vector products.

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arXiv2026

Mathematical methods of reinforcement learning

Reinforcement learning (RL) is increasingly grounded in tools from probability, optimization, and operator theory. This survey organizes the mathematical structures that underpin the design and analysis of modern algorithms in RL. We begin from Markov decision processes (MDPs) and the Bellman operators, emphasizing contraction mappings, monotonicity, and fixed-point theory that yield convergence guarantees and rates for value and policy iteration, and temporal-difference schemes. We then develop the optimization perspective: stochastic approximation and martingale methods, convex duality and the role of regularization linking mirror/proximal methods. Function approximation is treated through linear and non-linear settings, covering stabilization, error decomposition, and sample-complexity via concentration inequalities for dependent data and mixing processes. We further cover off-policy evaluation/learning, constrained RL and constrained MDPs (CMDPs). Throughout we unify algorithmic templates under common operator and variational lenses, highlighting both finite-sample bounds and asymptotic results. Our presentation is intended to provide a unified mathematical entry point for researchers in probability, optimization, and statistics interested in reinforcement learning.

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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 $\delta$ has a cost $c(\delta)$, and we minimize the total time $T_{\mathrm{total}} = \sum_{k=1}^{N} c(\delta_k)$, 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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arXiv2026

Decentralized Inexact Cubic Newton Method with Consensus Procedure

Distributed optimization is widely used in large-scale and privacy-preserving machine learning, where each agent stores a local objective and communicates only with its neighbors in a connected network. We study decentralized second-order optimization and focus on consensus procedures that approximately average local iterates, gradients, and Hessians through neighbor-to-neighbor communications. We propose a general Decentralized Cubic Newton method for convex optimization under $L_1$-smoothness of gradients and $L_2$-Lipschitz continuity of Hessians, and develop a theory that accurately tracks the inaccuracies caused by consensus and by disagreement between local iterates. Under these assumptions, the method matches the iteration complexity of the exact Cubic Newton method and requires only additional polylogarithmic communication-round overhead to reach the necessary consensus accuracy. We further propose an Accelerated Decentralized Cubic Newton method for strongly convex objectives and show that it matches the iteration complexity of the exact Accelerated Cubic Newton method, again with only additional polylogarithmic communication-round overhead. Finally, although the general method requires exchanging full $d \times d$ Hessian matrices, we show how it can be implemented for generalized linear models by transmitting only vectors, making the approach substantially more practical in high dimensions.

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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

SDG-MoE: Signed Debate Graph Mixture-of-Experts

Sparse MoE models achieve a good balance between capacity and compute by routing each token to a small subset of experts. However, in most MoE architectures, once a token is routed, the selected experts process it independently and their outputs are combined via a weighted sum. This leaves open whether enabling communication among them could improve performance. While prior work has raised this question, direct interaction among the active routed experts remains underexplored. In this paper, we propose SDG-MoE (Signed Debate Graph Mixture-of-Experts), a novel architecture that adds a lightweight, iterative deliberation step before final aggregation. SDG-MoE introduces three components: (i) two learned interaction matrices over the active experts, a support graph $A^+$ and a critique graph $A^-$, capturing reinforcing and corrective influences; (ii) a signed message-passing step that updates expert representations before aggregation; and (iii) a disagreement-gated Friedkin-Johnsen-style anchoring that controls deliberation strength while preventing expert drift. Together, these enable a structured deliberation process where interaction strength scales with disagreement and specialization is preserved. We also provide a theoretical analysis establishing stability conditions on expert states and showing that deliberation adds only low-order overhead over the active set. In controlled three-seed pretraining experiments, SDG-MoE improves validation perplexity over both an unsigned graph communication baseline and vanilla MoE, outperforming the strongest baseline by 19.8%, and gives the best external perplexity on WikiText-103, C4, and Paloma among the compared systems.

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arXiv2026

Stronger constraints for smooth min-max games

Saddle point problems with smooth convex-concave objective functions are often used to model min-max problems arising in machine learning. First-order methods are the standard paradigm for solving such problems. Therefore, it is important to know how those methods behave in the worst-case scenarios. In order to derive the guarantees, one would require the inequalities that appropriately constrain the iterates, gradients and function values. In this paper, we present stronger constraints for smooth convex-concave functions and show that they could allow tighter upper bounds for first-order methods.

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arXiv2026

Lower and upper bounds of the convergence rate of gradient methods with composite noise in gradient

We introduce a detailed analysis of the convergence of first-order methods with composite noise (sum of relative and absolute) in gradient for convex and smooth function minimization. This paper illustrates instances of practical problems where the utilization of inexact oracles becomes necessary, such as biased compressors, use of floating-point arithmetic and gradient-free optimization. We propose an algorithm that optimally accumulates absolute error, with intermediate convergence depending on the relative component of the noise. Usage of restart technique, regularization transformation, and stopping criteria has been demonstrated to yield results for various function classes. Also, gradient descent adaptive to relative error parameter is provided. For relative noise, lower bounds of convergence are given, confirming the dependence of the parameter of the noise on the condition number of the problem.

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arXiv2026

Exploring New Frontiers in Vertical Federated Learning: the Role of Saddle Point Reformulation

The objective of Vertical Federated Learning (VFL) is to collectively train a model using features available on different devices while sharing the same users. This paper focuses on the saddle point reformulation of the VFL problem via the classical Lagrangian function. We first demonstrate how this formulation can be solved using deterministic methods. More importantly, we explore various stochastic modifications to adapt to practical scenarios, such as employing compression techniques for efficient information transmission, enabling partial participation for asynchronous communication, and utilizing coordinate selection for faster local computation. We show that the saddle point reformulation plays a key role and opens up possibilities to use mentioned extension that seem to be impossible in the standard minimization formulation. Convergence estimates are provided for each algorithm, demonstrating their effectiveness in addressing the VFL problem. Additionally, alternative reformulations are investigated, and numerical experiments are conducted to validate performance and effectiveness of the proposed approach.

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arXiv2026

Decentralized Optimization with Mixed Affine Constraints

This paper considers decentralized optimization of convex functions with mixed affine equality constraints involving both local and global variables. Constraints on global variables may vary across different nodes in the network, while local variables are subject to coupled and node-specific constraints. Such problem formulations arise in machine learning applications, including federated learning and multi-task learning, as well as in resource allocation and distributed control. We analyze this problem under smooth and non-smooth assumptions, considering both strongly convex and general convex objective functions. Our main contribution is an optimal algorithm for the smooth, strongly convex regime, whose convergence rate matches established lower complexity bounds. We further provide near-optimal methods for the remaining cases.

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arXiv2026

SCOPE: Smooth Convex Optimization for Planned Evolution of Deformable Linear Objects

We present SCOPE, a fast and efficient framework for modeling and manipulating deformable linear objects (DLOs). Unlike conventional energy-based approaches, SCOPE leverages convex approximations to significantly reduce computational cost while maintaining smooth and physically plausible deformations. This trade-off between speed and accuracy makes the method particularly suitable for applications requiring real-time or near-real-time response. The effectiveness of the proposed framework is demonstrated through comprehensive simulation experiments, highlighting its ability to generate smooth shape trajectories under geometric and length constraints.

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arXiv2026

Quantum-Inspired Episode Selection for Monte Carlo Reinforcement Learning via QUBO Optimization

Monte Carlo (MC) reinforcement learning suffers from high sample complexity, especially in environments with sparse rewards, large state spaces, and correlated trajectories. We address these limitations by reformulating episode selection as a Quadratic Unconstrained Binary Optimization (QUBO) problem and solving it with quantum-inspired samplers. Our method, MC+QUBO, integrates a combinatorial filtering step into standard MC policy evaluation: from each batch of trajectories, we select a subset that maximizes cumulative reward while promoting state-space coverage. This selection is encoded as a QUBO, where linear terms favor high-reward episodes and quadratic terms penalize redundancy. We explore both Simulated Quantum Annealing (SQA) and Simulated Bifurcation (SB) as black-box solvers within this framework. Experiments in a finite-horizon GridWorld demonstrate that MC+QUBO outperforms vanilla MC in convergence speed and final policy quality, highlighting the potential of quantum-inspired optimization as a decision-making subroutine in reinforcement learning.

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arXiv2026

Gradient-Free Approaches is a Key to an Efficient Interaction with Markovian Stochasticity

This paper deals with stochastic optimization problems involving Markovian noise with a zero-order oracle. We present and analyze a novel derivative-free method for solving such problems in strongly convex smooth and non-smooth settings with both one-point and two-point feedback oracles. Using a randomized batching scheme, we show that when mixing time $\tau$ of the underlying noise sequence is less than the dimension of the problem $d$, the convergence estimates of our method do not depend on $\tau$. This observation provides an efficient way to interact with Markovian stochasticity: instead of invoking the expensive first-order oracle, one should use the zero-order oracle. Finally, we complement our upper bounds with the corresponding lower bounds. This confirms the optimality of our results.

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arXiv2025

Dual Smoothing for Decentralized Optimization

Decentralized optimization is widely used in different fields of study such as distributed learning, signal processing, and various distributed control problems. In these types of problems, nodes of the network are connected to each other and seek to optimize some objective function. In this article, we present a method for smoothing the non-smooth and non-strongly convex problems. This is done using the dual smoothing technique. We study two types of problems: consensus optimization of linear models and coupled constraints optimization. It is shown that these two problem classes are dual to each other.

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arXiv2025

Adaptive Regularized Newton Method with Inexact Hessian

Newton's method is the most widespread high-order method, demanding the gradient and the Hessian of the objective function. However, one of the main disadvantages of Newtons method is its lack of global convergence and high iteration cost. Both these drawbacks are critical for modern optimization motivated primarily by current applications in machine learning. In this paper, we introduce a novel algorithm to deal with these disadvantages. Our method can be implemented with various Hessian approximations, including methods that use only the first-order information. Thus, computational costs might be drastically reduced. Also, it can be adjusted to problems' geometries via the usage of different Bregman divergences. The proposed method converges for nonconvex and convex problems globally and it has the same rates as other well-known methods that lack mentioned properties. We present experiments validating our method performs according to the theoretical bounds and shows competitive performance among other Newton-based methods.

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arXiv2025

Speeding up the Goemans-Williamson randomized procedure by difference-of-convex optimization

We present a novel approach to accelerate the Goemans-Williamson (GW) randomized rounding procedure for quadratic unconstrained binary optimization (QUBO) problems. Instead of solving the conventional semi-definite programming (SDP) relaxation, which is computationally expensive, we employ a difference-of-convex (DC) optimization framework to efficiently approximate the SDP solution. The DC optimization produces candidate vectors that are then used within the GW randomized rounding scheme to generate high-quality binary solutions. Furthermore, we perform direct expectation minimization over manifolds of matrices with limited rank to further enhance the solution quality. Our method is benchmarked on real-world QUBO instances, including inverse kinematics problems, and compared against state-of-the-art solvers, such as quantum-inspired algorithms, demonstrating competitive approximation guarantees alongside substantial computational gains.

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