arXiv subjects

Anik Kumar Paul

Publications and source records attributed to Anik Kumar Paul.

8 recordsLinked to original sources

arXiv2026

Zeroth-Order Non-smooth Non-convex Optimization via Gaussian Smoothing

This paper addresses stochastic optimization of Lipschitz-continuous, nonsmooth and nonconvex objectives over compact convex sets, where only noisy function evaluations are available. While gradient-free methods have been developed for smooth nonconvex problems, extending these techniques to the nonsmooth setting remains challenging. The primary difficulty arises from the absence of a Taylor series expansion for Clarke subdifferentials, which limits the ability to approximate and analyze the behavior of the objective function in a neighborhood of a point. We propose a two time-scale zeroth-order projected stochastic subgradient method leveraging Gaussian smoothing to approximate Clarke subdifferentials. First, we establish that the expectation of the Gaussian-smoothed subgradient lies within an explicitly bounded error of the Clarke subdifferential, a result that extends prior analyses beyond convex/smooth settings. Second, we design a novel algorithm with coupled updates: a fast timescale tracks the subgradient approximation, while a slow timescale drives convergence. Using continuous-time dynamical systems theory and robust perturbation analysis, we prove that iterates converge almost surely to a neighborhood of the set of Clarke stationary points, with neighborhood size controlled by the smoothing parameter. To our knowledge, this is the first zeroth-order method achieving almost sure convergence for constrained nonsmooth nonconvex optimization problems.

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arXiv2026

Stochastic Mirror Descent under Iterate-Dependent Markov Noise: Analysis in the Asymptotic and Finite Time Regimes

We study a stochastic optimization problem in which the sampling distribution depends on the decision variable, and the available samples are generated through an iterate-dependent Markov chain. Such settings arise naturally in problems with decision-dependent uncertainty; however, they introduce bias and temporal dependence, which render standard techniques developed for i.i.d.\ noise inapplicable. In this work, we analyze the stochastic mirror descent algorithm under iterate-dependent Markov noise. We first establish almost sure convergence for both convex and non-convex problems under the mild assumption of Lipschitz continuity of the objective function, without requiring differentiability. We then derive finite-time concentration bounds for smooth objectives. In the convex setting, the resulting sample complexity matches the classical rate of stochastic mirror descent under i.i.d.\ noise. In the non-convex setting, we obtain a sample complexity bound in terms of the norm of the Riemannian gradient over the probability simplex. Overall, our results establish a unified convergence framework for stochastic mirror descent with state-dependent Markov noise, and highlight its behavior in both convex and non-convex regimes.

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arXiv2026

Adversary-Robust Learning from Fully Asynchronous Directional Derivative Estimates

We propose FAR-SIGN (Fully Asynchronous Robust optimization via SIGNed directional projections) for adversary-resilient learning in parameter-server--worker systems. FAR-SIGN achieves robustness through sign-based updates along carefully designed directions and mitigates the resulting bias via a two-timescale mechanism. It admits both first-order and zeroth-order implementations and enables fully asynchronous execution without requiring a private reference dataset at the server. We establish almost-sure convergence of FAR-SIGN to the set of stationary points for smooth, nonconvex objectives. Moreover, we prove the near-optimal rate of $O(n^{-1/4+ε})$ in the first-order setting and the standard $O(n^{-1/6+ε})$ in the zeroth-order setting, where $n$ is the iteration count and $ε>0$ can be chosen arbitrarily small. Experiments on MNIST show that FAR-SIGN outperforms robust aggregation-based methods in both accuracy and wall-clock time.

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arXiv2026

Stochastic Recursive Inclusions under Biased Perturbations: An Input-to-State Stability Perspective

This paper investigates the asymptotic behavior of stochastic recursive inclusions in the presence of non-zero, non-diminishing bias, a setting that frequently arises in zeroth-order optimization, stochastic approximation with iterate-dependent noise, and distributed learning with adversarial agents. The analysis is conducted through the lens of input-to-state stability of an associated differential inclusion, which serves as the continuous-time limit of the discrete recursion. We first establish that if the limiting differential inclusion is input-to-state stable and the iterates remain almost surely bounded, then the iterates converge almost surely to the neighborhood of desired equilibrium. We then provide a verifiable sufficient condition for almost sure boundedness by assuming that the underlying operator is single-valued and globally Lipschitz. Finally, we show that several zeroth-order variants of stochastic gradient naturally fit within this framework, and we demonstrate their input-to-state stability under standard conditions. Overall, the results provide a unified theoretical foundation for studying almost sure convergence of biased stochastic approximation schemes through the Input to State stability theory of differential inclusions.

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arXiv2025

Federated Learning: A Stochastic Approximation Approach

This paper considers the Federated learning (FL) in a stochastic approximation (SA) framework. Here, each client $i$ trains a local model using its dataset $\mathcal{D}^{(i)}$ and periodically transmits the model parameters $w^{(i)}_n$ to a central server, where they are aggregated into a global model parameter $\bar{w}_n$ and sent back. The clients continue their training by re-initializing their local models with the global model parameters. Prior works typically assumed constant (and often identical) step sizes (learning rates) across clients for model training. As a consequence the aggregated model converges only in expectation. In this work, client-specific tapering step sizes $a^{(i)}_n$ are used. The global model is shown to track an ODE with a forcing function equal to the weighted sum of the negative gradients of the individual clients. The weights being the limiting ratios $p^{(i)}=\lim_{n \to \infty} \frac{a^{(i)}_n}{a^{(1)}_n}$ of the step sizes, where $a^{(1)}_n \geq a^{(i)}_n, \forall n$. Unlike the constant step sizes, the convergence here is with probability one. In this framework, the clients with the larger $p^{(i)}$ exert a greater influence on the global model than those with smaller $p^{(i)}$, which can be used to favor clients that have rare and uncommon data. Numerical experiments were conducted to validate the convergence and demonstrate the choice of step-sizes for regulating the influence of the clients.

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arXiv2024

Almost Sure Convergence and Non-asymptotic Concentration Bounds for Stochastic Mirror Descent Algorithm

This letter investigates the convergence and concentration properties of the Stochastic Mirror Descent (SMD) algorithm utilizing biased stochastic subgradients. We establish the almost sure convergence of the algorithm's iterates under the assumption of diminishing bias. Furthermore, we derive concentration bounds for the discrepancy between the iterates' function values and the optimal value, based on standard assumptions. Subsequently, leveraging the assumption of Sub-Gaussian noise in stochastic subgradients, we present refined concentration bounds for this discrepancy.

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arXiv2024

Robust Analysis of Almost Sure Convergence of Zeroth-Order Mirror Descent Algorithm

This letter presents an almost sure convergence of the zeroth-order mirror descent algorithm. The algorithm admits non-smooth convex functions and a biased oracle which only provides noisy function value at any desired point. We approximate the subgradient of the objective function using Nesterov's Gaussian Approximation (NGA) with certain alternations suggested by some practical applications. We prove an almost sure convergence of the iterates' function value to the neighbourhood of optimal function value, which can not be made arbitrarily small, a manifestation of a biased oracle. This letter ends with a concentration inequality, which is a finite time analysis that predicts the likelihood that the function value of the iterates is in the neighbourhood of the optimal value at any finite iteration.

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arXiv2024

Convergence Analysis of Stochastic Saddle Point Mirror Descent Algorithm -- A Projected Dynamical View Point

Saddle point problems, ubiquitous in optimization, extend beyond game theory to diverse domains like power networks and reinforcement learning. This paper presents novel approaches to tackle saddle point problem, with a focus on continuous-time contexts. In this paper we propose a continuous time dynamics to tackle saddle point problem utilizing projected dynamical system in non-Euclidean domain. This involves computing the (sub/super) gradient of the min-max function within a Riemannian metric. Additionally, we establish viable Caratheodory solutions also prove the Lyapunov stability and asymptotic set stability of the proposed continuous time dynamical system. Next, we present the Stochastic Saddle Point Mirror Descent (SSPMD) algorithm and establish its equivalence with the proposed continuous-time dynamics. Leveraging stability analysis of the continuous-time dynamics, we demonstrate the almost sure convergence of the algorithm's iterates. Furthermore, we introduce the Zeroth-Order Saddle Point Mirror Descent (SZSPMD) algorithm, which approximates gradients using Nesterov's Gaussian Approximation, showcasing convergence to a neighborhood around saddle points. The analysis in this paper provides geometric insights into the mirror descent algorithm and demonstrates how these insights offer theoretical foundations for various practical applications of the mirror descent algorithm in diverse scenarios.

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