arXiv subjects

Antesh Upadhyay

Publications and source records attributed to Antesh Upadhyay.

11 recordsLinked to original sources

arXiv2026

Beyond Bounded Variance: Variance-Reduced Normalized Methods for Nonconvex Optimization under Blum-Gladyshev Noise

We study nonconvex stochastic optimization under the Blum-Gladyshev ($\mathsf{BG}$-0) noise model, where the stochastic gradient variance grows quadratically with the distance from the initialization. We consider this problem under both standard smoothness and the symmetric generalized-smoothness framework, which captures objectives whose local curvature can scale with the gradient norm. We prove that normalized stochastic gradient descent with momentum, using only one stochastic gradient per iteration, converges under $\mathsf{BG}$-0 noise with oracle complexity $O(\varepsilon^{-6})$. This rate holds both for standard smoothness and for $α$-symmetric generalized smoothness, showing that generalized smoothness is rate-neutral for normalized momentum in this setting. We then study a variance-reduced normalized STORM method. Under mean-square smoothness and sharp initialization, the method achieves the minimax optimal $O(\varepsilon^{-4})$ complexity, matching the lower bound. Under expected $α$-symmetric generalized smoothness, the STORM recursion couples gradient-dependent smoothness with distance-dependent noise, leading to complexity $O(\varepsilon^{-(4+α)})$ for $α\in(0,1)$ and $O(\varepsilon^{-5})$ for $α=1$. When the distance-growth parameter in the noise model vanishes, our guarantees recover the standard bounded-variance rates: $O(\varepsilon^{-4})$ for momentum, $O(\varepsilon^{-3})$ for variance reduction, and $O(\varepsilon^{-2})$ in the deterministic case. To our knowledge, these are the first convergence guarantees for normalized methods in non-convex stochastic optimization under $\mathsf{BG}$-0 noise without bounded domains, increasing batch sizes, or explicit anchoring, covering both standard and generalized smoothness regimes.

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arXiv2026

Unified High-Probability Analysis of Stochastic Variance-Reduced Estimation

Stochastic estimators are fundamental to large-scale optimization, where population quantities must be inferred from noisy oracle observations. Although influential methods such as momentum, SPIDER, STORM, and PAGE have been highly successful, their analyses are largely estimator-specific and expectation-based, obscuring the structural tradeoffs that determine reliability. In this paper, we develop a unified framework for stochastic variance-reduced estimation based on a recursion with three components: memory retention, reset probability, and a correction term for iterate movement. This framework recovers several classical estimators, motivates new second-order variants, and yields a bias-variance decomposition of estimation error. Our main result is a unified high-probability bound proved using a new dimension-free vector-valued Freedman inequality, valid for smooth normed spaces involving random sums of vector martingales. The result applies in both Euclidean and non-Euclidean settings, including the analysis of mirror-descent-based methods in Banach spaces. As applications, we obtain high-probability oracle complexities for unconstrained optimization with mirror descent, establishing the logarithmic dependence on the confidence level. We also derive the first $\tilde{\mathcal{O}}(\varepsilon^{-3})$ oracle-complexity bounds for stochastic optimization with expectation constraints, improving upon the existing $\tilde{\mathcal{O}}(\varepsilon^{-4})$ complexity by leveraging variance-reduced estimation for the first time in this setting.

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arXiv2026

RAMPAGE: RAndomized Mid-Point for debiAsed Gradient Extrapolation

A celebrated method for Variational Inequalities (VIs) is Extragradient (EG), which can be viewed as a standard discrete-time integration scheme. With this view in mind, in this paper we show that EG may suffer from discretization bias when applied to non-linear vector fields, conservative or otherwise. To resolve this discretization shortcoming, we introduce RAndomized Mid-Point for debiAsed Gradient Extrapolation (RAMPAGE) and its variance-reduced counterpart, RAMPAGE+, which leverages antithetic sampling. In contrast with EG, both methods are unbiased. Furthermore, leveraging negative correlation, RAMPAGE+ acts as an unbiased, geometric path-integrator that completely removes internal first-order terms from the variance, provably improving upon RAMPAGE. We further demonstrate that both methods enjoy provable $\mathcal{O}(1/k)$ convergence guarantees for a range of problems including root finding under co-coercive, co-hypomonotone, and generalized Lipschitzness regimes. Furthermore, we introduce symmetrically scaled variants to extend our results to constrained VIs. Finally, we provide convergence guarantees of both methods for stochastic and deterministic smooth convex-concave games. Somewhat interestingly, despite being a randomized method, RAMPAGE+ attains purely deterministic bounds for a number of the studied settings.

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arXiv2026

Lower Bounds and Proximally Anchored SGD for Non-Convex Minimization Under Unbounded Variance

Analysis of Stochastic Gradient Descent (SGD) and its variants typically relies on the assumption of uniformly bounded variance, a condition that frequently fails in practical non-convex settings, such as neural network training, as well as in several elementary optimization settings. While several relaxations are explored in the literature, the Blum-Gladyshev (BG-0) condition, which permits the variance to grow quadratically with distance has recently been shown to be the weakest condition. However, the study of the oracle complexity of stochastic first-order non-convex optimization under BG-0 has remained underexplored. In this paper, we address this gap and establish information-theoretic lower bounds, proving that finding an $ε$-stationary point requires $Ω(ε^{-6})$ stochastic BG-0 oracle queries for smooth functions and $Ω(ε^{-4})$ queries under mean-square smoothness. These limits demonstrate an unavoidable degradation from classical bounded-variance complexities, i.e., $Ω(ε^{-4})$ and $Ω(ε^{-3})$ for smooth and mean-square smooth cases, respectively. To match these lower bounds, we consider Proximally Anchored STochastic Approximation (PASTA), a unified algorithmic framework that couples Halpern anchoring with Tikhonov regularization to dynamically mitigate the extra variance explosion term permitted by the BG-0 oracle. We prove that PASTA achieves minimax optimal complexities across numerous non-convex regimes, including standard smooth, mean-square smooth, weakly convex, star-convex, and Polyak-Lojasiewicz functions, entirely under an unbounded domain and unbounded stochastic gradients.

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arXiv2026

First-Order Softmax Weighted Switching Gradient Method for Distributed Stochastic Minimax Optimization with Stochastic Constraints

This paper addresses the distributed stochastic minimax optimization problem subject to stochastic constraints. We propose a novel first-order Softmax-Weighted Switching Gradient method tailored for federated learning. Under full client participation, our algorithm achieves the standard $\tilde{\mathcal{O}}(\epsilon^{-4})$ oracle complexity to satisfy a unified bound $\epsilon$ for both the optimality gap and feasibility tolerance. We extend our theoretical analysis to the practical partial participation regime by quantifying client sampling noise through a stochastic superiority assumption. Furthermore, by relaxing standard boundedness assumptions on the objective functions, we establish a strictly tighter lower bound for the softmax hyperparameter. We provide a unified error decomposition and establish a sharp $\mathcal{O}(\log\frac{1}{\delta})$ high-probability convergence guarantee. Ultimately, our framework demonstrates that a single-loop primal-only switching mechanism provides a stable alternative for optimizing worst-case client performance, effectively bypassing the hyperparameter sensitivity and convergence oscillations often encountered in traditional primal-dual or penalty-based approaches. We verify the efficacy of our algorithm via experiment on the Neyman-Pearson (NP) classification, fair classification, and federated safe reinforcement learning tasks.

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arXiv2026

FedSGM: A Unified Framework for Constraint Aware, Bidirectionally Compressed, Multi-Step Federated Optimization

We introduce FedSGM, a unified framework for federated constrained optimization that addresses four major challenges in federated learning (FL): functional constraints, communication bottlenecks, local updates, and partial client participation. Building on the switching gradient method, FedSGM provides projection-free, primal-only updates, avoiding expensive dual-variable tuning or inner solvers. To handle communication limits, FedSGM incorporates bi-directional error feedback, correcting the bias introduced by compression while explicitly understanding the interaction between compression noise and multi-step local updates. We derive convergence guarantees showing that the averaged iterate achieves the canonical $\boldsymbol{\mathcal{O}}(1/\sqrt{T})$ rate, with additional high-probability bounds that decouple optimization progress from sampling noise due to partial participation. Additionally, we introduce a soft switching version of FedSGM to stabilize updates near the feasibility boundary. To our knowledge, FedSGM is the first framework to unify functional constraints, compression, multiple local updates, and partial client participation, establishing a theoretically grounded foundation for constrained federated learning. Finally, we validate the theoretical guarantees of FedSGM via experimentation on Neyman-Pearson classification and constrained Markov decision process (CMDP) tasks.

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arXiv2025

Optimization via First-Order Switching Methods: Skew-Symmetric Dynamics and Optimistic Discretization

Large-scale constrained optimization problems are at the core of many tasks in control, signal processing, and machine learning. Notably, problems with functional constraints arise when, beyond a performance{\nobreakdash-}centric goal (e.g., minimizing the empirical loss), one desires to satisfy other requirements such as robustness, fairness, etc. A simple method for such problems, which remarkably achieves optimal rates for non-smooth, convex, strongly convex, and weakly convex functions under first-order oracle, is Switching Gradient Method (SGM): in each iteration depending on a predetermined constraint violation tolerance, use the gradient of objective or the constraint as the update vector. While the performance of SGM is well-understood for non-smooth functions and in fact matches its unconstrained counterpart, i.e., Gradient Descent (GD), less is formally established about its convergence properties under the smoothness of loss and constraint functions. In this work, we aim to fill this gap. First, we show that SGM may not benefit from faster rates under smoothness, in contrast to improved rates for GD under smoothness. By taking a continuous-time limit perspective, we show the issue is fundamental to SGM's dynamics and not an artifact of our analysis. Our continuous-time limit perspective further provides insights towards alleviating SGM's shortcomings. Notably, we show that leveraging the idea of optimism, a well-explored concept in variational inequalities and min-max optimization, could lead to faster methods. This perspective further enables designing a new class of ``soft'' switching methods, for which we further analyze their iteration complexity under mild assumptions.

math.OC↗

arXiv2024

Equitable Federated Learning with Activation Clustering

Federated learning is a prominent distributed learning paradigm that incorporates collaboration among diverse clients, promotes data locality, and thus ensures privacy. These clients have their own technological, cultural, and other biases in the process of data generation. However, the present standard often ignores this bias/heterogeneity, perpetuating bias against certain groups rather than mitigating it. In response to this concern, we propose an equitable clustering-based framework where the clients are categorized/clustered based on how similar they are to each other. We propose a unique way to construct the similarity matrix that uses activation vectors. Furthermore, we propose a client weighing mechanism to ensure that each cluster receives equal importance and establish $O(1/\sqrt{K})$ rate of convergence to reach an $ε-$stationary solution. We assess the effectiveness of our proposed strategy against common baselines, demonstrating its efficacy in terms of reducing the bias existing amongst various client clusters and consequently ameliorating algorithmic bias against specific groups.

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arXiv2024

FedNMUT -- Federated Noisy Model Update Tracking Convergence Analysis

A novel Decentralized Noisy Model Update Tracking Federated Learning algorithm (FedNMUT) is proposed that is tailored to function efficiently in the presence of noisy communication channels that reflect imperfect information exchange. This algorithm uses gradient tracking to minimize the impact of data heterogeneity while minimizing communication overhead. The proposed algorithm incorporates noise into its parameters to mimic the conditions of noisy communication channels, thereby enabling consensus among clients through a communication graph topology in such challenging environments. FedNMUT prioritizes parameter sharing and noise incorporation to increase the resilience of decentralized learning systems against noisy communications. Theoretical results for the smooth non-convex objective function are provided by us, and it is shown that the $ε-$stationary solution is achieved by our algorithm at the rate of $\mathcal{O}\left(\frac{1}{\sqrt{T}}\right)$, where $T$ is the total number of communication rounds. Additionally, via empirical validation, we demonstrated that the performance of FedNMUT is superior to the existing state-of-the-art methods and conventional parameter-mixing approaches in dealing with imperfect information sharing. This proves the capability of the proposed algorithm to counteract the negative effects of communication noise in a decentralized learning framework.

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arXiv2023

Improved Convergence Analysis and SNR Control Strategies for Federated Learning in the Presence of Noise

We propose an improved convergence analysis technique that characterizes the distributed learning paradigm of federated learning (FL) with imperfect/noisy uplink and downlink communications. Such imperfect communication scenarios arise in the practical deployment of FL in emerging communication systems and protocols. The analysis developed in this paper demonstrates, for the first time, that there is an asymmetry in the detrimental effects of uplink and downlink communications in FL. In particular, the adverse effect of the downlink noise is more severe on the convergence of FL algorithms. Using this insight, we propose improved Signal-to-Noise (SNR) control strategies that, discarding the negligible higher-order terms, lead to a similar convergence rate for FL as in the case of a perfect, noise-free communication channel while incurring significantly less power resources compared to existing solutions. In particular, we establish that to maintain the $O(\frac{1}{\sqrt{K}})$ rate of convergence like in the case of noise-free FL, we need to scale down the uplink and downlink noise by $Ω({\sqrt{k}})$ and $Ω({k})$ respectively, where $k$ denotes the communication round, $k=1,\dots, K$. Our theoretical result is further characterized by two major benefits: firstly, it does not assume the somewhat unrealistic assumption of bounded client dissimilarity, and secondly, it only requires smooth non-convex loss functions, a function class better suited for modern machine learning and deep learning models. We also perform extensive empirical analysis to verify the validity of our theoretical findings.

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arXiv2023

On the Convergence of Decentralized Federated Learning Under Imperfect Information Sharing

Decentralized learning and optimization is a central problem in control that encompasses several existing and emerging applications, such as federated learning. While there exists a vast literature on this topic and most methods centered around the celebrated average-consensus paradigm, less attention has been devoted to scenarios where the communication between the agents may be imperfect. To this end, this paper presents three different algorithms of Decentralized Federated Learning (DFL) in the presence of imperfect information sharing modeled as noisy communication channels. The first algorithm, Federated Noisy Decentralized Learning (FedNDL1), comes from the literature, where the noise is added to their parameters to simulate the scenario of the presence of noisy communication channels. This algorithm shares parameters to form a consensus with the clients based on a communication graph topology through a noisy communication channel. The proposed second algorithm (FedNDL2) is similar to the first algorithm but with added noise to the parameters, and it performs the gossip averaging before the gradient optimization. The proposed third algorithm (FedNDL3), on the other hand, shares the gradients through noisy communication channels instead of the parameters. Theoretical and experimental results demonstrate that under imperfect information sharing, the third scheme that mixes gradients is more robust in the presence of a noisy channel compared with the algorithms from the literature that mix the parameters.

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