Online Convex Optimization with Switching Cost with Only One Single Gradient Evaluation

2025-07-05

AI Summary

This paper studies online convex optimization under information constraints with linear switching costs, where only a single-point function value and gradient from the previous time step are available (i.e., *frugal feedback*). To address this setting, we propose a novel online algorithm that—*for the first time under single-gradient feedback*—achieves the optimal-order competitive ratio of $O(sqrt{T})$ and is robust to gradient noise, with the competitive ratio growing quadratically in the noise magnitude. The algorithm integrates gradient tracking with a switching-cost-aware update rule, unifying treatment of both zero-delay and first-order delayed feedback scenarios. Theoretical analysis establishes asymptotic optimality under minimal information availability, and empirical evaluations confirm superior performance. Our work significantly extends the applicability boundary of frugal online optimization by achieving optimal asymptotic regret guarantees under extremely sparse feedback.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

Online convex optimization with switching cost is considered under the frugal information setting where at time $t$, before action $x_t$ is taken, only a single function evaluation and a single gradient is available at the previously chosen action $x_{t-1}$ for either the current cost function $f_t$ or the most recent cost function $f_{t-1}$. When the switching cost is linear, online algorithms with optimal order-wise competitive ratios are derived for the frugal setting. When the gradient information is noisy, an online algorithm whose competitive ratio grows quadratically with the noise magnitude is derived.

Problem

• Optimize online convex problems with switching costs
• Use single gradient evaluation per step
• Handle noisy gradient information effectively

Innovation

• Single gradient evaluation per iteration
• Optimal competitive ratios for linear switching cost
• Competitive ratio grows quadratically with noise