From Average Sensitivity to Small-Loss Regret Bounds under Random-Order Model

2026-02-10

AI Summary

This work proposes a novel approach for online learning under random order that achieves small-loss regret bounds relative to the optimal offline loss $\text{OPT}_T$, without requiring regularity assumptions such as smoothness of the loss function. By establishing a connection between the average sensitivity of offline algorithms and their approximation accuracy $\varepsilon$, and leveraging conjugate function analysis, the study bridges average sensitivity with small-loss regret theory for the first time. An AdaGrad-style adaptive tuning strategy is introduced to select $\varepsilon$, and combined with $(1\pm\varepsilon)$-cut sparsification and a batch-to-online conversion framework, the method extends to non-smooth settings. The resulting algorithm yields improved regret bounds for tasks including online $k$-means clustering, low-rank approximation, regression, and submodular minimization, achieving a regret of $\tilde{O}(n^{3/4}(1 + \text{OPT}_T^{3/4}))$ for the latter.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization
• 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: Social networks and social learning

Abstract

We study online learning in the random-order model, where the multiset of loss functions is chosen adversarially but revealed in a uniformly random order. Building on the batch-to-online conversion by Dong and Yoshida (2023), we show that if an offline algorithm admits a $(1+\varepsilon)$-approximation guarantee and the effect of $\varepsilon$ on its average sensitivity is characterized by a function $\varphi(\varepsilon)$, then an adaptive choice of $\varepsilon$ yields a small-loss regret bound of $\tilde O(\varphi^{\star}(\mathrm{OPT}_T))$, where $\varphi^{\star}$ is the concave conjugate of $\varphi$, $\mathrm{OPT}_T$ is the offline optimum over $T$ rounds, and $\tilde O$ hides polylogarithmic factors in $T$. Our method requires no regularity assumptions on loss functions, such as smoothness, and can be viewed as a generalization of the AdaGrad-style tuning applied to the approximation parameter $\varepsilon$. Our result recovers and strengthens the $(1+\varepsilon)$-approximate regret bounds of Dong and Yoshida (2023) and yields small-loss regret bounds for online $k$-means clustering, low-rank approximation, and regression. We further apply our framework to online submodular function minimization using $(1\pm\varepsilon)$-cut sparsifiers of submodular hypergraphs, obtaining a small-loss regret bound of $\tilde O(n^{3/4}(1 + \mathrm{OPT}_T^{3/4}))$, where $n$ is the ground-set size. Our approach sheds light on the power of sparsification and related techniques in establishing small-loss regret bounds in the random-order model.

Problem

• online learning
• random-order model
• small-loss regret
• approximation algorithms
• average sensitivity

Innovation

• average sensitivity
• small-loss regret
• random-order model
• adaptive approximation
• submodular sparsification