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

TL;DR

This work analyzes online learning under the random-order model by tying offline $(1+\varepsilon)$-approximation with online regret through a batch-to-online transformation governed by average sensitivity. The authors introduce an adaptive schedule for the approximation parameter $\varepsilon_t$, leveraging Fenchel duality to obtain small-loss regret bounds of the form $\tilde{O}(\varphi^{\star}(\mathrm{OPT}_T))$, where $\mathrm{OPT}_T$ is the offline optimum; this avoids requiring smoothness or other regularity on losses. They show this framework recovers and strengthens prior $(1+\varepsilon)$-regret results and applies to online $k$-means, online low-rank approximation, online regression, and online submodular function minimization using sparsifiers, yielding bounds that scale sublinearly with $\mathrm{OPT}_T$. A key technical contribution is the adaptive AdaGrad-like tuning of $\varepsilon_t$, which balances stability and accuracy to achieve first-order small-loss guarantees. The results highlight sparsification and coreset techniques as powerful tools for online learning in random-order settings and delineate the limits of small-loss bounds against adaptive adversaries.

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.

Theorems & Definitions (38)

• Definition 2.1: Total variation
• Definition 2.2: Average sensitivity
• Lemma 3.1: Single-step bound
• Lemma 3.2: Bound on sum of average optimal values
• Proposition 3.3: Regret decomposition
• Definition 4.1: Concave conjugate
• Lemma 4.2: Adaptive control of $\varepsilon_t$
• proof : Proof of Lemma \ref{['lem:exact-telescope']}
• Theorem 4.3: Small-loss regret bound
• proof