On the Universal Near Optimality of Hedge in Combinatorial Settings

• URL: http://arxiv.org/abs/2510.17099v2
• Date: Thu, 23 Oct 2025 22:55:03 GMT
• Title: On the Universal Near Optimality of Hedge in Combinatorial Settings
• Authors: Zhiyuan Fan, Arnab Maiti, Kevin Jamieson, Lillian J. Ratliff, Gabriele Farina,
• Abstract summary: We show that for any $X subseteq 0,1d$, Hedge is near-optimal-specifically, up to a $sqrtlog d$ factor--by establishing a lower bound of $Omegabig(sqrtT log(|X|/log dbig)$.<n>We also establish a near-optimal regularizer for online shortest-path problems in DAGs--a setting that subsumes a broad range of domains.
• Score: 40.84925385308883
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we study the classical Hedge algorithm in combinatorial settings. In each round, the learner selects a vector $\boldsymbol{x}_t$ from a set $X \subseteq \{0,1\}^d$, observes a full loss vector $\boldsymbol{y}_t \in \mathbb{R}^d$, and incurs a loss $\langle \boldsymbol{x}_t, \boldsymbol{y}_t \rangle \in [-1,1]$. This setting captures several important problems, including extensive-form games, resource allocation, $m$-sets, online multitask learning, and shortest-path problems on directed acyclic graphs (DAGs). It is well known that Hedge achieves a regret of $O\big(\sqrt{T \log |X|}\big)$ after $T$ rounds of interaction. In this paper, we ask whether Hedge is optimal across all combinatorial settings. To that end, we show that for any $X \subseteq \{0,1\}^d$, Hedge is near-optimal--specifically, up to a $\sqrt{\log d}$ factor--by establishing a lower bound of $\Omega\big(\sqrt{T \log(|X|)/\log d}\big)$ that holds for any algorithm. We then identify a natural class of combinatorial sets--namely, $m$-sets with $\log d \leq m \leq \sqrt{d}$--for which this lower bound is tight, and for which Hedge is provably suboptimal by a factor of exactly $\sqrt{\log d}$. At the same time, we show that Hedge is optimal for online multitask learning, a generalization of the classical $K$-experts problem. Finally, we leverage the near-optimality of Hedge to establish the existence of a near-optimal regularizer for online shortest-path problems in DAGs--a setting that subsumes a broad range of combinatorial domains. Specifically, we show that the classical Online Mirror Descent (OMD) algorithm, when instantiated with the dilated entropy regularizer, is iterate-equivalent to Hedge, and therefore inherits its near-optimal regret guarantees for DAGs.

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