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.

翻译：本文研究经典Hedge算法在组合场景中的表现。在每轮交互中，学习者从集合$X \subseteq \{0,1\}^d$中选择向量$\boldsymbol{x}_t$，观测完整的损失向量$\boldsymbol{y}_t \in \mathbb{R}^d$，并承受$\langle \boldsymbol{x}_t, \boldsymbol{y}_t \rangle \in [-1,1]$的损失。该设定涵盖多个重要问题，包括扩展式博弈、资源分配、$m$-集合、在线多任务学习以及有向无环图（DAG）上的最短路径问题。已知Hedge算法在$T$轮交互后能达到$O\big(\sqrt{T \log |X|}\big)$的遗憾界。本文探究Hedge算法是否在所有组合场景中均保持最优性。为此，我们证明对于任意$X \subseteq \{0,1\}^d$，Hedge算法具有近最优性——具体而言存在$\sqrt{\log d}$因子差距——通过建立适用于所有算法的$\Omega\big(\sqrt{T \log(|X|)/\log d}\big)$下界。随后我们识别出一类自然组合集合——即满足$\log d \leq m \leq \sqrt{d}$的$m$-集合——该下界对此类集合是紧致的，且Hedge算法在此场景下被证明恰好存在$\sqrt{\log d}$因子的次优性。同时，我们证明Hedge算法在经典$K$-专家问题推广而来的在线多任务学习场景中具有最优性。最后，我们利用Hedge算法的近最优性，为DAG上的在线最短路径问题——该设定涵盖广泛的组合领域——建立了近最优正则化器的存在性。具体而言，我们证明经典在线镜像下降（OMD）算法在采用扩张熵正则化器时与Hedge算法具有迭代等价性，因此继承了Hedge算法在DAG场景中的近最优遗憾保证。