Exploration by Optimization with Hybrid Regularizers: Logarithmic Regret with Adversarial Robustness in Partial Monitoring

• URL: http://arxiv.org/abs/2402.08321v1
• Date: Tue, 13 Feb 2024 09:34:22 GMT
• Title: Exploration by Optimization with Hybrid Regularizers: Logarithmic Regret with Adversarial Robustness in Partial Monitoring
• Authors: Taira Tsuchiya, Shinji Ito, Junya Honda
• Abstract summary: exploration by optimization (ExO) was proposed, which achieves the optimal bounds in adversarial environments. We first establish a novel framework and analysis for ExO with a hybrid regularizer. In particular, we derive a regret bound of $O(sum_a neq a* k2 log T / Delta_a)$ in which $a*$ is an optimal action, and $Delta_a$ is the suboptimality gap for action $a$.
• Score: 46.30750729936261
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Partial monitoring is a generic framework of online decision-making problems with limited observations. To make decisions from such limited observations, it is necessary to find an appropriate distribution for exploration. Recently, a powerful approach for this purpose, exploration by optimization (ExO), was proposed, which achieves the optimal bounds in adversarial environments with follow-the-regularized-leader for a wide range of online decision-making problems. However, a naive application of ExO in stochastic environments significantly degrades regret bounds. To resolve this problem in locally observable games, we first establish a novel framework and analysis for ExO with a hybrid regularizer. This development allows us to significantly improve the existing regret bounds of best-of-both-worlds (BOBW) algorithms, which achieves nearly optimal bounds both in stochastic and adversarial environments. In particular, we derive a stochastic regret bound of $O(\sum_{a \neq a^*} k^2 m^2 \log T / \Delta_a)$, where $k$, $m$, and $T$ are the numbers of actions, observations and rounds, $a^*$ is an optimal action, and $\Delta_a$ is the suboptimality gap for action $a$. This bound is roughly $\Theta(k^2 \log T)$ times smaller than existing BOBW bounds. In addition, for globally observable games, we provide a new BOBW algorithm with the first $O(\log T)$ stochastic bound.

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