Regret in Online Combinatorial Optimization
This work addresses regret minimization in online combinatorial optimization, providing theoretical insights for decision-making under uncertainty, but it is incremental as it builds on existing methods like Mirror Descent and INF.
The paper tackles online linear optimization with binary action vectors, aiming to determine the minimax regret under full information, semi-bandit, and bandit feedback models. It proves optimal regret bounds for the semi-bandit case, recovers optimal bounds for full information, and discusses a new lower bound and conjecture for the bandit case.
We address online linear optimization problems when the possible actions of the decision maker are represented by binary vectors. The regret of the decision maker is the difference between her realized loss and the best loss she would have achieved by picking, in hindsight, the best possible action. Our goal is to understand the magnitude of the best possible (minimax) regret. We study the problem under three different assumptions for the feedback the decision maker receives: full information, and the partial information models of the so-called "semi-bandit" and "bandit" problems. Combining the Mirror Descent algorithm and the INF (Implicitely Normalized Forecaster) strategy, we are able to prove optimal bounds for the semi-bandit case. We also recover the optimal bounds for the full information setting. In the bandit case we discuss existing results in light of a new lower bound, and suggest a conjecture on the optimal regret in that case. Finally we also prove that the standard exponentially weighted average forecaster is provably suboptimal in the setting of online combinatorial optimization.