Time-Dependent Blackwell Approachability and Application to Absorbing Games
This work provides an incremental extension to online learning tools for solving stochastic games, specifically targeting researchers in game theory and regret minimization.
The authors extended Blackwell's approachability framework to allow time-dependent outcome functions and inner products, establishing convergence guarantees for the adapted algorithm and applying it to construct ε-uniformly optimal strategies in absorbing games.
Blackwell's approachability (Blackwell, 1954, 1956) is a very general online learning framework where a Decision Maker obtains vector-valued outcomes, and aims at the convergence of the average outcome to a given ``target'' set. Blackwell gave a sufficient condition for the decision maker having a strategy guaranteeing such a convergence against an adversarial environment, as well as what we now call the Blackwell's algorithm, which then ensures convergence. Blackwell's approachability has since been applied to numerous problems, in regret minimization and game theory, in particular. We extend this framework by allowing the outcome function and the inner product to be time-dependent. We establish a general guarantee for the natural extension to this framework of Blackwell's algorithm. In the case where the target set is an orthant, we present a family of time-dependent inner products which yields different convergence speeds for each coordinate of the average outcome. We apply this framework to absorbing games (an important class of stochastic games) for which we construct $\varepsilon$-uniformly optimal strategies using Blackwell's algorithm in a well-chosen auxiliary approachability problem, thereby giving a novel illustration of the relevance of online learning tools for solving games.