Efficient Online-Bandit Strategies for Minimax Learning Problems
This work addresses efficiency in min-max optimization for machine learning practitioners, offering incremental improvements through structured algorithmic design.
The paper tackles the challenge of solving min-max problems in learning, such as distributional robust learning, by proposing efficient online-bandit strategies that leverage the structure of the constraint set. It provides high-probability convergence guarantees to minimax values for a specific family of sets, with concrete convergence rates implied by the guarantees.
Several learning problems involve solving min-max problems, e.g., empirical distributional robust learning or learning with non-standard aggregated losses. More specifically, these problems are convex-linear problems where the minimization is carried out over the model parameters $w\in\mathcal{W}$ and the maximization over the empirical distribution $p\in\mathcal{K}$ of the training set indexes, where $\mathcal{K}$ is the simplex or a subset of it. To design efficient methods, we let an online learning algorithm play against a (combinatorial) bandit algorithm. We argue that the efficiency of such approaches critically depends on the structure of $\mathcal{K}$ and propose two properties of $\mathcal{K}$ that facilitate designing efficient algorithms. We focus on a specific family of sets $\mathcal{S}_{n,k}$ encompassing various learning applications and provide high-probability convergence guarantees to the minimax values.