Subgame Perfect Methods in Nonsmooth Convex Optimization

Abstract: This paper considers nonsmooth convex optimization with either a subgradient or proximal operator oracle. In both settings, we identify algorithms that achieve the recently introduced game-theoretic optimality notion for algorithms known as subgame perfection. Subgame perfect algorithms meet a more stringent requirement than just minimax optimality. Not only must they provide optimal uniform guarantees on the entire problem class, but also on any subclass determined by information revealed during the execution of the algorithm. In the setting of nonsmooth convex optimization with a subgradient oracle, we show that the Kelley cutting plane-Like Method due to Drori and Teboulle [1] is subgame perfect. For nonsmooth convex optimization with a proximal operator oracle, we develop a new algorithm, the Subgame Perfect Proximal Point Algorithm, and establish that it is subgame perfect. Both of these methods solve a history-aware second-order cone program within each iteration, independent of the ambient problem dimension, to plan their next steps. This yields performance guarantees that are never worse than the minimax optimal guarantees and often substantially better.