Instance-Dependent Regret Bounds for Nonstochastic Linear Partial Monitoring

TL;DR

This paper introduces instance-dependent regret bounds for nonstochastic linear partial monitoring, generalizing linear bandits to more complex feedback structures with adaptive guarantees based on game difficulty.

Contribution

It provides the first instance-specific regret bounds for nonstochastic linear partial monitoring, capturing the influence of game structure on learning performance.

Findings

Achieves  regret in easy games

Achieves  regret in hard games

Bounds are tight in key cases

Abstract

In contrast to the classic formulation of partial monitoring, linear partial monitoring can model infinite outcome spaces, while imposing a linear structure on both the losses and the observations. This setting can be viewed as a generalization of linear bandits where loss and feedback are decoupled in a flexible manner. In this work, we address a nonstochastic (adversarial), finite-actions version of the problem through a simple instance of the exploration-by-optimization method that is amenable to efficient implementation. We derive regret bounds that depend on the game structure in a more transparent manner than previous theoretical guarantees for this paradigm. Our bounds feature instance-specific quantities that reflect the degree of alignment between observations and losses, and resemble known guarantees in the stochastic setting. Notably, they achieve the standard $\sqrt{T}$ rate in easy (locally observable) games and $T^{2/3}$ in hard (globally observable) games, where $T$ is the time horizon. We instantiate these bounds in a selection of old and new partial information settings subsumed by this model, and illustrate that the achieved dependence on the game structure can be tight in interesting cases.