Non-stationary Stochastic Optimization

TL;DR

This paper investigates non-stationary stochastic optimization, introducing a variation budget to measure environment change, and establishes conditions for optimal performance and the impact of non-stationarity on regret bounds.

Contribution

It introduces the variation budget concept, links adversarial and stochastic optimization paradigms, and derives bounds on the impact of non-stationarity on optimization performance.

Findings

Sharp conditions for long-run optimality

Connection between adversarial and stochastic paradigms

Tight bounds on minimax regret and the price of non-stationarity

Abstract

We consider a non-stationary variant of a sequential stochastic optimization problem, in which the underlying cost functions may change along the horizon. We propose a measure, termed variation budget, that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve long-run-average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: adversarial online convex optimization; and the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the "price of non-stationarity," which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one.