Non-stationary Stochastic Optimization
This work addresses the challenge of optimizing in changing environments for researchers and practitioners in machine learning and operations research, providing foundational insights that bridge adversarial online convex optimization and stochastic approximation.
The paper tackles the problem of sequential stochastic optimization in non-stationary environments where cost functions change over time, introducing a variation budget to control these changes and establishing sharp conditions for achieving optimal performance, including tight bounds on minimax regret to quantify the added complexity of non-stationarity.
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.