Constrained Online Two-stage Stochastic Optimization: Near Optimal Algorithms via Adversarial Learning

TL;DR

This paper introduces near-optimal online algorithms for two-stage stochastic optimization with long-term constraints, leveraging adversarial learning to achieve improved regret bounds under various stochastic and adversarial settings.

Contribution

The paper develops a unified adversarial learning-based framework for online two-stage stochastic optimization, achieving state-of-the-art regret bounds and robustness to adversarial corruptions.

Findings

Achieves $O( oot T)$ regret in stochastic settings.

Provides robustness to adversarial model parameter corruptions.

Develops algorithms with regret depending on prediction accuracy in non-stationary cases.

Abstract

We consider an online two-stage stochastic optimization with long-term constraints over a finite horizon of $T$ periods. At each period, we take the first-stage action, observe a model parameter realization and then take the second-stage action from a feasible set that depends both on the first-stage decision and the model parameter. We aim to minimize the cumulative objective value while guaranteeing that the long-term average second-stage decision belongs to a set. We develop online algorithms for the online two-stage problem from adversarial learning algorithms. Also, the regret bound of our algorithm cam be reduced to the regret bound of embedded adversarial learning algorithms. Based on our framework, we obtain new results under various settings. When the model parameter at each period is drawn from identical distributions, we derive \textit{state-of-art} $O(\sqrt{T})$ regret that improves previous bounds under special cases. Our algorithm is also robust to adversarial corruptions of model parameter realizations. When the model parameters are drawn from unknown non-stationary distributions and we are given machine-learned predictions of the distributions, we develop a new algorithm from our framework with a regret $O(W_T+\sqrt{T})$, where $W_T$ measures the total inaccuracy of the machine-learned predictions.