Leveraging Predictions in Smoothed Online Convex Optimization via Gradient-based Algorithms

TL;DR

This paper introduces RHIG, a gradient-based online algorithm for convex optimization with switching costs, which effectively balances short-term predictions and environment variation to improve performance.

Contribution

It proposes RHIG, a novel algorithm that uses limited multi-step predictions to mitigate long-term prediction errors in online convex optimization with switching costs.

Findings

RHIG achieves low dynamic regret depending on environment variation and prediction accuracy.

Theoretical bounds are derived for regret under stochastic prediction errors.

Numerical tests show RHIG's effectiveness in quadrotor tracking tasks.

Abstract

We consider online convex optimization with time-varying stage costs and additional switching costs. Since the switching costs introduce coupling across all stages, multi-step-ahead (long-term) predictions are incorporated to improve the online performance. However, longer-term predictions tend to suffer from lower quality. Thus, a critical question is: how to reduce the impact of long-term prediction errors on the online performance? To address this question, we introduce a gradient-based online algorithm, Receding Horizon Inexact Gradient (RHIG), and analyze its performance by dynamic regrets in terms of the temporal variation of the environment and the prediction errors. RHIG only considers at most $W$-step-ahead predictions to avoid being misled by worse predictions in the longer term. The optimal choice of $W$ suggested by our regret bounds depends on the tradeoff between the variation of the environment and the prediction accuracy. Additionally, we apply RHIG to a well-established stochastic prediction error model and provide expected regret and concentration bounds under correlated prediction errors. Lastly, we numerically test the performance of RHIG on quadrotor tracking problems.