Leveraging Predictions in Smoothed Online Convex Optimization via Gradient-based Algorithms
This work is significant for researchers and practitioners in online optimization and control, particularly those dealing with systems where switching costs and prediction errors are critical, by providing a method to balance prediction horizon and accuracy.
This paper addresses online convex optimization with switching costs and time-varying stage costs, where multi-step-ahead predictions are used to improve performance. The authors propose a gradient-based online algorithm, Receding Horizon Inexact Gradient (RHIG), which strategically uses only up to W-step-ahead predictions to mitigate the impact of lower-quality long-term predictions.
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.