Online Convex Optimization with Prediction Through Accelerated Gradient Descent

Junhui Zhang, Patrick Jaillet

Keywords: online optimization, accelerated gradient descent

Abstract:

We study online convex optimization with predictions, where, at each time step $t$, predictions about the next $k$ steps are available, and with coupled costs over time steps, where the cost function at time step $t$ depends on the decisions made between time $t-a$ and time $t+b$ for some nonnegative integers $a,b$.

We provide a general recipe to run synchronous update in an asynchronous fashion that respects the sequential revelation of information. Combined with existing convergence results for convex optimization using inexact first-order oracle, we show that acceleration is possible in this framework, where the dynamic regret can be reduced by a factor of $(1-O(\sqrt{\kappa}))^{\frac{k}{a+b}}$ through accelerated gradient descent, at a cost of an additive error term that depends on the prediction accuracy. This generalizes and improves the $(1-\kappa/4)^k$ factor obtained by Li & Li (2020) for $a+b = 1$. Our algorithm also has smaller dependency on longer-term prediction error. Moreover, our algorithm is the first gradient based algorithm which, when the strong-convexity assumption is relaxed, constructs a solution whose regret decays at the rate of $O(1/k^2)$, at a cost of an additive error term that depends on the prediction accuracy.

Primary Area: optimization

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Submission Number: 10871