Online Convex Optimization with Memory and Limited Predictions

We study the problem of online convex optimization with memory and predictions over a horizon $T$. At each time step, a decision maker is given some limited predictions of the cost functions from a finite window of future time steps, i.e., values of the cost function at certain decision points in the future. The decision maker then chooses an action and incurs a cost given by a convex function that depends on the actions chosen in the past. We propose an algorithm to solve this problem and show that the dynamic regret of the algorithm decays exponentially with the prediction window length. Our algorithm contains two general subroutines that work for wider classes of problems. The first subroutine can solve general online convex optimization with memory and bandit feedback with $\sqrt{T}$-dynamic regret with respect to $T$. The second subroutine is a zeroth-order method that can be used to solve general convex optimization problems with a linear convergence rate that matches the best achievable rate of first-order methods for convex optimization. The key to our algorithm design and analysis is the use of truncated Gaussian smoothing when querying the decision points for obtaining the predictions. We complement our theoretical results using numerical experiments.

Regret Bounds for Learning Decentralized Linear Quadratic Regulator with Partially Nested Information Structure

We study the problem of learning decentralized linear quadratic regulator under a partially nested information constraint, when the system model is unknown a priori. We propose an online learning algorithm that adaptively designs a control policy as new data samples from a single system trajectory become available. Our algorithm design uses a disturbance-feedback representation of state-feedback controllers coupled with online convex optimization with memory and delayed feedback. We show that our online algorithm yields a controller that satisfies the desired information constraint and enjoys an expected regret that scales as $\sqrt{T}$ with the time horizon $T$.