A Control-Theoretic Perspective on Optimal High-Order Optimization

We provide a control-theoretic perspective on optimal tensor algorithms for minimizing a convex function in a finite-dimensional Euclidean space. Given a function Phi: ℝd rightarrow ℝ that is convex and twice continuously differentiable, we study a closed-loop control system that is governed by the operators nabla Phi and nabla2 Phi together with a feedback control law λ(·) satisfying the algebraic equation (λ(t))p|nablaPhi(x(t))|p-1 = θ for some θ ∈ (0, 1). Our first contribution is to prove the existence and uniqueness of a local solution to this system via the Banach fixed-point theorem. We present a simple yet nontrivial Lyapunov function that allows us to establish the existence and uniqueness of a global solution under certain regularity conditions and analyze the convergence properties of trajectories. The rate of convergence is O(1/t(3p+1)/2) in terms of objective function gap and O(1/t3p) in terms of squared gradient norm. Our second contribution is to provide two algorithmic frameworks obtained from discretization of our continuous-time system, one of which generalizes the large-step A-HPE framework and the other of which leads to a new optimal p-th order tensor algorithm. While our discrete-time analysis can be seen as a simplification and generalization of~citetMonteiro-2013-Accelerated, it is largely motivated by the aforementioned continuous-time analysis, demonstrating the fundamental role that the feedback control plays in optimal acceleration and the clear advantage that the continuous-time perspective brings to algorithmic design. A highlight of our analysis is that we show that all of the p-th order optimal tensor algorithms that we discuss minimize the squared gradient norm at a rate of O(k-3p), which complements the recent analysis.