On a Combination of Alternating Minimization and Nesterov's Momentum
This provides a more efficient optimization method for practitioners in machine learning and related fields, though it is incremental as it builds on existing AM and acceleration techniques.
The paper tackles the problem of accelerating alternating minimization (AM) by combining it with Nesterov's momentum, resulting in an algorithm that achieves a $1/k^2$ convergence rate for convex problems and $1/k$ for non-convex problems, without requiring prior knowledge of problem parameters like convexity or Lipschitz constants.
Alternating minimization (AM) procedures are practically efficient in many applications for solving convex and non-convex optimization problems. On the other hand, Nesterov's accelerated gradient is theoretically optimal first-order method for convex optimization. In this paper we combine AM and Nesterov's acceleration to propose an accelerated alternating minimization algorithm. We prove $1/k^2$ convergence rate in terms of the objective for convex problems and $1/k$ in terms of the squared gradient norm for non-convex problems, where $k$ is the iteration counter. Our method does not require any knowledge of neither convexity of the problem nor function parameters such as Lipschitz constant of the gradient, i.e. it is adaptive to convexity and smoothness and is uniformly optimal for smooth convex and non-convex problems. Further, we develop its primal-dual modification for strongly convex problems with linear constraints and prove the same $1/k^2$ for the primal objective residual and constraints feasibility.