We consider a class of nonsmooth convex composite optimization problems, where the objective function is given by the sum of a continuously differentiable convex term and a potentially non-differentiable convex regularizer. In [1], the authors introduced the proximal augmented Lagrangian method and derived the resulting continuous-time primal-dual dynamics that converge to the optimal solution. In this paper, we extend these dynamics from continuous to discrete time via the forward Euler discretization. We prove explicit bounds on the exponential convergence rates of our proposed algorithm with a sufficiently small step size. Since a larger step size can improve the convergence speed, we further develop a linear matrix inequality (LMI) condition which can be numerically solved to provide rate certificates with general step size choices. In addition, we prove that a large range of step size values can guarantee exponential convergence. We close the paper by demonstrating the performance of the proposed algorithm via computational experiments.
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On the Exponential Convergence Rate of Proximal Gradient Flow Algorithms
Many modern large-scale and distributed optimization problems can be cast into a form in which the objective function is a sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via a proximal gradient method which generalizes standard gradient descent to a nonsmooth setup. In this paper, we leverage the tools from control theory to study global convergence of proximal gradient flow algorithms. We utilize the fact that the proximal gradient algorithm can be interpreted as a variable-metric gradient method on the forward-backward envelope. This continuously differentiable function can be obtained from the augmented Lagrangian associated with the original nonsmooth problem and it enjoys a number of favorable properties. We prove that global exponential convergence can be achieved even in the absence of strong convexity. Moreover, for in-network optimization problems, we provide a distributed implementation of the gradient flow dynamics based on the proximal augmented Lagrangian and prove global exponential stability for strongly convex problems.
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- Award ID(s):
- 1809833
- NSF-PAR ID:
- 10128663
- Date Published:
- Journal Name:
- 2018 IEEE Conference on Decision and Control (CDC)
- Page Range / eLocation ID:
- 4246 to 4251
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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The primal-dual gradient flow dynamics based on the proximal augmented Lagrangian were introduced in [1] to solve nonsmooth composite optimization problems with a linear equality constraint. We use a Lyapunov-based approach to demonstrate global exponential stability of the underlying dynamics when the differentiable part of the objective function is strongly convex and its gradient is Lipschitz continuous. This also allows us to determine a bound on the stepsize that guarantees linear convergence of the discretized algorithm.more » « less
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Abstract Convergence analysis of accelerated first-order methods for convex optimization problems are developed from the point of view of ordinary differential equation solvers. A new dynamical system, called Nesterov accelerated gradient (NAG) flow, is derived from the connection between acceleration mechanism and
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/CDC.2018.8618968
