Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization (1109.2415v2)

Abstract: We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the non-smooth term. We show that both the basic proximal-gradient method and the accelerated proximal-gradient method achieve the same convergence rate as in the error-free case, provided that the errors decrease at appropriate rates.Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems.

• The paper demonstrates that proximal-gradient methods achieve O(1/k) or O(1/k²) convergence when computational errors decrease at a sufficient rate.
• It outlines that both standard and accelerated methods maintain optimal convergence, even with inexact gradient and proximity evaluations under strong convexity.
• Numerical experiments on structured sparsity problems validate that careful error tuning is crucial for efficient optimization without sacrificing theoretical guarantees.

Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization

The paper "Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization" by Schmidt, Le Roux, and Bach addresses the optimization of composite functions wherein a smooth convex function is combined with a non-smooth convex component. This work specifically investigates the proximal-gradient techniques used to manage errors in gradient calculations and proximity operators. These methods are pivotal for efficiently handling structured sparsity problems, which are prevalent in various machine learning applications.

Proximal-Gradient Methods

Proximal-gradient methods are designed to exploit the structure of optimization problems characterized by a composite objective function: a smooth convex function $g$ and a potentially non-smooth convex function $h$. The method leverages fast convergence properties, achieving $O(1/k)$ for basic proximal-gradient and $O(1/k^2)$ for accelerated variants, provided there is no error in computations.

Inexact Proximal-Gradient Framework

The authors extend traditional proximal-gradient methods by accounting for errors in computation. These errors may originate in the evaluation of the gradient $g$ or the proximity operator of $h$. The paper examines both proximal-gradient and accelerated proximal-gradient methods and demonstrates that these methods can maintain their convergence rates as long as the errors decrease suitably.

Key Findings

1. Error Assumptions:
   • If the errors in the gradient and proximity operator computations decrease as $O(1/k^{1+\delta})$, $\delta > 0$, the basic proximal-gradient method achieves $O(1/k)$ convergence.
   • The accelerated proximal-gradient method reaches $O(1/k^2)$ convergence if errors decrease as $O(1/k^{2+\delta})$.
2. Strong Convexity:
   • In cases where $g$ is strongly convex, both methods demonstrate linear convergence rates, contingent on appropriately diminishing error sequences.
   • The convergence is characterized by the ratio $\gamma = \mu/L$, where $\mu$ is the strong convexity constant and $L$ is the Lipschitz constant.
3. Numerical Experiments:
   • The empirical evaluation on structured sparsity problems highlights that careful tuning of the error sequence is crucial for balancing convergence speed and computational efficiency. The paper also demonstrates that despite using approximate solutions, the accelerated proximal-gradient method can outperform the basic method under certain conditions.

Implications and Future Directions

This research impacts numerous applications in machine learning, particularly where exact calculations of proximity operators are infeasible. The findings offer practical guidance for employing inexact methods effectively without sacrificing theoretical convergence guarantees. Future exploration could involve adapting these methods to non-convex settings or integrating dynamic learning strategies for the error sequences, potentially extending their applicability and efficiency further.

In summary, this paper provides a comprehensive analysis of the convergence characteristics of inexact proximal-gradient methods, elaborating on conditions under which they retain optimal performance. The insights and numerical validations bridge gaps between theoretical advancements and practical implementation, rendering these techniques robust and versatile for a broad spectrum of optimization challenges in AI and beyond.