Catalyst Acceleration for First-order Convex Optimization: from Theory to Practice (1712.05654v2)

Abstract: We introduce a generic scheme for accelerating gradient-based optimization methods in the sense of Nesterov. The approach, called Catalyst, builds upon the inexact accelerated proximal point algorithm for minimizing a convex objective function, and consists of approximately solving a sequence of well-chosen auxiliary problems, leading to faster convergence. One of the keys to achieve acceleration in theory and in practice is to solve these sub-problems with appropriate accuracy by using the right stopping criterion and the right warm-start strategy. We give practical guidelines to use Catalyst and present a comprehensive analysis of its global complexity. We show that Catalyst applies to a large class of algorithms, including gradient descent, block coordinate descent, incremental algorithms such as SAG, SAGA, SDCA, SVRG, MISO/Finito, and their proximal variants. For all of these methods, we establish faster rates using the Catalyst acceleration, for strongly convex and non-strongly convex objectives. We conclude with extensive experiments showing that acceleration is useful in practice, especially for ill-conditioned problems.

• The paper introduces Catalyst, a generic acceleration scheme using accelerated proximal point algorithms to improve convergence of first-order optimization methods for convex functions.
• Catalyst employs Moreau envelopes to create smoothed objective functions and uses a nested optimization strategy with careful parameter choices and warm-starts.
• The method is applicable to various first-order algorithms, demonstrates improved convergence rates (e.g., from O(1/\varepsilon) to O(1/\sqrt{\varepsilon})), and shows practical effectiveness, especially on ill-conditioned problems.

Overview of Catalyst Acceleration for First-order Convex Optimization

The paper "Catalyst Acceleration for First-order Convex Optimization: From Theory to Practice" by Hongzhou Lin, Julien Mairal, and Zaid Harchaoui presents a generic acceleration scheme for gradient-based optimization algorithms, known as Catalyst. The approach leverages the principles of accelerated proximal point algorithms to improve convergence rates for a wide range of optimization methods applied to convex functions, including both smooth and non-smooth objectives.

Catalyst focuses on overcoming the limitations of traditional first-order methods which often exhibit slow convergence, especially on ill-conditioned problems. The key innovation involves solving a sequence of auxiliary sub-problems that approximate the original objective but with improved conditioning. This optimization strategy is significant for large-scale machine learning tasks that involve minimization of functions like regularized empirical risks.

Methodological Framework

The principal strategy of Catalyst is rooted in Nesterov's acceleration. This involves creating a "smooth" variant of the target function using Moreau envelopes—a known technique in convex optimization which essentially smooths out the objective by introducing a well-chosen quadratic regularization. The smoothed problem is then solved iteratively using a nested approach. Each inner loop performs an approximate minimization of an auxiliary problem with the help of a first-order method, while the outer loop constructs sequences that converge faster due to extrapolation techniques.

Key to Catalyst's effectiveness is the careful balance of computational efforts between these nested loops using a specific stopping criterion and warm-start strategies. The theoretical framework establishes guidelines on how to choose these parameters to ensure acceleration without unnecessarily excessive iteration counts.

Numerical and Theoretical Implications

Catalyst is applicable across a variety of algorithms including standard gradient descent, block coordinate descent, and incremental gradient methods such as SAG, SAGA, and SVRG. Importantly, the paper establishes improved convergence rates in scenarios involving both strongly convex and non-strongly convex functions. For example, under certain conditions, Catalyst can reduce the iteration complexity to achieve a particular accuracy from $O(1/\varepsilon)$ to $O(1/\sqrt{\varepsilon})$, a significant improvement for large-scale problems.

Theoretical analysis in the paper supports that the convergence acceleration comes not only from the extrapolation but also from constructing better conditioned auxiliary optimization problems. Furthermore, the implementation of Catalyst is examined through comprehensive experiments, confirming the utility of acceleration, particularly in scenarios with high condition numbers where traditional methods are less efficient.

Future Perspectives in AI Optimization

Catalyst's approach opens several pathways for future exploration. Optimizing the stochastic variants and exploring adaptive strategies for hyper-parameter selection could further enhance the versatility of this method. Additionally, integrating Catalyst with more advanced machine learning frameworks could lead to efficient optimization routines capable of handling real-world, large-scale datasets encountered in AI applications.

While the current focus of Catalyst is on convex functions, extending the principles to non-convex scenarios, perhaps by combining with other techniques like momentum methods or curvature correction, could be an exciting development. Such advancements could provide a sharper toolset for deep learning applications which require solving complex optimization problems.

In summary, Catalyst represents a versatile, theoretically sound, and practically effective framework for accelerating first-order optimization algorithms, offering substantial promise for large-scale machine learning tasks and beyond.