Stochastic model-based minimization of weakly convex functions (1803.06523v3)

Abstract: We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate $O(k^{-1/4})$. As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is that all algorithms under consideration can be interpreted as approximate descent methods on an implicit smoothing of the problem, given by the Moreau envelope. Specializing to classical circumstances, we obtain the long-sought convergence rate of the stochastic projected gradient method, without batching, for minimizing a smooth function on a closed convex set.

• The paper establishes complexity guarantees for minimizing weakly convex functions using stochastic prox-linear, proximal point, and subgradient methods.
• It leverages the Moreau envelope to enable convergence analysis, achieving a stationarity measure rate of O(k^{-1/4}).
• The proposed methods demonstrate practical efficiency for challenging optimization tasks in fields like robust phase retrieval and covariance estimation.

Stochastic Model-Based Minimization of Weakly Convex Functions

The paper "Stochastic model-based minimization of weakly convex functions" by Damek Davis and Dmitriy Drusvyatskiy addresses a crucial problem in the field of optimization: the minimization of weakly convex functions using stochastic methods. The focus of the research is on developing and analyzing algorithms that iteratively build simple stochastic models to approximate and minimize the objective function. This work provides new insights and complexity guarantees for several well-regarded stochastic optimization algorithms, including the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods, in the specific context of weakly convex functions.

Summary of Key Results

The paper makes a significant contribution by establishing complexity guarantees for minimizing compositions of convex functions with smooth maps, a class of functions termed as weakly convex. The authors show that under reasonable conditions regarding approximation quality and regularity, their proposed algorithms ensure that a natural measure of stationarity converges to zero at the rate of $O(k^{-1/4})$. This is a noteworthy achievement, considering that weakly convex functions generalize both convex and smooth functions.

One of the primary contributions is the complexity analysis of the stochastic prox-linear, proximal point, and proximal subgradient algorithms. These algorithms are viewed as approximate descent methods operating on the Moreau envelope, a smoothed version of the original problem that facilitates the analysis of nonsmooth and nonconvex problems.

Algorithmic Insights

1. Stochastic Proximal Point Method (SPPM): This algorithm is shown to minimize weakly convex functions using stochastic oracle models. The subproblems involved are proved to have explicit solutions, making the method computationally feasible.
2. Stochastic Prox-Linear Algorithm: A variant of the well-known Gauss-Newton method, this algorithm uses linear approximation models to solve each subproblem. It is demonstrated to have robust performance and empirically outperforms the basic stochastic subgradient algorithms.
3. Stochastic Subgradient Method: The classical subgradient method is extended to handle weakly convex functions by leveraging the Moreau envelope technique. Although more straightforward, it is shown to be less robust compared to the prox-linear and proximal point methods.

Theoretical Implications

An essential theoretical implication is the use of the Moreau envelope as a potential function, driving convergence analyses for all discussed algorithms. This insight extends traditional analyses beyond smooth or convex functions to a broader class of problems, potentially beneficial for various machine learning and data science applications. Furthermore, the paper rigorously establishes conditions under which these stochastic algorithms have provable convergence rates, even when the function regime is neither convex nor smooth.

Practical Implications

From a practical perspective, the results suggest that the developed algorithms can be applied effectively in real-world optimization tasks where the objective functions are weakly convex. This includes applications like robust phase retrieval, covariance matrix estimation, blind deconvolution, and more, as highlighted through various numerical experiments in the paper.

Future Directions

The findings open up several promising avenues for future research. One area of interest would be the exploration of improvements or adaptations of these algorithms for large-scale or distributed optimization scenarios. Additionally, extending these methods and theoretical guarantees to cover even broader classes of nonconvex, nonsmooth problems could be extremely advantageous, particularly in the context of modern machine learning challenges. Integrating these approaches with advanced techniques from statistical learning theory could also yield more efficient and robust algorithms.

In conclusion, Davis and Drusvyatskiy's work provides a substantial advancement in understanding and applying stochastic model-based approximation techniques within the field of weakly convex function minimization. The insights gained from their analysis not only expand the theoretical foundations but also enhance the practical toolkit available for tackling complex, high-dimensional optimization problems in various scientific and engineering domains.