Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming (1309.5549v1)

Abstract: In this paper, we introduce a new stochastic approximation (SA) type algorithm, namely the randomized stochastic gradient (RSG) method, for solving an important class of nonlinear (possibly nonconvex) stochastic programming (SP) problems. We establish the complexity of this method for computing an approximate stationary point of a nonlinear programming problem. We also show that this method possesses a nearly optimal rate of convergence if the problem is convex. We discuss a variant of the algorithm which consists of applying a post-optimization phase to evaluate a short list of solutions generated by several independent runs of the RSG method, and show that such modification allows to improve significantly the large-deviation properties of the algorithm. These methods are then specialized for solving a class of simulation-based optimization problems in which only stochastic zeroth-order information is available.

• The paper introduces the randomized stochastic gradient (RSG) method, establishing near-optimal convergence for both nonconvex and convex problems.
• It details complexity bounds, achieving an O(1/√N) rate for nonconvex problems and improved large-deviation performance via a two-phase approach.
• It further presents stochastic zeroth-order strategies employing Gaussian smoothing to approximate gradients in high-dimensional, simulation-based optimization tasks.

Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming

This essay presents an examination of "Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming" by Saeed Ghadimi and Guanghui Lan. The paper introduces robust stochastic approximation (SA) algorithms for addressing a broad class of nonlinear and potentially nonconvex stochastic programming (SP) problems. The research establishes the complexity and convergence properties of a newly proposed algorithm, the randomized stochastic gradient (RSG) method, as well as its variant for zeroth-order stochastic optimization problems.

Problem Context and Background

Traditional SA algorithms, notably the Robbins-Monro (1951) approach, are known for their efficacy in solving strongly convex SP problems by utilizing noisy gradient information. However, the convergence of classical SA methods hinges on the strong convexity of the problem and often underperforms in practice due to the reliance on specific stepsize policies that are challenging to implement. Notable enhancements, such as the Polyak-Ruppert averaging (Polyak, 1990; Polyak and Juditsky, 1992), highlight improved robustness and convergence rates, yet they still demand convexity for optimal performance. Extending SA methods to nonconvex problems, while ensuring convergence and efficiency, remains an open challenge.

Randomized Stochastic Gradient (RSG) Method

The RSG method marks a significant advancement by incorporating a probabilistic selection of the iteration count, thus addressing both convex and nonconvex SP problems. The algorithm's structure allows flexibility in stepsizes while maintaining convergence under nonconvexity constraints. The RSG method converges nearly optimally for convex problems when appropriately parameterized.

Convergence Analysis

The convergence properties of the RSG method are comprehensively analyzed. The method demonstrates that it can achieve the expectation properties:

• For nonconvex problems, the complexity to achieve $\mathcal{O}(1/\sqrt{N})$ iteration complexity is shown, with the expectation $E[\|\nabla f(x_R)\|^2] \le \epsilon$.
• For convex problems, it attains an expected convergence rate of $\mathcal{O}(1/\sqrt{N})$ in terms of the function values.

The research further establishes the rate of convergence and the nearly optimal complexity bounds within a probabilistic framework.

Improvements and Variants

To enhance large-deviation properties, the two-phase randomized stochastic gradient ($2$-RSG) method is introduced. This method evaluates several independent runs of the RSG algorithm in a post-optimization phase:

• It computes an $\epsilon$-solution with a complexity of $$\mathcal{O}[(\log(1/\Lambda) \sigma^2/\epsilon)(1/\epsilon + \log(1/\Lambda)/\Lambda)]$$.
• This approach significantly mitigates the large deviation problem, providing more reliable assurance of convergence within specified probabilistic bounds.

Stochastic Zeroth-order Methods

In situations where only zeroth-order information is available, typically in simulation-based optimization problems, the paper introduces the randomized stochastic gradient freer (RSGF) method. This method applies a Gaussian smoothing technique to enable the approximation of first-order information from zeroth-order data:

• The complexity bound for the RSGF method in nonconvex settings is derived to be $\mathcal{O}(n/\epsilon^2)$, demonstrating robust performance in high-dimensional spaces.

Furthermore, the $2$-RSGF method, akin to its first-order counterpart, incorporates a post-optimization phase, thus improving the large-deviation properties:

• The complexity for finding an $(\epsilon, \Lambda)$-solution with $2$-RSGF is reduced to $$\mathcal{O}[(n/\epsilon^2) \log(1/\Lambda) + (n/\epsilon) \log^2(1/\Lambda)/\Lambda]$$ under certain light-tail assumptions.

Implications and Future Work

The presented methods extend the frontier for solving nonconvex SP problems with positive implications for various fields including machine learning, where regularized loss functions often exhibit nonconvexity, and simulation-based optimization where explicit gradients are inaccessible.

The theoretical advancements pave the way for further practical implementations and augmentations, such as adaptive stepsize protocols and deeper exploration of convergence under different stochastic noise distributions. Future research may investigate tailored algorithms for specific nonconvex formulations, seeking to harness the full potential of these robust SA frameworks.

In conclusion, the paper substantiates significant theoretical advancements in stochastic optimization, offering practical tools for tackling complex nonconvex problems with noisy and zeroth-order data, ensuring broad applicability and enhanced reliability in diverse scientific and engineering domains.