[Paper Review] A Stochastic Successive Minimization Method for Nonsmooth Nonconvex Optimization with Applications to Transceiver Design in Wireless Communication Networks

Meisam Razaviyayn, Maziar Sanjabi+1 more|arXiv (Cornell University)|Jul 17, 2013

Sparse and Compressive Sensing Techniques47 references14 citations

TL;DR

This paper proposes the Stochastic Successive Upper-bound Minimization (SSUM) method for nonsmooth nonconvex optimization under expectation, where at each iteration a locally tight upper-bound approximation of the expected cost is minimized instead of the full sample average. The method ensures convergence and enables efficient online computation, with applications to wireless transceiver design and online dictionary learning, significantly outperforming existing methods in ergodic sum rate and extending stochastic gradient methods to nonsmooth nonconvex settings with convergence guarantees.

ABSTRACT

Consider the problem of minimizing the expected value of a cost function parameterized by a random variable. The classical sample average approximation (SAA) method for solving this problem requires minimization of an ensemble average of the objective at each step, which can be expensive. In this paper, we propose a stochastic successive upper-bound minimization method (SSUM) which minimizes an approximate ensemble average at each iteration. To ensure convergence and to facilitate computation, we require the approximate ensemble average to be a locally tight upper-bound of the expected cost function and be easily optimized. The main contributions of this work include the development and analysis of the SSUM method as well as its applications in linear transceiver design for wireless communication networks and online dictionary learning. Moreover, using the SSUM framework, we extend the classical stochastic (sub-)gradient (SG) method to the case of minimizing a nonsmooth nonconvex objective function and establish its convergence.

Motivation & Objective

• To address the high computational cost of the classical Sample Average Approximation (SAA) method in solving stochastic optimization problems with nonconvex and nonsmooth objectives.
• To develop a stochastic successive upper-bound minimization (SSUM) framework that replaces exact SAA subproblems with easily optimized upper-bound approximations at each iteration.
• To establish convergence of the SSUM method under mild assumptions, including locally tight upper-bounds and bounded gradients.
• To extend the classical stochastic (sub-)gradient method to nonsmooth nonconvex problems and prove its convergence using the SSUM framework.
• To demonstrate practical efficacy through applications in wireless communication network transceiver design and online dictionary learning.

Proposed method

• The SSUM method minimizes an approximate ensemble average at each iteration, where the approximation function $ \hat{g}(x, x^{i-1}, \xi^i) $ is a locally tight upper-bound of $ g(x, \xi^i) $ around the previous iterate $ x^{i-1} $.
• The algorithm updates the variable via $ x^r \leftarrow \arg\min_{x \in \mathcal{X}} \frac{1}{r} \sum_{i=1}^r \hat{g}(x, x^{i-1}, \xi^i) $, ensuring descent and convergence.
• Convergence is established by proving that the gap between the approximate and true objective functions vanishes as $ r \to \infty $, with $ \alpha^r = \hat{f}^r(x^r) - f^r(x^r) \to 0 $.
• The method leverages assumptions on Lipschitz continuity of gradients and boundedness of the approximation error to ensure $ \|x^{r+1} - x^r\| = \mathcal{O}(1/r) $.
• The framework generalizes the stochastic gradient method by allowing nonsmooth and nonconvex objectives, with convergence proven via a novel analysis of the approximation gap.
• Theoretical convergence relies on bounding the difference between successive iterates and using a telescoping argument on the approximation error.

Experimental results

Research questions

• RQ1 Can a stochastic successive upper-bound minimization method be designed to reduce computational cost in solving nonsmooth nonconvex stochastic optimization problems?
• RQ2 Does the SSUM method converge to a stationary point under mild assumptions, even when the objective function is nonconvex and nonsmooth?
• RQ3 Can the SSUM framework be used to extend the classical stochastic gradient method to nonsmooth nonconvex problems with convergence guarantees?
• RQ4 How does the performance of SSUM compare to existing algorithms in practical applications like wireless transceiver design and online dictionary learning?
• RQ5 What conditions ensure that the approximation error between the upper-bound and the true objective vanishes over iterations?

Key findings

• The SSUM method achieves convergence to a stationary point under mild assumptions, including locally tight upper-bounds and bounded gradients.
• The approximation error $ \alpha^r = \hat{f}^r(x^r) - f^r(x^r) $ vanishes as $ r \to \infty $, ensuring the algorithm tracks the true expected objective.
• The iterates satisfy $ \|x^{r+1} - x^r\| = \mathcal{O}(1/r) $, indicating diminishing step-sizes in the iterates.
• The classical stochastic gradient method is shown to be a special case of SSUM, and the framework extends it to nonsmooth nonconvex problems with convergence proof.
• Numerical results show that SSUM significantly outperforms existing algorithms in achieving higher ergodic sum rate in wireless beamforming networks.
• The method is successfully applied to online dictionary learning, demonstrating its practical utility in large-scale data-driven optimization.