[Paper Review] Primal-dual stochastic gradient method for convex programs with many functional constraints

Yangyang Xu|arXiv (Cornell University)|Feb 8, 2018

Stochastic Gradient Optimization Techniques34 references16 citations

TL;DR

This paper proposes a novel primal-dual stochastic gradient method for convex optimization with a large number of functional constraints, where only one randomly sampled constraint is evaluated per iteration. By leveraging an adaptive stochastic subgradient update based on the augmented Lagrangian and randomized dual updates, it achieves an optimal $O(1/ar{\sqrt{k}})$ convergence rate for convex problems and nearly optimal $O((\log k)/k)$ for strongly convex problems, with low per-iteration complexity.

ABSTRACT

Stochastic gradient method (SGM) has been popularly applied to solve optimization problems with objective that is stochastic or an average of many functions. Most existing works on SGMs assume that the underlying problem is unconstrained or has an easy-to-project constraint set. In this paper, we consider problems that have a stochastic objective and also many functional constraints. For such problems, it could be extremely expensive to project a point to the feasible set, or even compute subgradient and/or function value of all constraint functions. To find solutions of these problems, we propose a novel (adaptive) SGM based on the classical augmented Lagrangian function. Within every iteration, it inquires a stochastic subgradient of the objective, and a subgradient and the function value of one randomly sampled constraint function. Hence, the per-iteration complexity is low. We establish its convergence rate for convex problems and also problems with strongly convex objective. It can achieve the optimal $O(1/\sqrt{k})$ convergence rate for convex case and nearly optimal $O\big((\log k)/k\big)$ rate for strongly convex case. Numerical experiments on a sample approximation problem of the robust portfolio selection and quadratically constrained quadratic programming are conducted to demonstrate its efficiency.

Motivation & Objective

• Address the challenge of solving convex stochastic programs with a large number of functional constraints, where full projection or subgradient evaluation is computationally prohibitive.
• Develop a stochastic first-order method that maintains low per-iteration complexity despite the high number of constraints.
• Achieve optimal or near-optimal convergence rates for both convex and strongly convex problems under stochastic subgradient and function value access.
• Design an adaptive step-size strategy for the primal variable that scales with accumulated subgradient norms to improve convergence.
• Ensure convergence under minimal assumptions, including only access to stochastic subgradients of the objective and sampled constraint functions.

Proposed method

• Propose a primal-dual stochastic gradient method based on the augmented Lagrangian function, which combines the objective and constraint terms into a single saddle-point problem.
• At each iteration, sample one constraint function uniformly at random and query only its subgradient and function value, reducing per-iteration cost.
• Construct a stochastic subgradient of the augmented Lagrangian using a sampled constraint’s subgradient and function value, enabling primal update via projected subgradient descent.
• Use a diagonal preconditioning matrix $\mathbf{D}_k$ for the primal update, with two settings: constant step-size and adaptive step-size based on accumulated subgradient norms.
• Update the dual variable $\mathbf{z}$ via a randomized coordinate update rule that only modifies the component corresponding to the sampled constraint.
• Introduce an adaptive step-size strategy in $\mathbf{D}_k$ that incorporates the $\ell_2$-norm of past stochastic subgradients, improving convergence in ill-conditioned settings.

Experimental results

Research questions

• RQ1 Can a stochastic primal-dual method be designed to handle convex programs with many functional constraints while maintaining low per-iteration complexity?
• RQ2 What convergence rate can be achieved when only one constraint is sampled per iteration, and the objective and constraint subgradients are stochastic?
• RQ3 Can the method achieve the optimal $O(1/\sqrt{k})$ rate for convex problems and nearly optimal $O((\log k)/k)$ for strongly convex problems under such sampling and stochasticity?
• RQ4 How does adaptive step-size selection based on accumulated subgradient norms affect convergence in practice and theory?
• RQ5 Is the proposed method numerically efficient and scalable on large-scale problems such as robust portfolio selection and quadratically constrained quadratic programs?

Key findings

• The proposed method achieves an optimal $O(1/\sqrt{k})$ convergence rate for convex problems, matching the best-known rate for stochastic subgradient methods.
• For strongly convex problems, the method attains a nearly optimal convergence rate of $O((\log k)/k)$, which is tight up to a logarithmic factor.
• The convergence guarantees hold under standard assumptions, including bounded stochastic subgradient variance and bounded constraint function subgradients.
• The adaptive step-size strategy in $\mathbf{D}_k$ improves numerical stability and convergence speed compared to fixed step-sizes.
• Numerical experiments on robust portfolio selection and quadratically constrained quadratic programs confirm the method’s efficiency and scalability with increasing numbers of constraints.
• The method’s per-iteration cost remains low, as it only requires one sampled constraint’s subgradient and function value per iteration, making it suitable for large-scale problems.