[Paper Review] Stochastic inertial primal-dual algorithms

Lorenzo Rosasco, Silvia Villa+1 more|arXiv (Cornell University)|Jul 3, 2015

Sparse and Compressive Sensing Techniques40 references4 citations

TL;DR

This paper proposes a novel stochastic inertial primal-dual algorithm for solving composite optimization problems arising in machine learning, leveraging inertial dynamics and stochastic gradient estimates with convergence guarantees in a Hilbert space setting. The key contribution is a unified convergence analysis under general conditions, including bounded step-sizes and vanishing stochastic errors, with convergence of iterates to a solution for both deterministic and stochastic settings.

ABSTRACT

We propose and study a novel stochastic inertial primal-dual approach to solve composite optimization problems. These latter problems arise naturally when learning with penalized regularization schemes. Our analysis provide convergence results in a general setting, that allows to analyze in a unified framework a variety of special cases of interest. Key in our analysis is considering the framework of splitting algorithm for solving a monotone inclusions in suitable product spaces and for a specific choice of preconditioning operators.

Motivation & Objective

• To develop a stochastic and inertial extension of primal-dual methods for solving composite optimization problems in learning.
• To provide a unified convergence analysis for a broad class of optimization algorithms within a monotone inclusion framework in Hilbert spaces.
• To enable the use of stochastic gradient estimates and independent activation of proximity operators without matrix inversions.
• To establish convergence under bounded step-sizes and vanishing stochastic error conditions, improving numerical stability.
• To generalize existing primal-dual schemes by incorporating inertia and stochasticity in a single, principled framework.

Proposed method

• The method employs a stochastic inertial forward-backward splitting algorithm with preconditioning operators in a product Hilbert space setting.
• It uses inertial steps via extrapolation terms $ u_n = w_n + \alpha_n(w_n - w_{n-1}) $ and $ d_{k,n} = v_{k,n} + \alpha_n(v_{k,n} - v_{k,n-1}) $ to accelerate convergence.
• Proximity operators for regularization terms are computed independently using the resolvent of the conjugate functions: $ q_{k,n} = \text{prox}_{g_k^*}^{W_k^{-1}}(d_{k,n} + W_k D_k s_n) $.
• Stochastic gradient estimates $ \mathbf{a}_n $ are used in place of exact gradients, satisfying $ \mathbb{E}[\mathbf{a}_n | \mathcal{F}_n] = \nabla F(u_n) $ and summable error variance.
• The algorithm alternates between primal updates $ w_{n+1} = u_n - V\mathbf{a}_n - \sum_k D_k^* q_{k,n} $ and dual updates $ v_{k,n+1} = v_{k,n} + \lambda_n(q_{k,n} - v_{k,n}) $.
• Preconditioning matrices $ W_k $ and operator $ V $ are chosen to ensure convergence under the condition $ \sum_k \|W_k^{1/2} D_k V^{1/2}\|^2 < 1 $.

Experimental results

Research questions

• RQ1 Can inertial and stochastic extensions of primal-dual methods be unified under a single convergence framework?
• RQ2 Does the use of bounded step-sizes with vanishing stochastic errors lead to stable and convergent iterates in stochastic primal-dual schemes?
• RQ3 How does the inclusion of inertial terms improve convergence behavior in stochastic settings compared to standard methods?
• RQ4 Can the proposed algorithm handle non-separable regularization terms and general linear operators without requiring matrix inversions?
• RQ5 What are the conditions under which the iterates of the stochastic inertial primal-dual method converge weakly or strongly to a solution?

Key findings

• The algorithm achieves weak convergence of iterates $ w_n \rightharpoonup \overline{w} $ and $ v_{k,n} \rightharpoonup \overline{v}_k $ almost surely under the stated conditions.
• Strong convergence $ w_n \to \overline{w} $ is guaranteed if the objective function $ F $ is uniformly convex at the limit point $ \overline{w} $.
• Convergence is established under bounded step-sizes and summable stochastic error variance, which enhances numerical stability compared to vanishing step-size schemes.
• The method generalizes known primal-dual and inertial algorithms, including those in [13, 23, 9], by incorporating both stochasticity and inertia in a unified framework.
• Numerical experiments confirm convergence of the iterates and show accurate approximation of the regression function in a polynomial dictionary setting.
• The algorithm successfully solves a group lasso-type problem with non-separable penalties, achieving good regression performance with $ p=32 $, $ s=8 $, and $ N=48 $.