[Paper Review] A Randomized Block-Coordinate Primal-Dual Method for Large-scale Stochastic Saddle Point Problems

Erfan Yazdandoost Hamedani, Afrooz Jalilzadeh+1 more|arXiv (Cornell University)|Jul 8, 2019

Computational Geometry and Mesh Generation36 references4 citations

TL;DR

This paper proposes a doubly-randomized block-coordinate primal-dual algorithm for large-scale finite-sum convex-concave saddle point problems, combining randomized block updates with progressive variance reduction. It achieves an ergodic convergence rate of $\mathcal{O}(\log K / K)$, matching the best-known rate for deterministic methods, and establishes almost sure convergence to a saddle point.

ABSTRACT

We consider (stochastic) convex-concave saddle point (SP) problems with high-dimensional decision variables, arising in various applications including machine learning problems. To contend with the challenges in computing full gradients, we employ a randomized block-coordinate primal-dual scheme in which randomly selected primal and dual blocks of variables are updated. We consider both deterministic and stochastic settings, where deterministic partial gradients and their randomly sampled estimates are used, respectively, at each iteration. We investigate the convergence of the proposed method under different blocking strategies and provide the corresponding complexity results. While the best-known computational complexity result for computing a saddle point with $\varepsilon$ primal-dual gap for deterministic primal-dual methods using full gradients is $\mathcal O(\max\{m,n\}^2/\varepsilon)$, where $m$ and $n$ denote the dimensions of primal and dual variables, respectively, we show that our proposed randomized block-coordinate method achieves an improved complexity of $\mathcal O(mn/\varepsilon)$ assuming a coordinate-friendly structure on the problem. Moreover, for the stochastic setting where a mini-batch sample gradient is utilized, we show a computational complexity of $ ilde{\mathcal{O}}(m^2n^2/\varepsilon^2)$ through acceleration. Finally, almost sure convergence of the iterate sequence to a saddle point is established.

Motivation & Objective

• To address large-scale convex-concave saddle point problems arising in machine learning, such as robust classification and kernel matrix learning, with finite-sum structure.
• To develop an efficient algorithm that handles non-bilinear coupling functions and a large number of primal-dual block variables.
• To achieve deterministic convergence rates up to a logarithmic factor, even with stochastic gradient approximations.
• To establish almost sure convergence of the iterates to a saddle point under mild assumptions.
• To extend variance-reduction techniques to block-coordinate, non-bilinear saddle point problems.

Proposed method

• The algorithm uses a randomized block-coordinate scheme where at each iteration, a random pair of primal and dual blocks $(i_k, j_k)$ are selected for update.
• It employs progressive variance reduction by increasing the batch size of gradients over time, reducing variance in stochastic estimates.
• The method incorporates block-specific step-sizes $\tau_i$ and $\sigma_j$ to improve convergence and adaptivity.
• It applies a doubly-randomized framework: random selection of blocks and random sampling of component functions in the finite sum.
• The algorithm uses a proximal-like update with a Bregman divergence term to handle nonsmooth components $f_i$ and $h_j$.
• Convergence is analyzed via a Lyapunov function and a gap metric, leveraging the structure of the finite-sum, non-bilinear saddle point problem.

Experimental results

Research questions

• RQ1 Can a randomized block-coordinate primal-dual method achieve a non-asymptotic convergence rate of $\mathcal{O}(\log K / K)$ for finite-sum, non-bilinear saddle point problems?
• RQ2 Does progressive variance reduction via increasing batch sizes improve convergence in large-scale saddle point problems with block updates?
• RQ3 Can almost sure convergence of the iterates to a saddle point be established under random block selection and stochastic gradients?
• RQ4 Is the convergence rate of the proposed method comparable to deterministic primal-dual schemes, up to a logarithmic factor?
• RQ5 How do block-specific step-sizes and randomized block selection affect convergence in non-bilinear, large-scale problems?

Key findings

• The proposed algorithm achieves an ergodic convergence rate of $\mathcal{O}(\log K / K)$ using variance reduction, which is the first such rate result for a specialized method on finite-sum, non-bilinear saddle point problems.
• For the single-sample gradient case, a non-asymptotic convergence rate of $\mathcal{O}(\log K / \sqrt{K})$ is obtained, matching the best-known rate for stochastic methods.
• Almost sure convergence of the iterate sequence to a saddle point is proven under the assumption that the block selection process forms a single recurrent class in a Markov chain.
• The convergence analysis establishes that the gap metric diminishes at a rate of $\mathcal{O}(\log K / K)$, matching the best-known deterministic rates up to a logarithmic factor.
• The method is applicable to problems with a large number of constraints and non-bilinear coupling terms, such as those arising in robust classification and kernel matrix learning.
• The theoretical results are supported by a Lyapunov function and a gap-based analysis, with convergence proven under mild differentiability and convexity assumptions.