[Paper Review] A variational approach to stochastic minimization of convex functionals

Miroslav Bačák|arXiv (Cornell University)|May 11, 2016

Optimization and Variational Analysis25 references3 citations

TL;DR

This paper introduces a variational stochastic proximal point algorithm for minimizing convex integral functionals in locally compact Hadamard spaces, extending the stochastic proximal point method beyond Euclidean spaces. It establishes convergence under weaker growth conditions than prior work, enabling applications in non-smooth and non-Euclidean settings such as phylogenetic tree space, with closed-form resolvents for key problems like median estimation and least squares.

ABSTRACT

Stochastic methods for minimizing a convex integral functional, as initiated by Robbins and Monro in the early 1950s, rely on the evaluation of a gradient (or subgradient if the function is not smooth) and moving in the corresponding direction. In contrast, we use a variational technique resulting in an implicit stochastic minimization method, which has recently appeared in several diverse contexts. Such an approach is desirable whenever the underlying space does not have a differentiable structure and moreover it exhibits better stability properties which makes it preferable even in linear spaces. Our results are formulated in locally compact Hadamard spaces, but they are new even in Euclidean space, the main novelty being more general growth conditions on the functional. We verify that the assumptions of our convergence theorem are satisfied in a few classical minimization problems.

Motivation & Objective

• To develop a stochastic minimization method for convex integral functionals in locally compact Hadamard spaces, where traditional gradient-based methods fail due to lack of differentiable structure.
• To extend the stochastic proximal point algorithm (PPA) to general convex functionals with weaker growth conditions than existing literature, particularly in non-Euclidean spaces.
• To demonstrate the practical applicability of the method in classical optimization problems (e.g., median, least squares) and in advanced statistical models like phylogenetic inference in tree space.
• To prove convergence of the stochastic PPA under minimal assumptions, ensuring robustness and stability even in ill-posed or high-dimensional settings.
• To provide closed-form expressions for resolvents of marginal functions in key examples, enabling efficient implementation in practice.

Proposed method

• The method employs a variational formulation based on resolvent mappings: $ x_i = J_{\lambda_i}^{\xi_i} x_{i-1} $, where $ J_{\lambda}^{\xi}x $ minimizes $ f(y,\xi) + \frac{1}{2\lambda}d(x,y)^2 $.
• The algorithm uses independent random samples $ \xi_i \sim \mu $ to iteratively update the current estimate via implicit minimization, avoiding gradient computation.
• Convergence is established under weaker growth conditions on $ f(\cdot,\xi) $, including sublinear and quadratic growth, which exclude only pathological cases.
• The framework applies to both Euclidean and non-Euclidean Hadamard spaces, including CAT(0) cubical complexes like BHV tree space.
• The resolvent of each marginal function $ f(\cdot,\xi) $ is computed in closed form in key examples, such as $ \|x - b\| $ and $ \frac{1}{2}(\langle a,x\rangle - b)^2 $.
• The method is validated via theoretical convergence theorems and applied to real-world problems, including median estimation and Bayesian phylogenetic inference.

Experimental results

Research questions

• RQ1 Can the stochastic proximal point algorithm be extended to convex integral functionals in Hadamard spaces with weaker growth conditions than previously assumed?
• RQ2 Does the variational approach yield better stability and convergence than stochastic gradient methods in non-smooth or non-Euclidean settings?
• RQ3 Can the resolvents of marginal functions be computed in closed form for classical optimization problems such as median and least squares estimation?
• RQ4 Is the algorithm applicable to complex statistical models, such as posterior mean/median estimation in tree space for phylogenetic inference?
• RQ5 How do the new growth conditions compare to existing assumptions (e.g., uniform Lipschitz continuity) in terms of applicability to real-world problems?

Key findings

• The stochastic PPA converges to a minimizer of $ F(x) = \int_S f(x,\xi) d\mu(\xi) $ under weaker growth conditions than in prior works, including cases where marginal functions are not uniformly Lipschitz.
• The method is applicable even in non-Euclidean Hadamard spaces, such as BHV tree space, where gradient-based methods are inapplicable due to lack of differentiable structure.
• For median estimation ($ F(x) = \mathbb{E}\|x - b\| $), the resolvent is computable in closed form, enabling efficient implementation.
• In least squares problems ($ F(x) = \frac{1}{2}\mathbb{E}(\langle a,x\rangle - b)^2 $), the resolvent admits a simple analytical expression: $ J_\lambda^\xi x = x - \lambda \frac{\langle a,x\rangle - b}{1 + \lambda\|a\|^2}a $.
• The framework applies to regularization: minimizing $ F + \mu\|\cdot\|^2 $ remains tractable with closed-form resolvents, supporting overfitting control in statistical learning.
• The method enables direct minimization of posterior medians and means in tree space, as in phylogenetic inference, where resolvents are computable and convergence is guaranteed under the new conditions.