Decentralized Stochastic Constrained Optimization via Prox-Linearization

TL;DR

This paper introduces two decentralized stochastic optimization algorithms that efficiently handle non-convex objectives with constraints, achieving optimal convergence rates with minimal communication, and demonstrates their effectiveness in ocean trajectory planning.

Contribution

The paper develops two novel decentralized algorithms for constrained non-convex stochastic optimization that require only local gradients and minimal communication, matching optimal convergence rates.

Findings

Algorithms achieve $ ilde{O}( ext{epsilon}^{-3/2})$ oracle complexity.

Numerical experiments show improved performance over existing methods.

Methods effectively handle nonlinear constraints and non-convex objectives.

Abstract

This paper studies consensus-based decentralized stochastic optimization for minimizing possibly non-convex expected objectives with convex non-smooth regularizers and nonlinear functional inequality constraints. We reformulate the constrained problem using the exact-penalty model and develop two algorithms that require only local stochastic gradients and first-order constraint information. The first method, Decentralized Stochastic Momentum-based Prox-Linear Algorithm (D-SMPL), combines constraint linearization with a prox-linear step, resulting in a linearly constrained quadratic subproblem per iteration. Building on this approach, we propose a successive convex approximation (SCA) variant, Decentralized SCA Momentum-based Prox-Linear (D-SCAMPL), which handles additional objective structure through strongly convex surrogate subproblems while still allowing infeasible initialization. Both methods incorporate recursive momentum-based gradient estimators and a consensus mechanism requiring only two communication rounds per iteration. Under standard smoothness and regularity assumptions, both algorithms achieve an oracle complexity of $\mathcal{O}(\epsilon^{-3/2})$, matching the optimal rate known for unconstrained centralized stochastic non-convex optimization. Numerical experiments on energy-optimal ocean trajectory planning corroborate the theory and demonstrate improved performance over existing decentralized baselines.