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Decentralized Stochastic Constrained Optimization via Prox-Linearization

Shivangi Dubey Sharma, Basil M. Idrees, Lavish Arora, Ketan Rajawat

TL;DR

A successive convex approximation (SCA) variant is proposed, Decentralized SCA Momentum-based Prox-Linear (D-SCAMPL), which handles additional objective structure through strongly convex surrogate subproblems while still allowing infeasible initialization.

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 $\mathcal{O}(ε^{-3/2})$, 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.

Decentralized Stochastic Constrained Optimization via Prox-Linearization

TL;DR

A successive convex approximation (SCA) variant is proposed, Decentralized SCA Momentum-based Prox-Linear (D-SCAMPL), which handles additional objective structure through strongly convex surrogate subproblems while still allowing infeasible initialization.

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 , 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.
Paper Structure (32 sections, 13 theorems, 116 equations, 6 figures, 1 table, 2 algorithms)

This paper contains 32 sections, 13 theorems, 116 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations and Organization
  3. Motivating Example
  4. Algorithm Development
  5. Challenge
  6. Updates via Partial Linearization
  7. Updates via Successive Convex Approximation
  8. Assumptions
  9. Approximate optimality
  10. Analysis of D-SMPL (Algorithm \ref{['prox_linear']})
  11. Analysis of D-SCAMPL (Algorithm \ref{['alg:D-SCAMPL']})
  12. Experimental data and results
  13. Synthetic Example
  14. Effect of $\gamma$
  15. High-accuracy regime
  16. ...and 17 more sections

Key Result

Lemma 1

Under Assumption strongslater for any ${\mathbf{x}} \notin {\mathcal{X}}$, it holds that $\left\|{\mathbf{s}}({\mathbf{x}})\right\|\geq \varrho$ for all ${\mathbf{s}}({\mathbf{x}}) \in \partial \max_k\{[g_k({\mathbf{x}})]_+\}$.

Figures (6)

  • Figure 1: Navigation in the presence of ocean currents of a formation of 4 USVs. The straight-line path is not the energy-optimal path.
  • Figure 2: Objective function $f(x)$
  • Figure 3: (a) Evolution of $\Pi^t$ for different values of $\gamma$ (b) Iteration complexity $\tilde{T}_\epsilon$ vs. $\epsilon$ in the high-accuracy regime.
  • Figure 4: (a) Plot of iteration complexity $\tilde{T}_\epsilon$ vs. $n$ and (b) Evolution of $\Pi^t$ with the iterations for different value of spectral gap $\lambda$.
  • Figure 5: (a) Energy-efficient trajectory of 4 USVs in a formation and (b) evolution of the objective function with iterations.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 1
  • Theorem 2
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 3 more