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Safe Primal-Dual Optimization with a Single Smooth Constraint

Ilnura Usmanova, Kfir Yehuda Levy

TL;DR

This work tackles safe black-box optimization with a single smooth constraint by proposing SafePD, a primal–dual method that enforces safety in both primal and dual updates via controlled dual step-sizes and a safety region for primal transitions. The approach leverages Lagrangian duality, local dual smoothness and strong concavity to achieve fast convergence rates: $\tilde{O}(1/\varepsilon^2)$ for strongly convex, $\tilde{O}(1/\varepsilon^4)$ for convex, and $\tilde{O}(1/\varepsilon^6)$ for non-convex settings, with explicit sample-complexity bounds under stochastic feedback. A safe initialization and a guaranteed safe transition between iterates ensure zero safety violations throughout learning, and the method extends to multiple constraints via smoothing the max-constraint. Empirical results corroborate the theory, showing improved stability and safety over existing safe-first-order methods, especially under noise, and indicating practical relevance for robotics and autonomous systems.

Abstract

This paper addresses the problem of safe optimization under a single smooth constraint, a scenario that arises in diverse real-world applications such as robotics and autonomous navigation. The objective of safe optimization is to solve a black-box minimization problem while strictly adhering to a safety constraint throughout the learning process. Existing methods often suffer from high sample complexity due to their noise sensitivity or poor scalability with number of dimensions, limiting their applicability. We propose a novel primal-dual optimization method that, by carefully adjusting dual step-sizes and constraining primal updates, ensures the safety of both primal and dual sequences throughout the optimization. Our algorithm achieves a convergence rate that significantly surpasses current state-of-the-art techniques. Furthermore, to the best of our knowledge, it is the first primal-dual approach to guarantee safe updates. Simulations corroborate our theoretical findings, demonstrating the practical benefits of our method. We also show how the method can be extended to multiple constraints.

Safe Primal-Dual Optimization with a Single Smooth Constraint

TL;DR

This work tackles safe black-box optimization with a single smooth constraint by proposing SafePD, a primal–dual method that enforces safety in both primal and dual updates via controlled dual step-sizes and a safety region for primal transitions. The approach leverages Lagrangian duality, local dual smoothness and strong concavity to achieve fast convergence rates: for strongly convex, for convex, and for non-convex settings, with explicit sample-complexity bounds under stochastic feedback. A safe initialization and a guaranteed safe transition between iterates ensure zero safety violations throughout learning, and the method extends to multiple constraints via smoothing the max-constraint. Empirical results corroborate the theory, showing improved stability and safety over existing safe-first-order methods, especially under noise, and indicating practical relevance for robotics and autonomous systems.

Abstract

This paper addresses the problem of safe optimization under a single smooth constraint, a scenario that arises in diverse real-world applications such as robotics and autonomous navigation. The objective of safe optimization is to solve a black-box minimization problem while strictly adhering to a safety constraint throughout the learning process. Existing methods often suffer from high sample complexity due to their noise sensitivity or poor scalability with number of dimensions, limiting their applicability. We propose a novel primal-dual optimization method that, by carefully adjusting dual step-sizes and constraining primal updates, ensures the safety of both primal and dual sequences throughout the optimization. Our algorithm achieves a convergence rate that significantly surpasses current state-of-the-art techniques. Furthermore, to the best of our knowledge, it is the first primal-dual approach to guarantee safe updates. Simulations corroborate our theoretical findings, demonstrating the practical benefits of our method. We also show how the method can be extended to multiple constraints.
Paper Structure (40 sections, 26 theorems, 76 equations, 4 figures, 1 table, 3 algorithms)

This paper contains 40 sections, 26 theorems, 76 equations, 4 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Existing work
  4. Research motivation
  5. Challenge in safety of dual approaches.
  6. Our contribution
  7. Problem Formulation
  8. Feedback
  9. Assumptions
  10. Optimality criteria and sample complexity
  11. Preliminaries
  12. Duality
  13. Safety set
  14. Strongly-Convex Problem
  15. Internal algorithms
  16. ...and 25 more sections

Key Result

Lemma 4

Let Assumption def:beta hold for (problem), then $\lambda^* \leq \Lambda := \frac{\Delta_f}{\beta}$ , where $f(x)- f(x^*) \leq \Delta_f$ for all $x \in \mathcal{X}$. Additionally, $\lambda^* \leq \frac{f(x_0) - f(x^*) }{-g(x_0)}$ holds.

Figures (4)

  • Figure 1: Primal feasibility set in primal and dual spaces, for problem $\min_x f(x), \text{ s.t. }g(x)\leq 0$. For primal feasibility of dual updates, we restrict the dual step sizes. For feasibility of the primal updates, we ensure the next dual update $x_{t+1}$ lies within a safety set of $x_t$ (circle). Here, we denote $x_{\lambda_t}$ as $x_t$ for simplicity.
  • Figure 2: Primal feasibility of the dual iterates
  • Figure 3: Comparison of SafePD (SCSA, Algorithm \ref{['alg:base']}), and LB-SGD.
  • Figure 4: Comparison of SafePD (Alg. \ref{['alg:non-convex']}), and LB-SGD, on a non-convex problem.

Theorems & Definitions (26)

  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 9
  • Lemma 10
  • Lemma 11
  • Theorem 12
  • Theorem 13
  • Corollary 14
  • ...and 16 more