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Convergence rates for the moment-SoS hierarchy

Corbinian Schlosser, Matteo Tacchi, Alexey Lazarev

TL;DR

This paper develops a comprehensive framework to quantify convergence rates of the moment-SoS hierarchy when applied to generalized moment problems (GMPs), extending results from static polynomial optimization to broader GMP settings. The authors synthesize three core ingredients—an effective Putinar Positivstellensatz with explicit degree bounds, quantitative polynomial approximation, and an inward-pointing (Slater-type) condition—to produce explicit level bounds and convergence rates for the SOS relaxations. They demonstrate the method on dynamical-systems problems (optimal control and exit locations) and volume computation, achieving significantly improved rates over prior double-log bounds and deriving polynomial or near-polynomial rates under suitable regularity. The work offers practical guidelines for selecting relaxation levels to reach a desired accuracy and provides insights into how Stokes constraints and problem structure can further boost convergence, with broad applicability to GMP-driven applications in control, stochastic analysis, and geometry.

Abstract

We introduce a comprehensive framework for analyzing convergence rates for infinite dimensional linear programming problems (LPs) within the context of the moment-sum-of-squares hierarchy. Our primary focus is on extending the existing convergence rate analysis, initially developed for static polynomial optimization, to the more general and challenging domain of the generalized moment problem. We establish an easy-to-follow procedure for obtaining convergence rates. Our methodology is based on, firstly, a state-of-the-art degree bound for Putinar's Positivstellensatz, secondly, quantitative polynomial approximation bounds, and, thirdly, a geometric Slater condition on the infinite dimensional LP. We address a broad problem formulation that encompasses various applications, such as optimal control, volume computation, and exit location of stochastic processes. We illustrate the procedure at these three problems and, using a recent improvement on effective versions of Putinar's Positivstellensatz, we improve existing convergence rates.

Convergence rates for the moment-SoS hierarchy

TL;DR

This paper develops a comprehensive framework to quantify convergence rates of the moment-SoS hierarchy when applied to generalized moment problems (GMPs), extending results from static polynomial optimization to broader GMP settings. The authors synthesize three core ingredients—an effective Putinar Positivstellensatz with explicit degree bounds, quantitative polynomial approximation, and an inward-pointing (Slater-type) condition—to produce explicit level bounds and convergence rates for the SOS relaxations. They demonstrate the method on dynamical-systems problems (optimal control and exit locations) and volume computation, achieving significantly improved rates over prior double-log bounds and deriving polynomial or near-polynomial rates under suitable regularity. The work offers practical guidelines for selecting relaxation levels to reach a desired accuracy and provides insights into how Stokes constraints and problem structure can further boost convergence, with broad applicability to GMP-driven applications in control, stochastic analysis, and geometry.

Abstract

We introduce a comprehensive framework for analyzing convergence rates for infinite dimensional linear programming problems (LPs) within the context of the moment-sum-of-squares hierarchy. Our primary focus is on extending the existing convergence rate analysis, initially developed for static polynomial optimization, to the more general and challenging domain of the generalized moment problem. We establish an easy-to-follow procedure for obtaining convergence rates. Our methodology is based on, firstly, a state-of-the-art degree bound for Putinar's Positivstellensatz, secondly, quantitative polynomial approximation bounds, and, thirdly, a geometric Slater condition on the infinite dimensional LP. We address a broad problem formulation that encompasses various applications, such as optimal control, volume computation, and exit location of stochastic processes. We illustrate the procedure at these three problems and, using a recent improvement on effective versions of Putinar's Positivstellensatz, we improve existing convergence rates.
Paper Structure (32 sections, 38 theorems, 226 equations, 3 figures)

This paper contains 32 sections, 38 theorems, 226 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Context.
  3. Contribution.
  4. Outline.
  5. Preliminaries: the moment-SoS hierarchy
  6. Basic notations
  7. (Effective) Putinar's Positivstellensatz and polynomial operators
  8. Generalized Moment Problem
  9. The moment-SoS hierarchy
  10. The general case
  11. Method for convergence rates computation
  12. General method and function approximation
  13. Polynomial approximation (finding $w_\varepsilon$)
  14. Inward-pointing condition (finding $\tilde{w}$)
  15. Obtaining the convergence rates
  16. ...and 17 more sections

Key Result

Theorem 2.1

Let $r,n \in \mathbb{N}^\star$ be positive integers, $\mathbf{h} \in \mathbb{R}[\mathbf{x}]^r$ a family of $r$ polynomials in $n$ variables. Introduce the closed semialgebraic set ${\mathbf{X}} := \left\{\mathbf{x} \in \mathbb{R}^n \; ; \mathbf{h}(\mathbf{x}) \in \mathbb{R}_+^r\right\}$ as well as t If there exists $R \in \mathbb{Q}$ s.t. $R^2 - \mathbf{x}^\top\mathbf{x} \in \mathcal{Q}(\mathbf{h}

Figures (3)

  • Figure 1: The standard application of the moment-SoS hierarchy.
  • Figure 2: Left: Optimal control policies $u^\star(t)$ from \ref{['eq:non_smooth_ocp_u']} for initial values $y_0 \in \{-0.5,-0.1, -0.05, 0, 0.5, 0.8\}$. Right: corresponding trajectories $y(t)$.
  • Figure 3: Optimal value function $V^\star$ for \ref{['eq:OptContrNonSmooth']}.

Theorems & Definitions (101)

  • Theorem 2.1: Putinar's Positivstellensatz putinar1993positive
  • Remark 2.2: On the Archimedean condition
  • Theorem 2.3: Łojasiewicz exponent baldi2021moment
  • Theorem 2.4: Effective Putinar Positivstellensatz baldi2021moment
  • Remark 2.5: Farkas Lemma
  • Definition 2.6: Polynomial operator
  • Lemma 2.7
  • proof
  • Corollary 2.8: Action on bounded degree polynomials
  • proof
  • ...and 91 more