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Any-Dimensional Polynomial Optimization via de Finetti Theorems

Eitan Levin, Venkat Chandrasekaran

TL;DR

The paper develops a systematic framework to bound the limiting value $u_\infty$ of any-dimensional polynomial optimization problems by reducing to finite-dimensional POPs via a generalized de Finetti theory linked to representation stability. It formalizes freely-described and freely-symmetrized POPs on consistent dimension sequences and proves two generalized de Finetti theorems (coFS and dual) that connect the high-dimensional costs to lower-dimensional, tractable certificates. The main contribution is a hierarchy of convergent lower bounds $\ell_n$ with explicit rates (e.g., $u_\infty-\ell_n$ decays as $O(1/n)$ in many cases), enabling SOS-free certificates and practical numerics in applications like mean-field games, symmetric function inequalities, and graph densities. The work unifies probabilistic de Finetti ideas with category-theoretic representations to yield data-driven bounds and provides numerical evidence and software for reproducibility. Overall, it offers a principled, extensible path to proving dimension-spanning inequalities with certificates that are amenable to standard polynomial optimization relaxations.

Abstract

Polynomial optimization problems often arise in sequences indexed by dimension, and it is of interest to compute bounds on the optimal values of all problems in the sequence. Examples include certifying inequalities between symmetric functions or graph homomorphism densities that hold over vectors and graphs of all sizes, and computing the value of mean-field games viewed as limits of games with a growing number of players. In this paper, we study such any-dimensional polynomial problems using the theory of representation stability, and we develop a systematic framework to produce hierarchies of bounds for their limiting optimal values in terms of finite-dimensional polynomial optimization problems. In this paper, we study such any-dimensional polynomial problems using the theory of representation stability, and we develop a systematic framework to produce sequences of improving bounds on their limiting optimal values. Our bounds are obtained by solving finite-dimensional polynomial optimization problems (or their relaxations). These bounds converge at explicit rates, and they follow as a consequence of new de Finetti-type theorems pertaining to sequences of random arrays projecting onto each other in different ways. The proofs of these theorems are based on applying results from probability to representations of certain categories. We apply our framework to produce new bounds on problems arising in a number of application domains such as mean-field games, extremal graph theory, and symmetric function theory, and we illustrate our methods via numerical experiments.

Any-Dimensional Polynomial Optimization via de Finetti Theorems

TL;DR

The paper develops a systematic framework to bound the limiting value of any-dimensional polynomial optimization problems by reducing to finite-dimensional POPs via a generalized de Finetti theory linked to representation stability. It formalizes freely-described and freely-symmetrized POPs on consistent dimension sequences and proves two generalized de Finetti theorems (coFS and dual) that connect the high-dimensional costs to lower-dimensional, tractable certificates. The main contribution is a hierarchy of convergent lower bounds with explicit rates (e.g., decays as in many cases), enabling SOS-free certificates and practical numerics in applications like mean-field games, symmetric function inequalities, and graph densities. The work unifies probabilistic de Finetti ideas with category-theoretic representations to yield data-driven bounds and provides numerical evidence and software for reproducibility. Overall, it offers a principled, extensible path to proving dimension-spanning inequalities with certificates that are amenable to standard polynomial optimization relaxations.

Abstract

Polynomial optimization problems often arise in sequences indexed by dimension, and it is of interest to compute bounds on the optimal values of all problems in the sequence. Examples include certifying inequalities between symmetric functions or graph homomorphism densities that hold over vectors and graphs of all sizes, and computing the value of mean-field games viewed as limits of games with a growing number of players. In this paper, we study such any-dimensional polynomial problems using the theory of representation stability, and we develop a systematic framework to produce hierarchies of bounds for their limiting optimal values in terms of finite-dimensional polynomial optimization problems. In this paper, we study such any-dimensional polynomial problems using the theory of representation stability, and we develop a systematic framework to produce sequences of improving bounds on their limiting optimal values. Our bounds are obtained by solving finite-dimensional polynomial optimization problems (or their relaxations). These bounds converge at explicit rates, and they follow as a consequence of new de Finetti-type theorems pertaining to sequences of random arrays projecting onto each other in different ways. The proofs of these theorems are based on applying results from probability to representations of certain categories. We apply our framework to produce new bounds on problems arising in a number of application domains such as mean-field games, extremal graph theory, and symmetric function theory, and we illustrate our methods via numerical experiments.

Paper Structure

This paper contains 59 sections, 21 theorems, 130 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A First Illustration
  3. Our Contributions
  4. Formalizing any-dimensional POPs (Section \ref{['sec:any_dim_poly_probs']}).
  5. Generalizing de Finetti (Section \ref{['sec:deFinetti']}).
  6. Lower Bounds on Any-Dimensional POPs (Section \ref{['sec:free_poly_opts']}).
  7. Applications (Section \ref{['sec:construct_bds']}).
  8. Paper Organization.
  9. Related Work
  10. Polynomial Optimization.
  11. Representation Stability.
  12. Exchangeable Arrays.
  13. Graphons and Flag Algebras.
  14. Background on Representation Stability
  15. Consistent Sequences
  16. ...and 44 more sections

Key Result

Theorem 1.8

Consider any sequence of random vectors $(X^{(n)}\in\mathbb{R}^n)_n$ projecting onto each other in the following sense Any such sequence can be approximated arbitrarily well by sequences constructed as follows: For a finite random vector $Y\in\mathbb{R}^d$, and a uniformly random map $F_{n,d}\colon[d]\to[n]$ that is independent of $Y$, construct the sequence $(\tau(F_{n,d})Y\in\mathbb{R}^n)_{n\in

Figures (3)

  • Figure 1: Numerical results for mean-field game in Example \ref{['ex:mfg']}.
  • Figure 2: Example of graph quotient. Note that quotients of a simple graph need not be simple.
  • Figure 3: All refinements of $K_3$. Note that refinements of a simple graph are simple.

Theorems & Definitions (79)

  • Example 1.1: Inequalities in moments
  • Example 1.2: Inequalities in symmetric functions
  • Example 1.3: Inequalities in graph densities
  • Example 1.4: Mean-field games
  • Example 1.5: Symmetric functions
  • Example 1.6: Graph densities
  • Example 1.7: Graph numbers
  • Theorem 1.8: informal, special case; see Theorem \ref{['thm:dual_deFin_general']}
  • Theorem 1.9: informal, special case; see Theorem \ref{['thm:free_sym_bds']}
  • Definition 2.1: Consistent sequence
  • ...and 69 more