Any-Dimensional Polynomial Optimization via de Finetti Theorems

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 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 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.