An Optimized Franz-Parisi Criterion and its Equivalence with SQ Lower Bounds

TL;DR

This paper introduces a refined Franz-Parisi criterion that is proven to be equivalent to Statistical Query lower bounds, unifying and extending the understanding of computational hardness in various statistical inference problems.

Contribution

It proposes an optimized FP criterion and establishes its equivalence with SQ lower bounds under broad conditions, covering many models and simplifying prior derivations.

Findings

Equivalence between optimized FP criterion and SQ lower bounds established.

Unified framework simplifies derivation of known SQ lower bounds.

New SQ lower bounds derived for mixed sparse linear regression and convex truncation.

Abstract

Bandeira et al. (2022) introduced the Franz-Parisi (FP) criterion for characterizing the computational hard phases in statistical detection problems. The FP criterion, based on an annealed version of the celebrated Franz-Parisi potential from statistical physics, was shown to be equivalent to low-degree polynomial (LDP) lower bounds for Gaussian additive models, thereby connecting two distinct approaches to understanding the computational hardness in statistical inference. In this paper, we propose a refined FP criterion that aims to better capture the geometric ``overlap" structure of statistical models. Our main result establishes that this optimized FP criterion is equivalent to Statistical Query (SQ) lower bounds -- another foundational framework in computational complexity of statistical inference. Crucially, this equivalence holds under a mild, verifiable assumption satisfied by a broad class of statistical models, including Gaussian additive models, planted sparse models, as well as non-Gaussian component analysis (NGCA), single-index (SI) models, and convex truncation detection settings. For instance, in the case of convex truncation tasks, the assumption is equivalent with the Gaussian correlation inequality (Royen, 2014) from convex geometry. In addition to the above, our equivalence not only unifies and simplifies the derivation of several known SQ lower bounds -- such as for the NGCA model (Diakonikolas et al., 2017) and the SI model (Damian et al., 2024) -- but also yields new SQ lower bounds of independent interest, including for the computational gaps in mixed sparse linear regression (Arpino et al., 2023) and convex truncation (De et al., 2023).