The Franz-Parisi Criterion and Computational Trade-offs in High Dimensional Statistics

Abstract

Many high-dimensional statistical inference problems are believed to possess inherent computational hardness. Various frameworks have been proposed to give rigorous evidence for such hardness, including lower bounds against restricted models of computation (such as low-degree functions), as well as methods rooted in statistical physics that are based on free energy landscapes. This paper aims to make a rigorous connection between the seemingly different low-degree and free-energy based approaches. We define a free-energy based criterion for hardness and formally connect it to the well-established notion of low-degree hardness for a broad class of statistical problems, namely all Gaussian additive models and certain models with a sparse planted signal. By leveraging these rigorous connections we are able to: establish that for Gaussian additive models the "algebraic" notion of low-degree hardness implies failure of "geometric" local MCMC algorithms, and provide new low-degree lower bounds for sparse linear regression which seem difficult to prove directly. These results provide both conceptual insights into the connections between different notions of hardness, as well as concrete technical tools such as new methods for proving low-degree lower bounds.

Disordered systems insights on computational hardness

“…As our understanding of algorithms deepens, we hope to understand the universal characteristics that make problems easy or hard, unifying larger and larger classes of polynomial-time algorithms and connecting them rigorously with physical properties of the energy landscape. Very recently, [136] connected the lowdegree likelihood ratio with the Franz-Parisi potential, adding to the evidence that free energy barriers imply computational hardness. We will know much more in a few years than we know now.…”