Fall 2020

Low-Degree Hardness of Random Optimization Problems

Tuesday, December 15th, 2020 10:30 am11:15 am

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Alex Wein (New York University)

In high-dimensional statistical problems (including planted clique, sparse PCA, community detection, etc.), the class of "low-degree polynomial algorithms" captures many leading algorithmic paradigms such as spectral methods, approximate message passing, and local algorithms on sparse graphs. As such, lower bounds against low-degree algorithms constitute concrete evidence for average-case hardness of statistical problems. This method has been widely successful at explaining and predicting statistical-to-computational gaps in these settings. While prior work has understood the power of low-degree algorithms for problems with a "planted" signal, we consider here the setting of "random optimization problems" (with no planted signal), including the problem of finding a large independent set in a random graph, as well as the problem of optimizing the Hamiltonian of mean-field spin glass models. Focusing on the independent set problem, I will define low-degree algorithms in this setting, argue that they capture the best known algorithms, and explain new proof techniques that give sharp lower bounds against low-degree algorithms in this setting. The proof involves a generalization of the so-called "overlap gap property", which is a structural property of the solution space.

Based on arXiv:2004.12063 (joint with David Gamarnik and Aukosh Jagannath) and arXiv:2010.06563