The committee machine: Computational to statistical gaps in learning a two-layers neural network
This work addresses the computational-statistical gap in neural network learning, which is a foundational problem for machine learning researchers, though it is incremental as it builds on existing theoretical frameworks.
The authors rigorously justified statistical physics methods for analyzing a two-layer neural network (committee machine) and introduced an AMP algorithm for optimal learning, revealing regimes where low generalization error is information-theoretically possible but computationally infeasible, indicating a computational gap.
Heuristic tools from statistical physics have been used in the past to locate the phase transitions and compute the optimal learning and generalization errors in the teacher-student scenario in multi-layer neural networks. In this contribution, we provide a rigorous justification of these approaches for a two-layers neural network model called the committee machine. We also introduce a version of the approximate message passing (AMP) algorithm for the committee machine that allows to perform optimal learning in polynomial time for a large set of parameters. We find that there are regimes in which a low generalization error is information-theoretically achievable while the AMP algorithm fails to deliver it, strongly suggesting that no efficient algorithm exists for those cases, and unveiling a large computational gap.