High-dimensional manifold of solutions in neural networks: insights from statistical physics

• URL: http://arxiv.org/abs/2309.09240v1
• Date: Sun, 17 Sep 2023 11:10:25 GMT
• Title: High-dimensional manifold of solutions in neural networks: insights from statistical physics
• Authors: Enrico M. Malatesta
• Abstract summary: I review the statistical mechanics approach to neural networks, focusing on the paradigmatic example of the perceptron architecture with binary an continuous weights. I discuss some recent works that unveiled how the zero training error configurations are geometrically arranged.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In these pedagogic notes I review the statistical mechanics approach to neural networks, focusing on the paradigmatic example of the perceptron architecture with binary an continuous weights, in the classification setting. I will review the Gardner's approach based on replica method and the derivation of the SAT/UNSAT transition in the storage setting. Then, I discuss some recent works that unveiled how the zero training error configurations are geometrically arranged, and how this arrangement changes as the size of the training set increases. I also illustrate how different regions of solution space can be explored analytically and how the landscape in the vicinity of a solution can be characterized. I give evidence how, in binary weight models, algorithmic hardness is a consequence of the disappearance of a clustered region of solutions that extends to very large distances. Finally, I demonstrate how the study of linear mode connectivity between solutions can give insights into the average shape of the solution manifold.

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