Optimal storage capacity of neural networks at finite temperatures

Abstract

Gardner's analysis of the optimal storage capacity of neural networks is extended to study finite-temperature effects. The typical volume of the space of interactions is calculated for strongly-diluted networks as a function of the storage ratio $α$, temperature $T$, and the tolerance parameter $m$, from which the optimal storage capacity $α_c$ is obtained as a function of $T$ and $m$. At zero temperature it is found that $α_c = 2$ regardless of $m$ while $α_c$ in general increases with the tolerance at finite temperatures. We show how the best performance for given $α$ and $T$ is obtained, which reveals a first-order transition from high-quality performance to low-quality one at low temperatures. An approximate criterion for recalling, which is valid near $m=1$, is also discussed.