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Generalization is a central aspect of learning theory. Here, we propose aframework that explores an auxiliary task-dependent notion of generalization,and attempts to quantitatively answer the following question: given two sets ofpatterns with a given degree of dissimilarity, how easily will a network beable to "unify" their interpretation? This is quantified by the volume of theconfigurations of synaptic weights that classify the two sets in a similarmanner. To show the applicability of our idea in a concrete setting, we computethis quantity for the perceptron, a simple binary classifier, using theclassical statistical physics approach in the replica-symmetric ansatz. In thiscase, we show how an analytical expression measures the "distance-basedcapacity", the maximum load of patterns sustainable by the network, at fixeddissimilarity between patterns and fixed allowed number of errors. This curveindicates that generalization is possible at any distance, but with decreasingcapacity. We propose that a distance-based definition of generalization may beuseful in numerical experiments with real-world neural networks, and to explorecomputationally sub-dominant sets of synaptic solutions.

Generalization from correlated sets of patterns in the perceptron