Active online learning in the binary perceptron problem
This work addresses the problem of inefficient learning in neural networks for researchers in machine learning and statistical physics, offering incremental improvements in sample efficiency for a specific model.
The paper tackles the inefficiency of passive learning in the binary perceptron problem by exploring active online learning, where the student designs training patterns, and demonstrates that perfect inference can be achieved with as few as N patterns, though computationally unfeasible, and shows Bayesian protocols require 2.3N or 1.9N patterns while deductive reasoning uses N+log2N patterns.
The binary perceptron is the simplest artificial neural network formed by $N$ input units and one output unit, with the neural states and the synaptic weights all restricted to $\pm 1$ values. The task in the teacher--student scenario is to infer the hidden weight vector by training on a set of labeled patterns. Previous efforts on the passive learning mode have shown that learning from independent random patterns is quite inefficient. Here we consider the active online learning mode in which the student designs every new Ising training pattern. We demonstrate that it is mathematically possible to achieve perfect (error-free) inference using only $N$ designed training patterns, but this is computationally unfeasible for large systems. We then investigate two Bayesian statistical designing protocols, which require $2.3 N$ and $1.9 N$ training patterns, respectively, to achieve error-free inference. If the training patterns are instead designed through deductive reasoning, perfect inference is achieved using $N\!+\!\log_{2}\!N$ samples. The performance gap between Bayesian and deductive designing strategies may be shortened in future work by taking into account the possibility of ergodicity breaking in the version space of the binary perceptron.