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The binary perceptron is the simplest artificial neural network formed by $N$input units and one output unit, with the neural states and the synapticweights all restricted to $\pm 1$ values. The task in the teacher--studentscenario is to infer the hidden weight vector by training on a set of labeledpatterns. Previous efforts on the passive learning mode have shown thatlearning from independent random patterns is quite inefficient. Here weconsider the active online learning mode in which the student designs every newIsing training pattern. We demonstrate that it is mathematically possible toachieve perfect (error-free) inference using only $N$ designed trainingpatterns, but this is computationally unfeasible for large systems. We theninvestigate 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. Theperformance gap between Bayesian and deductive designing strategies may beshortened in future work by taking into account the possibility of ergodicitybreaking in the version space of the binary perceptron.

Active online learning in the binary perceptron problem