Spike-timing-dependent Hebbian learning as noisy gradient descent

TL;DR

This paper establishes a rigorous connection between Hebbian spike-timing-dependent plasticity and noisy gradient descent, showing that the learning rule can reliably identify the most active neuron with exponential convergence despite noise.

Contribution

It provides a novel theoretical framework linking Hebbian learning to noisy gradient descent and proves exponential convergence in a non-convex setting.

Findings

Hebbian learning corresponds to noisy gradient descent on a non-convex loss.

The learning rule identifies the most active neuron exponentially fast.

Convergence occurs despite constant noise injection, which is unusual.

Abstract

Hebbian learning is a key principle underlying learning in biological neural networks. We relate a Hebbian spike-timing-dependent plasticity rule to noisy gradient descent with respect to a non-convex loss function on the probability simplex. Despite the constant injection of noise and the non-convexity of the underlying optimization problem, one can rigorously prove that the considered Hebbian learning dynamic identifies the presynaptic neuron with the highest activity and that the convergence is exponentially fast in the number of iterations. This is non-standard and surprising as typically noisy gradient descent with fixed noise level only converges to a stationary regime where the noise causes the dynamic to fluctuate around a minimiser.