Local Online Backpropagation Based on Probabilistic Synapses: A Constructive Proof

Implementing backpropagation in physical continuous-time systems confronts the triple obstacles of signal delays, noise, and locality constraints, rendering instantaneous exact gradient descent physically unrealizable. This paper reformulates backpropagation as follows: it is recast as an equivalence relation between the expected dynamics of weights and gradient descent. Within this definitional framework, we present a constructive proof demonstrating that a spiking neural network composed of probabilistic synapses and Integrate-and-Fire neurons, with each synapse accessing only local signals, can drive the expected dynamics of effective weights to converge to the gradient descent direction via a stochastic differential equation update rule. The construction employs eligibility traces as local encoders of forward activation values, utilizes an isomorphic backward graph in conjunction with a dopamine baseline mechanism to propagate error signals, and achieves per-synapse learning under a locally fixed drift rate.