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Spike-timing-dependent Hebbian learning as noisy gradient descent

NeurIPS 2025 · Optimization › Implicit bias and generalization · approach: Edge-of-stability dynamics · 0 citations (OpenAlex)

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Shows that a pair-based STDP rule with spike-triggering probabilities reduces to noisy gradient descent (equivalently noisy entropic mirror descent) on a non-convex loss over the probability simplex, and proves that the output neuron aligns exponentially fast with the most active input with high probability.

Problem

Understanding the convergence of Hebbian spike-timing-dependent plasticity (STDP) with intrinsic spiking noise, which mathematical analysis has rarely covered beyond ensemble averages.

Approach

Rewrites a Hebbian STDP weight-update rule for one postsynaptic neuron as noisy gradient descent on a non-convex loss L(p) = -1/3 sum p_i^3 + 1/4 ||p||^4 over the probability simplex; under a constant learning rate, the noise is controlled by a martingale/Doob argument, giving linear convergence to the input with highest firing rate on an event of high probability.

Key results

• Theorem: with a constant learning rate, E[||p(k)-e_1||_1 1_Theta] decays exponentially in k, and Theta has probability at least 1-epsilon/2 for sufficiently small alpha.
• Gradient flow for the continuous-time version converges linearly to e_1 under p_1(0) >= max_{i>=2} p_i(0) + Delta.
• Simulations for d=3 show that the multi-output-neuron Algorithm (Sanger-like) aligns output neurons with inputs, with plateaus at Frobenius error 1, 2 indicating misalignments, mitigated by smaller learning rates.

Measured effects reported in the paper

Surprising findings

• Despite constant injected noise and non-convexity, the dynamics converge to a deterministic limit with a constant learning rate, contrary to typical fixed-noise noisy SGD behaviour.
• Simulated multi-neuron algorithm can plateau with one or more output neurons aligned with the wrong input.

Limitations

• Analysis restricted to a single postsynaptic neuron with a one-layer network of excitatory inputs; multi-output case lacks full theory.
• Weights grow unboundedly under the learning rule (weight explosion), requiring a separate homeostatic phase.
• Derivation relies on approximations: delayed weight updates, discarding O(alpha^2) and e^{t_k - t_{k+1}} terms, and approximate spike-triggering probabilities.

Experimental setup

Metrics: L1 distance to e_1 Frobenius error of probability matrix

Builds on: Hebbian learning noisy gradient descent for non-convex losses entropic mirror descent Sanger's rule

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