The Brain’s Algorithm: How Hebbian Learning and Multiscale Dynamics Solve a 70-Year-Old Convergence Problem
For decades, theorists have suspected that the brain performs dimensionality reduction not through backpropagation, but through local, biologically plausible learning rules that respect the arrow of time. A new paper published in Neural Computation finally provides the missing formal proof. The authors analyze a continuous-time neural network—the similarity matching network—derived from a min-max-min objective, whose dynamics unfold at three distinct timescales: fast neural activity, intermediate lateral synaptic plasticity, and slow feedforward synaptic learning. At each level, the cost function exhibits remarkable structure: strong convexity at the neural level, strong concavity at the lateral level, and a nonconvex, nonsmooth landscape at the feedforward level whose global minima the authors characterize explicitly. By leveraging a multilevel optimization framework, the team proves global exponential convergence for the first two levels and almost sure convergence to global minima for the third, bridging a gap between theoretical neuroscience and rigorous optimization theory that has persisted since Hebb’s original postulate.
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