Square the Error to Turn Convergence into Energy Dissipation
Series: Stochastic Approximation from First Principles · Part 16 of 22 The entry point to Dvoretzky’s proof is not the signed error $\Delta_k$, but the squared error: $$ h_k=\Delta_k^2. $$This transforms an error that can change direction into a nonnegative energy. The proof first shows that this energy converges, then uses persistent contraction to rule out any positive limit. Why Square the Error Start from: $$ \Delta_{k+1} = (1-\alpha_k)\Delta_k + \beta_k\eta_k. $$Directly tracking $\Delta_k$ is difficult because it can be positive, negative, and repeatedly pushed across zero by noise. ...