Martingale

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. ...

Series: Stochastic Approximation from First Principles · Part 22 of 22 Dvoretzky’s recursion supports two natural proof routes: $$ \Delta_{k+1} = (1-\alpha_k)\Delta_k + \beta_k\eta_k. $$One expands $\Delta_k$ and studies its expectation and variance. The other studies the conditional drift of the energy $h_k=\Delta_k^2$. The routes are complementary. The first explains why the Robbins–Monro conditions are reasonable. The second gives the almost-sure convergence result. The Shared Mechanism Both routes rely on the same ideas: ...