Tools from Stochastic Calculus 3
summary
Main lemma shows the stopping time τ when ∥∇f(x_t)∥ exceeds a fixed constant has probability ≥ 1/200 when Var(f) ≥ 1/100, via equality between expected quadratic variation and Var(f) and splitting before/after τ. A pseudo‑Brownian discretization defines x_t by dx_t = σ_t dB_t with σ_ii = √(1+ξ_t(1−ξ_t)) so σ_t slows near ±1. A level‑1 inequality (Gaussian isoperimetry) bounds ‖σ_t∇f‖, Dambis‑Dubins‑Schwarz reparametrizes martingales, and a comparison of quadratic variations bootstraps to Majority Is Stablest with improved bounds under small influences.
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chapters (6)
- 0:00Martingale proof approach
- 11:24Defining stochastic processes
- 24:37Majority Stables theorem
- 36:09Level one inequality
- 55:54Dambis-Dubins-Schwarz theorem
- 1:02:01Q&A and discussion
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