It remains to note that E[∥θk − θ⋆∥2p] ≤ 22p−1E[∥θ′ k − θ⋆∥2p] + 22p−1E[∥θk − θ′ k∥2p] ≤ C2p,1 exp − pµc0 4 (k + k0)1−γ (∥θ0 − θ⋆∥2p + σ2p 2p) + C2p,2σ2p 2pαp k

Third, for l ≥ 1 or j ≥ 2 (that is, 2l + j ≥ 2), we use Cauchy-Schwartz inequality together with A7, A1 |⟨θk−1−θ′ k−1, ∇f(θk−1)−∇f(θ′ k−1)+g(θk−1, ξk)−g(θ′ k−1, ξk)⟩j| ≤ ∥ θk−1−

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Gaussian Approximation and Multiplier Bootstrap for Stochastic Gradient Descent

stat.ML · 2025-02-10 · unverdicted · novelty 6.0

Proves the first fully non-asymptotic bound on the accuracy of multiplier bootstrap for constructing confidence sets from Polyak-Ruppert SGD iterates, achieving convex-distance rates up to 1/sqrt(n) under regularity conditions.

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Gaussian Approximation and Multiplier Bootstrap for Stochastic Gradient Descent stat.ML · 2025-02-10 · unverdicted · none · ref 62
Proves the first fully non-asymptotic bound on the accuracy of multiplier bootstrap for constructing confidence sets from Polyak-Ruppert SGD iterates, achieving convex-distance rates up to 1/sqrt(n) under regularity conditions.

It remains to note that E[∥θk − θ⋆∥2p] ≤ 22p−1E[∥θ′ k − θ⋆∥2p] + 22p−1E[∥θk − θ′ k∥2p] ≤ C2p,1 exp − pµc0 4 (k + k0)1−γ (∥θ0 − θ⋆∥2p + σ2p 2p) + C2p,2σ2p 2pαp k

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