G h not accessible iid sample of noisy observations h

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Unformatted text preview: i - Foundations of Machine Learning page 34 Mean Estimation Theorem: Let X be a random variable taking values in [0, 1] and let x0 , . . . , xm be i.i.d. values of X. Define µm the sequence(µm )m∈N by µm+1 = (1 − αm)￿+ αm xm ￿ with µ0 = x0 , αm ∈ [0, 1] , αm = +∞ and α2 < +∞. m m≥ 0 m≥ 0 Then, a.s µm −→ E[X ]. − Proof: By the independence assumption, for m ≥ 0 , • Var[µm+1 ] = (1 − αm )2 Var[µm ] + α2 Var[xm ] m ≤ (1 − αm )Var[µm ] + α2 . m ￿ We have αm → 0 since m≥0 α2 < +∞. m Mehryar Mohri - Foundations of Machine Learning page 35 Mean Estimation 0 and suppose • Let ￿ >m ≥ N Var[µ ] ≥there exists N ∈ N such that ￿ for all which implies , m . Then, for m ≥ N , Var[µm+1 ] ≤ Var[µm ] − αm ￿ + α2 , m ￿m+N ￿ m +N 2 Var[µm+N ] ≤ Var[µN ] − ￿ n=N αn + n=N αn , ￿ ￿￿ ￿ →−∞ when m→∞ contradicting Var[µm+N ] ≥ 0 . • Thus, for all N ∈ N there exists m ≥ N such that 0 Var[µm0 ] < ￿. Choose N large enough so that ∀m ≥ N, αm ≤ ￿. Then, Var[µm0 +1 ] ≤ (1 − αm0)￿ + ￿αm0= ￿. • Therefore, µ m ≤￿ Mehryar Mohri - Foundations of Machine Learning for all m ≥ m0 (L2 convergence). page 36 Notes 1 special case: αm = m . Strong law of large numbers. • Connection with stochastic approximation. Mehryar Mohri - Foundations of Machine Learning page 37 Stochastic Approximation Problem: find solution of x = H (x) with x ∈ RN while H (x) cannot be computed, e.g., H not accessible; i.i.d sample of noisy observations H (xi )+ wi , available, i ∈ [1, m] , with E[w] = 0. • • Idea: algorithm based on iterative technique: xt+1 = (1 − αt )xt + αt [H (xt ) + wt ] = xt + αt [H (xt ) + wt − xt ]. • more generally x t+1 Mehryar Mohri - Foundations of Machine Learning = xt + αt D(xt , wt ) . page 3...
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