A Fixed Point Theorem for Iterative Random Contraction Operators over Banach Spaces
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Consider a contraction operator T over a Banach space X with a fixed point x^. Assume that one can approximate the operator T by a random operator T̂^N using N∈ independent and identically distributed samples of a random variable. Consider the sequence (X̂^N_k)_k∈, which is generated by X̂^N_k+1 = T̂^N(X̂^N_k) and is a random sequence. In this paper, we prove that under certain conditions on the random operator, (i) the distribution of X̂^N_k converges to a unit mass over x^ as k and N goes to infinity, and (ii) the probability that X̂^N_k is far from x^ as k goes to infinity can be made arbitrarily small by an appropriate choice of N. We also find a lower bound on the probability that X̂^N_k is far from x^ as k→∞. We apply the result to study probabilistic convergence of certain randomized optimization and value iteration algorithms.
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