Bounds on Query Convergence

الأصل · EN

The problem of finding an optimum using noisy evaluations of a smooth cost function arises in many contexts, including economics, business, medicine, experiment design, and foraging theory. We derive an asymptotic bound E[(xₜ - x*)²] >= O(1/sqrt(t)) on the rate of convergence of a sequence (x₀, x₁, >...) generated by an unbiased feedback process observing noisy evaluations of an unknown quadratic function maximised at x*. The bound is tight, as the proof leads to a simple algorithm which meets it. We further establish a bound on the total regret, E[sumᵢ₌₁..ₜ (xᵢ - x*)²] >= O(sqrt(t)) These bounds may impose practical limitations on an agent's performance, as O(eps-4) queries are made before the queries converge to x* with eps accuracy.