Bounds on Query Convergence

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)] ≥ O(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 [ ∑t τ=1(xτ − x) ] ≥ O(t). These bounds may impose practical limitations on an agent’s performance, as O(ǫ) queries are made before the queries converge to x∗ with ǫ accuracy.