Bounds on Query Convergence
<|reference_start|>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[ (x_t - x*)^2 ] >= O(1/sqrt(t)) on the rate of convergence of a sequence (x_0, x_1, >...) 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[ sum_{i=1..t} (x_i - 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.<|reference_end|>
@article{pearlmutter2005bounds,
    title={Bounds on Query Convergence},
    author={Barak A. Pearlmutter},
    journal={arXiv preprint arXiv:cs/0511088},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0511088},
    primaryClass={cs.LG}
}