Tight Bounds for Blind Search on the Integers

TL;DR

This paper establishes tight bounds on the expected time for a random process to reach zero in a bounded interval, using optimal probability distributions, with implications for blind optimization strategies.

Contribution

It introduces a novel potential function method to derive tight bounds for the expected hitting time in a blind search process on integers.

Findings

Expected time is Θ((log n)^2) for optimal distribution.

Introduces a new potential function technique for analysis.

Results inform blind optimization of continuous functions.

Abstract

We analyze a simple random process in which a token is moved in the interval $A=\{0,...,n\$: Fix a probability distribution $\mu$ over $\{1,...,n\$. Initially, the token is placed in a random position in $A$. In round $t$, a random value $d$ is chosen according to $\mu$. If the token is in position $a\geq d$, then it is moved to position $a-d$. Otherwise it stays put. Let $T$ be the number of rounds until the token reaches position 0. We show tight bounds for the expectation of $T$ for the optimal distribution $\mu$. More precisely, we show that $\min_\mu\{E_\mu(T)\=\Theta((\log n)^2)$. For the proof, a novel potential function argument is introduced. The research is motivated by the problem of approximating the minimum of a continuous function over $[0,1]$ with a ``blind'' optimization strategy.