[Paper Review] When Does Hillclimbing Fail on Monotone Functions: An entropy compression argument

Johannes Lengler, Anders Martinsson+1 more|arXiv (Cornell University)|Aug 3, 2018

Metaheuristic Optimization Algorithms Research4 citations

TL;DR

This paper resolves a long-standing open question in evolutionary computation by proving that the (1+1) EA with mutation rate $ c/n $ for $ c \leq 1+\varepsilon $ optimizes all monotone pseudo-Boolean functions in $ O(n\log^2 n) $ expected steps, using an entropy compression argument inspired by Moser's algorithmic proof of the Lovász Local Lemma. The result shows no phase transition at $ c=1 $, extending the efficiency of hillclimbing beyond the previously suspected threshold.

ABSTRACT

Hillclimbing is an essential part of any optimization algorithm. An important benchmark for hillclimbing algorithms on pseudo-Boolean functions $f: \{0,1\}^n o \mathbb{R}$ are (strictly) montone functions, on which a surprising number of hillclimbers fail to be efficient. For example, the $(1+1)$-Evolutionary Algorithm is a standard hillclimber which flips each bit independently with probability $c/n$ in each round. Perhaps surprisingly, this algorithm shows a phase transition: it optimizes any monotone pseudo-boolean function in quasilinear time if $c<1$, but there are monotone functions for which the algorithm needs exponential time if $c>2.2$. But so far it was unclear whether the threshold is at $c=1$. In this paper we show how Moser's entropy compression argument can be adapted to this situation, that is, we show that a long runtime would allow us to encode the random steps of the algorithm with less bits than their entropy. Thus there exists a $c_0 > 1$ such that for all $0

Motivation & Objective

• To resolve the open question of whether a phase transition occurs at $ c=1 $ in the runtime of the (1+1) EA on monotone functions.
• To extend the known quasilinear runtime regime beyond $ c<1 $, showing efficiency persists for $ c>1 $.
• To apply Moser's entropy compression argument to evolutionary computation, demonstrating its power in analyzing randomized search heuristics.
• To prove that the expected number of search points visited is $ O(n) $, even under adversarial initialization.
• To establish that the runtime remains efficient for all $ c \leq 1+\varepsilon $, with $ \varepsilon>0 $, for any strictly monotone function.

Proposed method

• Adapts Moser's entropy compression argument to the setting of randomized hillclimbing on monotone functions.
• Uses information-theoretic contradiction: a long runtime would allow compressing the random trajectory below its entropy, which is impossible.
• Defines phases based on the number of zero-bits, from $ 2^k $ to $ 2^{k-1} $, to analyze progress per phase.
• Applies conditional expectation bounds to show that the expected number of updates per phase is $ O(2^k \log n) $.
• Uses Markov chain analysis on the number of zero-bits to bound the expected time to reduce the number of zeros.
• Employs a coupling argument to show that conditioning on survival (no bit flip to zero) does not increase the expected number of bit flips.

Experimental results

Research questions

• RQ1 Does the (1+1) EA exhibit a phase transition in runtime at $ c=1 $ when optimizing monotone functions?
• RQ2 Can entropy compression techniques be applied to prove efficient convergence of randomized search heuristics on monotone functions?
• RQ3 Is the runtime quasilinear for mutation rates $ c>1 $, or does it become exponential?
• RQ4 What is the largest $ c>1 $ for which the (1+1) EA still achieves $ O(n\log^2 n) $ expected runtime on all monotone functions?
• RQ5 Does the algorithm remain efficient even when the initial search point is chosen adversarially?

Key findings

• There exists $ \varepsilon>0 $ such that for all $ 0<c\leq 1+\varepsilon $, the (1+1) EA optimizes any monotone function in $ O(n\log^2 n) $ expected steps.
• The expected number of search points visited is $ O(n) $, even if the initial point is chosen adversarially.
• The runtime bound holds uniformly for all monotone functions, including the so-called HotTopic functions that are known to be hard for $ c>2.2 $.
• The entropy compression argument successfully rules out long runtimes by showing that such trajectories could be compressed below their entropy, a contradiction.
• The result implies that no phase transition occurs at $ c=1 $, contradicting earlier speculation in the literature.
• The analysis shows that the expected number of update steps per phase is $ O(2^k \log n) $, and since there are $ O(\log n) $ phases, the total runtime is $ O(n\log^2 n) $.