Hardest Monotone Functions for Evolutionary Algorithms

• URL: http://arxiv.org/abs/2311.07438v1
• Date: Mon, 13 Nov 2023 16:13:30 GMT
• Title: Hardest Monotone Functions for Evolutionary Algorithms
• Authors: Marc Kaufmann and Maxime Larcher and Johannes Lengler and Oliver Sieberling
• Abstract summary: We conjecture that for the self-adjusting $(1,lambda)$-EA, Adrial Dynamic BinVal (ADBV) is the hardest dynamic monotone function to optimize. We introduce the function Switching Dynamic BinVal (SDBV) which coincides with ADBV whenever the number of remaining zeros in the search point is strictly less than $n/2$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: The study of hardest and easiest fitness landscapes is an active area of research. Recently, Kaufmann, Larcher, Lengler and Zou conjectured that for the self-adjusting $(1,\lambda)$-EA, Adversarial Dynamic BinVal (ADBV) is the hardest dynamic monotone function to optimize. We introduce the function Switching Dynamic BinVal (SDBV) which coincides with ADBV whenever the number of remaining zeros in the search point is strictly less than $n/2$, where $n$ denotes the dimension of the search space. We show, using a combinatorial argument, that for the $(1+1)$-EA with any mutation rate $p \in [0,1]$, SDBV is drift-minimizing among the class of dynamic monotone functions. Our construction provides the first explicit example of an instance of the partially-ordered evolutionary algorithm (PO-EA) model with parameterized pessimism introduced by Colin, Doerr and F\'erey, building on work of Jansen. We further show that the $(1+1)$-EA optimizes SDBV in $\Theta(n^{3/2})$ generations. Our simulations demonstrate matching runtimes for both static and self-adjusting $(1,\lambda)$ and $(1+\lambda)$-EA. We further show, using an example of fixed dimension, that drift-minimization does not equal maximal runtime.

Related papers

• Proven Runtime Guarantees for How the MOEA/D Computes the Pareto Front From the Subproblem Solutions [9.044970217182117]
  decomposition-based multi-objective evolutionary algorithm (MOEA/D) does not directly optimize a given multi-objective function $f$, but instead optimize $N + 1$ single-objective subproblems of $f$ in a co-evolutionary manner. We analyze for the first time how the MOEA/D with only standard mutation operators computes the whole Pareto front of the OneMinMax benchmark. Our overall bound for power-law suggests that the MOEA/D performs best for $N = O(nbeta - 1)$, resulting in an $O(n
  arXiv  Detail & Related papers  (2024-05-02T05:32:19Z)
• Kernelized Normalizing Constant Estimation: Bridging Bayesian Quadrature and Bayesian Optimization [51.533164528799084]
  We show that to estimate the normalizing constant within a small relative error, the level of difficulty depends on the value of $lambda$. We find that this pattern holds true even when the function evaluations are noisy.
  arXiv  Detail & Related papers  (2024-01-11T07:45:09Z)
• Statistical Learning under Heterogeneous Distribution Shift [71.8393170225794]
  Ground-truth predictor is additive $mathbbE[mathbfz mid mathbfx,mathbfy] = f_star(mathbfx) +g_star(mathbfy)$.
  arXiv  Detail & Related papers  (2023-02-27T16:34:21Z)
• Improved Dynamic Regret for Online Frank-Wolfe [54.690867216880356]
  We investigate the dynamic regret of online Frank-Wolfe (OFW), which is an efficient projection-free algorithm for online convex optimization. In this paper, we derive improved dynamic regret bounds for OFW by extending the fast convergence rates of FW from offline optimization to online optimization.
  arXiv  Detail & Related papers  (2023-02-11T07:19:51Z)
• Hard Problems are Easier for Success-based Parameter Control [0.0]
  Self-adjusting parameters in evolutionary algorithms can match or outperform the best fixed parameters on discrete problems. We show that self-adjustment works as intended in the absence of easy slopes. We discuss implications for the everywhere hard function LeadingOnes and a new class OneMaxBlocks of everywhere hard functions with tunable difficulty.
  arXiv  Detail & Related papers  (2022-04-12T13:56:29Z)
• Self-adjusting Population Sizes for the $(1, \lambda)$-EA on Monotone Functions [7.111443975103329]
  We study the $(1,lambda)$-EA with mutation rate $c/n$ for $cle 1$, where the population size is adaptively controlled with the $(1:s+1)$-success rule. We show that this setup with $c=1$ is efficient on onemax for $s1$, but inefficient if $s ge 18$.
  arXiv  Detail & Related papers  (2022-04-01T15:46:12Z)
• Private Stochastic Convex Optimization: Optimal Rates in $\ell_1$ Geometry [69.24618367447101]
  Up to logarithmic factors the optimal excess population loss of any $(varepsilon,delta)$-differently private is $sqrtlog(d)/n + sqrtd/varepsilon n.$ We show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $sqrtlog(d)/n + (log(d)/varepsilon n)2/3.
  arXiv  Detail & Related papers  (2021-03-02T06:53:44Z)
• Runtime analysis of the (mu+1)-EA on the Dynamic BinVal function [0.0]
  We study evolutionary algorithms in a dynamic setting, where for each generation a different fitness function is chosen. In particular, the hardest region for optimization is NOT around the optimum.
  arXiv  Detail & Related papers  (2020-10-26T08:55:53Z)
• Streaming Complexity of SVMs [110.63976030971106]
  We study the space complexity of solving the bias-regularized SVM problem in the streaming model. We show that for both problems, for dimensions of $frac1lambdaepsilon$, one can obtain streaming algorithms with spacely smaller than $frac1lambdaepsilon$.
  arXiv  Detail & Related papers  (2020-07-07T17:10:00Z)
• Optimal Mutation Rates for the $(1+\lambda)$ EA on OneMax [1.0965065178451106]
  We extend the analysis of optimal mutation rates to two variants of the OneMax problem. We compute for all population sizes $lambda in 2i mid 0 le i le 18$ which mutation rates minimize the expected running time. Our results do not only provide a lower bound against which we can measure common evolutionary approaches.
  arXiv  Detail & Related papers  (2020-06-20T01:23:14Z)
• A Simple Convergence Proof of Adam and Adagrad [74.24716715922759]
  We show a proof of convergence between the Adam Adagrad and $O(d(N)/st)$ algorithms. Adam converges with the same convergence $O(d(N)/st)$ when used with the default parameters.
  arXiv  Detail & Related papers  (2020-03-05T01:56:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.