Tight Runtime Bounds for Static Unary Unbiased Evolutionary Algorithms on Linear Functions

• URL: http://arxiv.org/abs/2302.12338v3
• Date: Sat, 28 Sep 2024 14:32:08 GMT
• Title: Tight Runtime Bounds for Static Unary Unbiased Evolutionary Algorithms on Linear Functions
• Authors: Carola Doerr, Duri Andrea Janett, Johannes Lengler,
• Abstract summary: Witt showed that the (1+1) Evolutionary Algorithm with standard bit mutation needs time to find the optimum of any linear function. We investigate how this result generalizes if standard bit mutation is replaced by an arbitrary unbiased mutation operator.
• Score: 0.44241702149260353
• License:
• Abstract: In a seminal paper in 2013, Witt showed that the (1+1) Evolutionary Algorithm with standard bit mutation needs time $(1+o(1))n \ln n/p_1$ to find the optimum of any linear function, as long as the probability $p_1$ to flip exactly one bit is $\Theta(1)$. In this paper we investigate how this result generalizes if standard bit mutation is replaced by an arbitrary unbiased mutation operator. This situation is notably different, since the stochastic domination argument used for the lower bound by Witt no longer holds. In particular, starting closer to the optimum is not necessarily an advantage, and OneMax is no longer the easiest function for arbitrary starting positions. Nevertheless, we show that Witt's result carries over if $p_1$ is not too small, with different constraints for upper and lower bounds, and if the number of flipped bits has bounded expectation~$\chi$. Notably, this includes some of the heavy-tail mutation operators used in fast genetic algorithms, but not all of them. We also give examples showing that algorithms with unbounded $\chi$ have qualitatively different trajectories close to the optimum.

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)
• Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path [80.60592344361073]
  We study the Shortest Path (SSP) problem with a linear mixture transition kernel. An agent repeatedly interacts with a environment and seeks to reach certain goal state while minimizing the cumulative cost. Existing works often assume a strictly positive lower bound of the iteration cost function or an upper bound of the expected length for the optimal policy.
  arXiv  Detail & Related papers  (2024-02-14T07:52:00Z)
• Gradient-free optimization of highly smooth functions: improved analysis and a new algorithm [87.22224691317766]
  This work studies problems with zero-order noisy oracle information under the assumption that the objective function is highly smooth. We consider two kinds of zero-order projected gradient descent algorithms.
  arXiv  Detail & Related papers  (2023-06-03T17:05:13Z)
• Deterministic Nonsmooth Nonconvex Optimization [94.01526844386977]
  We show that randomization is necessary to obtain a dimension-free dimension-free algorithm. Our algorithm yields the first deterministic dimension-free algorithm for optimizing ReLU networks.
  arXiv  Detail & Related papers  (2023-02-16T13:57:19Z)
• Runtime Analysis for Permutation-based Evolutionary Algorithms [9.044970217182117]
  We show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order $mTheta(m)$ on jump functions with jump size $m$. A short empirical analysis confirms these findings, but also reveals that small implementation details like the rate of void mutations can make an important difference.
  arXiv  Detail & Related papers  (2022-07-05T15:49:29Z)
• Near-Optimal Lower Bounds For Convex Optimization For All Orders of Smoothness [26.71898403195793]
  We study the complexity of optimizing highly smooth convex functions. For a positive integer $p$, we want to find an $epsilon$-approximate minimum of a convex function $f$. We prove a new lower bound that matches this bound (up to log factors), and holds not only for randomized algorithms, but also for quantum algorithms.
  arXiv  Detail & Related papers  (2021-12-02T10:51:43Z)
• A first-order primal-dual method with adaptivity to local smoothness [64.62056765216386]
  We consider the problem of finding a saddle point for the convex-concave objective $min_x max_y f(x) + langle Ax, yrangle - g*(y)$, where $f$ is a convex function with locally Lipschitz gradient and $g$ is convex and possibly non-smooth. We propose an adaptive version of the Condat-Vu algorithm, which alternates between primal gradient steps and dual steps.
  arXiv  Detail & Related papers  (2021-10-28T14:19:30Z)
• 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 $sqrtlog(d)/n + (log(d)/varepsilon n)2/3.
  arXiv  Detail & Related papers  (2021-03-02T06:53:44Z)
• Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations [54.42518331209581]
  We find an algorithm which finds. epsilon$-approximate stationary point (with $|nabla F(x)|le epsilon$) using. $(epsilon,gamma)$surimate random random points. Our lower bounds here are novel even in the noiseless case.
  arXiv  Detail & Related papers  (2020-06-24T04:41:43Z)
• 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)
• The $(1+(\lambda,\lambda))$ Genetic Algorithm for Permutations [0.0]
  We show that the $(lambda,lambda)$ genetic algorithm finds the optimum in $O(n2)$ fitness queries. We also present the first analysis of this algorithm on a permutation-based problem called Ham.
  arXiv  Detail & Related papers  (2020-04-18T17:04:57Z)

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.