Optimal robust expensive optimization is tractable

Following a number of recent papers investigating the possibility of optimal comparison-based optimization algorithms for a given distribution of probability on fitness functions, we (i) discuss the comparison-based constraints (ii) choose a setting in which theoretical tight bounds are known (iii) develop a careful implementation using billiard algorithms, Upper Confidence trees and (iv) experimentally test the tractability of the approach. The results, on still very simple cases, show that the approach, yet still preliminary, could be tested successfully until dimension 10 and horizon 50 iterations within a few hours on a standard computer, with convergence rate far better than the best algorithms.

[1]  Karl Johan Åström,et al.  Optimal control of Markov processes with incomplete state information , 1965 .

[2]  G. Unter Rudolph.  Convergence Rates of Evolutionary Algorithms for a Class of Convex Objective Functions , 1997 .

[3]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[4]  Colin Campbell,et al.  Bayes Point Machines , 2001, J. Mach. Learn. Res..

[5]  Peter Auer,et al.  Using Confidence Bounds for Exploitation-Exploration Trade-offs , 2003, J. Mach. Learn. Res..

[6]  Yew-Soon Ong,et al.  Hierarchical surrogate-assisted evolutionary optimization framework , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[7]  Anne Auger,et al.  Convergence results for the (1, lambda)-SA-ES using the theory of phi-irreducible Markov chains , 2005, Theor. Comput. Sci..

[8]  Jens Jägersküpper,et al.  Rigorous runtime analysis of a (μ+1)ES for the sphere function , 2005, GECCO '05.

[9]  A. Auger.  Convergence results for the ( 1 , )-SA-ES using the theory of-irreducible Markov chains , 2005 .

[10]  Stefan Droste,et al.  Not all linear functions are equally difficult for the compact genetic algorithm , 2005, GECCO '05.

[11]  Thomas Philip Runarsson.  Ordinal Regression in Evolutionary Computation , 2006, PPSN.

[12]  Olivier Teytaud,et al.  General Lower Bounds for Evolutionary Algorithms , 2006, PPSN.

[13]  Edmund K. Burke,et al.  Parallel Problem Solving from Nature - PPSN IX: 9th International Conference, Reykjavik, Iceland, September 9-13, 2006, Proceedings , 2006, PPSN.

[14]  S. Gelly,et al.  Comparison-based algorithms: worst-case optimality, optimality w.r.t a bayesian prior, the intraclass-variance minimization in EDA, and implementations with billiards , 2006 .

[15]  Rémi Coulom,et al.  Efficient Selectivity and Backup Operators in Monte-Carlo Tree Search , 2006, Computers and Games.

[16]  Csaba Szepesvári,et al.  Bandit Based Monte-Carlo Planning , 2006, ECML.

[17]  C. Witt,et al.  Runtime Analysis of a (μ +1) ES for the Sphere Function , 2006 .

[18]  David Silver,et al.  Combining online and offline knowledge in UCT , 2007, ICML '07.

[19]  Rémi Coulom,et al.  Computing "Elo Ratings" of Move Patterns in the Game of Go , 2007, J. Int. Comput. Games Assoc..

[20]  Sylvain Gelly,et al.  Modifications of UCT and sequence-like simulations for Monte-Carlo Go , 2007, 2007 IEEE Symposium on Computational Intelligence and Games.

[21]  Eric Walter,et al.  Global optimization based on noisy evaluations: An empirical study of two statistical approaches , 2008 .

[22]  Olivier Teytaud,et al.  Continuous Lunches Are Free Plus the Design of Optimal Optimization Algorithms , 2010, Algorithmica.

[23]  Olivier Teytaud,et al.  Lower Bounds for Evolution Strategies Using VC-Dimension , 2008, PPSN.

[24]  H. Jaap van den Herik,et al.  Progressive Strategies for Monte-Carlo Tree Search , 2008 .

[25]  Olivier Teytaud,et al.  On the Parallelization of Monte-Carlo planning , 2008, ICINCO 2008.

[26]  Eric Walter,et al.  An informational approach to the global optimization of expensive-to-evaluate functions , 2006, J. Glob. Optim..

[27]  T. L. Lai Andherbertrobbins.  Asymptotically Efficient Adaptive Allocation Rules , 2022 .