Gradient search in the space of permutations: an application for the linear ordering problem

Gradient search is a classical technique for optimizing differentiable functions that has gained much relevance recently due to its application on Neural Network training. Despite its popularity, the application of gradient search has been limited to the continuous optimization and its usage in the combinatorial case is confined to a few works, all which tackle the binary search space. In this paper, we present a new approach for applying Gradient Search to the space of permutations. The idea consists of optimizing the expected objective value of a random variable defined over permutations. Such a random variable is distributed according to the Plackett-Luce model, and a gradient search over its continuous parameters is performed. Conducted experiments on a benchmark of the linear ordering problem confirm that the Gradient Search performs better than its counterpart Estimation of Distribution Algorithm: the Plackett-Luce EDA. Moreover, results reveal that the scalability of the Gradient Search is better than that of the PL-EDA.

[1]  Alexander Mendiburu,et al.  The Plackett-Luce ranking model on permutation-based optimization problems , 2013, 2013 IEEE Congress on Evolutionary Computation.

[2]  F. Mosteller.  Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations , 1951 .

[3]  C. L. Mallows.  NON-NULL RANKING MODELS. I , 1957 .

[4]  M. Fligner,et al.  Multistage Ranking Models , 1988 .

[5]  Valentino Santucci,et al.  Using pairwise precedences for solving the linear ordering problem , 2020, Appl. Soft Comput..

[6]  R. A. Bradley,et al.  RANK ANALYSIS OF INCOMPLETE BLOCK DESIGNS THE METHOD OF PAIRED COMPARISONS , 1952 .

[7]  Harry Joe,et al.  On the Babington Smith Class of Models for Rankings , 1993 .

[8]  Xin Yao,et al.  Parallel Problem Solving from Nature PPSN VI , 2000, Lecture Notes in Computer Science.

[9]  Matteo Matteucci,et al.  Towards the geometry of estimation of distribution algorithms based on the exponential family , 2011, FOGA '11.

[10]  A. Berny,et al.  Statistical machine learning and combinatorial optimization , 2001 .

[11]  Joseph S. Verducci,et al.  Probability Models and Statistical Analyses for Ranking Data , 1992 .

[12]  Borja Calvo,et al.  scmamp: Statistical Comparison of Multiple Algorithms in Multiple Problems , 2016, R J..

[13]  Ekhine Irurozki,et al.  An R package for permutations, Mallows and Generalized Mallows models , 2014 .

[14]  Joseph S. Verducci,et al.  Probability models on rankings. , 1991 .

[15]  Sebastian Ruder,et al.  An overview of gradient descent optimization algorithms , 2016, Vestnik komp'iuternykh i informatsionnykh tekhnologii.

[16]  Anne Auger,et al.  Information-Geometric Optimization Algorithms: A Unifying Picture via Invariance Principles , 2011, J. Mach. Learn. Res..

[17]  Alfredo Milani,et al.  Variable neighborhood algebraic Differential Evolution: An application to the Linear Ordering Problem with Cumulative Costs , 2020, Inf. Sci..

[18]  Novi Quadrianto,et al.  Low-variance Black-box Gradient Estimates for the Plackett-Luce Distribution , 2019, AAAI.

[19]  R. Duncan Luce,et al.  Individual Choice Behavior , 1959 .

[20]  Mauro Birattari,et al.  Model-Based Search for Combinatorial Optimization: A Critical Survey , 2004, Ann. Oper. Res..

[21]  Nikolaus Hansen,et al.  The CMA Evolution Strategy: A Tutorial , 2016, ArXiv.

[22]  Marco Zaffalon,et al.  Time for a change: a tutorial for comparing multiple classifiers through Bayesian analysis , 2016, J. Mach. Learn. Res..

[23]  R. Plackett.  The Analysis of Permutations , 1975 .

[24]  Arnaud Berny.  Boltzmann Machine for Population-Based Incremental Learning , 2002, ECAI.

[25]  L. Thurstone.  A law of comparative judgment. , 1994 .

[26]  Arnaud Berny.  Selection and Reinforcement Learning for Combinatorial Optimization , 2000, PPSN.

[27]  Josu Ceberio.  Solving permutation problems with estimation of distribution algorithms and extensions thereof , 2014 .

[28]  M. Fligner,et al.  Distance Based Ranking Models , 1986 .

[29]  Ekhiñe Irurozqui Arrieta,et al.  Sampling and learning distance-based probability models for permutation spaces , 2014 .

[30]  Alfredo Milani,et al.  MOEA/DEP: An Algebraic Decomposition-Based Evolutionary Algorithm for the Multiobjective Permutation Flowshop Scheduling Problem , 2018, EvoCOP.

[31]  J. A. Lozano,et al.  PerMallows: An R Package for Mallows and Generalized Mallows Models , 2016 .

[32]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[33]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decisionmaking , 1988, IEEE Trans. Syst. Man Cybern..

[34]  Alfredo Milani,et al.  An algebraic framework for swarm and evolutionary algorithms in combinatorial optimization , 2020, Swarm Evol. Comput..

[35]  Tom Schaul,et al.  Natural Evolution Strategies , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).