Active Learning of Combinatorial Features for Interactive Optimization

We address the problem of automated discovery of preferred solutions by an interactive optimization procedure. The algorithm iteratively learns a utility function modeling the quality of candidate solutions and uses it to generate novel candidates for the following refinement. We focus on combinatorial utility functions made of weighted conjunctions of Boolean variables. The learning stage exploits the sparsity-inducing property of 1-norm regularization to learn a combinatorial function from the power set of all possible conjunctions up to a certain degree. The optimization stage uses a stochastic local search method to solve a weighted MAX-SAT problem. We show how the proposed approach generalizes to a large class of optimization problems dealing with satisfiability modulo theories. Experimental results demonstrate the effectiveness of the approach in focusing towards the optimal solution and its ability to recover from suboptimal initial choices.

[1]  F. Glover,et al.  Handbook of Metaheuristics , 2019, International Series in Operations Research & Management Science.

[2]  Filip Radlinski,et al.  Active exploration for learning rankings from clickthrough data , 2007, KDD '07.

[3]  Michael R. Lyu,et al.  Direct Zero-Norm Optimization for Feature Selection , 2008, 2008 Eighth IEEE International Conference on Data Mining.

[4]  Alexander Gammerman,et al.  Ridge Regression Learning Algorithm in Dual Variables , 1998, ICML.

[5]  Toby Walsh,et al.  Elicitation strategies for soft constraint problems with missing preferences: Properties, algorithms and experimental studies , 2010, Artif. Intell..

[6]  Sayan Mukherjee,et al.  Choosing Multiple Parameters for Support Vector Machines , 2002, Machine Learning.

[7]  Chiranjib Bhattacharyya,et al.  Structured learning for non-smooth ranking losses , 2008, KDD.

[8]  Cesare Tinelli,et al.  Satisfiability Modulo Theories , 2021, Handbook of Satisfiability.

[9]  Burr Settles,et al.  Active Learning Literature Survey , 2009 .

[10]  Jason Weston,et al.  Gene Selection for Cancer Classification using Support Vector Machines , 2002, Machine Learning.

[11]  Rajeev Alur,et al.  A Temporal Logic of Nested Calls and Returns , 2004, TACAS.

[12]  Mauro Brunato,et al.  Reactive Search Optimization: Learning While Optimizing , 2018, Handbook of Metaheuristics.

[13]  Francesca Rossi,et al.  Interval-valued soft constraint problems , 2010, Annals of Mathematics and Artificial Intelligence.

[14]  Bernhard Schölkopf,et al.  Use of the Zero-Norm with Linear Models and Kernel Methods , 2003, J. Mach. Learn. Res..

[15]  Alberto Griggio,et al.  Satisfiability Modulo the Theory of Costs: Foundations and Applications , 2010, TACAS.

[16]  Yi Zhang,et al.  Incorporating Diversity and Density in Active Learning for Relevance Feedback , 2007, ECIR.

[17]  Albert Oliveras,et al.  On SAT Modulo Theories and Optimization Problems , 2006, SAT.

[18]  Mauro Brunato,et al.  Reactive Search and Intelligent Optimization , 2008 .

[19]  Greg Nelson,et al.  Simplification by Cooperating Decision Procedures , 1979, TOPL.

[20]  Nikolaj Bjørner,et al.  Satisfiability Modulo Theories: An Appetizer , 2009, SBMF.

[21]  Trevor Hastie,et al.  Discussion of Boosting Papers , 2003 .

[22]  Robert Tibshirani,et al.  1-norm Support Vector Machines , 2003, NIPS.

[23]  Albert Oliveras,et al.  DPLL(T) with Exhaustive Theory Propagation and Its Application to Difference Logic , 2005, CAV.

[24]  Rocco A. Servedio,et al.  Efficiency versus Convergence of Boolean Kernels for On-Line Learning Algorithms , 2001, NIPS.

[25]  Bruno Dutertre,et al.  A Fast Linear-Arithmetic Solver for DPLL(T) , 2006, CAV.

[26]  Peter Ingwersen,et al.  Developing a Test Collection for the Evaluation of Integrated Search , 2010, ECIR.

[27]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[28]  Toby Walsh,et al.  Handbook of satisfiability , 2009 .