{"type":"ScholarlyArticle","title":"Approximating Pseudo-Boolean Functions on Non-Uniform Domains","authors":[{"name":"Robert F. Lax","slug":"robert-f-lax","given":"Robert F.","family":"Lax","affiliation":""},{"name":"Guoli Ding","slug":"guoli-ding","given":"Guoli","family":"Ding","affiliation":""},{"name":"Peter P. Chen","slug":"peter-p-chen","given":"Peter P.","family":"Chen","affiliation":""},{"name":"Jianhua Chen","slug":"jianhua-chen","given":"Jianhua","family":"Chen","affiliation":""}],"abstract":"In Machine Learning (ML) and Evolutionary Computation (EC), it is often beneficial to approximate a complicated function by a simpler one, such as a linear or quadratic function, for computational efficiency or feasibility reasons (cf. [Jin, 2005]). A complicated function (the target function in ML or the fitness function in EC) may require an exponential amount of computation to learn/evaluate, and thus approximations by simpler functions are needed. We consider the problem of approximating pseudo-Boolean functions by simpler (e.g., linear) functions when the instance space is associated with a probability distribution. We consider {0, 1}n as a sample space with a (possibly nonuniform) probability measure on it, thus making pseudo-Boolean functions into random variables. This is also in the spirit of the PAC learning framework of Valiant [Valiant, 1984] where the instance space has a probability distribution on it. The best approximation to a target function f is then defined as the function g (from all possible approximating functions of the simpler form) that minimizes the expected distance to f . In an example, we use methods from linear algebra to find, in this more general setting, the best approximation to a given pseudo-Boolean function by a linear function.","doi":null,"url":"https://ijcai.org/proceedings/2005/256","pdf":"https://ijcai.org/proceedings/2005/0256.pdf","markdown":"https://ijcai.org/proceedings/2005/256.md","bibtex":"https://ijcai.org/proceedings/2005/bibtex/256","volume":{"slug":"2005","title":"Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2005","url":"https://ijcai.org/proceedings/2005/","year":2005,"location":"Edinburgh, Scotland, UK","dates":"","editors":"","isbn":"","publisher":"International Joint Conferences on Artificial Intelligence","doi_prefix":""},"section":null,"track":"","number":256,"pages":{"first":1754,"last":1754},"keywords":[],"videos":[],"references":[],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}