Pavel F. Chernavin, Nikolai P. Chernavin, Fedor P. Chernavin

Ural federal university

The XOR problem is a classic problem in machine learning. Most machine learning methods yield unsatisfactory results when addressing the XOR problem. Attempts to improve the quality metrics of decision rules by transitioning from linear separators to polynomial ones, or by increasing the depth and number of trees, reduce the interpretability of the solution and can lead to overfitting. Therefore, this problem is usually solved using neural networks with the backpropagation method. Currently, there is a significant interest in finding alternatives to the neural network approach for solving classification tasks and the backpropagation method for training neural networks. This article proposes solving the XOR problem based on committee constructions in the form of mathematical programming tasks. Their effectiveness largely depends on the structure of the original data, which can only be understood in the process of solving the problem. Classical committees of unanimity, majority, and seniority only achieve high-quality metrics of the decision rule when the data structure is relatively simple. Therefore, it is proposed to augment the set of committee constructions with an XOR committee. The article presents its geometric interpretation, mathematical model, program listing in IBM ILOG CPLEX package codes, and a comparison with the quality metrics of solutions by other machine learning methods. Translating the process of building committee constructions into mathematical programming tasks provides additional opportunities for controlling the quality of the decision rule during its construction, rather than post facto, as occurs when using programs from standard libraries for machine learning tasks.

machine learning, XOR problem, mathematical programming, classification, committee constructions