QUBO-Based Calibration for Regression Trees

TL;DR

This paper reformulates the split-selection problem in CART regression trees as a QUBO problem, enabling the use of advanced solvers to improve split quality and potentially enhance predictive performance.

Contribution

It introduces a novel QUBO-based approach for solving the fractional optimization problem in CART, offering higher-quality splits compared to greedy heuristics.

Findings

QUBO-based trees achieve comparable predictive performance to standard CART.

QUBO formulation yields higher-quality split solutions.

Potential for hybrid classical-quantum implementations in tree learning.

Abstract

Tree-based regression models are widely used in supervised learning, with the Classification and Regression Tree (CART) algorithm serving as a standard reference. CART construction involves solving a sequence of split-selection optimization problems. For categorical predictors, this problem can be formulated as a combinatorial fractional optimization problem. This structure makes the exact optimization computationally challenging and leads to standard implementations that rely on greedy heuristics, which may result in suboptimal splits. In this work, we reformulate this fractional problem and apply Dinkelbach (1967) algorithm to convert it into a Quadratic Unconstrained Binary Optimization (QUBO) problem. Using state-of-the-art QUBO solvers, we obtain QUBO-based regression trees with predictive performance comparable to standard CART while yielding higher-quality split solutions. These results highlight the potential of QUBO formulations for improving tree-based learning methods and open perspectives for future hybrid classical--quantum implementations.