Differentiable Genetic Programming for High-dimensional Symbolic Regression

Symbolic regression (SR) is the process of discovering hidden relationships from data with mathematical expressions, which is considered an effective way to reach interpretable machine learning (ML). Genetic programming (GP) has been the dominator in solving SR problems. However, as the scale of SR problems increases, GP often poorly demonstrates and cannot effectively address the real-world high-dimensional problems. This limitation is mainly caused by the stochastic evolutionary nature of traditional GP in constructing the trees. In this paper, we propose a differentiable approach named DGP to construct GP trees towards high-dimensional SR for the first time. Specifically, a new data structure called differentiable symbolic tree is proposed to relax the discrete structure to be continuous, thus a gradient-based optimizer can be presented for the efficient optimization. In addition, a sampling method is proposed to eliminate the discrepancy caused by the above relaxation for valid symbolic expressions. Furthermore, a diversification mechanism is introduced to promote the optimizer escaping from local optima for globally better solutions. With these designs, the proposed DGP method can efficiently search for the GP trees with higher performance, thus being capable of dealing with high-dimensional SR. To demonstrate the effectiveness of DGP, we conducted various experiments against the state of the arts based on both GP and deep neural networks. The experiment results reveal that DGP can outperform these chosen peer competitors on high-dimensional regression benchmarks with dimensions varying from tens to thousands. In addition, on the synthetic SR problems, the proposed DGP method can also achieve the best recovery rate even with different noisy levels. It is believed this work can facilitate SR being a powerful alternative to interpretable ML for a broader range of real-world problems.