Characterizing Genetic Programming Error Through Extended Bias and Variance Decomposition

An error function can be used to select between candidate models but it does not provide a thorough understanding of the behavior of a model. A greater understanding of an algorithm can be obtained by performing a bias-variance decomposition. Splitting the error into bias and variance is effective for understanding a deterministic algorithm such as $k$ -nearest neighbor, which provides the same predictions when performed multiple times using the same data. However, simply splitting the error into bias and variance is not sufficient for nondeterministic algorithms, such as genetic programming (GP), which potentially produces a different model each time it is run, even when using the same data. This article presents an extended bias-variance decomposition that decomposes error into bias, external variance (error attributable to limited sampling of the problem), and internal variance (error due to random actions performed in the algorithm itself). This decomposition is applied to GP to expose the three components of error, providing a unique insight into the role of maximum tree depth, number of generations, size/complexity of function set, and data standardization in influencing predictive performance. The proposed tool can be used to inform targeted improvements for reducing specific components of model error.