Improving the Generalisation of Genetic Programming for Symbolic Regression

Gradient-based local optimization has been shown to improve results of genetic programming (GP) for symbolic regression. Several state-of-the-art GP implementations use iterative nonlinear least squares (NLS) algorithms such as the Levenberg-Marquardt algorithm for local optimization. The effectiveness of NLS algorithms depends on appropriate scaling and conditioning of the optimization problem. This has so far been ignored in symbolic regression and GP literature. In this study we use a singular value decomposition of NLS Jacobian matrices to determine the numeric rank and the condition number. We perform experiments with a GP implementation and six different benchmark datasets. Our results show that rank-deficient and ill-conditioned Jacobian matrices occur frequently and for all datasets. The issue is less extreme when restricting GP tree size and when using many non-linear functions in the function set.

Generalization ability, which reflects the prediction ability of a learned model, is an important property in genetic programming (GP) for symbolic regression. Structural risk minimization (SRM) is a framework providing a reliable estimation of the generalization performance of prediction models. Introducing the framework into GP has the potential to drive the evolutionary process toward models with good generalization performance. However, this is tough due to the difficulty in obtaining the Vapnik–Chervonenkis (VC) dimension of nonlinear models. To address this difficulty, this paper proposes an SRM-driven GP approach, which uses an experimental method (instead of theoretical estimation) to measure the VC dimension of a mixture of linear and nonlinear regression models for the first time. The experimental method has been conducted using uniform and nonuniform settings. The results show that our method has impressive generalization gains over standard GP and GP with the 0.632 bootstrap, and that the proposed method using the nonuniform setting has further improvement than its counterpart using the uniform setting. Further analyzes reveal that the proposed method can evolve more compact models, and that the behavioral difference between these compact models and the target models is much smaller than their counterparts evolved by the other GP methods.

Transfer learning has attracted more attention in the machine-learning community recently. It aims to improve the learning performance on the domain of interest with the help of the knowledge acquired from a similar domain(s). However, there is only a limited number of research on tackling transfer learning in genetic programming for symbolic regression. This article attempts to fill this gap by proposing a new instance weighting framework for transfer learning in genetic programming-based symbolic regression. In the new framework, differential evolution is employed to search for optimal weights for source-domain instances, which helps genetic programming to identify more useful source-domain instances and learn from them. Meanwhile, a density estimation method is used to provide good starting points to help the search for the optimal weights while discarding some irrelevant or less important source-domain instances before learning regression models. The experimental results show that compared with genetic programming and support vector regression that learn only from the target instances, and learning from a mixture of instances from the source and target domains without any transfer learning component, the proposed method can evolve regression models which not only achieve notably better cross-domain generalization performance in stability but also reduce the trend of overfitting effectively. Meanwhile, these models are generally much simpler than those generated by the other GP methods.

Genetic Programming (GP) has been around for over two decades and has been used in a wide range of practical applications producing human competitive results in several domains. In this paper we present a discussion and a proposal of a GP algorithm that could be conveniently implemented on an embedded system, as part of a broader research project that pursues the implementation of a complete GP system in a Field Programmable Gate Array (FPGA). Motivated by the significant time savings associated with such a platform, as well as low power consumption, low maintenance requirements, small size of the system and the possibility of performing several parallel processes. The proposal is focused on the Geometric Semantic Genetic Programming (GSGP) approach that has been recently introduced with promising results. GSGP induces a unimodal fitness landscape, simplifying the search process. The experimental work considers five variants of GSGP, that incorporate local search strategies, optimal mutations and alignment in error space. Best results were obtained by a simple variant that uses both the optimal mutation step and the standard geometric semantic mutation, using three difficult real-world problems to evaluate the methods, outperforming the original GSGP formulation in terms of fitness and empirical convergence.

Interpretable policies for reinforcement learning by genetic programming

This paper introduces Multi-Region Symbolic Regression (MR-SR), a general framework that divides the original input data space of symbolic regression problems into subspaces (regions), generates different solutions to fit these regions and then combines them. MR-SR has three main components: (1) a strategy for finding the different regions of the data input space; (2) a method for generating the functions for each region; and (3) a strategy for combining the models found by (2). The main contribution of this paper is on how we generate the functions for each region. We model the function generation problem following a multi-objective approach, where each objective corresponds to the quality of the evolved function in a region, and the number of objectives is equal to the number of regions of the data input space. We test MR-SR in two scenarios with different objectives. In the first, we used the new approach to solve the symbolic regression problem with standard GP, with the main objective of reducing error rate. In the second, we took advantage of this method for a different purpose: to reduce the dimensionality of the semantic space of a variation of GP, namely Geometric Semantic Genetic Programming (GSGP). Results in 10 datasets showed that the method using clustering k-means and a model switching strategy—which makes predictions using the best evolved function for the region of interest—obtained better results in 5 out of 10 datasets for GP with 2 regions. For GSGP the framework was less effective due to the lack of diversity of the solutions evolved.

Geometric Semantic Genetic Programming (GSGP) is a recently proposed form of Genetic Programming (GP) that searches directly the space of the underlying semantics of the programs. The fitness landscape seen by the GSGP variation operators is unimodal with a linear slope by construction and, consequently, easy to search. Despite this advantage, the offspring produced by these operators grow very quickly. A new implementation of the same operators was proposed that computes the semantics of the offspring without having to explicitly build their syntax. This allowed GSGP to be used for the first time in real-life multidimensional datasets. GSGP presented a surprisingly good generalization ability, which was justified by some properties of the geometric semantic operators. In this paper, we show that the good generalization ability of GSGP was the result of a small implementation deviation from the original formulation of the mutation operator, and that without it the generalization results would be significantly worse. We explain the reason for this difference, and then we propose two variants of the geometric semantic mutation that deterministically and optimally adapt the mutation step. They reveal to be more efficient in learning the training data, and they also achieve a competitive generalization in only a single operator application. This provides a competitive alternative when performing semantic search, particularly since they produce small individuals and compute fast.

Most often, climate change signals are slow moving (in human terms), low amplitude changes embedded in high amplitude, noisy data. There are many techniques for extracting such signals. This paper introduces a technique which in contrast with empirical methods produces a decomposition of a time series into a set of equations plus a This is referred to as stepwise symbolic secomposition (SSD). The extraction of symbolic equations in lieu of empirical functions assists with characterization of time series. This paper takes two examples of climate data, and applies two different techniques for characterising the low amplitude, slow change embedded therein. The record of CO2 levels at Mauna-Loa since March 1958, is used to demonstrate and contrast empirical mode decomposition (EMD), and SSD. The mean monthly tidal gauge records from the small number of gauges which have more than 120 years of data are analysed in more detail by SSD, then three techniques are used to characterize the residual. The techniques are (a) LOESS smoothing, (b) EMD, and (c) high order polynomial regression. SSD uses a genetic programming system called Eureqa from Cornell Creative Machines Lab guided by an information metric, to extract the most compact informative function it can at each step; the process is repeated on the residuals until no sufficiently informative function is found. EMD and SSD are in stark contrast in the order in which signals are decomposed. EMD extracts high frequency components first. SSD extracts components based on a mixture of parsimonious representation and variance explained. EMD leaves a low frequency filtrate of the signal in its residue, SSD tends to operate as a broad band filter, leaving high frequency noise plus a possible low frequency signal. EMD, LOESS and polynomial fitting all serve to extract low frequency components of the signal from this SSD residual. For the set of tidal gauge data, in the absence of a change in global sea level rise, the SSD procedure should randomize all segments of residual signals equally - the residual should be whitened with respect to the initial signal. It is shown however that late 20th century portions of the residuals show behaviour that is consistent with accelerating sea level rise in keeping with the bulk of the literature.

Genetic Programming (GP) for symbolic regression is often prone to overfitting the training data, causing poor performance on unseen data. A number of recent works in the field have been devoted to regulating this problem by investigating both the structural and functional complexity of GP individuals during the evolutionary process. This work uses the Rademacher complexity and incorporates it into the fitness function of GP, utilising it as a means of controlling the functional complexity of GP individuals. The experiment results confirm that the new GP method has a notable generalization gain compared to the standard GP and Support Vector Regression (SVR) in most of the considered problems. Further investigations also show that the new GP method generates symbolic regression models that could not only release the overfitting trend in standard GP but also are significantly smaller in size compared to their counterparts in standard GP.

Population diversity plays an important role in the evolutionary dynamics of genetic programming (GP). In this paper we use structural and semantic similarity measures to investigate the evolution of diversity in three GP algorithmic flavors: standard GP, offspring selection GP (OS-GP), and age-layered population structure GP (ALPS-GP). Empirical measurements on two symbolic regression benchmark problems reveal important differences between the dynamics of the tested configurations. In standard GP, after an initial decrease, population diversity remains almost constant until the end of the run. The higher variance of the phenotypic similarity values suggests that small changes on individual genotypes have significant effects on their corresponding phenotypes. By contrast, strict offspring selection within the OS-GP algorithm causes a significantly more pronounced diversity loss at both genotypic and, in particular, phenotypic levels. The pressure for adaptive change increases phenotypic robustness in the face of genotypic perturbations, leading to less genotypic variability on the one hand, and very low phenotypic diversity on the other hand. Finally, the evolution of similarities in ALPS-GP follows a periodic pattern marked by the time interval when the bottom layer is reinitialized with new individuals. This pattern is easily noticed in the lower layers characterized by shorter migration intervals, and becomes less and less noticeable on the upper layers.

<p><strong>The Cyber-Marine project aims to maximise the utility of commercial fish by using Artificial Intelligence (AI) supported tools and non-invasive chemical methods to assess their nutrient content. Vibrational Spectroscopy has proven to be a reliable method to determine molecular characteristics of a product. Particularly, it has gained prominence in evaluating the nutrient level of produce through statistical modelling techniques, such as Partial Least Squared regression (PLSR). This thesis investigates the use of Machine Learning (ML) to relate spectral data of fish with their nutrient levels. Within ML algorithms, Genetic Programming for symbolic regression (GPSR) emerges as a notable approach, producing models that exhibit generalisability and interpretability. As an extension of GPSR, multi-tree GP (MTGP), provides a platform that can simultaneously train models for several nutrients or bioactive components. In this thesis, significant enhancements and modifications are made to key aspects of MTGP, including its evolutionary process, by the introduction of new genetic operator, and the evaluation function, through the use of a distance metric. These are designed not only to leverage the relationship between the spectral information and bioactive components, but also to optimise the interplay between different target variables thereby maximising the capabilities of the algorithm.</strong></p>

Symbolic regression is a process to find a mathematical expression that represents the relationship between a set of explanatory variables and a measured variable. It has become a best-known problem for GP (genetic programming), as GP can use the tree representation to represent solutions as expression trees. Since the success of memetic algorithms (MAs (Memetic algorithms (MAs) can be regarded as a class of methods that combine population-based global search and local search [6, 30])) has proved the importance of local search in augmenting the global search ability of GP, GP with local search is investigated to solve symbolic regression tasks in this work. An important design issue of MAs is the balance between the global exploration of GP and the local exploitation, which has a great influence on the performance and efficiency of MAs. This work proposes a GP-based memetic algorithm for symbolic regression, termed as aMeGP (a daptive Me metic GP), which can balance global exploration and local exploitation adaptively. Compared with GP, two improvements are made in aMeGP to invoke and stop local search adaptively during evolution. The proposed aMeGP is compared with GP-based and nonGP-based symbolic regression methods on both benchmark test functions and real-world applications. The results show that aMeGP is generally better than both GP-based and nonGP-based reference methods with its evolved solutions achieving lower root mean square error (RMSE) for most test cases. Moreover, aMeGP outperforms the reference GP-based methods in the convergence ability, which can converge to lower RMSE values with faster or similar speeds.

When performing symbolic regression using genetic programming, overfitting and bloat can negatively impact generalizability and interpretability of the resulting equations as well as increase computation times. A Bayesian fitness metric is introduced and its impact on bloat and overfitting during population evolution is studied and compared to common alternatives in the literature. The proposed approach was found to be more robust to noise and data sparsity in numerical experiments, guiding evolution to a level of complexity appropriate to the dataset. Further evolution of the population resulted not in overfitting or bloat, but rather in slight simplifications in model form. The ability to identify an equation of complexity appropriate to the scale of noise in the training data was also demonstrated. In general, the Bayesian model selection algorithm was shown to be an effective means of regularization which resulted in less bloat and overfitting when any amount of noise was present in the training data.

Networks are continuously growing in complexity, which creates challenges for determining their most important characteristics. While analytical bounds are often too conservative, the computational effort of algorithmic approaches does not scale well with network size. This work uses Cartesian Genetic Programming for symbolic regression to evolve mathematical equations that relate network properties directly to the eigenvalues of network adjacency and Laplacian matrices. In particular, we show that these eigenvalues are powerful features to evolve approximate equations for the network diameter and the isoperimetric number, which are hard to compute algorithmically. Our experiments indicate a good performance of the evolved equations for several real-world networks and we demonstrate how the generalization power can be influenced by the selection of training networks and feature sets.

There are two important limitations of standard tree-based genetic programming (GP). First, GP tends to evolve unnecessarily large programs, what is referred to as bloat. Second, it uses inefficient search operators that operate at the syntax level. The first problem has been the subject of a fair amount of research over the years. Regarding the second problem, one approach is to use alternative search operators, for instance geometric semantic operators. However, another approach is to introduce greedy local search strategies, combining the syntactic search performed by standard GP with local search strategies for solution tuning, which is a simple strategy that has comparatively received much less attention. This work combines a recently proposed bloat-free GP called neat-GP with a local search strategy. One benefit of using a bloat-free GP is that it reduces the size of the parameter space confronted by the local searcher, offsetting some of the added computational cost. The algorithm is validated on a real-world problem with promising results.