Hod Lipson

How does Eureqa’s performance, in terms of predictive accuracy and simplicity, compare to other machine learning methods, such as Neural Networks, Support Vector Machines, Decision Trees, and simple Linear Regression?

To answer this question we did a simple comparison. We ran Eureqa on seven test-cases for which data is publically available, and compared performance to four standard machine learning methods. The implementations used were the WEKA codes, with settings optimized for best performance:

**Linear Regression**: Fit a linear equation of the form y=a_{1}x_{1}+ a_{2}x_{2}+ a_{3}x_{3}… using least squares method. This approach is the traditional regression method used in many statistical regression software packages**Decision trees (DT)**: This process tries to find multiple linear regression models, each for a different part of the dataset. The dataset is portioned using conditions on the input variables.**Neural Networks (NN)**: A classic multi-layer perceptron network attempts to learn to predict the output from the input using back-propagation learning method. Early-stopping using validation set is used, with a single hidden layer whose size is optimized automatically.**Support vector machines (SVM)**: Model the data as a combination of a few, selected data points (called support vectors).

We ran the tests on five datasets, obtained from the UCI Dataset repository. They included the Auto MPG Benchmark, the Challenger O-Ring Benchmark, the Concrete Compressive Strength Benchmark, the Solar Flare Benchmark, and the Coil 2000 Benchmark.

Each algorithm produced a result in a different format: Linear regression produced a hyperplane, while a neural network produced a connectivity weight matrix and Eureqa produced an analytical expression. One example result can be seen to the right. It is clear that some solutions are more complex than others. The more complex solutions involve more free parameters, or just take more ink to write down. Some solutions were more accurate than others: They produced less error when tested in a separate test dataset. Of course, we’d like to have a machine learning algorithm that produces models that are both accurate and simple, but that isn’t always the case.

We plotted the average performance of all five algorithms at a location corresponding to the average complexity and accuracy of the models they produced. In a complexity versus accuracy chart, we can see several regions. The top left region is where we would see algorithms that produced models that are fairly accurate, but have many free parameters, The bottom right region is where we would see algorithms that produce very simple solutions, even if they are somewhat less accurate. The top right region of the chart is the worst region to be in, where models are both complicated and not so accurate. And the bottom left region is where we find algorithms that produce models that are at the same time both simple and accurate.

It appears that Eureqa’s use of symbolic regression produces models that are both more accurate and simpler than other machine learning methods, but what’s the catch?

There is no free lunch. Symbolic regression is substantially more computationally intensive when compared to neural networks, SVMs and Linear regression. Luckily, however, while accuracy and simplicity are priceless, computational power can be bought on-demand with platforms like Amazon EC2.

Topics: hod lipson, symbolic regression