Algebraic Machine Learning for Small-to-Medium Datasets Is Competitive against Strong Standard Baselines

Symbolic methods are generally not considered competitive with strong modern learners on realistic supervised tasks. We evaluate Algebraic Machine Learning (AML), a framework that learns through subdirect decomposition of algebraic structure rather than numerical optimization, against standard baselines on image and tabular classification across varying training-set sizes. We find that AML trained only on training data without using validation or cross-validation outperforms a family of cross-validated baseline methods including CNNs on small to medium image datasets (50--2000 training examples). On tabular datasets in the same size range, XGBoost is overall the best performing method, but AML is nonetheless comparable to methods incorporating task-specific biases such as LightGBM and random forests. AML achieves this competitive performance across two very different types of datasets using a generic algebraic inductive bias, rather than the modality-specific biases built into standard baselines like CNNs for images or XGBoost for tabular data, and requires no cross validation because it has no task-dependent hyperparameters to tune.

Algebraic Machine Learning: Learning as computing an algebraic decomposition of a task

Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the task and the data are encoded as axioms of an algebra, and a model is obtained where only these axioms and their logical consequences hold. Although this is not a generalizing model, we show that selecting specific subsets of its breakdown into algebraic atoms obtained via subdirect decomposition gives a model that generalizes. We validate this new learning principle on standard datasets such as MNIST, FashionMNIST, CIFAR-10, and medical images, achieving performance comparable to optimized multilayer perceptrons. Beyond data-driven tasks, the new learning principle extends to formal problems, such as finding Hamiltonian cycles from their specifications and without relying on search. This algebraic foundation offers a fresh perspective on machine intelligence, featuring direct learning from training data without the need for validation dataset, scaling through model additivity, and asymptotic convergence to the underlying rule in the data.