Spectral Regularization Allows Data-frugal Learning over Combinatorial Spaces
This work addresses data-frugal learning challenges in fields like biology, chemistry, and physics, providing theoretical insights into spectral regularization, but it is incremental as it builds on prior empirical studies.
The paper tackles the problem of learning over combinatorial spaces with limited labeled data by regularizing the spectral representation of pseudo-Boolean functions, showing that this approach reshapes the loss landscape and achieves statistically optimal generalization under certain conditions, with empirical results demonstrating better performance compared to baselines in data-scarce real-world problems.
Data-driven machine learning models are being increasingly employed in several important inference problems in biology, chemistry, and physics which require learning over combinatorial spaces. Recent empirical evidence (see, e.g., [1], [2], [3]) suggests that regularizing the spectral representation of such models improves their generalization power when labeled data is scarce. However, despite these empirical studies, the theoretical underpinning of when and how spectral regularization enables improved generalization is poorly understood. In this paper, we focus on learning pseudo-Boolean functions and demonstrate that regularizing the empirical mean squared error by the L_1 norm of the spectral transform of the learned function reshapes the loss landscape and allows for data-frugal learning, under a restricted secant condition on the learner's empirical error measured against the ground truth function. Under a weaker quadratic growth condition, we show that stationary points which also approximately interpolate the training data points achieve statistically optimal generalization performance. Complementing our theory, we empirically demonstrate that running gradient descent on the regularized loss results in a better generalization performance compared to baseline algorithms in several data-scarce real-world problems.