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Data-driven machine learning models are being increasingly employed inseveral important inference problems in biology, chemistry, and physics whichrequire learning over combinatorial spaces. Recent empirical evidence (see,e.g., [1], [2], [3]) suggests that regularizing the spectral representation ofsuch models improves their generalization power when labeled data is scarce.However, despite these empirical studies, the theoretical underpinning of whenand how spectral regularization enables improved generalization is poorlyunderstood. In this paper, we focus on learning pseudo-Boolean functions anddemonstrate that regularizing the empirical mean squared error by the L_1 normof the spectral transform of the learned function reshapes the loss landscapeand allows for data-frugal learning, under a restricted secant condition on thelearner's empirical error measured against the ground truth function. Under aweaker quadratic growth condition, we show that stationary points which alsoapproximately interpolate the training data points achieve statisticallyoptimal generalization performance. Complementing our theory, we empiricallydemonstrate that running gradient descent on the regularized loss results in abetter generalization performance compared to baseline algorithms in severaldata-scarce real-world problems.

Spectral Regularization Allows Data-frugal Learning over Combinatorial Spaces