ScoreStop: Gradient-based early stopping using functional score tests

Gradient-flow optimization is usually viewed as an algorithmic procedure for minimizing empirical loss, with training duration selected by validation or heuristic early stopping rules. We develop a statistical inference framework for gradient-flow training. We show that whenever fitted values evolve through a time-invariant positive semidefinite training operator, the output at each time is equivalent to the best linear unbiased predictor under a corresponding random-effects model. Training time then becomes a variance-component parameter governing variance reallocation from residual noise to structured signal. This turns two training decisions into inferential problems: whether training is needed becomes a variance-component test for signal beyond initialization, and how long to train becomes restricted maximum likelihood (REML) estimation of the training-time variance component. We show that the REML-guided early stopping rule selects the time at which optimized spectral losses become decorrelated from the training-operator eigenvalues. The asymptotic prediction optimality of the REML-guided early stopping time is established for fixed-design in-sample risk and random-design out-of-sample risk. Deep learning models in fixed-kernel gradient regimes provide canonical instantiations for our results. Numerical experiments and a UK Biobank proteomics application show competitive accuracy of the REML-guided early stopping time with reduced reliance on validation splits and repeated checkpoint evaluation.

Risk scores are an interpretable and actionable class of machine learning models with applications in medicine, insurance, and risk management. Unlike most computational methods, risk scores are designed to be computed by a human by attributing points to a data sample based on a limited set of criteria. The most common approaches for generating risk scores use linear regressions to estimate the effect of selected variables. We propose a simple and effective approach towards building compact and predictive risk scores. We provide an algorithm based on gradient boosting that is capable of modeling nonlinear effects, along with a C++ implementation with Python and R bindings. Through extensive empirical evaluation on twelve tabular datasets spanning regression, classification, and time-to-event tasks, we show that our method achieves competitive predictive performance while producing substantially more compact scores than regression-based alternatives, with 60% fewer rules for classification tasks and 16% fewer rules for time-to-event tasks on average, compared to AutoScore.

A fundamental problem in machine learning is understanding the effect of early stopping on the parameters obtained and the generalization capabilities of the model. Even for linear models, the effect is not fully understood for arbitrary learning rates and data. In this paper, we analyze the dynamics of discrete full batch gradient descent for linear regression. With minimal distributional assumptions, we characterize the trajectory of the parameters and the expected excess risk. Using this characterization, we show that when training with any learning rate schedule and finite time horizon, the early stopped solution is equivalent to the minimum norm solution for a generalized ridge regression problem. We also prove that early stopping is beneficial for generic data with arbitrary spectrum and for a wide variety of learning rate schedules. We provide an estimate for the optimal stopping time and empirically demonstrate the accuracy of our estimate.