Theory-Guided Instance-Dependent Early Stopping via Second-Order Loss Dynamics

Binary neural networks are very attractive for constrained deployment, enabling small footprint and low-power inference. For binary activations, the dot products become sign-controlled additions or subtractions, but the number of operations is unchanged. Indeed, every neuron or output channel still accumulates all of its input, even though only the sign will be retained, which is often wasteful. As the accumulation progresses, the running partial sum frequently drifts so far from zero that its final sign becomes highly predictable long before the last term is reached; every contribution evaluated after that point changes the value of the sum but not the final output activation. This paper turns this observation into a post-training early-stopping mechanism. We characterize the behavior of the running accumulations on the training dataset and use this information to predict the final sign as soon as possible. No model parameter is retrained. We count the number of operations under an idealized ordering of weights. On VGG11 applied to the CIFAR-10 dataset, the method removes $86.6\%$ of the accumulation terms of the deepest convolution for a $0.37$-point accuracy drop, and $25\%$ of the full-network arithmetic when used on the three deepest convolutions simultaneously, for a $1.36$-point drop.

Gradient boosted decision trees require a stopping rule to avoid overfitting. The standard rule monitors a validation loss and stops if the loss fails to improve for a fixed patience period. However, the patience parameter has no interpretable scale and validation losses can be noisy or implicitly defined by a user-specified gradient. We propose ScoreStop, a gradient-based early-stopping rule that casts the stopping decision at each iteration as a test of the null hypothesis that the current predictor is the population risk minimizer. We use a functional score test, computed on validation data, with a statistic that is scale-invariant in the update direction, with a known asymptotic distribution under the null. Because our test uses gradients rather than loss values, the same construction applies to implicit losses such as LambdaRank, and data-dependent losses such as Cox regression via influence functions. In synthetic experiments and real-data benchmarks, we show that ScoreStop is competitive with loss-based methods.

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.