Threshold-Based Early Stopping of Accumulations in Neural Networks with Binary Activation

Training binary neural networks (BNNs) from scratch is dominated by the straight-through estimator (STE), whose forward/backward mismatch produces severe accuracy degradation as networks deepen. We study an orthogonal axis: when and where binarization is enforced during training. We introduce StoMPP (Stochastic Masked Partial Progressive Binarization), which gradually replaces clipped weights and activations with their hard binary counterparts layer by layer from input to output, using stochastic partial masks with soft refresh. StoMPP delivers two complementary benefits. As a standalone training rule, it provides a fully STE-free procedure that improves over vanilla STE with gains that grow with depth (ResNet-50 BNN: +18.0/+13.5/+3.8 on CIFAR-10/100/ImageNet), and the pattern holds across ResNet-18/34/50, MobileNetV2, and BERT fine-tuning. Composed with surrogate gradients by applying STE only to frozen entries, it reaches +27.1/+19.8/+17.7 over vanilla STE on the same setting. Underlying both regimes is a single mechanistic finding: progression order is decisive. Forward layerwise progression prevents depth collapse, reverse progression collapses to near-chance, and binary-weight networks (without binary activations) are insensitive to order. We trace this asymmetry to activation-induced gradient blockades: a committed binary activation severs gradient flow upstream, and ordering controls when these blockades form. To isolate the progression's contribution from any benefit conferred by STE, we conduct all ablations in the STE-free regime; the resulting characterization (schedule, refresh, ordering, dynamics) thus reflects the progression itself rather than its interaction with surrogate gradients.

Early-exiting neural networks enable adaptive inference by allowing inputs to exit at intermediate classifiers, reducing computation for easy samples while maintaining high accuracy. In practice, exits can be trained sequentially by incrementally adding them to a shared backbone; however, this sequential training can cause newly introduced exits to interfere with previously learned ones, degrading the performance of earlier classifiers. We address this problem by retaining the knowledge embedded in existing exits while allowing new ones to specialize. We propose two alternative approaches that operate at different levels of the model. The first constrains learning by protecting parameters that are important for previously trained exits, while the second preserves the output distributions of earlier exits as the network adapts. These alternatives directly reflect the stability-plasticity trade-off studied in continual learning. Accordingly, we leverage \textit{Elastic Weight Consolidation} to constrain critical weights and \textit{Learning without Forgetting} to preserve output distributions. Experiments on standard benchmarks show that our approaches consistently improve early-exit performance, achieving higher accuracy over existing sequential training methods and significant performance speedups at low computational budgets.

We study compute reduction in neural networks through a unified partial versus full computation view, captured by one-shot magnitude pruning in the static regime and early exit in the adaptive regime. In an asymptotic single-neuron model, we prove a concentration theorem for one-shot magnitude pruning with explicit rates. We also introduce the conditional perceptron for early exit and show that its excess generalization error decays as a power of the compute gap, with an exponent that grows to infinity as the alignment between partial and full computations tends to one. We then extend the analysis to deep networks, characterizing how pruning-induced distortions accumulate with depth and deriving a corresponding compute-accuracy tradeoff for frozen-backbone early exit under a neural network Gaussian process model. Numerical simulations corroborate the predicted scaling laws.