Layer-local self-supervised objectives remove the backward pass between layers, but almost all still compute their loss over a batch of samples and are therefore local in space but not in time. We introduce Temporal Regularized Learning (TeReL), a spatially and temporally local learning rule for self-supervised representation learning from sequential input streams. The key idea is to instantiate a VICReg-style objective with temporally adjacent positives and non-differentiated running population statistics, yielding a per-neuron online loss with bounded state. The objective is a soft form of Slow Feature Analysis, and its gradient is a Hebbian update of the feedforward weights and an anti-Hebbian update of one lateral matrix per layer. The aim of the experiments is to show that the rule works in a small network, not to compete with backpropagation. On a class-chunked MNIST stream, a two-layer fully connected network trained with batch size one outperforms random features (96.48% versus 95.09% linear-probe accuracy) and classical SFA, and a variant that relaxes both locality constraints is comparable to supervised backpropagation. Because training and inference use the same computation, the network can keep learning from unlabeled data during inference without loss of accuracy.
Elastic Weight Consolidation (EWC) is a widely used regularization-based technique for continual learning, protecting performance on old tasks with a quadratic penalty whose empirical Fisher is usually justified through probabilistic curvature. We instead introduce Continual Learning as an Optimization Constraint (CLOC): new training must preserve chosen functionals of learned behavior. EWC's empirical Fisher is exactly the first-order form of this constraint when the preserved functional is each sample's loss. This derivation exposes a flaw: EWC implicitly weights each sample by the scale of its loss gradient, so well-fitted samples lose importance. EWC-DR observed this failure in classification; CLOC identifies its general cause. Improved Elastic Weight Consolidation (IEWC) penalizes output changes according to the first-order approximation of each old sample's loss level set, removing the weighting for any differentiable loss and output geometry, and prevents corrupt samples from dominating the importance matrix. Across classification, segmentation, language, forecasting, and diffusion, IEWC variants match or improve on EWC and outperform EWC-DR. In a held-out evaluation with validation-selected hyperparameters, uniform IEWC improves ImageNet-R accuracy over matched EWC by 2.00 percentage points (95% CI 0.93-3.07).