We study logistic regression for continual learning with linear predictors in a high-dimensional statistical setting. Existing theoretical work has largely focused on continual linear regression with variants of L2 regularization, motivated by their analytical simplicity and connection to the kernel regime of neural networks. We first derive the optimal L2 regularization strength for continual linear classification and characterize its generalization performance, recovering behavior analogous to prior results for continual linear regression. Next, we examine the effect of L2 regularization in deeper models using a minimal two-layer diagonal linear network. In the single-task setting, L2 regularization on the weights of such models is known to promote sparsity in the resulting predictor, yielding a rich, sparse feature-learning regime beyond the kernel regime (a similar effect arises in two-layer ReLU networks). However, in continual learning, where L2 regularization is typically centered on previous-task weights, this sparsity bias is no longer preserved. We therefore propose a weighted variant of L2 regularization that extends this beneficial sparsity bias to continual learning. We show theoretically and empirically that, for a sufficiently sparse teacher, our variant converges substantially faster than directly learning a linear predictor with conventional continual L2 or L1 regularization, trading some long-horizon bias for markedly improved performance over moderate task horizons.
Equivariant neural networks incorporate known symmetries of the data, such as permutations, rotations, or translations, directly into their architecture, often improving sample efficiency and generalization. However, strict equivariance can make optimization difficult, for example, when the data only approximately satisfy the assumed symmetry, or when the equivariance constraints induce a challenging loss landscape. Recent studies have shown that relaxing equivariance during training by introducing non-equivariant dense linear layers can ease optimization in such cases and improve the performance of the resulting strictly equivariant model, after removing the relaxation layers at test time. While effective, this approach incurs a significant computational overhead, limiting its applicability in scalable architectures. In this work, we propose a more efficient constraint-relaxation method based on positional encoding (PE). Our framework injects a positional encoding with learnable scale that breaks the model's original symmetry. This positional signal is gradually annealed to zero, recovering the original equivariant model by the end of training. Empirically, we demonstrate significant improvements for translational and rotational equivariant models with only marginal compute and memory overhead.