Incremental Learning in Mirror Flows

We study the max-margin solutions reached by mirror flow in deep neural networks with homogeneous activation functions. Extending classical results on gradient flow, we derive a novel balance equation for mirror flow from convex duality, enabling a characterization of the horizon function governing the induced margin. We further establish max-margin characterizations together with convergence rates and norm growth estimates. Finally, we support our theory through experiments on synthetic datasets and standard vision tasks. Concretely, we show that: (1) distinct non-homogeneous mirror maps can induce the same max-margin solution; (2) convergence can be extremely slow, including exponentially slow regimes; and (3) although all considered mirror maps exhibit feature learning, they can produce markedly different representations, ranging from sparse to dense neuron activations. Together, these results provide a unified perspective on sparse and dense feature learning in homogeneous neural networks, highlighting how mirror maps shape both optimization dynamics and the geometry of the learned classifiers.

In this work we introduce a novel approach to domain incremental learning, adapting models over time to evolving, non-stationary data. In contrast to other works, we do not attempt to avoid catastrophic forgetting, but rather allow it and exploit it. Our model combines a main task head with a self-supervised masked autoencoder (MAE) head. We then learn domain-specific LoRA adapters during incremental training. Each adapter specializes to its domain, naturally inducing forgetting on other domains in both heads. At inference, we perform online test-time training on the self-supervised MAE head to identify which LoRAs best matches the current input, so the model can `remember' the domain again. Our scheme is especially well-suited to real-world streaming data, such as video, where consecutive samples are highly correlated and domain shifts are gradual. We demonstrate our method on domain-incremental action recognition and semantic segmentation tasks.

Sequence convergence to a boundary Karush--Kuhn--Tucker (KKT) point has long remained unclear for nonconvex mirror descent with Legendre kernels. The difficulty arises from the blow-up of the gradient of the Legendre kernel at the boundary. Recent work~\cite{dingtoh2026nonkkt} shows that mirror descent can accumulate at non-KKT boundary points despite decreasing objective values, precluding a convergence guarantee to KKT points in general. Despite this negative result, mirror descent remains effective in many real applications. Motivated by this contrast, we address the boundary difficulty directly and establish KKT convergence of mirror descent for a broad class of structured nonconvex problems. We analyze mirror descent in reparameterized variables, where the Hessian metric is flattened and remains nondegenerate as the boundary is approached. Under extension and definability conditions jointly coupling the objective, the Legendre kernel, and the feasible region, the reparameterized sequence has finite length and converges, thereby recovering convergence to a KKT point of the original sequence. Our general framework applies to some concrete instances: Shannon entropy, Fermi--Dirac entropy, and power kernels on polyhedron.