We study online and transductive online learning in settings where the learner can interact with the concept class only via Empirical Risk Minimization (ERM) or weak consistency oracles on arbitrary subsets of the instance domain. This contrasts with standard online models, where the learner has full knowledge of the concept class. The ERM oracle returns a hypothesis that minimizes the loss on a given subset, while the weak consistency oracle returns only a binary signal indicating whether the subset is realizable by a concept in the class. The learner's performance is measured by the number of mistakes and oracle calls. In the standard online setting with ERM access, we establish tight lower bounds in both the realizable and agnostic cases: $\Omega(2^{d_\mathrm{LD}})$ mistakes and $\Omega(\sqrt{T 2^{d_\mathrm{LD}}})$ regret, respectively, where $T$ is the number of timesteps and $d_\mathrm{LD}$ is the Littlestone dimension of the class. We further show how existing results for online learning with ERM access translate to the setting with a weak consistency oracle, at the cost of increasing the number of oracle calls by $O(T)$. We then consider the transductive online model, where the instance sequence is known in advance but labels are revealed sequentially. For general Littlestone classes, we show that the optimal mistake bound in the realizable case and in the agnostic case can be achieved using $O(T^{d_\mathrm{VC}+1})$ weak consistency oracle calls, where $d_\mathrm{VC}$ is the VC dimension of the class. On the negative side, we show that $\Omega(T)$ weak consistency queries are necessary for transductive online learnability, and that $\Omega(T)$ ERM queries are necessary to avoid exponential dependence on the Littlestone dimension.

Federated Learning (PFL) has emerged, where personalized models are trained for individual clients.

Relevance modeling in e-commerce search remains challenged by semantic gaps in term-matching methods (e.g., BM25) and neural models' reliance on the scarcity of domain-specific hard samples. We propose ADORE, a self-sustaining framework that synergizes three innovations: (1) A Rule-aware Relevance Discrimination module, where a Chain-of-Thought LLM generates intent-aligned training data, refined via Kahneman-Tversky Optimization (KTO) to align with user behavior; (2) An Error-type-aware Data Synthesis module that auto-generates adversarial examples to harden robustness; and (3) A Key-attribute-enhanced Knowledge Distillation module that injects domain-specific attribute hierarchies into a deployable student model. ADORE automates annotation, adversarial generation, and distillation, overcoming data scarcity while enhancing reasoning. Large-scale experiments and online A/B testing verify the effectiveness of ADORE. The framework establishes a new paradigm for resource-efficient, cognitively aligned relevance modeling in industrial applications.

Federated Learning (PFL) has emerged, where personalized models are trained for individual clients.

We propose an online learning framework for forecasting nonlinear spatio-temporal signals (fields). The method integrates (i) dimensionality reduction, here, a simple proper orthogonal decomposition (POD) projection; (ii) a generalized autoregressive model to forecast reduced dynamics, here, a reservoir computer; (iii) online adaptation to update the reservoir computer (the model), here, ensemble sequential data assimilation. We demonstrate the framework on a wake past a cylinder governed by the Navier-Stokes equations, exploring the assimilation of full flow fields (projected onto POD modes) and sparse sensors. Three scenarios are examined: a naïve physical state estimation; a two-fold estimation of physical and reservoir states; and a three-fold estimation that also adjusts the model parameters. The two-fold strategy significantly improves ensemble convergence and reduces reconstruction error compared to the naïve approach. The three-fold approach enables robust online training of partially-trained reservoir computers, overcoming limitations of a priori training. By unifying data-driven reduced order modelling with Bayesian data assimilation, this work opens new opportunities for scalable online model learning for nonlinear time series forecasting.

Continual reinforcement learning (CRL) refers to a naturalistic setting where an agent needs to endlessly evolve, by trial and error, to solve multiple tasks that are presented sequentially. One of the largest obstacles to CRL is that the agent may forget how to solve previous tasks when learning a new task, known as catastrophic forgetting. In this paper, we propose to address this challenge by planning with online world models. Specifically, we learn a Follow-The-Leader shallow model online to capture the world dynamics, in which we plan using model predictive control to solve a set of tasks specified by any reward functions. The online world model is immune to forgetting by construction with a proven regret bound of $\mathcal{O}(\sqrt{K^2D\log(T)})$ under mild assumptions. The planner searches actions solely based on the latest online model, thus forming a FTL Online Agent (OA) that updates incrementally. To assess OA, we further design Continual Bench, a dedicated environment for CRL, and compare with several strong baselines under the same model-planning algorithmic framework. The empirical results show that OA learns continuously to solve new tasks while not forgetting old skills, outperforming agents built on deep world models with various continual learning techniques.