We study streaming data with categorical features where the vocabulary of categorical feature values is changing and can even grow unboundedly over time. Feature hashing is commonly used as a pre-processing step to map these categorical values into a feature space of fixed size before learning their embeddings. While these methods have been developed and evaluated for offline or batch settings, in this paper we consider online settings. We show that deterministic embeddings are sensitive to the arrival order of categories and suffer from forgetting in online learning, leading to performance deterioration. To mitigate this issue, we propose a probabilistic hash embedding (PHE) model that treats hash embeddings as stochastic and applies Bayesian online learning to learn incrementally from data. Based on the structure of PHE, we derive a scalable inference algorithm to learn model parameters and infer/update the posteriors of hash embeddings and other latent variables. Our algorithm (i) can handle an evolving vocabulary of categorical items, (ii) is adaptive to new items without forgetting old items, (iii) is implementable with a bounded set of parameters that does not grow with the number of distinct observed values on the stream, and (iv) is invariant to the item arrival order. Experiments in classification, sequence modeling, and recommendation systems in online learning setups demonstrate the superior performance of PHE while maintaining high memory efficiency (consumes as low as 2~4 memory of a one-hot embedding table). Supplementary materials are at https://github.com/aodongli/probabilistic-hash-embeddings

We study online convex optimization on $\ell_p$-balls in $\mathbb{R}^d$ for $p > 2$. While always sub-linear, the optimal regret exhibits a shift between the high-dimensional setting ($d > T$), when the dimension $d$ is greater than the time horizon $T$ and the low-dimensional setting ($d \leq T$). We show that Follow-the-Regularised-Leader (FTRL) with time-varying regularisation which is adaptive to the dimension regime is anytime optimal for all dimension regimes. Motivated by this, we ask whether it is possible to obtain anytime optimality of FTRL with fixed non-adaptive regularisation. Our main result establishes that for separable regularisers, adaptivity in the regulariser is necessary, and that any fixed regulariser will be sub-optimal in one of the two dimension regimes. Finally, we provide lower bounds which rule out sub-linear regret bounds for the linear bandit problem in sufficiently high-dimension for all $\ell_p$-balls with $p \geq 1$.

The growth of Massive Open Online Courses (MOOCs) presents significant challenges for personalized learning, where concept recommendation is crucial. Existing approaches typically rely on heterogeneous information networks or knowledge graphs to capture conceptual relationships, combined with knowledge tracing models to assess learners' cognitive states. However, these methods face significant limitations due to their dependence on high-quality structured knowledge graphs, which are often scarce in real-world educational scenarios. To address this fundamental challenge, this paper proposes CLLMRec, a novel framework that leverages Large Language Models through two synergistic technical pillars: Semantic Alignment and Prerequisite Knowledge Distillation. The Semantic Alignment component constructs a unified representation space by encoding unstructured textual descriptions of learners and concepts. The Prerequisite Knowledge Distillation paradigm employs a teacher-student architecture, where a large teacher LLM (implemented as the Prior Knowledge Aware Component) extracts conceptual prerequisite relationships from its internalized world knowledge and distills them into soft labels to train an efficient student ranker. Building upon these foundations, our framework incorporates a fine-ranking mechanism that explicitly models learners' real-time cognitive states through deep knowledge tracing, ensuring recommendations are both structurally sound and cognitively appropriate. Extensive experiments on two real-world MOOC datasets demonstrate that CLLMRec significantly outperforms existing baseline methods across multiple evaluation metrics, validating its effectiveness in generating truly cognitive-aware and personalized concept recommendations without relying on explicit structural priors.

Educational simulations have long been recognized as powerful tools for enhancing learning outcomes, yet their creation has traditionally required substantial resources and technical expertise. This paper introduces MicroSims a novel framework for creating lightweight, interactive educational simulations that can be rapidly generated using artificial intelligence, universally embedded across digital learning platforms, and easily customized without programming knowledge. MicroSims occupy a unique position at the intersection of three key innovations: (1) standardized design patterns that enable AI-assisted generation, (2) iframe-based architecture that provides universal embedding and sandboxed security, and (3) transparent, modifiable code that supports customization and pedagogical transparency. We present a comprehensive framework encompassing design principles, technical architecture, metadata standards, and development workflows. Drawing on empirical research from physics education studies and meta-analyses across STEM disciplines, we demonstrate that interactive simulations can improve conceptual understanding by up to 30-40\% compared to traditional instruction. MicroSims extend these benefits while addressing persistent barriers of cost, technical complexity, and platform dependence. This work has significant implications for educational equity, and low-cost intelligent interactive textbooks that enabling educators worldwide to create customized, curriculum-aligned simulations on demand. We discuss implementation considerations, present evidence of effectiveness, and outline future directions for AI-powered adaptive learning systems built on the MicroSim foundation.

We study online learning with the general notion of transductive regret, that is regret with modification rules applying to expert sequences (as opposed to single experts) that are representable by weighted finite-state transducers. We show how transductive regret generalizes existing notions of regret, including: (1) external regret; (2) internal regret; (3) swap regret; and (4) conditional swap regret. We present a general and efficient online learning algorithm for minimizing transductive regret. We further extend that to design efficient algorithms for the time-selection and sleeping expert settings. A by-product of our study is an algorithm for swap regret, which, under mild assumptions, is more efficient than existing ones, and a substantially more efficient algorithm for time selection swap regret.

We develop a nonparametric and online learning algorithm that estimates the triggering functions of a multivariate Hawkes process (MHP).

Online learning is a general framework for sequential prediction.

The algorithm's objective is to minimize regret, which is the total loss of all rounds relative to

We study the problem of completing a binary matrix in an online learning setting. On each trial we predict a matrix entry and then receive the true entry.

UP A is widely used in practice.