Online Learning

Understanding minimal assumptions that enable learning and generalization is perhaps the central question of learning theory. Several celebrated results in statistical learning theory, such as the VC theorem and Littlestone's characterization of online learnability, establish conditions on the hypothesis class that allow for learning under independent data and adversarial data, respectively. Building upon recent work bridging these extremes, we study sequential decision making under distributional adversaries that can adaptively choose data-generating distributions from a fixed family $U$ and ask when such problems are learnable with sample complexity that behaves like the favorable independent case. We provide a near complete characterization of families $U$ that admit learnability in terms of a notion known as generalized smoothness i.e. a distribution family admits VC-dimension-dependent regret bounds for every finite-VC hypothesis class if and only if it is generalized smooth. Further, we give universal algorithms that achieve low regret under any generalized smooth adversary without explicit knowledge of $U$. Finally, when $U$ is known, we provide refined bounds in terms of a combinatorial parameter, the fragmentation number, that captures how many disjoint regions can carry nontrivial mass under $U$. These results provide a nearly complete understanding of learnability under distributional adversaries. In addition, building upon the surprising connection between online learning and differential privacy, we show that the generalized smoothness also characterizes private learnability under distributional constraints.

Continuous stress forecasting could potentially contribute to lifestyle interventions. This paper presents a novel, explainable, and individualized approach for stress prediction using physiological data from consumer-grade smartwatches. We develop a time series forecasting model that leverages multivariate features, including heart rate variability, activity patterns, and sleep metrics, to predict stress levels across 16 temporal horizons (History window: 3, 5, 7, 9 days; forecasting window: 1, 3, 5, 7 days). Our evaluation involves 16 participants monitored for 10-15 weeks. We evaluate our approach across 16 participants, comparing against state-of-the-art time series models (Informer, TimesNet, PatchTST) and traditional baselines (CNN, LSTM, CNN-LSTM) across multiple temporal horizons. Our model achieved performance with an MSE of 0.053, MAE of 0.190, and RMSE of 0.226 in optimal settings (5-day input, 1-day prediction). A comparison with the baseline models shows that our model outperforms TimesNet, PatchTST, CNN-LSTM, LSTM, and CNN under all conditions, representing improvements of 36.9%, 25.5%, and 21.5% over the best baseline. According to the explanability analysis, sleep metrics are the most dominant and consistent stress predictors (importance: 1.1, consistency: 0.9-1.0), while activity features exhibit high inter-participant variability (0.1-0.2). Most notably, the model captures individual-specific patterns where identical features can have opposing effects across users, validating its personalization capabilities. These findings establish that consumer wearables, combined with adaptive and interpretable deep learning, can deliver relevant stress assessment adapted to individual physiological responses, providing a foundation for scalable, continuous, explainable mental health monitoring in real-world settings.

As humans increasingly rely on multiround conversational AI for high stakes decisions, principled frameworks are needed to ensure such interactions reliably improve decision quality. We adopt a human centric view governed by two principles: counterfactual harm, ensuring the AI does not undermine human strengths, and complementarity, ensuring it adds value where the human is prone to err. We formalize these concepts via user defined rules, allowing users to specify exactly what harm and complementarity mean for their specific task. We then introduce an online, distribution free algorithm with finite sample guarantees that enforces the user-specified constraints over the collaboration dynamics. We evaluate our framework across two interactive settings: LLM simulated collaboration on a medical diagnostic task and a human crowdsourcing study on a pictorial reasoning task. We show that our online procedure maintains prescribed counterfactual harm and complementarity violation rates even under nonstationary interaction dynamics. Moreover, tightening or loosening these constraints produces predictable shifts in downstream human accuracy, confirming that the two principles serve as practical levers for steering multi-round collaboration toward better decision quality without the need to model or constrain human behavior.

Omniprediction is a learning problem that requires suboptimality bounds for each of a family of losses $\mathcal{L}$ against a family of comparator predictors $\mathcal{C}$. We initiate the study of omniprediction in a multiclass setting, where the comparator family $\mathcal{C}$ may be infinite. Our main result is an extension of the recent binary omniprediction algorithm of [OKK25] to the multiclass setting, with sample complexity (in statistical settings) or regret horizon (in online settings) $\approx \varepsilon^{-(k+1)}$, for $\varepsilon$-omniprediction in a $k$-class prediction problem. En route to proving this result, we design a framework of potential broader interest for solving Blackwell approachability problems where multiple sets must simultaneously be approached via coupled actions.

We investigate the recently introduced model of learning with improvements, where agents are allowed to make small changes to their feature values to be warranted a more desirable label. We extensively extend previously published results by providing combinatorial dimensions that characterize online learnability in this model, by analyzing the multiclass setup, learnability in a bandit feedback setup, modeling agents' cost for making improvements and more.

This paper proposes a multi-class online fuzzy classifier for dynamic environments. A fuzzy classifier comprises a set of fuzzy if-then rules where human users determine the antecedent fuzzy sets beforehand. In contrast, the consequent real values are determined by learning from training data. In an online framework, not all training dataset patterns are available beforehand. Instead, only a few patterns are available at a time step, and the subsequent patterns become available at the following time steps. The conventional online fuzzy classifier considered only two-class problems. This paper investigates the extension to the conventional fuzzy classifiers for multi-class problems. We evaluate the performance of the multi-class online fuzzy classifiers through numerical experiments on synthetic dynamic data and also several benchmark datasets.

Immersive applications such as virtual and augmented reality impose stringent requirements on frame rate, latency, and synchronization between physical and virtual environments. To meet these requirements, an edge server must render panoramic content, predict user head motion, and transmit a portion of the scene that is large enough to cover the user viewport while remaining within wireless bandwidth constraints. Each portion produces two feedback signals: prediction feedback, indicating whether the selected portion covers the actual viewport, and transmission feedback, indicating whether the corresponding packets are successfully delivered. Prior work models this problem as a multi-armed bandit with two-level bandit feedback, but fails to exploit the fact that prediction feedback can be retrospectively computed for all candidate portions once the user head pose is observed. As a result, prediction feedback constitutes full-information feedback rather than bandit feedback. Motivated by this observation, we introduce a two-level hybrid feedback model that combines full-information and bandit feedback, and formulate the portion selection problem as an online learning task under this setting. We derive an instance-dependent regret lower bound for the hybrid feedback model and propose AdaPort, a hybrid learning algorithm that leverages both feedback types to improve learning efficiency. We further establish an instance-dependent regret upper bound that matches the lower bound asymptotically, and demonstrate through measurements on an end-to-end testbed that AdaPort outperforms state-of-the-art learning-based baselines as well as the heuristic minimum scene delivery scheme.

This article considers an online version of conformal inference, called adaptive conformal inference [ACI] and introduced by Gibbs and Candès (2021): prediction sets are issued sequentially, after observing features and before the outcomes are revealed. These sets are evaluated both in terms of validity (the fraction of rounds where the outcome was lying in the prediction set) and efficiency (the average lengths of the prediction sets). The two criteria point to different directions (validity favors larger sets). We also target a wide range of scenarios, with exchangeable data and arbitrary data (lack of any stochastic guarantees) as two extremes. A series of existing strategies for ACI typically guarantee that empirical coverage converges to the desired level for arbitrary sequences, but they generally lack simultaneous efficiency guarantees. To provide a unified study, we first formulate ACI as a repeated two-player game with finite action sets and vector-valued payoffs encoding validity and efficiency. Building on this reformulation, we introduce a strategy based on Blackwell approachability and on its opportunistic extension by Bernstein et al. (2014) that ensures validity while adapting the efficiency of the prediction intervals to the underlying degree of stochasticity of the opponent player. The resulting guarantee is "best of many worlds": it recovers the relevant efficiency guarantees in exchangeable and adversarial settings, and provides guarantees in intermediate settings that arise in typical applications such as the forecasting of time series.

MOOCs, Smart Teaching (ST), and AI-assisted learning support different stages of higher education, yet a unified quantitative basis for analyzing their contributions, course design, and resource allocation remains limited. This study develops a mathematical model integrating MOOC-based pre-class learning, ST-based in-class adaptation, and AI-assisted post-class personalization. Learner mastery is represented as a bounded multidimensional state, with a common exponential learning-response function describing how instructional resources reduce remaining knowledge gaps. The stages differ in their allocation rules: predefined course-content emphasis for MOOCs, feedback-driven class-level adaptation for ST, and individualized gap-based allocation for AI-assisted learning. Simulations across 100 independently generated classes demonstrate stable cumulative progression and quantify stage-wise gains, while budget analysis reveals diminishing returns from additional AI support. For a fixed learner and learning mechanism, varying course-content emphasis shows a strong association between course-learner alignment and post-MOOC mastery. An optimal AI allocation is also derived under a fixed budget: supported components reach a common residual mastery gap, while components below this threshold receive no resources. A controlled comparison yields over 14% greater learning gain than proportional allocation. These analyses make the instructional process quantitatively analyzable and provide a basis for examining course-learner fit and coordinating limited learning resources. The model offers an analytical foundation for instructional decisions, with practical application requiring empirical estimation of learner states and calibration of learning-response parameters.

We study budget pacing in repeated first-price auctions when an advertiser's private-value distributions change over time and the stationary competing-bid distribution is unknown. We ask how a feasible expenditure plan should enter online bid shading, learning, and hard budget control. We establish a plan-to-performance decomposition for a plan-driven projected-dual policy. The policy uses any feasible expenditure plan as a soft target, learns an unknown stationary competing-bid CDF from thresholds revealed after each auction, and enforces the campaign budget on every sample path. Against a distribution-informed expected-budget fluid benchmark, the uniform-plan reward gap is $O(\sqrt T)+O(\mathcal W_T)$, where $\mathcal W_T$ measures heterogeneity in private-value distributions. With a supplied feasible plan, the global gap decomposes into a one-sided $O(\sqrt{T})$ fixed-plan execution term and a plan-mismatch term bounded by $(b/2a)PlanError$. The same analysis provides guarantees for strict and relaxed period-cap comparators, exact recovery of the global benchmark under a specific allowance vector, and separate lower bounds establishing the necessity of the temporal-heterogeneity and Plan Error terms. An upstream planner can translate forecasts or managerial priorities into a feasible spending trajectory, while the online controller adapts bids using realized thresholds and expenditures. The guarantee is modular: it evaluates the final normalized or projected plan through $PlanError$. A specific forecasting model can be linked to the guarantee by establishing how its primitive estimation errors propagate to this plan-quality metric.

We initiate the study of structured Stackelberg games, a novel form of strategic interaction between a leader and a follower where contextual information can be predictive of the follower's (unknown) type. Motivated by applications such as security games and AI safety, we show how this additional structure can help the leader learn a utility-maximizing policy in both the online and distributional settings. In the online setting, we first prove that standard learning-theoretic measures of complexity do not characterize the difficulty of the leader's learning task. Notably, we find that there exists a learning-theoretic measure of complexity, analogous to the Littlestone dimension in online classification, that tightly characterizes the leader's instance-optimal regret. We term this the Stackelberg-Littlestone dimension, and leverage it to provide a provably optimal online learning algorithm. In the distributional setting, we provide analogous results by showing that two new dimensions control the sample complexity upper- and lower-bound.

In this book, I introduce the concepts of online learning through a modern view based on convex optimization. Here, online learning refers to the framework of regret minimization under worst-case assumptions. I attempted to unify all the literature as instantiations of Online Mirror Descent and Follow-the-Regularized-Leader (and their variants). I paid particular attention to the issue of tuning the parameters of the algorithms, through adaptive and parameter-free online learning algorithms. The bandit setting is also briefly discussed, touching on the problem of adversarial and stochastic multi-armed bandits. Building on fundamental algorithms and concepts, I also cover advanced topics, including black-box reductions, saddle-point optimization, sequential investment, and non-stationary forms of regret analysis. Finally, I conclude with a selection of applications of online learning to domains far from it, such as generalization theory and concentration inequalities. I attempted to maintain an informal, yet mathematically rigorous, tone throughout the book. Moreover, all the included proofs have been carefully chosen to be as simple and as short as possible. This also means that sometimes I have added one or two additional assumptions, just to simplify the proofs.