General ML · online learning

We study the active learning problem of fixed-confidence top-$k$ identification from noisy pairwise comparisons. In this problem, an algorithm sequentially chooses pairs of items to compare, observes the outcomes, and stops when it can return the set of top-$k$ items with error probability at most $\delta$. The objective is to design such a *$\delta$-correct* procedure that minimizes the expected number of comparisons (the sample complexity). This problem falls within the broader literature on fixed-confidence pure exploration in bandit models, where a common target is asymptotic optimality: the algorithm's expected sample complexity matches the information theoretic lower bound as $\delta \to 0$. Asymptotically optimal procedures have been developed for a range of fixed-confidence pure-exploration problems, however to the best of our knowledge, for top-$1$, or more generally top-$k$ identification from pairwise comparisons under latent utility models an asymptotically optimal algorithm has not been established. In this setting, we develop such an algorithm. We characterize the structure of the lower bound and formulate it as a saddle-point problem. This structure enables a computationally efficient primal--dual procedure that learns the asymptotically optimal comparison allocation online. We then construct an adaptive comparison-allocation algorithm that tracks the allocation learned by the primal--dual procedure and prove it is asymptotically optimal.

We consider an online learning problem in environments with multiple change points. In contrast to the single change point problem that is widely studied using classical "high confidence" detection schemes, the multiple change point environment presents new learning-theoretic and algorithmic challenges. Specifically, we show that classical methods may exhibit catastrophic failure (high regret) due to a phenomenon we refer to as endogenous confounding. To overcome this, we propose a new class of learning algorithms dubbed Anytime Tracking CUSUM (ATC). These are horizon-free online algorithms that implement a selective detection principle, balancing the need to ignore "small" (hard-to-detect) shifts, while reacting "quickly" to significant ones. We prove that the performance of a properly tuned ATC algorithm is nearly minimax-optimal; its regret is guaranteed to closely match a novel information-theoretic lower bound on the achievable performance of any learning algorithm in the multiple change point problem. Experiments on synthetic as well as real-world data validate the aforementioned theoretical findings.

We derive a robust update rule for the online infinite hidden Markov model (iHMM) for when the streaming data contains outliers and the model is misspecified. Leveraging recent advances in generalised Bayesian inference, we define robustness via the posterior influence function (PIF), and provide conditions under which the online iHMM has bounded PIF. Imposing robustness inevitably induces an adaptation lag for regime switching. Our method, which is called Batched Robust iHMM (BR-iHMM), balances adaptivity and robustness with two additional tunable parameters. Across limit order book data, hourly electricity demand, and a synthetic high-dimensional linear system, BR-iHMM reduces one-step-ahead forecasting error by up to 67% relative to competing online Bayesian methods. Together with theoretical guarantees of bounded PIF, our results highlight the practicality of our approach for both forecasting and interpretable online learning.

We give a very general and simple framework to incorporate predictions on requests for online covering problems in a rigorous and black-box manner. Our framework turns any online algorithm with competitive ratio $\rho(k, \cdot)$ depending on $k$, the number of arriving requests, into an algorithm with competitive ratio of $\rho(\eta, \cdot)$, where $\eta$ is the prediction error. With accurate enough prediction, the resulting competitive ratio breaks through the corresponding worst-case online lower bounds, and smoothly degrades as the prediction error grows. This framework directly applies to a wide range of well-studied online covering problems such as facility location, Steiner problems, set cover, parking permit, etc., and yields improved and novel bounds.

This paper studies \emph{preference-based reinforcement learning} (PbRL), where agents learn from comparative, trajectory-level feedback rather than numeric rewards. While PbRL has seen rapid empirical and theoretical progress, existing analyses are largely confined to restricted settings and fail to jointly capture the outcome-based and comparison-based nature of preference feedback. We prove that under a broad \emph{general function approximation} framework, PbRL admits a $\sqrt{T}$ regret guarantee. In particular, we introduce a simple and provably efficient algorithm, \emph{Recursive Trajectory-based Preference Q-Learning} (RTPQ), and establish its regret bound while explicitly accounting for the trajectory-level and comparative structure of preferences. Our analysis is characterized by a new complexity measure, the \emph{Dual Episodic Eluder Dimension} (DEED), which quantifies the intrinsic difficulty of PbRL. We show that for linear MDPs, the DEED scales as $\mathcal{O}(dH)$, yielding a regret bound of $\tilde{\mathcal{O}}(dH\sqrt{T}\max(H^{3/2},\,1/\kappa))$, where $\kappa$ is a problem-dependent constant. This bound is near-optimal up to horizon- and problem-dependent factors when compared to standard reward-based linear MDPs. In addition, our framework recovers the best-known regret bounds in the special cases of dueling bandits and standard outcome-based reinforcement learning. Overall, our results provide a general regret guarantee for PbRL with outcome-based preference feedback and broad function approximation.

Online reinforcement learning in non-episodic, finite-horizon MDPs remains underexplored and is challenged by the need to estimate returns to a fixed terminal time. Existing infinite-horizon methods, which often rely on discounted contraction, do not naturally account for this fixed-horizon structure. We introduce a modified Q-function: rather than targeting the full-horizon, we learn a K-step lookahead Q-function that truncates planning to the next K steps. To further improve sample efficiency, we introduce a thresholding mechanism: actions are selected only when their estimated K-step lookahead value exceeds a time-varying threshold. We provide an efficient tabular learning algorithm for this novel objective, proving it achieves fast finite-sample convergence: it achieves minimax optimal constant regret for $K=1$ and $\mathcal{O}(\max((K-1),C_{K-1})\sqrt{SAT\log(T)})$ regret for any $K \geq 2$. We numerically evaluate the performance of our algorithm under the objective of maximizing reward. Our implementation adaptively increases K over time, balancing lookahead depth against estimation variance. Empirical results demonstrate superior cumulative rewards over state-of-the-art tabular RL methods across synthetic MDPs and RL environments: JumpRiverswim, FrozenLake and AnyTrading. Code is provided on \href{https://github.com/jamie01713/K-Step-Lookahead}{github}.

Modern recommender systems are typically trained offline and deployed with parameters held fixed between periodic refreshes, yet user behavior can evolve substantially during deployment. This can cause ranking utility to degrade over time and makes it difficult to provide formal guarantees about recommendation quality. We propose **CARE**, an adaptive calibration framework that wraps an arbitrary backbone recommender and outputs variable-size recommendation sets with finite-sample performance guarantees over interaction streams. CARE combines (i) a loss-based monitoring module that localizes behavioral changes and triggers threshold recalibration, and (ii) an online aggregation rule that promotes compact recommendation sets by dynamically reweighting candidate set predictors. We provide theoretical results establishing finite-sample guarantees for utility-based risk control and bounds on the expected set size relative to the best constituent predictor. Experiments across multiple datasets and backbone models demonstrate that CARE improves robustness and maintains compact recommendation sets while preserving the desired statistical guarantees. The code and implementation are available in https://github.com/kalpiree/CARE.

Test-time adaptation (TTA) offers a compelling remedy for machine learning (ML) models that degrade under domain shifts, improving generalisation *on-the-fly* with only unlabelled samples. This flexibility suits real deployments, yet conventional evaluations unrealistically assume unbounded processing time, overlooking the accuracy-latency trade-off. As ML increasingly underpins latency-sensitive and user-facing use-cases, temporal pressure constrains the viability of adaptable inference; predictions arriving too late to act on are futile. We introduce *Tempora*, a framework for evaluating TTA under this pressure. It consists of temporal scenarios that model deployment constraints, evaluation protocols that operationalise measurement, and time-contingent utility metrics that quantify the accuracy-latency trade-off. We instantiate the framework with three such metrics: (1) *discrete* utility for asynchronous streams with hard deadlines, (2) *continuous* utility for interactive settings where value decays with latency, and (3) *amortised* utility for budget-constrained deployments. By applying Tempora to 11 TTA methods, we find that *rank instability* persists across 750+ temporal evaluations spanning diverse datasets, models, and hardware platforms; *i.e.,* conventional rankings do not predict rankings under temporal pressure. The highest-utility method varies with the shift and temporal pressure, with no clear winner. By enabling systematic evaluation across diverse temporal constraints for the first time, Tempora reveals when and why rankings change, offering practitioners a lens for method selection and researchers a target for deployable adaptation. Code: https://github.com/sudotensor/tempora.

Reliable decision-making with streaming data requires principled uncertainty quantification of online methods. While first-order methods enable efficient iterate updates, their inference procedures still require updating proper (covariance) matrices, incurring $O(d^2)$ time and memory complexity, and are sensitive to ill-conditioning and noise heterogeneity of the problem. This costly inference task offers an opportunity for more robust second-order methods, which are, however, bottlenecked by solving Newton systems with $O(d^3)$ complexity. In this paper, we address this gap by studying an online Newton method with Hessian averaging, where the Newton direction at each step is approximately computed using a *sketch-and-project solver with Nesterov's acceleration*, matching $O(d^2)$ complexity of first-order methods. For the proposed method, we quantify its uncertainty arising from both random data and randomized computation. Under standard smoothness and moment conditions, we establish global almost-sure convergence, prove asymptotic normality of the last iterate with a limiting covariance characterized by a Lyapunov equation, and develop a fully online covariance estimator with non-asymptotic convergence guarantees. We also connect the resulting uncertainty quantification to that of exact and sketched Newton methods without Nesterov's acceleration. Extensive experiments on regression models demonstrate the superiority of the proposed method for online inference.

Reinforcement learning (RL) in real-world tasks often suffers from the performance degradation due to the distribution shift between training and deployment environments. Distributionally Robust RL (DR-RL) addresses this issue by optimizing the worst-case performance over an uncertainty set of transition dynamics, providing an optimized baseline performance upon deployment. However, existing methods typically require strong data access assumptions (e.g., a generative model or comprehensive offline datasets) and mostly focus on tabular settings. In this paper, we introduce a purely online DR-RL algorithm with general function approximation that learns a robust policy directly from interaction, without any prior knowledge or pre-collected data. Our method uses a dual-based fitted robust Bellman update to jointly learn the value function and the robust backup operator. We establish the first regret guarantee for online DR-RL in terms of an intrinsic complexity measure—the robust Bellman–Eluder (BE) dimension, for general $\phi$-divergence uncertainty sets. Our regret bound is sublinear and independent of $|\mathcal{S}|$ and $|\mathcal{A}|$, and recovers sharp rates in structured regimes, providing a scalable method for practical DR-RL.

Bagging-based ensembles, most notably Adaptive Random Forests, are among the strongest performers for learning from data streams. A common denominator across these methods is their reliance on Hoeffding Trees as base learners, which grow decision trees incrementally by testing whether a candidate split is significantly better than its alternatives using concentration inequalities. Despite their empirical success, existing Hoeffding Trees variants lack valid statistical guarantees. Current analyses rely on fixed-sample concentration bounds, while split decisions are made using data-dependent stopping rules, which invalidates their guarantees and can drive the probabilty of incorrect splits to one. We introduce a principled alternative based on \emph{anytime-valid inference}. Our method provides: (i) anytime-valid control of false splits under arbitrary data streams, including non-stationary settings; (ii) finite commitment time under a predictive advantage; and (iii) under stationary i.i.d.\ data, risk is monotone decreasing and strictly improves at every split. Empirically, we evaluate both standalone trees and their use within Adaptive Random Forests on non-stationary streams. Our method improves performance while producing substantially smaller trees.

We study reinforcement learning with delayed state observation, where the agent observes the current state after some random number of time steps. We propose an algorithm that combines the augmentation method and the upper confidence bound approach. For tabular Markov decision processes (MDPs), we derive a regret bound of $\tilde{\mathcal{O}}(H \sqrt{D_{\max} SAK})$, where $S$ and $A$ are the cardinalities of the state and action spaces, $H$ is the time horizon, $K$ is the number of episodes, and $D_{\max}$ is the maximum length of the delay. We also provide a matching lower bound up to logarithmic factors, showing the optimality of our approach. Our analytical framework formulates this problem as a special case of a broader class of MDPs, where their transition dynamics decompose into a known component and an unknown but structured component. We establish general results for this abstract setting, which may be of independent interest.

Speculative decoding has emerged as a widely adopted paradigm for accelerating large language model inference, where a lightweight draft model rapidly generates candidate tokens that are then verified in parallel by a larger target model. However, due to limited model capacity, drafts often struggle to approximate the target distribution, resulting in shorter acceptance lengths and diminished speedup. A key yet under-explored observation is that speculative decoding inherently provides *verification feedback* that quantifies the deviation between the draft and target models at no additional cost. This process naturally forms an iterative "draft commits--feedback provides--draft adapts" evolving loop, which precisely matches the *online learning* paradigm. Motivated by this connection, we propose OnlineSPEC, a unified framework that systematically leverages interactive feedback to continuously evolve draft models. Grounded in *dynamic regret minimization*, we establish a formal link between online learning performance and speculative system's acceleration rate, and develop novel algorithms via modern online learning techniques, including optimistic online learning that adaptively reuses historical gradients as predictive update hints, and online ensemble learning that dynamically maintains multiple draft models. Our algorithms are equipped with theoretical justifications and improved acceleration rates, achieving up to 24% speedup over seven benchmarks and five foundation models.

Online Continual Learning (OCL) aims to learn from endless non\text{-}stationary data streams, yet most existing methods assume a flat label space and overlook the hierarchical organization of real\text{-}world concepts that evolves both horizontally (sibling classes) and vertically (coarse or fine categories). To better reflect this context, we introduce a new problem setting, DHOCL (Online Continual Learning from Dynamic Hierarchies), where taxonomies evolve across granularities and each sample provides supervision at a single hierarchical level. In this setting, we find two fundamental issues: (i) partial supervision under mixed granularities provides only point-wise signals over an evolving path-wise hierarchy, which constrains plasticity and undermines cross-level semantic consistency, and (ii) the dynamically evolving hierarchies induce granularity-dependent interference, destabilizing popular replay and regularization mechanisms and thereby exacerbating catastrophic forgetting. To tackle these issues, we propose HALO (Hierarchical Adaptive Learning with Organized Prototypes), which adaptively combines complementary classification heads, regularized by organized learnable hierarchical prototypes, enabling rapid adaptation, hierarchical consistency, and structured knowledge consolidation as the taxonomy evolves. Extensive experiments on multiple benchmarks demonstrate that HALO consistently outperforms existing methods across hierarchical accuracy, mistake severity, and continual performance.

We introduce Box Thirding (B3), a flexible and efficient algorithm for Best Arm Identification (BAI) under fixed budget constraints. It is designed for both anytime BAI and scenarios with large $N$, where the number of arms is too large for exhaustive evaluation within a limited budget $T$. The algorithm employs a Remedian Estimation strategy: in each iteration, three arms are compared—the best-performing arm is explored further, the median is retained for future comparisons, and the weakest is discarded. Even without prior knowledge of $T$, B3 achieves an $\epsilon$ -best arm misidentification probability comparable to Sequential Halving, which requires $T$ as a prior, applied to a randomly selected subset of $c_0$ arms that fit within the budget. Empirical results show that B3 outperforms existing methods for the limited budget constraint in terms of simple regret, as demonstrated on the New Yorker Cartoon Caption Contest dataset.

Existing auto-bidding algorithms in digital advertising often treat the value of an ad opportunity as the revenue obtained when an ad is shown and/or clicked, and bid accordingly. This can lead to wasteful spending because the true value is the marginal gain from paid exposure: even without winning a sponsored slot, an advertiser may still earn revenue via an organic search result (e.g., on Google or Amazon). Motivated by recent work, we model ad value as a treatment effect—the outcome difference between winning and losing the auction—and study online learning for bidding in second-price (Vickrey) auctions under this causal perspective. We develop algorithms that attain rate-optimal regret under several feedback models. A key ingredient exploits the information revealed by the second-price payment rule, which strictly improves regret relative to analogous learning problems in first-price auctions.

Online learning for the Cox model is challenging because its partial likelihood is non-decomposable, with each risk set requiring a summation over all samples. We propose Collaborative Operation of Linked Survival Analysis (COLSA), an online learning framework that replaces the partial likelihood with the full likelihood using sieve approximation of the baseline hazard. The proposed likelihood function is decomposable and eliminates the need to store historical data in memory, enabling efficient online updates. Moreover, COLSA maintains sufficient statistics for a higher-order basis and employs data-driven basis projection to adaptively scale model complexity to the effective sample size. Unlike existing online Cox methods, COLSA achieves asymptotic normality and attains the same statistical efficiency as the pooled-data partial likelihood estimator, without accessing full data and only requiring constant memory. Simulation studies and application to kidney transplant data demonstrate that COLSA outperforms existing online methods and matches the performance of full-data estimation.

We study contextual bandits with paid observations, where the learner actively chooses which actions to observe at a given cost in each round, with the goal of minimizing total regret that jointly accounts for learning loss and observation expenditure. We develop a near-optimal algorithm for adversarial environments and show that even small observation costs fundamentally raise the minimax regret order. We further uncover a novel phase transition under a free observation budget: below a critical threshold, free observations only reduce total cost without improving the regret rate; above it, asymptotic improvements become possible. To exploit this phenomenon, we design a meta-controller that adaptively switches between strategies to achieve near-optimal performance across all budget regimes. To handle large or infinite policy spaces, we also propose an oracle-efficient algorithm under a function approximation framework that maintains rigorous guarantees with computational efficiency. Our analysis also connects to related problems including switching costs, budgeted constraints, model misspecification, and knapsack bandits. Numerical experiments validate our theoretical findings.

Online convex optimization in non-stationary environments has garnered considerable attention in the literature. Recently, Pasteris et al. (2024) investigate online convex optimization with the optimal switching regret, defined as the sum of the static regret over each segment, where the segmentation is an arbitrary partition of the entire time horizon. For general convex functions, their work has established an optimal switching regret bound. However, it remains open whether similar bounds are attainable for other types of convex functions, such as exponentially concave or strongly convex functions. In this paper, we affirmatively answer this question by proposing a novel meta-algorithm, termed IRESET, which is used to aggregate the decisions from a group of experts. The essence of our method lies in running multiple experts over a set of intervals, and then employing a meta-algorithm equipped with second-order bounds to sequentially combine their decisions. We leverage the segment tree structure to analyze the switching regret over the entire time horizon, and offer new insights into utilizing recursive equations over the segment tree. By choosing appropriate expert-algorithms for IRESET, our methods achieve logarithmic switching regret bounds for exponentially concave or strongly convex functions, respectively.

*Learning-augmented algorithms* have received significant attention in recent years, particularly in the context of online optimization. Motivated by the high computational cost of generating predictions, a growing line of work studies the tradeoff between performance guarantees and the number of predictions used in learning-augmented algorithms for problems such as caching and metrical task systems. In this paper, we extend this line of research to *online metric matching* by developing *parsimonious* learning-augmented algorithms and establishing lower bounds on their performance. Our approach extends the Follow-the-Prediction framework to the parsimonious setting by filling in a *virtual prediction* in the absence of an actual prediction, using an online metric matching algorithm that maintains good intermediate matchings throughout its execution. We complement our theoretical results with an empirical evaluation, demonstrating the practical effectiveness of our approach.