To address the sequential and evolving nature of time series, the Online Time Series Forecasting (OTSF) task has been extensively studied in multiple domains. Existing research focuses on adapting to non-stationary environments by employing memory buffer-based retrieval strategies. However, we observe that such frameworks struggle with long-term adaptation and fail to generalize to unseen patterns. To this end, we introduce CoSPOT, an LLM-based online time series forecasting framework that leverages a pre-trained LLM as the backbone online forecaster, motivated by its strong few-shot capabilities. For efficient online adaptation, CoSPOT keeps the LLM frozen and employs compositional spectral prompts grounded in frequency-domain bases to guide the model with the overall distribution of the input, thereby substantially reducing the number of parameters updated during the online phase. Specifically, CoSPOT decomposes time series into frequency bases and composes the corresponding spectral basis prompts according to their amplitudes, allowing unseen patterns to be represented as new combinations of learned basis prompts. Our extensive experiments on real-world datasets demonstrate the superiority and practicality of CoSPOT across challenging online scenarios, including extended online phases and cross-dataset settings with substantial distribution shifts. Our code is available at https://github.com/seungyoon-Choi/CoSPOT.
An LLM application often sells or internally allocates several service products: a small or premium model, a short or long token cap, and possibly multiple posted prices. The operational decision is not merely which model answers a prompt. A price changes purchase probability, a token cap changes both user value and the tail of resource consumption, and accepted requests compete for shared compute and premium-model capacity. Demand and output length are initially uncertain, while an offline model may provide useful but imperfect predictions. We formulate sequential pricing and admission with stochastic resource consumption. Each arriving request belongs to an observable segment. The platform chooses a product--price pair or makes no offer; purchase, revenue, and resource use are then random. An offline predictor supplies a uniform, validated error radius for every segment--product cell. We propose Prediction-Clipped UCB (PCUCB), which intersects the offline prediction interval with an online confidence interval, evaluates products using resource shadow prices, and reserves a sample-path envelope before commitment. The prior gives a fast start when accurate, while online learning protects the platform when predictions are coarse. The analysis is modular. On a simultaneous confidence event, regret against a buffered fluid benchmark is bounded by a pacing term plus the cumulative diameter of the intersected intervals. For $J$ segment-product cells and prediction radius $\varepsilon$, this yields \[ \widetilde O\left( \sqrt{T}+(1+\barΛ) \min\{T\varepsilon,\sqrt{JT}\} \right), \] where $\barΛ$ bounds operational shadow prices. Thus the algorithm smoothly interpolates between an almost full-information regime and learning from scratch. Hard feasibility holds on every sample path through reservation envelopes.
In online retailing, when a product sells out, a retailer often sees only the units sold, not how many customers would have bought it had inventory been available. However, the inventory level determines how much demand is revealed, and this information can influence subsequent decisions and future profits. We study an online selling problem in which, in each round, the seller observes a market context and then makes pricing and stocking decisions based on censored sales data from previous rounds. The challenge is to learn a context-dependent pricing and stocking policy without assuming a particular formula for demand or observing realized profit. To overcome this difficulty, we propose a Mean-Calibrated Kernel UCB (MCK-UCB) algorithm that turns each incomplete sales record into a reliable guide for both inventory and price decisions, using data from past rounds with similar market conditions. This design allows us to learn while serving customers, without a separate exploration phase or the need to recover all demand hidden by stockouts. We prove the minimax optimality of the proposed algorithm, with strictly faster rates when expected profit varies more smoothly with price. Comprehensive numerical experiments have been conducted to confirm the effectiveness of the proposed algorithm.
Worst-case online classification is governed by sequential complexity, such as Littlestone dimension, and can be impossible even for statistically simple classes, such as thresholds of VC dimension one. We study a preview model in which an oblivious adversary fixes an entire labeled sequence of length $T$, a uniformly random subset of size $pT$ is revealed before prediction begins, and the remaining $(1-p)T$ examples are then presented in their original adversarial order. Against the best full-sequence hypothesis evaluated on the unrevealed examples, we characterize the dependence on the preview rate $p$: for binary classes of VC dimension $d$, the optimal excess loss is $Θ(d/p+\sqrt{dT})$, up to the trivial cap at $T$; for multiclass classes we obtain the corresponding $\widetilde O(d_{\rm DS}/p+\sqrt{d_{\rm Nat}T})$ bound with no dependence on the number of labels. Thus a random preview can replace worst-case sequential complexity by classical statistical dimensions without randomizing the online order. To achieve the sharp binary bound, our ChainedPrediction algorithm uses an online analogue of chaining, implemented as a multiscale aggregation algorithm rather than only as an analytic argument.
Knowledge distillation trains a small student model to reproduce the outputs of a large teacher model, and its progress is typically monitored through the teacher--student discrepancy. The quantity of ultimate interest, however, is the student's error with respect to the true task. We study the relation between these two objectives in a minimal three-party model, a true teacher (generative model), a teacher, and a student, all soft committee machines, in which the true teacher contains a shared latent factor that the teacher cannot represent, with mismatch strength controlled by a single scalar $\dmiss$. Within an order-parameter description of online distillation, and exploiting closed-form (arcsine-type) expressions for all errors under error-function activations, we prove that the learning dynamics and the distillation error $\Ets$ are exactly invariant to $\dmiss$, whereas the true error $\Etzs$ and the gap $Δ=\Etzs-\Ets$ are strictly increasing in $\dmiss$, with a rate that is amplified linearly by the complexity $M_0$ of the true teacher. Numerical phase diagrams over the plane spanned by true-teacher complexity and student capacity confirm the predicted deformation: the contours of $\Ets$ do not move while the landscape of $\Etzs$ rises systematically, and a teacher-miss regime, where mimicry succeeds but the task fails, expands with $\dmiss$. The results give a quantitative warning against evaluating distillation solely through teacher-mimicry metrics and identify the gap $Δ$ as a minimal diagnostic for distinguishing teacher-miss from capacity-limited failure.
Predictive Process Monitoring (PPM) models are increasingly deployed in dynamic environments where concept drift causes the underlying process distribution to shift over time. While recent work has moved toward online continual learning, existing methods train compact, task-specific networks entirely from scratch, leaving a persistent cold-start problem. Foundation Models (FMs) offer a compelling solution to this problem, but their continual fine-tuning in the process mining domain remains unexplored. We propose COMPASS (Continual Online foundation Model-based PPM with Adaptive SubSpaces), the first framework for online continual fine-tuning of FMs for PPM. COMPASS adapts loss-plateau drift detection to autonomously identify task boundaries in event streams and maintains a unified knowledge subspace including both pre-trained and task-specific directions. We evaluate our approach on nine event streams covering synthetic and real-world concept drift scenarios, across task-free and task-aware settings with multiple backbones and with consistent hyperparameter tuning across all methods. Our approach outperforms three SOTA non-FM competitors and two update strategy baselines, with particularly strong gains on streams exhibiting recurrent drift and complex, long-running cases, while incurring acceptable computational overhead compared to the non-FM competitors.
Exact Bayes prediction enjoys fast predictive regret guarantees, but exact posterior updating or representation may be too costly for online use. We study when these statistical guarantees are preserved by computational approximations. We show that the cumulative price of posterior approximation can be governed by the interaction between the contraction radius of the exact Gibbs posterior and the Wasserstein distance between the approximate and exact posteriors. Our general theorem shows that whenever exact Bayes prediction achieves a fast regret bound, any approximate posterior method that tracks the exact posterior with sufficient accuracy inherits the same fast regret, up to an additive term determined by the approximation error. Three online learning examples are developed. For linear models with strongly convex regularized losses, a projected Langevin algorithm yields an approximate posterior that achieves logarithmic regret. For an infinite-dimensional canonical exponential family sequence model over a Sobolev ellipsoid, a prior-preserving truncation method attains the minimax predictive regret rate with sublinear memory and constant update cost per observation. For random-design Gaussian process (GP) regression, a sparse variational posterior with inducing variables achieves the same predictive regret rate as the exact GP, but at substantially lower computational cost.
Optimization of hyperparameters is a critical factor to obtain optimal model performance. While existing research has predominantly concentrated on batch-learning scenarios, addressing the complexities inherent in data streams presents a challenge. The deployment of sophisticated methodologies to manage data streams becomes highly important. Consequently, the capacity for self-adjusting hyperparameters during on-line learning phases emerges as a goal. Many hyperparameters exhibit constraints and are confined within bounded search spaces, rendering specific solutions unacceptable upon applying optimization operators. To solve this issue, employing boundary constraint- handling techniques becomes imperative to rectify invalid solutions. This paper presents strategies for effectively managing boundary constraints within constrained numerical optimization problems. Recent methodologies, including heuristic and evolutionary-based optimization, employ a "boundary" strategy, wherein values that surpass boundary thresholds for a given hyperparameter are realigned to the respective limits. Our study introduces four strategies to navigate boundary constraints in online optimization algorithms. Through empirical investigations conducted on established datasets, we demonstrate that adopting boundary strategies outperforms the "boundary" strategy.
Predictive Coding (PC) is a neural learning paradigm that enables parallelizable neural network layer updates. However, the main bottleneck of PC Networks (PCN) is the sequential backwards error propagation. To tackle this, we introduce a training technique that pairs a Generative PCN with a support Encoding PCN. The two PCNs are trained in parallel to match their neural activations, without sequential propagation. We apply this to time series anomaly detection and show that our approach results in more stable, continuous, online learning.
Anagha Gokul, Jason Hartline, Lunjia Hu +2cs.DS cs.GT cs.LG
Calibration requires probabilistic reports to be conditionally unbiased and reliably interpretable as probabilities. A calibration measure assigns numerical error to miscalibrated reports. Haghtalab et al. (2024) proposed an approximately truthful calibration measure for online prediction, leaving open whether exact truthfulness is compatible with completeness and soundness. We resolve this question negatively for sequential binary prediction: exact truthfulness is incompatible with completeness and soundness, even for independent outcomes. We then show that this impossibility is specific to exact truthfulness. We give two general reductions from a base calibration measure, producing additively and multiplicatively approximately truthful calibration measures, respectively. Applying the multiplicative reduction, for every $0 < \varepsilon < 1$ we construct a sound and complete calibration measure that is $(1+\exp(-T^{(1-\varepsilon)/2}/2))$-multiplicatively truthful. This improves the approximate-truthfulness guarantee of Haghtalab et al. (2024).
Production machine learning systems degrade under concept drift, yet practitioners have little principled guidance on when to retrain. Retraining is costly, retraining budgets are finite, and a retrained model does not take effect instantly: training and deployment latency leave a stale model serving predictions while the data continues to move. We present a controlled empirical study of three practical model-refresh policies (periodic retraining, error-threshold triggering, and statistical drift-triggered retraining with ADWIN) against a no-retrain baseline, evaluated under a unified system model that makes retraining budgets and training-plus-deployment latency explicit. Across 3,933 experiment runs spanning three drift regimes, three budget levels, up to five latency levels, three datasets, and two learning modes, we find that the single most consequential design decision is not the retraining policy but whether the deployed model learns incrementally. With per-sample incremental updates, and for the linear online learner with immediate labels studied here, no policy differs from the no-retrain baseline by a practically significant margin in any of 54 paired comparisons, even at extreme latency. Without incremental updates, policy choice separates outcomes by 15-55 percentage points of post-drift accuracy, and simple periodic retraining significantly outperforms both reactive policies under abrupt and gradual drift, while reactive policies retain an advantage only under recurring drift. We document systematic failure modes of reactive policies and a latency-budget queueing interaction that silently halves effective retraining budgets, and release the full simulator, dataset pipelines, and per-run artifacts for reproducibility.
Huibo Xu, Shi Fu, Qixin Zhang +1stat.ML cs.LG stat.AP
In high-dimensional online prediction, the best predictor may depend on only a few features, so regret should scale with sparsity rather than the ambient dimension. Feature priming pursues this goal by estimating feature weights from past data and refitting a minimum-norm predictor on the rescaled design. Warmuth and Amid asked at COLT 2023 whether any of three such rules admits a competitive online regret guarantee. Using the natural Moore--Penrose protocol based only on past data, we give a negative answer to the sparse-logarithmic form of this COLT open problem. Our analysis identifies a common obstruction: cheap nuisance interpolation causes the refit to underweight the truly predictive coordinate. An exact target-mass identity and a two-sign argument turn this effect into clipped prediction loss. Hadamard constructions force $Ω(\min\{T,\sqrt{d}\})$ regret for all three rules against a zero-loss one-sparse comparator, with extensions to fixed prime powers and selectors among the rules. Conversely, regret is controlled by data rank, and a Euclidean-normalized triangular construction matches this dependence for powered univariate priming, even under nonnegative second-stage ridge regularization; a paired ridge construction also covers all three powered rules. Exploratory diagnostics on frozen language-model activations exhibit the same relation among nuisance interpolation, target weight, and loss. The exact multivariate and Pearson frontiers remain open.
Regularized sparse regression has been extensively studied in the offline setting, but online formulation remains relatively under-explored. This gap stems from four key challenges: (i) the infeasibility of dynamically updating the regularization parameter in every online round, (ii) managing storage and memory complexity, (iii) enabling real-time computation via closed-form updates rather than solving full optimization problems at each round, and (iv) achieving optimal statistical guarantees under realistic assumptions. In this paper, we propose an online generalized-sparsity-constrained regression framework, focusing on online cardinality-constrained linear regression and low-rank matrix sensing. Unlike online regularized regression, our constrained formulation eliminates the need for dynamic parameter tuning. We introduce an efficient online hard-thresholding algorithm that performs closed-form updates and requires storing only summary statistics, making it computationally, memory, and storage efficient. Despite the inherent nonconvexity and combinatorial nature of the formulation, our algorithm achieves global convergence at the optimal statistical rate under realistic assumptions, provided that the projection set is properly overparameterized. Numerical experiments demonstrate that our method consistently outperforms state-of-the-art alternatives.
Motivated by modern marketplaces, where the platform or the seller routinely gathers detailed user profiles, we study a novel learning theoretic model that simultaneously involves information and mechanism design. Specifically, we consider the economic setting recently introduced by Bergemann et al. (2022), where in addition to the menu of quality-price pairs, the seller offers information on the value of the match between product quality and buyer's taste via a signaling scheme. We relax the assumption that the seller knows the buyers' belief about the distribution of tastes and study the sample requirements of designing a revenue maximizing scheme. We consider both the batch setting where we have access to data from a set of i.i.d. buyers and an online demand query model where we observe the buyers' behaviors to seller's schemes. Despite the apparent non-convexity of the problem, we also give the first FPTAS to compute a scheme that maximizes the revenue within an arbitrarily small additive loss, which was left open by Bergemann et al. (2022). Overall, this brings a new learning perspective in asymmetric economic settings where buyers and sellers know different types of information.
We study online probabilistic forecasting of binary outcomes chosen by an adaptive adversary. Given an online learning algorithm for a weak hypothesis class $H$, we would like to efficiently obtain two incomparable guarantees that existing online boosting techniques provide separately. Online gradient boosting competes in Brier score with the best predictor induced by the span of $H$ on every sequence, but promises nothing when the span does not contain an accurate predictor. Online weak-to-strong boosting drives classification error to zero under a weak-learning condition, but promises little when that condition fails. We give a simple defensive forecasting algorithm, the Defensive Booster, that obtains both guarantees. On every adaptive sequence, its Brier score is competitive with the best prediction induced by the span of $H$ at the same rate as online gradient boosting; simultaneously, whenever the realized transcript satisfies the smooth weak-learning condition, its Brier score and randomized classification error satisfy the same rate guarantee as online classification boosting. This is achieved by operationalizing the "dual view" of boosting: When the algorithm's randomized classification error is persistently high, its mistake weights form a smooth reweighting on which every weak hypothesis has low edge, yielding an ex-post hard-core certificate that the weak-learning condition fails. We also develop a strongly adaptive variant, which satisfies both guarantees on every time interval. The Defensive Booster is very efficient: it accesses just one weak-class learner, whereas the prior online boosting methods we compare against maintain large weak-learner ensembles. Experiments on synthetic and real data streams demonstrate its strong predictive performance (sometimes substantially improving over all prior baselines) coupled with orders-of-magnitude faster runtime.
Betting-based sequential tests and Blackwell approachability are linked by a rate-explicit reduction through support-function residuals. For a compact convex target $S$ and vector observations $r_t$, an OCO learner selects a predictable normal $w_t$ and produces $q_t=\langle w_t,r_t\rangle-h_S(w_t)$. We prove the exact pathwise identity $$ \dist(\bar r_T,S) =\frac1T\sum_{t=1}^Tq_t+\frac{\Reg_T}{T}. $$ When $|q_t|\leq B$, composing this identity with one-sided betting yields a finite-time transfer: if the OCO and log-wealth regrets are at most $a_T$ and $\ell_T$, respectively, then a target gap exceeding \[ \frac{a_T}{T} +2B\sqrt{\frac{\log(1/α)+\ell_T}{T}} \] forces rejection by time $T$, while non-rejection certifies the converse radius. We then formulate a controlled stochastic experiment in which an action selected after $w_t$ satisfies Blackwell's supporting-halfspace condition for every null mean payoff. The resulting wealth is an e-process under adaptive nulls; sublinear OCO regret gives stochastic approachability, whereas persistent mean separation under an alternative gives exponential wealth at rate at least $δ^2/(4B^2)$. Deterministic Blackwell games and passive tests are, respectively, the noise-free and singleton-action cases of this protocol. Bounded two-sample means, kernel MMD, and active heterogeneous data sources instantiate the reduction. The resulting connection is exact algebraically, quantitative at finite time, and operational when experiments are controlled.
Score-driven filters multiply a scaled log-likelihood score by a gain that controls the update magnitude. We treat this gain as a decision variable and study its online learning. Conditional on the current state, observation, score, and scaling rule, each admissible gain induces a reachable next state and a one-step-ahead predictive density: scalar gains govern distance along a line, while diagonal gains govern coordinatewise transmission. Gain selection is therefore a conditional predictive decision problem with a Kullback-Leibler objective. For a scalar unscaled gain, the negative raw product of consecutive scores is the stochastic gradient of this loss; positive aGAS scaling only rescales the effective step. Monotone differentiable gain links induce mirror-descent geometries on bounded gain domains, while persistence yields a Bregman pull towards a reference gain. Under convexity, compactness, and regularity conditions, we establish dynamic-regret bounds for projected and discounted mirror updates relative to time-varying, current-information comparators. Simulations illustrate the roles of scaling, link geometry, persistence, and coordinatewise transmission rates. An out-of-sample panel of equity-index volatilities shows that the bounded mirror gain generally matches or outperforms a constant gain while avoiding the extreme spikes of a nominally unbounded exponential link, with the strongest improvements observed in multi-crisis markets.
Uncertainty quantification is essential when deploying machine learning models in safety-critical applications. Online conformal prediction (OCP) provides theoretically principled uncertainty quantification for arbitrary black-box classifiers and non-i.i.d. data streams by constructing prediction sets that are guaranteed to contain the true label at a user-specified frequency. OCP usually updates prediction sets using feedback from previously deployed predictions. We instead study an OCP setting beyond feedback: on each round, the learner can either output a prediction set or query the correct label, but not both. Thus, no deployed prediction is ever evaluated directly. We reduce this problem to a partial monitoring game in which prediction actions return no observation and a separate query action reveals the label. The reward function is constructed in a way that encourages the learner to output small prediction sets while ensuring that the correct label is covered with a sufficiently high probability. To solve this game, we develop OCP with queries (OCPQ) by adapting the label efficient forecaster of Cesa-Bianchi, Lugosi, and Stoltz (2004) to our setting. For any black box classifier and any (non-i.i.d.) oblivious data stream of length $T$, OCPQ has $O(T^{2/3})$ expected regret and expected coverage at least $β-O(T^{-1/3})$ for a user-defined $β$, while querying only an expected $T^{-1/3}$ fraction of rounds. This provides coverage comparable to bandit-based OCP methods while requiring no feedback from deployed prediction sets. Experiments on real-world datasets further demonstrate the effectiveness of our approach.
Can one forecaster attain the optimal regret rate for every bounded proper loss and also adapt to every smooth proper loss? Recent work answered this up to a dimension gap. Its self-concordant perturbation gives roughly $K^{5/4}\sqrt{T}$ worst-case regret and incurs an additional $β\sqrt{K}\log K$ for $β$-smooth losses. We close both gaps with a one-line forecaster. After observing class counts $c_{t-1}$, draw the next prediction from $\operatorname{Dir}(c_{t-1})$, on the face of classes seen so far. This is a fresh Bayesian bootstrap of the outcomes. The analysis rests on an exact identity: averaging any bounded proper loss under $\operatorname{Dir}(α)$ equals a discrete derivative of its Dirichlet-averaged Bayes risk. The identity makes the be-the-perturbed-leader term telescope to a nonpositive Jensen gap. A one-count likelihood ratio then bounds stability by the inverse square root of that class's count. The resulting single, horizon-free algorithm satisfies $\sup_{\ell}\mathbb{E}\operatorname{Reg}_{\ell}\leq 4\sqrt{S_T T}\leq 4\sqrt{K T}$ and $\mathbb{E}\operatorname{Reg}_{\ell}\leq \frac{5}{2}β(1+\log T)$ for every $β$-smooth proper loss. Here $S_T$ is the number of observed classes. Known lower bounds show that both rates are optimal in their nontrivial regimes. The proof covers nondifferentiable losses and changes of the active simplex face.
Neural networks that can grow or both grow and shrink during learning, referred to as growing neural networks and elastic neural networks, respectively, have recently been explored in offline continual learning with a particular focus on catastrophic forgetting. Driven by the observations that 1) online continual learning closely resembles how animals learn; 2) loss of plasticity---the progressive decline in a learning network's ability to learn---is another crucial challenge facing continual learning; and 3) incremental introduction of randomly initialized hidden units was recently shown to help preserve plasticity, in this paper, we study the plasticity of several foundational growing and elastic networks in online continual learning. Our experiments in supervised learning settings show that adaptive growing networks, which incrementally incorporate new, randomly initialized units to the network while keeping all existing connections adaptive, can maintain high prediction accuracy without losing plasticity despite the continuous increase in the dead hidden unit proportion. Furthermore, we demonstrate that adaptive elastic networks, which in addition to progressively adding new hidden units also prune estimated dead hidden units at the beginning of each new task, can achieve excellent accuracy without loss of plasticity while simultaneously maintaining a near-constant, compact size. Our results suggest that growing and elastic networks, which exhibit the ability to adapt its structure to the relevant learning objectives, can be a promising class of algorithms also for preserving high plasticity in online continual learning.
Adaptive conformal inference (ACI) of Gibbs and Cand{è}s and its variants are the standard approach to online conformal prediction under distribution shift, but they suffer from three fundamental limitations. First, their guarantees control only the \emph{signed} long-run coverage error: persistent miscoverage in one direction can be masked by compensating errors later, so a method can satisfy the theoretical guarantee while being badly wrong for extended periods. Second, existing guarantees say nothing about prediction-set size, so validity can be achieved trivially at the cost of unduly wide prediction sets. Third, the efficiency guarantees that do exist compare against a \emph{fixed} predictor chosen in hindsight, a benchmark that becomes increasingly less meaningful once the data-generating distribution shifts, since the very notion of an optimal threshold then changes over time. We consider a unified online learning framework that simultaneously controls absolute, non-cancelling coverage violation and prediction-set efficiency against a dynamically evolving benchmark for three important models. In the fully adversarial setting, exploiting the fact that the standard ACI update is exactly projected online gradient descent on the pinball loss, we derive simultaneous coverage and efficiency guarantees for arbitrary monotone Lipschitz efficiency objectives, with no distributional or {\it convexity} assumptions. In the stochastic setting with full-score feedback, we propose a sliding-window quantile tracker and establish a matching minimax lower bound showing our algorithm is rate-optimal. In the covariate-dependent stochastic setting, we develop a partitioned ACI algorithm that tracks a function-valued oracle threshold, and derive simultaneous coverage and efficiency guarantees.
Lunjia Hu, Kevin Tian, Chutong Yangstat.ML cs.DS cs.LG
We study the problem of recalibrating an online predictor [KE17, OKS24]: given an arbitrary "hint" sequence of forecasts, the learner must output new predictions that are calibrated while incurring small excess error relative to the original forecasts, under a proper loss. We give an online algorithm that achieves $(\varepsilon, \varepsilon^2)$-recalibration for Lipschitz proper losses in $T \approx \varepsilon^{-3}$ rounds, using an imbalanced extension of the recent simultaneous Blackwell approachability reduction framework of [HTY26]. We show that this tradeoff is optimal by proving a matching lower bound for recalibrating against the squared loss. We also prove a companion $\mathcal{K}_2$-recalibration theorem that obtains the same tradeoffs up to a logarithmic factor. As our main application, we show how our recalibration algorithms can be combined with the online refinement method of [FH23] to obtain simultaneous $\varepsilon$-calibration and $\varepsilon^2$-calibeating for smooth proper losses at the same asymptotic rate, improving upon prior works that achieved these properties separately or with a worse $\varepsilon$ dependence. In particular, the $\mathcal{K}_2$ variant answers a question of [CHJL26] on simultaneously achieving near-optimal calibeating and calibration rates. We also derive extensions to settings with multiple hint sequences. Finally, we empirically evaluate our algorithms on a classification dataset undergoing distribution shift.
Zhiwei Lei, Benedict Jun Ma, Ilya Jacksoncs.LG cs.AI
Retail demand forecasting remains difficult when demand shifts faster than static forecasting models can be retrained, especially in early demand cycles where newly observed labels are sparse. To address this, this study aims to improve adaptive retail forecasting by proposing a predict-then-correct (PtC) framework that retains a first-stage machine learning (ML) forecast and applies a few-shot continuous contextual bandit correction policy with similar-SKUs augmentation and top-p masked updating. Across Walmart retail data and an exclusive beverage dataset, PtC delivers statistically significant reductions in MAPE, MAE, and RMSE across stable & high volume, stable & low volume, and erratic & intermittent demand patterns, improves average RMSE by 9.52% over the ML-only baseline in the ablation study, and yields lower inventory costs than base-stock, proximal policy optimization, and soft actor-critic policies under the tested lead-time settings. These findings show that online forecast correction can bridge offline demand learning and real-time retail decision-making by adapting to sparse feedback without fully retraining the base forecasting model.
Mobile crowdsensing (MC) recruits mobile users to perform sensing tasks using their smartphones, enabling large-scale applications such as traffic monitoring and environmental sensing. A fundamental challenge is online worker recruitment under uncertainty, where the platform must learn workers' sensing performance while operating with a limited budget. Existing learning-based MC recruitment methods typically assume that each worker's sensing quality is stationary with a fixed mean over time. In practice, however, worker performance often improves with experience and eventually stabilizes, while the incurred sensing cost can be unknown in advance due to time-varying device and context states. In this paper, we study a budget-constrained online recruitment problem in which the platform selects one worker in each round, observes the sensing quality and incurred cost, where the expected sensing quality of each worker increases with experience and eventually converges to a plateau, and repeats until the budget is exhausted. We formulate this problem as a structured bandit model where each worker's expected reward evolves according to an unknown increasing-then-converging function of its participation count, and each worker has an unknown expected cost. We develop a cost-aware online learning framework that jointly learns evolving reward trajectories and heterogeneous costs, detects performance saturation, and allocates the limited budget to maximize long-term sensing utility. We provide theoretical performance guarantees and validate the proposed approach through extensive experiments, demonstrating consistent improvements over baselines that ignore experience-driven dynamics or assume known costs.
We study the online binary sequential calibration problem. A recent breakthrough by \citet{dagan2024breaking} overcomes the classical \(T^{2/3}\) barrier for calibration error. Building on this result, we present an efficient randomized forecaster that achieves an expected calibration error \(O(T^{2/3-\varepsilon})\) for some constant \(\varepsilon>0\). Our forecaster combines the \textsc{SPR-Calibration} procedure \citep{dagan2024breaking} with an outer Blackwell-style correction layer. The \textsc{SPR-Calibration} procedure controls calibration with respect to a surrogate sequence of conditional-mean estimates, while the correction layer controls the additional error incurred when these surrogates are used to approximate the true outcomes. The analysis decomposes the total calibration error into the surrogate calibration error and the residual discrepancy between the surrogate sequence and the true outcomes. The former is bounded by the \textsc{SPR-Calibration} guarantee in \citet{dagan2024breaking}, and the latter is controlled using a quadratic potential argument together with the sparsity of the \textsc{SPR-Calibration} forecaster.
Intelligent systems should not only solve tasks but also adapt under real-world constraints. Autonomous adaptation via self-supervised learning, sequential adaptation via online learning, and memory-efficient implementation via perturbation-based learning are important requirements for such systems. However, these requirements are generally in tension for high-dimensional systems, because perturbation-based learning suffers from variance that grows with the dimension of the perturbed variables. In this study, we focus on echo state networks (ESNs), where this tension naturally arises in large reservoirs. We propose a perturbation-based learning rule for online self-supervised learning in ESNs. The proposed rule is derived from an orthogonal decomposition of the self-supervised learning cost, which separates an input-dependent component from a redundant component determined by the fixed ESN parameters. By perturbing only the input-dependent component, the effective perturbation dimension is reduced from the reservoir dimension to the input dimension. Thus, the proposed method preserves self-supervised adaptation, online learning, and scalar-feedback perturbation learning, while avoiding reservoir-size-dependent variance growth. This suggests a design principle for scalable and hardware-compatible learning: online learning should be restricted to the dynamically necessary low-dimensional component of the objective.
The Continual Learning (CL) literature has long been driven by the goal of mitigating catastrophic forgetting. This objective rests on a pervasive, often unstated assumption: that a lifelong learner should approximate the Joint-Task Learning (JTL) solution and retain all previously acquired knowledge. We challenge this retention-centered premise, arguing that in non-stationary environments prioritizing retention can impede real-time adaptation. Shifting the focus to the Average Lifelong Error (ALE), we formalize CL as an online optimization problem governed by the interaction between environmental and learning dynamics. We introduce Transfer Efficiency as a quantitative measure of the tension between Instability, the bias inherited from conflicting past experience, and Transient Error, the optimization cost of learning new tasks from scratch. Under mild convergence conditions, holding across linear and neural network models, this decomposition yields a Critical Task Duration: a closed-form threshold beyond which historical knowledge transitions from a warm-start advantage to an optimization liability whenever retention induces a positive stationary bias. We validate these theoretical predictions on continual image classification and reinforcement learning benchmarks. Finally, by connecting continual learning to the online learning framework of predictable sequences, we show that JTL is only one instance of a broader family of objectives, and we propose a new general class of continual learning algorithms, which we call Predictive Continual Learning. Predictive CL algorithms optimize expected future performance under an explicit, dynamically updated model of future tasks. As a proof of concept, we analyze a Window algorithm that interpolates between JTL and Independent-Task Learning (ITL), outperforming both under controlled distributional drift.
We study estimation and inference for online quantile regression under a one-report user-level $\eps$-locally differentially private ($\eps$-LDP) protocol. The main difficulty is that the standard quantile-regression estimating-equation contribution couples covariates with a residual comparison, so a server that receives only privatized reports cannot form the usual online update. We address this by developing a finite-alphabet channel in which each user computes the contribution locally, applies support-aware stochastic quantization and randomized response to one selected-block category, and sends one report. A public decoder corrects the randomized-response distortion and reconstructs a server-side estimating-equation input with the correct conditional mean. These decoded inputs are then used in projected Polyak-Ruppert averaging. For fixed finite channel designs, we establish local privacy, decoder unbiasedness, consistency, asymptotic normality, and Hessian-free self-normalized inference for prespecified scalar contrasts. Simulations and a New York City taxi-trip illustration show that the private trajectory approaches the nonprivate online reference as the privacy budget grows and outperforms direct Laplace and face-exponential geometric releases in the reported regimes.
Tensor Train (TT) decomposition is a powerful technique for analyzing high-dimensional data. Existing algorithms for computing TT decompositions can be categorized into two main types: conventional batch-based approaches and recursive online methods. In the context of streaming data, batch methods typically achieve higher reconstruction accuracy but often suffer from memory exhaustion, while online methods provide greater computational efficiency. In this work, we introduce Online TT-ALS (Alternating Least Squares), an algorithm that sequentially enforces orthogonality constraints. This approach allows for efficient and exact updates of the core tensor while maintaining high reconstruction accuracy. Theoretically, we prove that enforcing these orthogonal gauge constraints guarantees monotonic decrease of the local objective function and temporal smoothness. Computationally, our deterministic single-sweep update reduces the rank dependence from quadratic to linear, achieving an overall complexity of $\mathcal{O}(I^{n-1} r)$. Experimental results demonstrate that the proposed method outperforms existing online techniques not only in terms of mathematical approximation accuracy but also in human perception-based video quality metrics. Furthermore, compared to recent deep learning-based paradigms, our algebraic approach achieves speedups of several orders of magnitude. Consequently, our method exhibits high computational efficiency and is suitable for low-latency real-time processing applications.
Bayesian and multiplicative-weights updates reweight experts, models, or actions from sequential feedback. We show that the regret of any such update obeys an exact information-accounting identity. On each round, the learner's excess loss to any chosen comparator is the sum of an immediate cost for the uncertainty exposed by the round and a reduction in the information distance from the learner's current weights to the comparator. The cumulative cost defines a pathwise uncertainty clock, the intrinsic time of the realized sequence. Summing one-step balances yields two exact adaptive decompositions of cumulative regret, one for each natural way of composing the update across rounds. Because the decompositions are exact, favorable stochastic or low-noise regimes appear as self-bounding properties of the realized intrinsic time. The accounting also fixes a learning rate, inverse in the square root of intrinsic time. That schedule is competitive with adaptive baselines in selected online-learning settings. The same calculus covers Hedge, optimistic and side-information variants, continuous priors, boosting, online convex optimization, contextual bandits, and repeated games: the pathwise account is the same in every case.