Maintaining predictive accuracy in non-stationary environments requires online model selection to adapt autonomously to unknown distribution shifts. However, existing tuning-free algorithms face a fundamental trade-off between robustness and agility. Specifically, to ensure dynamic regret bounds, they must restrict learning rates to small constants (e.g., $O(1)$). This restriction inevitably causes significant adaptation lag during abrupt changes. To resolve this, we propose a novel optimistic online mirror descent that utilizes safeguarded large learning rates up to $Θ(T)$, where $T$ is the number of rounds. Our key technical contribution is a post-hoc penalty mechanism that dynamically monitors unstable updates and excludes learning rates incurring excessive regret, eliminating the need for restrictive a priori constraints. We show that the cumulative penalty remains $O(\log T)$, allowing our algorithm to match near-optimal worst-case guarantees while achieving superior rates in benign cases. Empirical evaluations on synthetic and eleven diverse real-world datasets demonstrate that our approach reduces the adaptation lag from hundreds of rounds to a few rounds, consistently outperforming tuning-free baselines.

Predictive models trained on observational data often fail to generalise to the distributions they encounter when deployed, especially when the training data is a product of the system being optimised. Recommender systems are a canonical example: they are trained on interaction logs confounded by the deployed policy, past user behaviour, and platform filtering. As a result, the training distribution differs substantially from the candidate distribution scored at serving time, a gap that makes offline metrics unreliable predictors of online performance. We address the distribution shift problem with a method motivated by causal representation learning (CRL). We propose an information-theoretic disentanglement criterion and prove that its optimum depends only on the causal components of the input. We then derive a tractable variational lower bound that makes the criterion optimisable from finite observational data alone. The scope of our method is narrower than that of much of the CRL literature, in that we target better generalisation under distribution shift, not full identification of all latent causal factors. This narrower target is what makes the method practical, requiring only the existing confounded logs, applying to any standard supervised model, and adding no inference-time cost. Our headline evaluation is an A/B test with millions of users on Spotify, applied to a production ranker for personalised playlist generation. A capacity-matched CRL variant performed on par offline but delivered substantial online gains in listener engagement. Complementary evidence on the public KuaiRand recommendation dataset and a synthetic benchmark with known causal structure shows the same pattern: offline parity with baseline, gains under distribution shift. Across all three settings, adding our causal disentanglement objective yields meaningfully better out-of-distribution generalisation.

Reliable deployment of machine learning models requires reasoning under epistemic uncertainty--the ability to recognize when the operating distribution has shifted beyond the scope of what was encountered during training. This challenge is central to test-time adaptation (TTA), a paradigm in which a model pretrained on source distribution Ps receives unlabeled data from a target distribution Pt = Ps at deployment time. Existing TTA methods (Wang et al., 2021; Niu et al., 2023; Zhang et al., 2022a; Yuan et al., 2023; Su et al., 2022) improve accuracy under distribution shift by adapting model parameters using statistics computed from test batches, but they provide no formal guarantees about when predictions should be trusted or how much risk degrades as a function of shift magnitude. This gap is particularly concerning in safety-critical applications such as autonomous driving, medical imaging, and financial risk assessment, where a model that silently degrades under distribution shift can cause significant harm. The inability to quantify how wrong a model's predictions might be in an unseen environment fundamentally limits its trustworthy deployment.

We address out-of-distribution (OOD) detection across the full spectrum of distribution shifts -- global domain changes, semantic divergence, texture differences, and covariate corruptions -- through a multi-encoder fusion of per-encoder representation-space diffusion models (RDMs). We statistically identify each encoder's sensitivity to specific shift types from ID data alone and introduce EncMin2L -- an encoder-agnostic two-level $\min(\cdot)$-gate that combines and calibrates per-encoder diffusion-based likelihood detectors without OOD labels, outperforming monolithic multi-encoder baselines at $2.3\times$ lower parameter cost. Two ID-data diagnostics: $η^2$ (class-conditional F-test) and $Δμ$ (log-likelihood shift under synthetic corruptions) -- quantify encoder specialization, while a Tippett minimum $p$-value combination aggregates per-encoder scores into a single, calibration-stable OOD signal. EncMin2L achieves $\geq 0.94$ AUROC across all four shift types simultaneously, outperforming the state-of-the-art representation-space diffusion OOD detectors across overlapping benchmarks.

Split conformal prediction provides finite-sample marginal coverage under exchangeability, but this guarantee averages over the random calibration sample. We study instead the law of the calibration-conditional coverage induced by a realized conformal threshold. In the continuous i.i.d. setting this law is exactly $Beta(k,n+1-k)$, so the usual marginal guarantee corresponds to its mean. We take this beta law as a finite-sample reference object and quantify departures from it using Wasserstein distances on $[0,1]$. The framework yields direct bounds on marginal coverage gaps and on bad-calibration probabilities, and separates different sources of non-i.i.d. behavior according to how they deform the beta reference: test-side shift acts through a transport map on the coverage scale, while calibration dependence changes the order-statistic law itself. We instantiate the framework in scale-shift, clustered, and stationary mixing settings, where the induced deformations can be characterized explicitly or through Berry-Esseen approximations. Simulations on dependent processes confirm that the first-order approximation tracks the empirical Wasserstein distance even at moderate sample sizes.

We introduce FLUXtrapolation, a benchmark for extrapolating ecosystem fluxes under progressively harder distribution shifts. Ecosystem fluxes are central to understanding the carbon, water, and energy cycles, yet they can only be measured directly at sparsely located measurement towers. Producing global flux estimates therefore requires training models on observed sites using globally available covariates and predicting in unobserved regions, that is, upscaling. Flux upscaling is a challenging domain generalization problem that is affected by a shift in covariate distribution across climates, ecosystem types, and environmental conditions, as well as by conditional shift: important drivers remain unobserved at global scale. We provide a quantitative analysis of both these shifts in $P_X$ and $P_{Y\mid X}$. FLUXtrapolation is designed based on domain expertise on flux upscaling: it defines temporal, spatial, and temperature-based extrapolation scenarios and evaluates performance across held-out domains, temporal aggregations, and tail errors. In a pilot study, we find that baselines perform similarly under median hourly RMSE, but separate under the proposed tail-focused and multi-scale evaluation. FLUXtrapolation therefore poses a realistic and thus relevant challenge for machine learning methods under distribution shift; at the same time, progress on this benchmark would directly support the scientific goal of improving flux upscaling.

Offline-to-online learning aims to improve online decision-making by leveraging offline logged data. A central challenge in this setting is the distribution shift between offline and online environments. While some existing works attempt to leverage shifted offline data, they largely rely on UCB-type algorithms. Thompson sampling (TS) represents another canonical class of bandit algorithms, well known for its strong empirical performance and naturally suited to offline-to-online learning through its Bayesian formulation. However, unlike UCB indices, posterior samples in TS are not guaranteed to be optimistic with respect to the true arm means. This makes indices constructed from purely online and hybrid data difficult to compare and complicates their use. To address this issue, we propose sample-mean anchored TS (Anchor-TS), which introduces a novel median-based anchoring rule that defines the arm index as the median of an online posterior sample, a hybrid posterior sample, and the online sample mean. The median anchoring systematically corrects bias induced by distribution shift by mitigating over-estimation for suboptimal arms and under-estimation for optimal arms, while exploiting offline information to obtain more accurate estimates when the shift is small. We establish theoretical guarantees showing that the proposed algorithm safely leverages offline data to accelerate online learning, and quantifying how the degree of distribution shift and the size of offline data affect the resulting regret reduction. Extensive experiments demonstrate consistent improvements of our algorithm over baselines.

We provide a theoretical analysis of Adam under non-stationary stochastic objectives, separating two regimes: Euclidean tracking under adaptive strong monotonicity of the Adam-preconditioned mean-gradient operator, and high-probability projected stationarity guarantees under general $L$-smooth objectives. In the tracking regime, we derive finite-time expected and high-probability bounds that decompose sharply into four components: initialization, objective drift, a first-moment tracking error governed by $β_1$, and a preconditioner perturbation governed by $β_2$. We characterize the burn-in time to reach Adam's irreducible tracking floor under constant and step-decay schedules. We also prove a high-probability bound on the average projected stationarity gap for Adam under distribution shift. Across both analyses, our bounds reveal a noise--drift tradeoff: in noise-dominated regimes, first-moment averaging and adaptive preconditioning can improve the high-probability error, whereas in drift-dominated regimes, stale first-moment information and preconditioner perturbations can compound the cost of nonstationarity, allowing vanilla SGD to achieve a smaller tracking floor. Our explicit $(β_1,β_2,ε)$-dependent bounds delineate when adaptive step-sizing is beneficial versus harmful, and provide a theoretical mechanism for Adam's empirical instability and stabilization under distribution shift.

We study long-horizon deployment of a frozen predictor under dynamic covariate shift. A time-domain Poincaré inequality reduces temporal risk volatility to derivative energy, and a Jacobian-velocity theorem identifies directional tangent energy along the deployment path as the governing quantity under explicit along-path regularity and domination assumptions. Under low-rank drift, that quantity reduces to directional Jacobian energy in the drift subspace, motivating drift-aligned tangent regularization (DTR) and a matched monitoring proxy. Rather than smoothing the network isotropically, DTR penalizes sensitivity only along estimated drift directions. We validate the theorem-to-method pipeline in four experiments: a synthetic benchmark for the time-domain inequality, a controlled synthetic comparison against isotropic Jacobian regularization, and two frozen-deployment studies on the UCI Air Quality and Tetouan power-consumption datasets. DTR reduces risk volatility and directional gain in the controlled low-rank regime, beats isotropic smoothing there, and gives validation-selected deployment gains on both real datasets when the Air Quality drift subspace is estimated from target-orthogonal sensor motion. Moderate drift-subspace misspecification is tolerable while orthogonal misspecification largely removes the benefit.

Contextual MDPs are powerful tools with wide applicability in areas from biostatistics to machine learning. However, specializing them to offline datasets has been challenging due to a lack of robust, theoretically backed methods. Our work tackles this problem by introducing a new approach towards adaptive estimation and cost optimization of contextual MDPs. This estimator, to the best of our knowledge, is the first of its kind, and is endowed with strong optimality guarantees. We achieve this by overcoming the key technical challenges evolving from the endogenous properties of contextual MDPs; such as non-stationarity, or model irregularity. Our guarantees are established under complete generality by utilizing the relatively recent and powerful statistical technique of $T$-estimation (Baraud, 2011). We first provide a procedure for selecting an estimator given a sample from a contextual MDP and use it to derive oracle risk bounds under two distinct, but nevertheless meaningful, loss functions. We then consider the problem of determining the optimal control with the aid of the aforementioned density estimate and provide finite sample guarantees for the cost function.