Conformal Bayes combines Bayesian posterior predictives with conformal calibration to produce prediction sets that are both statistically valid and geometrically efficient. We study conformal Bayes under label shift from a unified perspective, identifying two complementary approaches that restore nominal target-domain coverage through importance-weighted conformal calibration but operate through independent mechanisms. \emph{Post-hoc calibration} tilts the posterior predictive toward the target domain and corrects the conformal threshold via an importance-weighted quantile, leaving the parameter posterior unchanged. \emph{In-training adaptation} tilts the parameter posterior itself to the target domain, producing a corrected predictive whose highest predictive density region serves as the highest predictive density (HPD)-based prediction set under the fitted target predictive; efficiency is model-dependent and does not imply finite-sample conditional optimality. Two controlled experiments isolate the regime-dependence of each strategy: in the low-dimensional, well-estimated regime Strategy~A produces the narrowest valid intervals, while in the high-dimensional, underdetermined regime Strategy~B achieves up to $43\%$ width reduction at unchanged coverage, under the stated source-sampling and label-shift assumptions.

In small-batch scientific deployments, labeled target outcomes may be too scarce for reliable shift estimation even when unlabeled target inputs are available. We address the complementary setting where the practitioner has a pre-specified label-shift correction from domain knowledge and asks whether incoming labeled outcomes support it. We show that the per-observation likelihood ratio between a label-shift-corrected predictive and the source predictive is a conditional e-value, so its running product is a nonnegative martingale and Ville's inequality yields an anytime-valid confirmation rule. The log martingale equals the cumulative negative log-predictive density (NLPD) gap between the source and the corrected predictive, converting routine model monitoring into a formal sequential test. Rejection means the incoming data support the posited correction relative to the source predictive, but it is not a precise estimate of the degree of shift. Closed forms are available for GP sources with Gaussian label-shift ratios. GP regression simulations validate Type I control, finite-sample power, miscalibration sensitivity, and the small-batch advantage of a reliable prior over label-based re-estimation.

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.

Dataset distillation, a training-aware data compression technique, has recently attracted increasing attention as an effective tool for mitigating costs of optimization and data storage. However, progress remains largely empirical. Mechanisms underlying the extraction of task-relevant information from the training process and the efficient encoding of such information into synthetic data points remain elusive. In this paper, we theoretically analyze practical algorithms of dataset distillation applied to the gradient-based training of two-layer neural networks with width $L$. By focusing on a non-linear task structure called multi-index model, we prove that the low-dimensional structure of the problem is efficiently encoded into the resulting distilled data. This dataset reproduces a model with high generalization ability for a required memory complexity of $\tildeΘ$$(r^2d+L)$, where $d$ and $r$ are the input and intrinsic dimensions of the task. To the best of our knowledge, this is one of the first theoretical works that include a specific task structure, leverage its intrinsic dimensionality to quantify the compression rate and study dataset distillation implemented solely via gradient-based algorithms.

In this paper,we propose aprobabilisticmeta-learning algorithm that can sample models for a new task from a modeldistribution.

Thecorrespondingfm (x) where means element-wise Rn.

To extend this and adapt other results to our setting, we could now apply the Simulation Lemma [1]to bound the value difference given the model error,or alternatively, develop the theory in the direction of[55]andrelated work. Code is available at https://github.com/spitis/mocoda Forexample, in2d Navigation,themaskfunction was implementedasfollows: def Mask2dNavigation(input_tensor): """ accepts B x num_sa_features, and returns B x num_parents x num_children """ # base local mask mask = torch.tensor( Theadvantageofthisapproach isthat we can easily do conditional sampling incase of overlapping parent sets. The CQL implementation uses SAC [17].

For regression, we model output noise as a zero-mean Gaussian: N(0,σ2) where σ2 is the varianceofthenoise,treatedasahyperparameter. Neal[21] shows that in the regression setting, the isotropic Gaussian prior for a BNN with a single hidden layer approaches aGaussian process prior asthe number ofhidden units tends toinfinity,solong as the chosen activation function is bounded. We will use this prior in the baseline BNN for our experiments. In the context of BNNs, our Markov chain is a sequence ofrandomparametersW(1),W(2),... definedoverW,whichweconstruct bydefining thetransitionkernel. BBB is scalable and fast, and therefore can be applied to high-dimensional and large datasets in real-life applications.

The ability to encode informative functional beliefs in BNN priors can significantly reduce the bias and uncertainty of the posterior predictive, especially in regions of input space sparsely coveredbytraining data[27].

For each word, we use WordNet [7] to find its synonyms and build a list of word sets. Inaddition, toavoidreplacement clash, wedonotallowanyword to appear in more than word set. Eventually, top 50 semantically matching pairs are retained for CATER. Since the training data of the victim model is unknown to the malicious users, we randomly select 5M sentences from common crawl data as thebenigncorpus. Numbers in parentheses are resultsofcleandata.