Learning-to-Defer (L2D) methods route each query either to a predictive model or to external experts. While existing work studies this problem in batch settings, real-world deployments require handling streaming data, changing expert availability, and shifting expert distribution. We introduce the first online L2D algorithm for multiclass classification with bandit feedback and a dynamically varying pool of experts. Our method achieves regret guarantees of $O((n+n_e)T^{2/3})$ in general and $O((n+n_e)\sqrt{T})$ under a low-noise condition, where $T$ is the time horizon, $n$ is the number of labels, and $n_e$ is the number of distinct experts observed across rounds. The analysis builds on novel $\mathcal{H}$-consistency bounds for the online framework, combined with first-order methods for online convex optimization. Experiments on synthetic and real-world datasets demonstrate that our approach effectively extends standard Learning-to-Defer to settings with varying expert availability and reliability.

Inferring latent dynamics from multivariate time-series defined over topological cell complexes is crucial for capturing the complex, higher-order interactions inherent in real-world systems such as in water, sensor, and transportation networks. However, reconstructing these latent states is challenging because the signals are coupled across higher-order topologies, while high dimensionality, nonlinear observations, and unknown structures increase the difficulty. To address this, we propose a topology-aware state space framework derived from stochastic partial differential equations on cell complexes. State evolution follows heat-like topological diffusion, with perturbations propagating along boundary operators. Under partial observability, we model observations using a cell complex convolution of latent states coupled with a nonlinear mapping. We perform recursive state estimation via an Extended Kalman Filter, simultaneously learning model parameters and uncertainties through an online Expectation-Maximization algorithm. Finally, for scenarios where only lower-order topological structure is known, e.g., nodes and edges, as in critical infrastructure networks, we introduce a heuristic cell identification algorithm to explicitly infer the second-order cell structures. Validations on synthetic and real datasets from water, sensor and transportation networks demonstrate that our approach yields reliable estimates under partial observability and successfully recovers the underlying topological structures.

Many bandit deployments (recommendation, clinical dosing, ad targeting) share two facts prior work handles only in isolation: rewards live on a low-dimensional latent subspace, and that subspace drifts. Stationary low-rank bandits exploit rank but break under subspace change; non-stationary linear bandits adapt to drift but pay ambient rate $\widetilde{O}(d\sqrt{T})$. We study piecewise-stationary low-rank linear contextual bandits with scalar feedback: $θ_t = B_k^\star w_t$ with rank-$r$ factor $B_k^\star\in\mathbb{R}^{d\times r}$ constant within each of $K$ unknown segments and able to shift at boundaries. Our results are tight along three axes. (i) Identification boundary. With single-play scalar rewards, the moving subspace is recoverable through quadratic functionals of rewards iff three probe-side conditions hold: known noise variance, bounded state-noise coupling, and full-dimensional probe support. Each is necessary in the unrestricted-second-moment problem, and jointly they are sufficient, characterizing the boundary of the solvable region. (ii) Algorithm and dynamic regret. SPSC interleaves isotropic probes with windowed projected ridge-UCB exploitation inside the learned $r$-dimensional subspace; a CUSUM-style variant discovers segment boundaries online. The costed dynamic regret is $\widetilde{O}(r\sqrt{T})+\widetilde{O}(T^{2/3})+O(W\,V_{\mathrm{in}})$, replacing the ambient $d\sqrt{T}$ rate with the intrinsic rank. (iii) Empirics. On eleven benchmarks spanning synthetic, UCI/MovieLens, semi-synthetic clinical, and ZOZOTOWN production-log data, SPSC outperforms non-stationary and low-rank baselines whenever $d-r\gtrsim T^{1/6}$, matching the analytical crossover. To our knowledge, this is the first work to characterize the identification boundary and attain the intrinsic-rank dynamic-regret rate in this setting.

A local specialist LLM, fine-tuned with reinforcement learning from verifiable rewards (RLVR) on operator-local data, is installed in a regulated organization with per-deployment error budget $α$. The operator needs a safety certificate for this deployment's stream at every round: no pooling across deployments, no waiting for a long-run average. Existing wrappers cannot deliver this on adaptive, online-updated streams: offline conformal-risk methods require exchangeability; online-conformal methods bound only long-run averages; non-exchangeable extensions are marginally valid; and the closest anytime wrapper, A-RCPS, controls marginal rather than selective risk. Using a (test statistic, validity guarantee, deployment rule) framework, we identify one empty cell forced by deployment requirements: e-process per threshold, selective risk, anytime-pathwise validity, max-certified-threshold rule. Conformal Selective Acting (CSA) fills it as a per-round wrapper maintaining a Ville-type e-process per threshold on a Bonferroni grid, evaluated against the RLVR filtration. Under predictable updates and isotonic-calibrated monotone risk we prove (i) an anytime-pathwise selective-risk bound $R_T^{\mathrm{act}}\leα+O(N_T^{-1/2})$, (ii) rate-optimal certification matching $Θ(\barη^{-2}\log(1/δ))$, and (iii) a horizon-independent release-rate gap. Across eight specialist benchmarks ($480$ streams), sixteen adversarial distribution-shift cells ($160$ streams), and five live Expert-Iteration RLVR cells with online LoRA over four base models in three architecture families ($10{,}300$ rounds), CSA is the only method among ten compared that satisfies pathwise validity and non-refusing deployment on every cell. We do not propose a new LLM, training algorithm, or policy class; CSA is the deployment-side complement, orthogonal to the model, for operators who cannot use a frontier API.

Identifying latent variables and the causal structure involving them is essential across various scientific fields. While many existing works fall under the category of constraint-based methods (with e.g. conditional independence or rank deficiency tests), they may face empirical challenges such as testing-order dependency, error propagation, and choosing an appropriate significance level. These issues can potentially be mitigated by properly designed score-based methods, such as Greedy Equivalence Search (GES) (Chickering, 2002) in the specific setting without latent variables. Yet, formulating score-based methods with latent variables is highly challenging. In this work, we develop score-based methods that are capable of identifying causal structures containing causally-related latent variables with identifiability guarantees. Specifically, we show that a properly formulated scoring function can achieve score equivalence and consistency for structure learning of latent variable causal models. We further provide a characterization of the degrees of freedom for the marginal over the observed variables under multiple structural assumptions considered in the literature, and accordingly develop both exact and continuous score-based methods. This offers a unified view of several existing constraint-based methods with different structural assumptions. Experimental results validate the effectiveness of the proposed methods.

Effective medication management in Parkinson's Disease (PD) is challenging due to heterogeneous disease progression, variable patient response, and medication side effects. While AI models can forecast levodopa equivalent daily dose (LEDD) as a measure of medication needs, standard uncertainty quantification often fails to communicate the reliability of these predictions, treating high and low confidence clinical decisions identically. We introduce CASCADE (Calibrated Adaptive Scaling via Conformal And Distributional Estimation), a novel conformal prediction framework that propagates epistemic uncertainty from a screening classifier to adapt downstream predictions. Unlike standard conformal methods that rely on auxiliary residual regression, we leverage epistemic uncertainty from a primary classification task (identifying whether a medication change is needed) to dynamically scale the prediction intervals of a secondary regression task (predicting how much change). By mapping Venn-Abers multi-probabilistic uncertainty directly to non-conformity scores, our framework achieves continuous risk adaptation. We demonstrate that this ``cascade effect'' produces highly efficient intervals for confident patients (38.9% narrower than standard conformal baselines) while automatically expanding intervals to ensure robust coverage for uncertain cases, bridging the gap between discrete clinical decision-making and continuous dose forecasting in PD.

While deep neural networks have achieved remarkable success across a wide range of domains, their underlying mechanisms remain poorly understood, and they are often regarded as black boxes. This gap between empirical performance and theoretical understanding poses a challenge analogous to the pre-axiomatic stage of classical geometry. In this work, we introduce the Pursuit of Subspaces (PoS) hypothesis, an axiomatic framework that formulates neural network behavior through a set of geometric postulates. These axioms, together with their derived consequences, provide a unified perspective on representation, computation, and generalization in both shallow and deep architectures. We show that this framework yields geometric explanations for fundamental questions in deep learning, including representation structure, architectural mechanisms, and generalization behavior, offering a principled step toward a coherent theoretical foundation.

Transfer learning is an essential technique for many machine learning/AI models of complex structures such as large language models and generative AI. The essence of transfer learning is to leverage knowledge from resolved source tasks for a new target task, especially when the sample size $m$ of the training data for the latter is low. In this work, we rigorously analyze the potential benefit of transfer learning in terms of sample efficiency. Specifically, taking an optimal transport viewpoint of transfer learning, we find that when the data dimension $d$ is higher than $3$, the sample complexity for transfer learning is $O(m^{-(α+1)/d})$, with $α$ indicating the smoothness of the data distribution, as opposed to the $O(m^{-p/d})$ sample complexity for direct learning with $p$ indicating the smoothness of the optimal target model. Our finding theoretically supports a better sample efficiency for transfer learning, when the target task is optimizing over a family of not-so-smooth models (i.e., highly complex networks with the possible use of non-smooth activation functions). Using image classification as an example, we numerically demonstrate the sample efficiency for transfer learning, that is, in the data hungry regime, the model performance can be significantly improved by transfer learning.

A related situation arises when an auxiliary process is introduced to aid training but modelling its dynamics at generation time is unnecessary or difficult, as in Billera et al. [2025b] and Kim et al. [2025]. In each of these works, the projection result and its associated loss are derived on a case-by-case basis, and all theorems are restricted to marginalization over a discrete component of the extended state space. We introduce a general framework that removes these restrictions: given a time-inhomogeneous Feller process (Yt)0 t 1 on an arbitrary state space Y and a map Φ: Y X, one may learn a linear parametrisation of the generator of a Feller process on X whose one-time marginals coincide with those of (Φ(Yt))0 t 1. For Y = X Z and Φthe projection onto the first coordinate, this subsumes these prior works as special cases, allowing for a general class of latent processes (Zt)0 t 1 in a nearly arbitrary state space Z, using the formalism of generator matching to allow for continuous, discrete, or manifold-valued processes. In particular, the learnt process at t = 1 samples from the distribution of Φ(Y1), which is the desired data distribution. We give sufficient conditions for a loss function to be valid in this general setting, recovering the results of the works cited above as corollaries. This result has broad applicability, enabling the construction of a wide array of new flow matching schemes by allowing for a more general class of latent spaces. As a concrete new application, we outline a non-projection Φ: Y X with manifold-valued latents for protein structure generation that separates chain-level rigid-body motion from internal flexibility ( 4), where the particular chain-level versus residue-level or internal state is latent, and the model only sees the world state, which we plan to implement in future work. 2 EARLIERWORK Several recent generative models train with the aid of a latent stochastic process that is marginalised out at generation time.

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this paper, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each recommended item is a node and its expected rating is similar to its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose two algorithms for solving our problem that scale linearly in this dimension. Our experiments on real-world content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens nodes evaluations.