The Inference Report

Aligning two models without paired data requires a distance that both share, yet existing methods use Euclidean or spherical geometry. This is either an arbitrary choice or imposed by their loss function. We show that, under a precise form of the manifold hypothesis, the pullback of the data metric gives such a distance for any two models of one data-generating process. This holds whether the models are encoders, decoders or both, and whatever their architecture, loss, prior or training data. The alignment map preserves this distance and the distribution of representations, and is identifiable if and only if the data distribution breaks every symmetry of the data manifold. We show that this symmetry breaking holds generically. Identifiability makes the alignment coupling the unique optimum of Gromov--Wasserstein and of its weighted combinations with a linear program. The linear program alone has the same optimum if almost every data point has a distinct distribution of distances. The coupling, and with it the alignment map, is recovered exactly from permuted samples, even without identifiability, and consistently from independent ones. Our method, Neural Gromov--Monge Alignment (NGMA), uses no pairs or labels and outperforms every unpaired and label-informed baseline. It relies on pullback distances to align image models with different priors and single-cell models trained on disjoint cells, where Euclidean distances fail.

Amortized Bayesian experimental design (BED) enables real-time design strategies by shifting acquisition cost offline, but existing methods remain tied to the prior and task distribution they were trained on and cannot exploit additional information available at deployment. We introduce in-context amortized BED with unified knowledge encoding (IMBUE), a BED framework that admits external knowledge at deployment through a unified in-context interface. IMBUE accepts two forms of such knowledge without retraining, namely user-specified prior knowledge and observations from previous instances of the experiment. Both are encoded as additional input tokens and processed in-context with the accumulated observations by a shared Transformer, and a learned reliability filter scores each external token against the observations and excludes inconsistent ones. Across four benchmarks, IMBUE accelerates early-stage information acquisition when the external knowledge is reliable, and remains close to the no-external baseline when it is not.

User sequence modeling is widely used in recommendation and advertising. Search ranking presents an additional challenge: the current query expresses immediate intent, while a candidate product determines which parts of a user's long-term history are relevant. We present a production sequence framework for sponsored-product ranking that augments recent behavior with two retrieval channels over long-term history: candidate-nearest-neighbor retrieval and query-nearest-neighbor retrieval. The selected actions are encoded together with dedicated query and candidate context tokens, while candidate-aware early fusion injects the candidate into every history token. Offline ablations show that dual retrieval improves major ranking metrics relative to the production baseline. In an online A/B test on sponsored-product search traffic, the model increased CTR, attributed conversion rate, and orders per impression.

This work proposes the Adaptive Federated Hamiltonian Monte Carlo (AFHMC) for Bayesian federated learning (FL) tasks. The prior work on HMC applications for FL achieves $O(\sqrt{d}/ε)$ communication cost under log-convex distributions, already demonstrating its advantage over the Langevin dynamic-based counterpart. Based on a refined analysis, this paper further reveals the benefit of HMC for the FL regime by incorporating a control mechanism for local node shifts. We prove that AFHMC at most requires $O(\{d/ε^2\}^{1/3})$ communication cost up to a logarithmic term, which is significantly better than the existing results. Moreover, the proposed method adapts to client heterogeneity. Under low-heterogeneity settings (i.e., the target sampling precision $ε^2$ is higher than the heterogeneity level), the communication rate of AFHMC can be as good as $O(\log(d/ε^2))$, resembling the communication costs for FL optimization tools. To the best of our knowledge, this provides the first logarithmic communication complexity result for federated sampling, even though it requires low heterogeneity. Numerically, we also justify AFHMC's advantage under the low-heterogeneity regime. Our code is available at https://github.com/laaaarrywang/Adaptive-Federated-HMC

Predict-then-optimize systems convert contextual forecasts into decisions, yet conformal robust optimization typically calibrates prediction residuals rather than downstream loss. For linear optimization, this parameter-space geometry can be badly misaligned with decision risk: large objective perturbations may preserve the optimizer, while small boundary-crossing perturbations may cause substantial regret. We introduce Conformal Regret Sets, a post-hoc, model-agnostic framework that conformalizes an optimization-derived regret surrogate and uses its calibrated level set for minimax-regret refinement. We first show that directly conformalizing exact regret, although sufficient to certify the nominal predict-then-optimize decision, yields a degenerate robust problem: absent an exogenous bounded support, every alternative decision has unbounded worst-case regret. We therefore use SPO+, a pointwise upper bound on exact regret, as the conformal score, obtaining distribution-free finite-sample marginal regret guarantees under exchangeability. For contextual linear programs, the resulting set decomposes into polyhedral pieces aligned with the normal fan of the feasible polytope. A vertex is active if and only if its predicted suboptimality gap is at most half the calibrated threshold $η$, enabling local, output-sensitive generation of all relevant pieces. Moreover, if $Γ$ is the predicted gap to the closest competing vertex, the nominal and refined decisions admit the sharper worst-case-regret certificate $(η-2Γ)_+$, with an exact early exit when $η\leq 2Γ$. Experiments on contextual fractional knapsack validate the geometry and certificates: continuous refinement reduces the $0.95$-quantile and $\operatorname{CVaR}_{0.95}$ of regret by $9.1\%$ and $13.1\%$ relative to nominal PTO, while vertex-restricted refinement changes only $10.0\%$ of nominal decisions.

Many problems across the sciences compare two stochastic processes through the sequences they generate. Reinforcement learning asks what a new policy would return from the trajectories of an old one; molecular simulation asks how much more a ligand binds from samples of the unbound state; climate attribution asks how a heat wave shifted under warming; and language-model post-training asks what a deployed agent will do from logged interactions. In every case, one quantity would answer all these questions at once: the ratio of the two probability laws over their generated sequences, the trajectory density ratio. This ratio exists only under strong conditions. When one process reaches states the other cannot, the ratio does not exist; when their overlap is thin, its variance grows without control with the sequence length; and when they concentrate on different lower-dimensional sets, the ratio is undefined altogether. Weak support of this kind is the norm for sequence data, not a sampling accident, and its price is paid in identifiability rather than in noise alone. This survey organizes the field around a three-level diagnosis of weak support and a map of the nine retreats that methods take when the full ratio is out of reach, from direct estimation and reweighting through staged decomposition, stationarization, and partial-identification bounds to smoothing, transport couplings, operator spectra, and direct estimation of a single functional. Each retreat keeps part of what the ratio offers and pays for the rest, and each application domain has standardized the retreat its data force. The survey closes with open problems: testable diagnostics at every level, partial identification at the trajectory level, a rate theory of time-amplified error, and benchmarks with ground truth.