Retrospective causal questions ask what would have happened to an observed individual had they received a different treatment. We study the problem of estimating $μ(x,y)=\mathbb{E}[Y(1)\mid X=x,Y(0)=y]$, the expected counterfactual outcome for an individual with covariates $x$ and observed outcome $y$, and constructing valid prediction intervals under the Neyman-Rubin superpopulation model. This quantity is generally not identified without additional assumptions. To link the observed and unobserved potential outcomes, we work with a cross-world correlation $ρ(x)=cor(Y(1),Y(0)\mid X=x)$; plausible bounds on $ρ(x)$ enable a principled approach to this otherwise unidentified problem. We introduce retrospective counterfactual estimators $\hatμ_ρ(x,y)$ and prediction intervals $C_ρ(x,y)$ that asymptotically satisfy $P[Y(1)\in C_ρ(x,y)\mid X=x, Y(0)=y]\ge1-α$ under standard causal assumptions. Many common baselines implicitly correspond to endpoint choices $ρ=0$ or $ρ=1$ (ignoring the factual outcome or treating the counterfactual as a shifted factual outcome). Interpolating between these cases through cross-world dependence yields substantial gains in both theory and practice.

Predictive inference is a fundamental task in statistics, traditionally addressed using parametric assumptions about the data distribution and detailed analyses of how models learn from data. In recent years, conformal prediction has emerged as a rapidly growing alternative framework that is particularly well suited to modern applications involving high-dimensional data and complex machine learning models. Its appeal stems from being both distribution-free -- relying mainly on symmetry assumptions such as exchangeability -- and model-agnostic, treating the learning algorithm as a black box. Even under such limited assumptions, conformal prediction provides exact finite-sample guarantees, though these are typically of a marginal nature that requires careful interpretation. This paper explains the core ideas of conformal prediction and reviews selected methods. Rather than offering an exhaustive survey, it aims to provide a clear conceptual entry point and a pedagogical overview of the field.

The existence of such a subgraph is guaranteed (See Lemma 3). Let x,y Rn be vectors of dimensionn. We choose a random embeddingZP Das thepivot (line 2), based on which wesplittheremaining embeddings intofourgroups (lines3-8). Thisprocess continues recursively on each group (lines 9-10) till a partition gets empty (line 1). D.3 k-NNquery k-NN utilizes the same bounds from Alg. 2 to prune and prioritize the search space.

Identifying truly predictive covariates while strictly controlling false discoveries remains a fundamental challenge in nonlinear, highly correlated, and low signal-to-noise regimes, where deep learning based feature selection methods are most attractive. We propose Group Regularization Importance Persistence in 2 Dimensions (GRIP2), a deep knockoff feature importance statistic that integrates first-layer feature activity over a two-dimensional regularization surface controlling both sparsity strength and sparsification geometry. To approximate this surface integral in a single training run, we introduce efficient block-stochastic sampling, which aggregates feature activity magnitudes across diverse regularization regimes along the optimization trajectory. The resulting statistics are antisymmetric by construction, ensuring finite-sample FDR control. In extensive experiments on synthetic and semi-real data, GRIP2 demonstrates improved robustness to feature correlation and noise level: in high correlation and low signal-to-noise ratio regimes where standard deep learning based feature selectors may struggle, our method retains high power and stability. Finally, on real-world HIV drug resistance data, GRIP2 recovers known resistance-associated mutations with power better than established linear baselines, confirming its reliability in practice.

Glioblastoma (GBM) remains the most aggressive tumor, urgently requiring novel therapeutic strategies. Here, we present a dry-to-wet framework combining generative modeling and experimental validation to optimize peptides targeting ATP5A, a potential peptide-binding protein for GBM. Our framework introduces the first lead-conditioned generative model, which focuses exploration on geometrically relevant regions around lead peptides and mitigates the combinatorial complexity of de novo methods. Specifically, we propose POTFlow, a \underline{P}rior and \underline{O}ptimal \underline{T}ransport-based \underline{Flow}-matching model for peptide optimization. POTFlow employs secondary structure information (e.g., helix, sheet, loop) as geometric constraints, which are further refined by optimal transport to produce shorter flow paths. With this design, our method achieves state-of-the-art performance compared with five popular approaches. When applied to GBM, our method generates peptides that selectively inhibit cell viability and significantly prolong survival in a patient-derived xenograft (PDX) model. As the first lead peptide-conditioned flow matching model, POTFlow holds strong potential as a generalizable framework for therapeutic peptide design.

Time-critical data aggregation in Internet of Things (IoT) networks demands efficient, collision-free scheduling to minimize latency for applications like smart cities and industrial automation. Traditional heuristic methods, with two-phase tree construction and scheduling, often suffer from high computational overhead and suboptimal delays due to their static nature. To address this, we propose a novel Q-learning framework that unifies aggregation tree construction and scheduling, modeling the process as a Markov Decision Process (MDP) with hashed states for scalability. By leveraging a reward function that promotes large, interference-free batch transmissions, our approach dynamically learns optimal scheduling policies. Simulations on static networks with up to 300 nodes demonstrate up to 10.87% lower latency compared to a state-of-the-art heuristic algorithm, highlighting its robustness for delay-sensitive IoT applications. This framework enables timely insights in IoT environments, paving the way for scalable, low-latency data aggregation.

We explore the use of expert-guided bandit learning, which we refer to as online mixture-of-experts (OMoE). In this setting, given a context, a candidate committee of experts must determine how to aggregate their outputs to achieve optimal results in terms of aggregate accuracy. We propose two algorithms to address this problem. The first algorithm combines aggregate voting with UCB-driven successive elimination, efficiently pruning suboptimal exploration actions. The second algorithm employs an online weighted-majority-voting mechanism, leveraging the respective voting power of each expert proportional to their predictive power. We derive theoretical guarantees for the regret properties in the bandit setting under ideal circumstances, and empirical results are provided accordingly. As a modern study on applications, these methods are applied to the online fine-tuning of a set of expert large language models (LLMs), where after each response, the generative LLM dynamically reweighs its set of experts and/or selects the optimal committee of experts to generate the most accurate response. Our results introduce new methodologies and no-regret guarantees for combining multiple experts to improve on the performance of the an aggregate model overall.

Projection methods aim to reduce the dimensionality of the optimization instance, thereby improving the scalability of high-dimensional problems. Recently, Sakaue and Oki proposed a data-driven approach for linear programs (LPs), where the projection matrix is learned from observed problem instances drawn from an application-specific distribution of problems. We analyze the generalization guarantee for the data-driven projection matrix learning for convex quadratic programs (QPs). Unlike in LPs, the optimal solutions of convex QPs are not confined to the vertices of the feasible polyhedron, and this complicates the analysis of the optimal value function. To overcome this challenge, we demonstrate that the solutions of convex QPs can be localized within a feasible region corresponding to a special active set, utilizing Caratheodory's theorem. Building on such observation, we propose the unrolled active set method, which models the computation of the optimal value as a Goldberg-Jerrum (GJ) algorithm with bounded complexities, thereby establishing learning guarantees. We then further extend our analysis to other settings, including learning to match the optimal solution and input-aware setting, where we learn a mapping from QP problem instances to projection matrices.

Gradient Boosting Decision Trees (GBDTs) are widely used in industry and academia for their high accuracy and efficiency, particularly on structured data. However, watermarking GBDT models remains underexplored compared to neural networks. In this work, we present the first robust watermarking framework tailored to GBDT models, utilizing in-place fine-tuning to embed imperceptible and resilient watermarks. We propose four embedding strategies, each designed to minimize impact on model accuracy while ensuring watermark robustness. Through experiments across diverse datasets, we demonstrate that our methods achieve high watermark embedding rates, low accuracy degradation, and strong resistance to post-deployment fine-tuning.