Feature selection is a classical problem in statistics and machine learning, and it continues to remain an extremely challenging problem especially in the context of unknown non-linear relationships with dependent features. On the other hand, Shapley values are a classic solution concept from cooperative game theory that is widely used for feature attribution in general non-linear models with highly-dependent features. However, Shapley values are not naturally suited for feature selection since they tend to capture both direct effects from each feature to the response and indirect effects through other features. In this paper, we combine the advantages of Shapley values and adapt them to feature selection by proposing \emph{MinShap}, a modification of the Shapley value framework along with a suite of other related algorithms. In particular for MinShap, instead of taking the average marginal contributions over permutations of features, considers the minimum marginal contribution across permutations. We provide a theoretical foundation motivated by the faithfulness assumption in DAG (directed acyclic graphical models), a guarantee for the Type I error of MinShap, and show through numerical simulations and real data experiments that MinShap tends to outperform state-of-the-art feature selection algorithms such as LOCO, GCM and Lasso in terms of both accuracy and stability. We also introduce a suite of algorithms related to MinShap by using the multiple testing/p-value perspective that improves performance in lower-sample settings and provide supporting theoretical guarantees.

Since their introduction by Breiman (2001), permutation-based feature importance measures have been widely adopted. However, randomly permuting the entries of a dataset may create new points far from the original data or even "impossible data." In a permuted dataset, we may find children who are retired or individuals who graduated from high school before they were born (Mase et al. 2022, p. 1). Forcing ML models to make predictions at these points causes them to extrapolate, making explanations unreliable (Hooker et al. 2021). Every non-trivial permutation-based variable importance measure, including SHAP (Lundberg and Lee 2017), Knockoffs (Barber and Candés 2015), conditional model reliance (Fisher et al. 2019), and accumulated local effect (ALE) plots (Apley and Zhu 2020) suffer from this. We propose and compare three new strategies to address extrapolation issues. The first combines conditional model reliance from Fisher et al. (2019) with a Gaussian transformation. By mapping data quantiles to a Gaussian distribution and back, we adjust only the quantiles of point values, significantly reducing extrapolation. Under a Gaussian copula assumption for the feature distribution, we prove that the new data points follow the same probability distribution as the original data.

Causal representation learning seeks to uncover causal relationships among high-level latent variables from low-level, entangled, and noisy observations. Existing approaches often either rely on deep neural networks, which lack interpretability and formal guarantees, or impose restrictive assumptions like linearity, continuous-only observations, and strong structural priors. These limitations particularly challenge applications with a large number of discrete latent variables and mixed-type observations. To address these challenges, we propose discrete causal representation learning (DCRL), a generative framework that models a directed acyclic graph among discrete latent variables, along with a sparse bipartite graph linking latent and observed layers. This design accommodates continuous, count, and binary responses through flexible measurement models while maintaining interpretability. Under mild conditions, we prove that both the bipartite measurement graph and the latent causal graph are identifiable from the observed data distribution alone. We further propose a three-stage estimate-resample-discovery pipeline: penalized estimation of the generative model parameters, resampling of latent configurations from the fitted model, and score-based causal discovery on the resampled latents. We establish the consistency of this procedure, ensuring reliable recovery of the latent causal structure. Empirical studies on educational assessment and synthetic image data demonstrate that DCRL recovers sparse and interpretable latent causal structures.

We investigate contextual graph matching in the Gaussian setting, where both edge weights and node features are correlated across two networks. We derive precise information-theoretic thresholds for exact recovery, and identify conditions under which almost exact recovery is possible or impossible, in terms of graph and feature correlation strengths, the number of nodes, and feature dimension. Interestingly, whereas an all-or-nothing phase transition is observed in the standard graph-matching scenario, the additional contextual information introduces a richer structure: thresholds for exact and almost exact recovery no longer coincide. Our results provide the first rigorous characterization of how structural and contextual information interact in graph matching, and establish a benchmark for designing efficient algorithms.

Can latent actions and environment dynamics be recovered from offline trajectories when actions are never observed? We study this question in a setting where trajectories are action-free but tagged with demonstrator identity. We assume that each demonstrator follows a distinct policy, while the environment dynamics are shared across demonstrators and identity affects the next observation only through the chosen action. Under these assumptions, the conditional next-observation distribution $p(o_{t+1}\mid o_t,e)$ is a mixture of latent action-conditioned transition kernels with demonstrator-specific mixing weights. We show that this induces, for each state, a column-stochastic nonnegative matrix factorization of the observable conditional distribution. Using sufficiently scattered policy diversity and rank conditions, we prove that the latent transitions and demonstrator policies are identifiable up to permutation of the latent action labels. We extend the result to continuous observation spaces via a Gram-determinant minimum-volume criterion, and show that continuity of the transition map over a connected state space upgrades local permutation ambiguities to a single global permutation. A small amount of labeled action data then suffices to fix this final ambiguity. These results establish demonstrator diversity as a principled source of identifiability for learning latent actions and dynamics from offline RL data.

We study best-arm identification in stochastic dueling bandits under the sole assumption that a Condorcet winner exists, i.e., an arm that wins each noisy pairwise comparison with probability at least $1/2$. We introduce a new identification procedure that exploits the full gap matrix $Δ_{i,j}=q_{i,j}-\tfrac12$ (where $q_{i,j}$ is the probability that arm $i$ beats arm $j$), rather than only the gaps between the Condorcet winner and the other arms. We derive high-probability, instance-dependent sample-complexity guarantees that (up to logarithmic factors) improve the best known ones by leveraging informative comparisons beyond those involving the winner. We complement these results with new lower bounds which, to our knowledge, are the first for Condorcet-winner identification in stochastic dueling bandits. Our lower-bound analysis isolates the intrinsic cost of locating informative entries in the gap matrix and estimating them to the required confidence, establishing the optimality of our non-asymptotic bounds. Overall, our results reveal new regimes and trade-offs in the sample complexity that are not captured by asymptotic analyses based only on the expected budget.

We show in this paper that all existing implementations of gradient boosting face the following statistical issue.

Black-box optimisation is one of the important areas in optimisation.

However, developing models that can process weight-space objects is challenging due to their high dimensional nature.