Reinforcement learning (RL) typically models the interaction between the agent and environment as a Markov decision process (MDP), where the rewards that guide the agent's behavior are always observable. However, in many real-world scenarios, rewards are not always observable, which can be modeled as a monitored Markov decision process (Mon-MDP). Prior work on Mon-MDPs have been limited to simple, tabular cases, restricting their applicability to real-world problems. This work explores Mon-MDPs using function approximation (FA) and investigates the challenges involved. We show that combining function approximation with a learned reward model enables agents to generalize from monitored states with observable rewards, to unmonitored environment states with unobservable rewards. Therefore, we demonstrate that such generalization with a reward model achieves near-optimal policies in environments formally defined as unsolvable. However, we identify a critical limitation of such function approximation, where agents incorrectly extrapolate rewards due to overgeneralization, resulting in undesirable behaviors. To mitigate overgeneralization, we propose a cautious police optimization method leveraging reward uncertainty. This work serves as a step towards bridging this gap between Mon-MDP theory and real-world applications.

-- We study policy distillation under privileged information, where a student policy with only partial observations must learn from a teacher with full-state access. A key challenge is information asymmetry: the student cannot directly access the teacher's state space, leading to distributional shifts and policy degradation. Existing approaches either modify the teacher to produce realizable but sub-optimal demonstrations or rely on the student to explore missing information independently, both of which are inefficient. Our key insight is that the student should strategically interact with the teacher --querying only when necessary and resetting from recovery states --to stay on a recoverable path within its own observation space. We introduce two methods: (i) an imitation learning approach that adaptively determines when the student should query the teacher for corrections, and (ii) a reinforcement learning approach that selects where to initialize training for efficient exploration. The project website is available here. Robots operating in the real world must learn to act effectively despite partial observations and limited ability to explore. Unlike in simulation, where policies have access to privileged state information, real-world policies must make decisions based on incomplete inputs [1]-[3].

Counterfactuals are widely used in AI to explain how minimal changes to a model's input can lead to a different output. However, established methods for computing counterfactuals typically focus on one-step decision-making, and are not directly applicable to sequential decision-making tasks. This paper fills this gap by introducing counterfactual strategies for Markov Decision Processes (MDPs). During MDP execution, a strategy decides which of the enabled actions (with known probabilistic effects) to execute next. Given an initial strategy that reaches an undesired outcome with a probability above some limit, we identify minimal changes to the initial strategy to reduce that probability below the limit. We encode such counterfactual strategies as solutions to non-linear optimization problems, and further extend our encoding to synthesize diverse counterfactual strategies. We evaluate our approach on four real-world datasets and demonstrate its practical viability in sophisticated sequential decision-making tasks.

--The recovery of block-sparse signals with unknown structural patterns remains a fundamental challenge in structured sparse signal reconstruction. By proposing a variance transformation framework, this paper unifies existing pattern-based block sparse Bayesian learning methods, and introduces a novel space power prior based on undirected graph models to adaptively capture the unknown patterns of block-sparse signals. By combining the EM algorithm with high-order equation root-solving, we develop a new structured sparse Bayesian learning method, SPP-SBL, which effectively addresses the open problem of space coupling parameter estimation in pattern-based methods. We further demonstrate that learning the relative values of space coupling parameters is key to capturing unknown block-sparse patterns and improving recovery accuracy. Experiments validate that SPP-SBL successfully recovers various challenging structured sparse signals (e.g., chain-structured signals and multi-pattern sparse signals) and real-world multi-modal structured sparse signals (images, audio), showing significant advantages in recovery accuracy across multiple metrics. Index T erms --Compressed sensing, Space-Power prior, block sparsity, sparse Bayesian learning, expectation-maximization. P ARSE recovery through Compressed Sensing (CS) has garnered significant attention due to its robust theoretical foundation and wide-ranging applications [1], particularly for its efficacy in reconstructing sparse vectors from a substantially reduced number of linear measurements.

We study the classical binary classification problem for hypothesis spaces of Deep Neural Networks (DNNs) with ReLU activation under Tsybakov's low-noise condition with exponent $q>0$, and its limit-case $q\to\infty$ which we refer to as the "hard-margin condition". We show that DNNs which minimize the empirical risk with square loss surrogate and $\ell_p$ penalty can achieve finite-sample excess risk bounds of order $\mathcal{O}\left(n^{-α}\right)$ for arbitrarily large $α>0$ under the hard-margin condition, provided that the regression function $η$ is sufficiently smooth. The proof relies on a novel decomposition of the excess risk which might be of independent interest.

Censored response variables--where outcomes are only partially observed due to known bounds--arise in numerous scientific domains and present serious challenges for regression analysis. The Tobit model, a classical solution for handling left-censoring, has been widely used in economics and beyond. However, with the increasing prevalence of high-dimensional data, where the number of covariates exceeds the sample size, traditional Tobit methods become inadequate. While frequentist approaches for high-dimensional Tobit regression have recently been developed, notably through Lasso-based estimators, the Bayesian literature remains sparse and lacks theoretical guarantees. In this work, we propose a novel Bayesian framework for high-dimensional Tobit regression that addresses both censoring and sparsity. Our method leverages the Horseshoe prior to induce shrinkage and employs a data augmentation strategy to facilitate efficient posterior computation via Gibbs sampling. We establish posterior consistency and derive concentration rates under sparsity, providing the first theoretical results for Bayesian Tobit models in high dimensions. Numerical experiments show that our approach outperforms favorably with the recent Lasso-Tobit method. Our method is implemented in the R package tobitbayes, which can be found on Github.

Network interference occurs when treatments assigned to some units affect the outcomes of others. Traditional approaches often assume that the observed network correctly specifies the interference structure. However, in practice, researchers frequently only have access to proxy measurements of the interference network due to limitations in data collection or potential mismatches between measured networks and actual interference pathways. In this paper, we introduce a framework for estimating causal effects when only proxy networks are available. Our approach leverages a structural causal model that accommodates diverse proxy types, including noisy measurements, multiple data sources, and multilayer networks, and defines causal effects as interventions on population-level treatments. Since the true interference network is latent, estimation poses significant challenges. To overcome them, we develop a Bayesian inference framework. We propose a Block Gibbs sampler with Locally Informed Proposals to update the latent network, thereby efficiently exploring the high-dimensional posterior space composed of both discrete and continuous parameters. We illustrate the performance of our method through numerical experiments, demonstrating its accuracy in recovering causal effects even when only proxies of the interference network are available.

We propose a hybrid generative model for efficient sampling of high-dimensional, multimodal probability distributions for Bayesian inference. Traditional Monte Carlo methods, such as the Metropolis-Hastings and Langevin Monte Carlo sampling methods, are effective for sampling from single-mode distributions in high-dimensional spaces. However, these methods struggle to produce samples with the correct proportions for each mode in multimodal distributions, especially for distributions with well separated modes. To address the challenges posed by multimodality, we adopt a divide-and-conquer strategy. We start by minimizing the energy function with initial guesses uniformly distributed within the prior domain to identify all the modes of the energy function. Then, we train a classifier to segment the domain corresponding to each mode. After the domain decomposition, we train a diffusion-model-assisted generative model for each identified mode within its support. Once each mode is characterized, we employ bridge sampling to estimate the normalizing constant, allowing us to directly adjust the ratios between the modes. Our numerical examples demonstrate that the proposed framework can effectively handle multimodal distributions with varying mode shapes in up to 100 dimensions. An application to Bayesian inverse problem for partial differential equations is also provided.

Statistical inference of model parameters from empirical observations is a fundamental challenge in scientific research, enabling researchers to derive meaningful insights from complex simulation models. These parameters govern the behavior of simulators that replicate real-world phenomena, providing a bridge between theoretical constructs and empirical observations [Lavin et al., 2021]. Calibrating these parameters to ensure that simulator outputs align with observed data constitutes an inverse problem, formally defined within the framework of simulation-based inference (SBI) [Cranmer et al., 2020]. Solving this inverse problem involves addressing uncertainties arising from model stochasticity and potential multi-valuedness, where different sets of parameter values can produce similar observations or similar parameters may lead to varied outputs. Additionally, parameter inference becomes increasingly complex when simulators operate as'black boxes' with intractable likelihood functions, rendering traditional likelihood-based Bayesian methods impractical [Sisson et al., 2018].

In this paper, we study the concentration properties of quadratic forms associated with Markov chains using the martingale decomposition method introduced by Atchadé and Cattaneo (2014). In particular, we derive concentration inequalities for the overlapped batch mean (OBM) estimators of the asymptotic variance for uniformly geometrically ergodic Markov chains. Our main result provides an explicit control of the $p$-th moment of the difference between the OBM estimator and the asymptotic variance of the Markov chain with explicit dependence upon $p$ and mixing time of the underlying Markov chain.