We consider learning the structures of Gaussian latent tree models with vector observations when a subset of them are arbitrarily corrupted. First, we present the sample complexities of Recursive Grouping (RG) and Chow-Liu Recursive Grouping (CLRG) without the assumption that the effective depth is bounded in the number of observed nodes, significantly generalizing the results in Choi et al. (2011). We show that Chow-Liu initialization in CLRG greatly reduces the sample complexity of RG from being exponential in the diameter of the tree to only logarithmic in the diameter for the hidden Markov model (HMM).

Much of reinforcement learning theory is built on top of oracles that are computationally hard to implement. Specifically for learning near-optimal policies in Partially Observable Markov Decision Processes (POMDPs), existing algorithms either need to make strong assumptions about the model dynamics (e.g.

We introduce a framework for inference in general state-space hidden Markov models (HMMs) under likelihood misspecification. In particular, we leverage the loss-theoretic perspective of Generalized Bayesian Inference (GBI) to define generalised filtering recursions in HMMs, that can tackle the problem of inference under model misspecification. In doing so, we arrive at principled procedures for robust inference against observation contamination by utilising the $\beta$-divergence. Operationalising the proposed framework is made possible via sequential Monte Carlo methods (SMC), where the standard particle methods, and their associated convergence results, are readily adapted to the new setting. We demonstrate our approach to object tracking and Gaussian process regression problems, and observe improved performance over standard filtering algorithms.

We study the problem of off-policy evaluation (OPE) for episodic Partially Observable Markov Decision Processes (POMDPs) with continuous states. Motivated by the recently proposed proximal causal inference framework, we develop a non-parametric identification result for estimating the policy value via a sequence of so-called V-bridge functions with the help of time-dependent proxy variables. We then develop a fitted-Q-evaluation-type algorithm to estimate V-bridge functions recursively, where a non-parametric instrumental variable (NPIV) problem is solved at each step. By analyzing this challenging sequential NPIV estimation, we establish the finite-sample error bounds for estimating the V-bridge functions and accordingly that for evaluating the policy value, in terms of the sample size, length of horizon and so-called (local) measure of ill-posedness at each step. To the best of our knowledge, this is the first finite-sample error bound for OPE in POMDPs under non-parametric models.

We study Reinforcement Learning for partially observable systems using function approximation. We propose a new PO-bilinear framework, that is general enough to include models such as undercomplete tabular Partially Observable Markov Decision Processes (POMDPs), Linear Quadratic Gaussian (LQG), Predictive State Representations (PSRs), as well as a newly introduced model Hilbert Space Embeddings of POMDPs. Under this framework, we propose an actor-critic style algorithm that is capable to performing agnostic policy learning. Given a policy class that consists of memory based policies (i.e., policy that looks at a fixed-length window of recent observations), and a value function class that consists of functions taking both memory and future observations as inputs, our algorithm learns to compete against the best memory-based policy among the policy class. For certain examples such as undercomplete POMDPs and LQGs, by leveraging their special properties, our algorithm is even capable of competing against the globally optimal policy without paying an exponential dependence on the horizon.

Establishing central limit theorems (CLTs) for ergodic averages of Markov chains is a fundamental problem in probability and its applications. Since the seminal work~\cite{MR834478}, a vast literature has emerged on the sufficient conditions for such CLTs. To counterbalance this, the present paper provides verifiable necessary conditions for CLTs of ergodic averages of Markov chains on general state spaces. Our theory is based on drift conditions, which also yield lower bounds on the rates of convergence to stationarity in various metrics. The validity of the ergodic CLT is of particular importance for sampling algorithms, where it underpins the error analysis of estimators in Bayesian statistics and machine learning. Although heavy-tailed sampling is of central importance in applications, the characterisation of the CLT and the convergence rates are theoretically poorly understood for almost all practically-used Markov chain Monte Carlo (MCMC) algorithms. In this setting our results provide sharp conditions on the validity of the ergodic CLT and establish convergence rates for large families of MCMC sampling algorithms for heavy-tailed targets. Our study includes a rather complete analyses for random walk Metropolis samplers (with finite- and infinite-variance proposals), Metropolis-adjusted and unadjusted Langevin algorithms and the stereographic projection sampler (as well as the independence sampler). By providing these sharp results via our practical drift conditions, our theory offers significant insights into the problems of algorithm selection and comparison for sampling heavy-tailed distributions (see short YouTube presentations~\cite{YouTube_talk} describing our \href{https://youtu.be/m2y7U4cEqy4}{\underline{theory}} and \href{https://youtu.be/w8I_oOweuko}{\underline{applications}}).

Sampling from multimodal distributions is a central challenge in Bayesian inference and machine learning. In light of hardness results for sampling -- classical MCMC methods, even with tempering, can suffer from exponential mixing times -- a natural question is how to leverage additional information, such as a warm start point for each mode, to enable faster mixing across modes. To address this, we introduce Reweighted ALPS (Re-ALPS), a modified version of the Annealed Leap-Point Sampler (ALPS) that dispenses with the Gaussian approximation assumption. We prove the first polynomial-time bound that works in a general setting, under a natural assumption that each component contains significant mass relative to the others when tilted towards the corresponding warm start point. Similarly to ALPS, we define distributions tilted towards a mixture centered at the warm start points, and at the coldest level, use teleportation between warm start points to enable efficient mixing across modes. In contrast to ALPS, our method does not require Hessian information at the modes, but instead estimates component partition functions via Monte Carlo. This additional estimation step is crucial in allowing the algorithm to handle target distributions with more complex geometries besides approximate Gaussian. For the proof, we show convergence results for Markov processes when only part of the stationary distribution is well-mixing and estimation for partition functions for individual components of a mixture. We numerically evaluate our algorithm's mixing performance compared to ALPS on a mixture of heavy-tailed distributions.

Autoregressive models (ARMs) currently constitute the dominant paradigm for large language models (LLMs). Energy-based models (EBMs) represent another class of models, which have historically been less prevalent in LLM development, yet naturally characterize the optimal policy in post-training alignment. In this paper, we provide a unified view of these two model classes. Taking the chain rule of probability as a starting point, we establish an explicit bijection between ARMs and EBMs in function space, which we show to correspond to a special case of the soft Bellman equation in maximum entropy reinforcement learning. Building upon this bijection, we derive the equivalence between supervised learning of ARMs and EBMs. Furthermore, we analyze the distillation of EBMs into ARMs by providing theoretical error bounds. Our results provide insights into the ability of ARMs to plan ahead, despite being based on the next-token prediction paradigm.

Markov Chain Monte Carlo (MCMC), Laplace approximation (LA) and variational inference (VI) methods are popular approaches to Bayesian inference, each with trade-offs between computational cost and accuracy. However, a theoretical understanding of these differences is missing, particularly when both the sample size $n$ and the dimension $d$ are large. LA and Gaussian VI are justified by Bernstein-von Mises (BvM) theorems, and recent work has derived the characteristic condition $n\gg d^2$ for their validity, improving over the condition $n\gg d^3$. In this paper, we show for linear, logistic and Poisson regression that for $n\gtrsim d$, MCMC attains the same complexity scaling in $n$, $d$ as first-order optimization algorithms, up to sub-polynomial factors. Thus MCMC is competitive with LA and Gaussian VI in complexity, under a scaling between $n$ and $d$ more general than BvM regimes. Our complexities apply to appropriately scaled priors that are not necessarily Gaussian-tailed, including Student-$t$ and flat priors, with log-posteriors that are not necessarily globally concave or gradient-Lipschitz.

We present a framework using variational inference with normalizing flows (VI-NFs) to generate proposals of reversible jump Markov chain Monte Carlo (RJMCMC) for efficient trans-dimensional Bayesian inference. Unlike transport reversible jump methods relying on forward KL minimization with pilot MCMC samples, our approach minimizes the reverse KL divergence which requires only samples from a base distribution, eliminating costly target sampling. The method employs RealNVP-based flows to learn model-specific transport maps, enabling construction of both between-model and within-model proposals. Our framework provides accurate marginal likelihood estimates from the variational approximation. This facilitates efficient model comparison and proposal adaptation in RJMCMC. Experiments on illustrative example, factor analysis and variable selection tasks in linear regression show that TRJ designed by VI-NFs achieves faster mixing and more efficient model space exploration compared to existing baselines. The proposed algorithm can be extended to conditional flows for amortized vairiational inference across models. Code is available at https://github.com/YinPingping111/TRJ_VINFs.