A core goal in systems neuroscience and neuroethology is to understand how neural circuits generate naturalistic behavior. One foundational idea is that complex naturalistic behavior may be composed of sequences of stereotyped behavioral syllables, which combine to generate rich sequences of actions. To investigate this, a common approach is to use autoregressive hidden Markov models (ARHMMs) to segment video into discrete behavioral syllables. While these approaches have been successful in extracting syllables that are interpretable, they fail to account for other forms of behavioral variability, such as differences in speed, which may be better described as continuous in nature. To overcome these limitations, we introduce a class of warped ARHMMs (WARHMM). As is the case in the ARHMM, behavior is modeled as a mixture of autoregressive dynamics.

Partial observability is a common challenge in many reinforcement learning applications, which requires an agent to maintain memory, infer latent states, and integrate this past information into exploration. This challenge leads to a number of computational and statistical hardness results for learning general Partially Observable Markov Decision Processes (POMDPs). This work shows that these hardness barriers do not preclude efficient reinforcement learning for rich and interesting subclasses of POMDPs. In particular, we present a sample-efficient algorithm, OOM-UCB, for episodic finite undercomplete POMDPs, where the number of observations is larger than the number of latent states and where exploration is essential for learning, thus distinguishing our results from prior works. OOM-UCB achieves an optimal sample complexity of $\tilde{\mathcal{O}}(1/\varepsilon^2)$ for finding an $\varepsilon$-optimal policy, along with being polynomial in all other relevant quantities. As an interesting special case, we also provide a computationally and statistically efficient algorithm for POMDPs with deterministic state transitions.

We consider the best-of-both-worlds problem for learning an episodic Markov Decision Process through $T$ episodes, with the goal of achieving $\widetilde{\mathcal{O}}(\sqrt{T})$ regret when the losses are adversarial and simultaneously $\mathcal{O}(\log T)$ regret when the losses are (almost) stochastic. Recent work by [Jin and Luo, 2020] achieves this goal when the fixed transition is known, and leaves the case of unknown transition as a major open question. In this work, we resolve this open problem by using the same Follow-the-Regularized-Leader (FTRL) framework together with a set of new techniques. Specifically, we first propose a loss-shifting trick in the FTRL analysis, which greatly simplifies the approach of [Jin and Luo, 2020] and already improves their results for the known transition case. Then, we extend this idea to the unknown transition case and develop a novel analysis which upper bounds the transition estimation error by the regret itself in the stochastic setting, a key property to ensure $\mathcal{O}(\log T)$ regret.

The vast majority of work in self-supervised learning have focused on assessing recovered features by a chosen set of downstream tasks. While there are several commonly used benchmark datasets, this lens of feature learning requires assumptions on the downstream tasks which are not inherent to the data distribution itself. In this paper, we present an alternative lens, one of parameter identifiability: assuming data comes from a parametric probabilistic model, we train a self-supervised learning predictor with a suitable parametric form, and ask whether the parameters of the optimal predictor can be used to extract the parameters of the ground truth generative model.Specifically, we focus on latent-variable models capturing sequential structures, namely Hidden Markov Models with both discrete and conditionally Gaussian observations. We focus on masked prediction as the self-supervised learning task and study the optimal masked predictor. We show that parameter identifiability is governed by the task difficulty, which is determined by the choice of data model and the amount of tokens to predict. Technique-wise, we uncover close connections with the uniqueness of tensor rank decompositions, a widely used tool in studying identifiability through the lens of the method of moments.

Existing Graph Neural Network (GNN) methods that learn graph representations focus on learning node and edge representations by predicting observed edges in the graph. Although such approaches have shown advances in downstream node classification tasks, they are ineffective in jointly representing larger k-node sets, k{>}2. We propose MHM-GNN, an inductive unsupervised graph representation approach that combines joint k-node representations with energy-based models (hypergraph Markov networks) and GNNs. To address the intractability of the loss that arises from this combination, we endow our optimization with a loss upper bound using a finite-sample unbiased Markov Chain Monte Carlo estimator. Our experiments show that the unsupervised joint k-node representations of MHM-GNN produce better unsupervised representations than existing approaches from the literature.

We consider a high-dimensional mean estimation problem over a binary hidden Markov model, which illuminates the interplay between memory in data, sample size, dimension, and signal strength in statistical inference. In this model, an estimator observes $n$ samples of a $d$-dimensional parameter vector $\theta_{*}\in\mathbb{R}^{d}$, multiplied by a random sign $ S_i $ ($1\le i\le n$), and corrupted by isotropic standard Gaussian noise. The sequence of signs $\{S_{i}\}_{i\in[n]}\in\{-1,1\}^{n}$ is drawn from a stationary homogeneous Markov chain with flip probability $\delta\in[0,1/2]$. As $\delta$ varies, this model smoothly interpolates two well-studied models: the Gaussian Location Model for which $\delta=0$ and the Gaussian Mixture Model for which $\delta=1/2$. Assuming that the estimator knows $\delta$, we establish a nearly minimax optimal (up to logarithmic factors) estimation error rate, as a function of $\|\theta_{*}\|,\delta,d,n$. We then provide an upper bound to the case of estimating $\delta$, assuming a (possibly inaccurate) knowledge of $\theta_{*}$. The bound is proved to be tight when $\theta_{*}$ is an accurately known constant. These results are then combined to an algorithm which estimates $\theta_{*}$ with $\delta$ unknown a priori, and theoretical guarantees on its error are stated.

This work studies the problem of learning episodic Markov Decision Processes with known transition and bandit feedback. We develop the first algorithm with a ``best-of-both-worlds'' guarantee: it achieves O(log T) regret when the losses are stochastic, and simultaneously enjoys worst-case robustness with \tilde{O}(\sqrt{T}) regret even when the losses are adversarial, where T is the number of episodes. More generally, it achieves \tilde{O}(\sqrt{C}) regret in an intermediate setting where the losses are corrupted by a total amount of C. Our algorithm is based on the Follow-the-Regularized-Leader method from Zimin and Neu (2013), with a novel hybrid regularizer inspired by recent works of Zimmert et al. (2019a, 2019b) for the special case of multi-armed bandits. Crucially, our regularizer admits a non-diagonal Hessian with a highly complicated inverse. Analyzing such a regularizer and deriving a particular self-bounding regret guarantee is our key technical contribution and might be of independent interest.

Bandits with knapsacks (BwK) is an influential model of sequential decision-making under uncertainty that incorporates resource consumption constraints. In each round, the decision-maker observes an outcome consisting of a reward and a vector of nonnegative resource consumptions, and the budget of each resource is decremented by its consumption. In this paper we introduce a natural generalization of the stochastic BwK problem that allows non-monotonic resource utilization. In each round, the decision-maker observes an outcome consisting of a reward and a vector of resource drifts that can be positive, negative or zero, and the budget of each resource is incremented by its drift. Our main result is a Markov decision process (MDP) policy that has constant regret against a linear programming (LP) relaxation when the decision-maker knows the true outcome distributions. We build upon this to develop a learning algorithm that has logarithmic regret against the same LP relaxation when the decision-maker does not know the true outcome distributions. We also present a reduction from BwK to our model that shows our regret bound matches existing results.

The study of Markov processes and broadcasting on trees has deep connections to a variety of areas including statistical physics, graphical models, phylogenetic reconstruction, Markov Chain Monte Carlo, and community detection in random graphs. Notably, the celebrated Belief Propagation (BP) algorithm achieves Bayes-optimal performance for the reconstruction problem of predicting the value of the Markov process at the root of the tree from its values at the leaves.Recently, the analysis of low-degree polynomials has emerged as a valuable tool for predicting computational-to-statistical gaps. In this work, we investigate the performance of low-degree polynomials for the reconstruction problem on trees. Perhaps surprisingly, we show that there are simple tree models with $N$ leaves and bounded arity where (1) nontrivial reconstruction of the root value is possible with a simple polynomial time algorithm and with robustness to noise, but not with any polynomial of degree $N^{c}$ for $c > 0$ a constant depending only on the arity, and (2) when the tree is unknown and given multiple samples with correlated root assignments, nontrivial reconstruction of the root value is possible with a simple Statistical Query algorithm but not with any polynomial of degree $N^c$. These results clarify some of the limitations of low-degree polynomials vs. polynomial time algorithms for Bayesian estimation problems. They also complement recent work of Moitra, Mossel, and Sandon who studied the circuit complexity of Belief Propagation. As a consequence of our main result, we are able to prove a result of independent interest regarding the performance of RBF kernel ridge regression for learning to predict the root coloration: for some $c' > 0$ depending only on the arity, $\exp(N^{c'})$ many samples are needed for the kernel regression to obtain nontrivial correlation with the true regression function (BP). We pose related open questions about low-degree polynomials and the Kesten-Stigum threshold.

Pretrained language models have achieved state-of-the-art performance when adapted to a downstream NLP task. However, theoretical analysis of these models is scarce and challenging since the pretraining and downstream tasks can be very different. We propose an analysis framework that links the pretraining and downstream tasks with an underlying latent variable generative model of text -- the downstream classifier must recover a function of the posterior distribution over the latent variables. We analyze head tuning (learning a classifier on top of the frozen pretrained model) and prompt tuning in this setting. The generative model in our analysis is either a Hidden Markov Model (HMM) or an HMM augmented with a latent memory component, motivated by long-term dependencies in natural language. We show that 1) under certain non-degeneracy conditions on the HMM, simple classification heads can solve the downstream task, 2) prompt tuning obtains downstream guarantees with weaker non-degeneracy conditions, and 3) our recovery guarantees for the memory-augmented HMM are stronger than for the vanilla HMM because task-relevant information is easier to recover from the long-term memory. Experiments on synthetically generated data from HMMs back our theoretical findings.