Reinforcement learning (RL) algorithms face two distinct challenges: learning effective representations of past and present observations, and determining how actions influence future returns. Both challenges involve modeling long-term dependencies. The Transformer architecture has been very successful to solve problems that involve long-term dependencies, including in the RL domain. However, the underlying reason for the strong performance of Transformer-based RL methods remains unclear: is it because they learn effective memory, or because they perform effective credit assignment? After introducing formal definitions of memory length and credit assignment length, we design simple configurable tasks to measure these distinct quantities. Our empirical results reveal that Transformers can enhance the memory capability of RL algorithms, scaling up to tasks that require memorizing observations $1500$ steps ago. However, Transformers do not improve long-term credit assignment. In summary, our results provide an explanation for the success of Transformers in RL, while also highlighting an important area for future research and benchmark design. Our code is open-sourced at https://github.com/twni2016/Memory-RL

We study the consequences of mode-collapse of normalizing flows in the context of lattice field theory. Normalizing flows allow for independent sampling. For this reason, it is hoped that they can avoid the tunneling problem of local-update MCMC algorithms for multi-modal distributions. In this work, we first point out that the tunneling problem is also present for normalizing flows but is shifted from the sampling to the training phase of the algorithm. Specifically, normalizing flows often suffer from mode-collapse for which the training process assigns vanishingly low probability mass to relevant modes of the physical distribution. This may result in a significant bias when the flow is used as a sampler in a Markov-Chain or with Importance Sampling. We propose a metric to quantify the degree of mode-collapse and derive a bound on the resulting bias. Furthermore, we propose various mitigation strategies in particular in the context of estimating thermodynamic observables, such as the free energy.

Injecting structure into neural networks enables learning functions that satisfy invariances with respect to subsets of inputs. For instance, when learning generative models using neural networks, it is advantageous to encode the conditional independence structure of observed variables, often in the form of Bayesian networks. We propose the Structured Neural Network (StrNN), which injects structure through masking pathways in a neural network. The masks are designed via a novel relationship we explore between neural network architectures and binary matrix factorization, to ensure that the desired independencies are respected. We devise and study practical algorithms for this otherwise NP-hard design problem based on novel objectives that control the model architecture. We demonstrate the utility of StrNN in three applications: (1) binary and Gaussian density estimation with StrNN, (2) real-valued density estimation with Structured Autoregressive Flows (StrAFs) and Structured Continuous Normalizing Flows (StrCNF), and (3) interventional and counterfactual analysis with StrAFs for causal inference. Our work opens up new avenues for learning neural networks that enable data-efficient generative modeling and the use of normalizing flows for causal effect estimation.

Quantile regression is a powerful tool for inferring how covariates affect specific percentiles of the response distribution. Existing methods either estimate conditional quantiles separately for each quantile of interest or estimate the entire conditional distribution using semi- or non-parametric models. The former often produce inadequate models for real data and do not share information across quantiles, while the latter are characterized by complex and constrained models that can be difficult to interpret and computationally inefficient. Further, neither approach is well-suited for quantile-specific subset selection. Instead, we pose the fundamental problems of linear quantile estimation, uncertainty quantification, and subset selection from a Bayesian decision analysis perspective. For any Bayesian regression model, we derive optimal and interpretable linear estimates and uncertainty quantification for each model-based conditional quantile. Our approach introduces a quantile-focused squared error loss, which enables efficient, closed-form computing and maintains a close relationship with Wasserstein-based density estimation. In an extensive simulation study, our methods demonstrate substantial gains in quantile estimation accuracy, variable selection, and inference over frequentist and Bayesian competitors. We apply these tools to identify the quantile-specific impacts of social and environmental stressors on educational outcomes for a large cohort of children in North Carolina.

Under model misspecification, it is known that Bayesian posteriors often do not properly quantify uncertainty about true or pseudo-true parameters. Even more fundamentally, misspecification leads to a lack of reproducibility in the sense that the same model will yield contradictory posteriors on independent data sets from the true distribution. To define a criterion for reproducible uncertainty quantification under misspecification, we consider the probability that two confidence sets constructed from independent data sets have nonempty overlap, and we establish a lower bound on this overlap probability that holds for any valid confidence sets. We prove that credible sets from the standard posterior can strongly violate this bound, particularly in high-dimensional settings (i.e., with dimension increasing with sample size), indicating that it is not internally coherent under misspecification. To improve reproducibility in an easy-to-use and widely applicable way, we propose to apply bagging to the Bayesian posterior ("BayesBag"'); that is, to use the average of posterior distributions conditioned on bootstrapped datasets. We motivate BayesBag from first principles based on Jeffrey conditionalization and show that the bagged posterior typically satisfies the overlap lower bound. Further, we prove a Bernstein--Von Mises theorem for the bagged posterior, establishing its asymptotic normal distribution. We demonstrate the benefits of BayesBag via simulation experiments and an application to crime rate prediction.

Reinforcement learning from Human Feedback (RLHF) learns from preference signals, while standard Reinforcement Learning (RL) directly learns from reward signals. Preferences arguably contain less information than rewards, which makes preference-based RL seemingly more difficult. This paper theoretically proves that, for a wide range of preference models, we can solve preference-based RL directly using existing algorithms and techniques for reward-based RL, with small or no extra costs. Specifically, (1) for preferences that are drawn from reward-based probabilistic models, we reduce the problem to robust reward-based RL that can tolerate small errors in rewards; (2) for general arbitrary preferences where the objective is to find the von Neumann winner, we reduce the problem to multiagent reward-based RL which finds Nash equilibria for factored Markov games with a restricted set of policies. The latter case can be further reduced to adversarial MDP when preferences only depend on the final state. We instantiate all reward-based RL subroutines by concrete provable algorithms, and apply our theory to a large class of models including tabular MDPs and MDPs with generic function approximation. We further provide guarantees when K-wise comparisons are available.

We establish conditions under which latent causal graphs are nonparametrically identifiable and can be reconstructed from unknown interventions in the latent space. Our primary focus is the identification of the latent structure in measurement models without parametric assumptions such as linearity or Gaussianity. Moreover, we do not assume the number of hidden variables is known, and we show that at most one unknown intervention per hidden variable is needed. This extends a recent line of work on learning causal representations from observations and interventions. The proofs are constructive and introduce two new graphical concepts -- imaginary subsets and isolated edges -- that may be useful in their own right. As a matter of independent interest, the proofs also involve a novel characterization of the limits of edge orientations within the equivalence class of DAGs induced by unknown interventions. These are the first results to characterize the conditions under which causal representations are identifiable without making any parametric assumptions in a general setting with unknown interventions and without faithfulness.

Deep reinforcement learning (RL) works impressively in some environments and fails catastrophically in others. Ideally, RL theory should be able to provide an understanding of why this is, i.e. bounds predictive of practical performance. Unfortunately, current theory does not quite have this ability. We compare standard deep RL algorithms to prior sample complexity bounds by introducing a new dataset, BRIDGE. It consists of 155 deterministic MDPs from common deep RL benchmarks, along with their corresponding tabular representations, which enables us to exactly compute instance-dependent bounds. We choose to focus on deterministic environments because they share many interesting properties of stochastic environments, but are easier to analyze. Using BRIDGE, we find that prior bounds do not correlate well with when deep RL succeeds vs. fails, but discover a surprising property that does. When actions with the highest Q-values under the random policy also have the highest Q-values under the optimal policy (i.e. when it is optimal to be greedy on the random policy's Q function), deep RL tends to succeed; when they don't, deep RL tends to fail. We generalize this property into a new complexity measure of an MDP that we call the effective horizon, which roughly corresponds to how many steps of lookahead search would be needed in that MDP in order to identify the next optimal action, when leaf nodes are evaluated with random rollouts. Using BRIDGE, we show that the effective horizon-based bounds are more closely reflective of the empirical performance of PPO and DQN than prior sample complexity bounds across four metrics. We also find that, unlike existing bounds, the effective horizon can predict the effects of using reward shaping or a pre-trained exploration policy. Our code and data are available at https://github.com/cassidylaidlaw/effective-horizon

We study reinforcement learning (RL) with linear function approximation. For episodic time-inhomogeneous linear Markov decision processes (linear MDPs) whose transition probability can be parameterized as a linear function of a given feature mapping, we propose the first computationally efficient algorithm that achieves the nearly minimax optimal regret $\tilde O(d\sqrt{H^3K})$, where $d$ is the dimension of the feature mapping, $H$ is the planning horizon, and $K$ is the number of episodes. Our algorithm is based on a weighted linear regression scheme with a carefully designed weight, which depends on a new variance estimator that (1) directly estimates the variance of the optimal value function, (2) monotonically decreases with respect to the number of episodes to ensure a better estimation accuracy, and (3) uses a rare-switching policy to update the value function estimator to control the complexity of the estimated value function class. Our work provides a complete answer to optimal RL with linear MDPs, and the developed algorithm and theoretical tools may be of independent interest.

This paper introduces a novel approach for multi-task regression that connects Kernel Machines (KMs) and Extreme Learning Machines (ELMs) through the exploitation of the Random Fourier Features (RFFs) approximation of the RBF kernel. In this sense, one of the contributions of this paper shows that for the proposed models, the KM and the ELM formulations can be regarded as two sides of the same coin. These proposed models, termed RFF-BLR, stand on a Bayesian framework that simultaneously addresses two main design goals. On the one hand, it fits multitask regressors based on KMs endowed with RBF kernels. On the other hand, it enables the introduction of a common-across-tasks prior that promotes multioutput sparsity in the ELM view. This Bayesian approach facilitates the simultaneous consideration of both the KM and ELM perspectives enabling (i) the optimisation of the RBF kernel parameter $\gamma$ within a probabilistic framework, (ii) the optimisation of the model complexity, and (iii) an efficient transfer of knowledge across tasks. The experimental results show that this framework can lead to significant performance improvements compared to the state-of-the-art methods in multitask nonlinear regression.