Should sequences be modeled autoregressively---one symbol at a time? How much computation is needed to predict the next symbol? While local normalization is cheap, this also limits its power. We point out that some probability distributions over discrete sequences cannot be well-approximated by any autoregressive model whose runtime and parameter size grow polynomially in the sequence length---even though their unnormalized sequence probabilities are efficient to compute exactly. Intuitively, the probability of the next symbol can be expensive to compute or approximate (even via randomized algorithms) when it marginalizes over exponentially many possible futures, which is in general $\mathrm{NP}$-hard. Our result is conditional on the widely believed hypothesis that $\mathrm{NP} \nsubseteq \mathrm{P/poly}$ (without which the polynomial hierarchy would collapse at the second level). This theoretical observation serves as a caution to the viewpoint that pumping up parameter size is a straightforward way to improve autoregressive models (e.g., in language modeling). It also suggests that globally normalized (energy-based) models may sometimes outperform locally normalized (autoregressive) models, as we demonstrate experimentally for language modeling.

A central goal of probabilistic programming languages (PPLs) is to separate modelling from inference. However, this goal is hard to achieve in practice. Users are often forced to re-write their models in order to improve efficiency of inference or meet restrictions imposed by the PPL. Conditional independence (CI) relationships among parameters are a crucial aspect of probabilistic models that captures a qualitative summary of the specified model and can facilitate more efficient inference. We present an information flow type system for probabilistic programming that captures conditional independence (CI) relationships, and show that, for a well-typed program in our system, the distribution it implements is guaranteed to have certain CI-relationships. Further, by using type inference, we can statically \emph{deduce} which CI-properties are present in a specified model. As a practical application, we consider the problem of how to perform inference on models with mixed discrete and continuous parameters. Inference on such models is challenging in many existing PPLs, but can be improved through a workaround, where the discrete parameters are used \textit{implicitly}, at the expense of manual model re-writing. We present a source-to-source semantics-preserving transformation, which uses our CI-type system to automate this workaround by eliminating the discrete parameters from a probabilistic program. The resulting program can be seen as a hybrid inference algorithm on the original program, where continuous parameters can be drawn using efficient gradient-based inference methods, while the discrete parameters are drawn using variable elimination. We implement our CI-type system and its example application in SlicStan: a compositional variant of Stan.

A new class of Markov chain Monte Carlo (MCMC) algorithms, based on simulating piecewise deterministic Markov processes (PDMPs), have recently shown great promise: they are non-reversible, can mix better than standard MCMC algorithms, and can use subsampling ideas to speed up computation in big data scenarios. However, current PDMP samplers can only sample from posterior densities that are differentiable almost everywhere, which precludes their use for model choice. Motivated by variable selection problems, we show how to develop reversible jump PDMP samplers that can jointly explore the discrete space of models and the continuous space of parameters. Our framework is general: it takes any existing PDMP sampler, and adds two types of trans-dimensional moves that allow for the addition or removal of a variable from the model. We show how the rates of these trans-dimensional moves can be calculated so that the sampler has the correct invariant distribution. Simulations show that the new samplers can mix better than standard MCMC algorithms. Our empirical results show they are also more efficient than gradient-based samplers that avoid model choice through use of continuous spike-and-slab priors which replace a point mass at zero for each parameter with a density concentrated around zero.

Variational Bayes (VB) is a popular scalable alternative to Markov chain Monte Carlo for Bayesian inference. We study a mean-field spike and slab VB approximation of widely used Bayesian model selection priors in sparse high-dimensional logistic regression. We provide non-asymptotic theoretical guarantees for the VB posterior in both $\ell_2$ and prediction loss for a sparse truth, giving optimal (minimax) convergence rates. Since the VB algorithm does not depend on the unknown truth to achieve optimality, our results shed light on effective prior choices. We confirm the improved performance of our VB algorithm over common sparse VB approaches in a numerical study.

We present a framework to derive bounds on the test loss of randomized learning algorithms for the case of bounded loss functions. This framework leads to bounds that depend on the conditional information density between the the output hypothesis and the choice of the training set, given a larger set of data samples from which the training set is formed. If the conditional information density is bounded uniformly in the sizenof the training set, our bounds decay as1/n, which is referred to as a fast rate. This is in contrast with the tail bounds involving conditional information measures available in the literature, which have a less benign 1/ n dependence. We demonstrate the usefulness of our tail bounds by showing that they lead to estimates of the test loss achievable with several neural network architectures trained on MNIST and Fashion-MNIST that match the state-of-the-art bounds available in the literature. In recent years, there has been a surge of interest in the use of information-theoretic techniques for bounding the loss of learning algorithms. While the first results of this flavor can be traced to the probably approximately correct (PAC)-Bayesian approach (McAllester, 1998; Catoni, 2007) (see also (Guedj, 2019) for a recent review), the connection between loss bounds and classical information-theoretic measures was made explicit in the works of Russo & Zou (2016) and Xu & Raginsky (2017), where bounds on the average population loss were derived in terms of the mutual information between the training data and the output hypothesis. Since then, these average loss bounds have been tightened (Bu et al., 2019; Asadi et al., 2018; Negrea et al., 2019). Furthermore, the information-theoretic framework has also been successfully applied to derive tail probability bounds on the population loss (Bassily et al., 2018; Esposito et al., 2019; Hellström & Durisi, 2020a). Of particular relevance to the present paper is the random-subset setting, introduced by Steinke & Zakynthinou (2020) and further studied in (Hellström & Durisi, 2020b; Haghifam et al., 2020).

Accurate estimation of aleatoric and epistemic uncertainty is crucial to build safe and reliable systems. Traditional approaches, such as dropout and ensemble methods, estimate uncertainty by sampling probability predictions from different submodels, which leads to slow uncertainty estimation at inference time. Recent works address this drawback by directly predicting parameters of prior distributions over the probability predictions with a neural network. While this approach has demonstrated accurate uncertainty estimation, it requires defining arbitrary target parameters for in-distribution data and makes the unrealistic assumption that out-of-distribution (OOD) data is known at training time. In this work we propose the Posterior Network (PostNet), which uses Normalizing Flows to predict an individual closed-form posterior distribution over predicted probabilites for any input sample. The posterior distributions learned by PostNet accurately reflect uncertainty for in- and out-of-distribution data -- without requiring access to OOD data at training time. PostNet achieves state-of-the art results in OOD detection and in uncertainty calibration under dataset shifts.

Approximate Bayesian computation (ABC) is a likelihood-free inference method that has been employed in various applications. However, ABC is sensitive to outliers, which is caused by an inappropriate choice of the data discrepancy measure. In this paper, we propose to use a nearest-neighbor-based $\gamma$-divergence estimator as a data discrepancy measure. We show that our estimator possesses a suitable robustness property called the redescending property. In addition, our estimator enjoys various desirable properties such as high flexibility, asymptotic unbiasedness, almost sure convergence, and linear time complexity. Through experiments, we demonstrate that our method achieves significantly higher robustness than existing discrepancy measures.

Offline reinforcement learning seeks to utilize offline (observational) data to guide the learning of (causal) sequential decision making strategies. The hope is that offline reinforcement learning coupled with function approximation methods (to deal with the curse of dimensionality) can provide a means to help alleviate the excessive sample complexity burden in modern sequential decision making problems. However, the extent to which this broader approach can be effective is not well understood, where the literature largely consists of sufficient conditions. This work focuses on the basic question of what are necessary representational and distributional conditions that permit provable sample-efficient offline reinforcement learning. Perhaps surprisingly, our main result shows that even if: i) we have realizability in that the true value function of \emph{every} policy is linear in a given set of features and 2) our off-policy data has good coverage over all features (under a strong spectral condition), then any algorithm still (information-theoretically) requires a number of offline samples that is exponential in the problem horizon in order to non-trivially estimate the value of \emph{any} given policy. Our results highlight that sample-efficient offline policy evaluation is simply not possible unless significantly stronger conditions hold; such conditions include either having low distribution shift (where the offline data distribution is close to the distribution of the policy to be evaluated) or significantly stronger representational conditions (beyond realizability).

Discrete latent spaces in variational autoencoders have been shown to effectively capture the data distribution for many real-world problems such as natural language understanding, human intent prediction, and visual scene representation. However, discrete latent spaces need to be sufficiently large to capture the complexities of real-world data, rendering downstream tasks computationally challenging. For instance, performing motion planning in a high-dimensional latent representation of the environment could be intractable. We consider the problem of sparsifying the discrete latent space of a trained conditional variational autoencoder, while preserving its learned multimodality. As a post hoc latent space reduction technique, we use evidential theory to identify the latent classes that receive direct evidence from a particular input condition and filter out those that do not. Experiments on diverse tasks, such as image generation and human behavior prediction, demonstrate the effectiveness of our proposed technique at reducing the discrete latent sample space size of a model while maintaining its learned multimodality.

Specifying a Bayesian prior is notoriously difficult for complex models such as neural networks. Reasoning about parameters is made challenging by the high-dimensionality and over-parameterization of the space. Priors that seem benign and uninformative can have unintuitive and detrimental effects on a model's predictions. For this reason, we propose predictive complexity priors: a functional prior that is defined by comparing the model's predictions to those of a reference model. Although originally defined on the model outputs, we transfer the prior to the model parameters via a change of variables. The traditional Bayesian workflow can then proceed as usual. We apply our predictive complexity prior to high-dimensional regression, reasoning over neural network depth, and sharing of statistical strength for few-shot learning.