We study the problem of policy evaluation with linear function approximation and present efficient and practical algorithms that come with strong optimality guarantees. We begin by proving lower bounds that establish baselines on both the deterministic error and stochastic error in this problem. In particular, we prove an oracle complexity lower bound on the deterministic error in an instance-dependent norm associated with the stationary distribution of the transition kernel, and use the local asymptotic minimax machinery to prove an instance-dependent lower bound on the stochastic error in the i.i.d. observation model. Existing algorithms fail to match at least one of these lower bounds: To illustrate, we analyze a variance-reduced variant of temporal difference learning, showing in particular that it fails to achieve the oracle complexity lower bound. To remedy this issue, we develop an accelerated, variance-reduced fast temporal difference algorithm (VRFTD) that simultaneously matches both lower bounds and attains a strong notion of instance-optimality. Finally, we extend the VRFTD algorithm to the setting with Markovian observations, and provide instance-dependent convergence results that match those in the i.i.d. setting up to a multiplicative factor that is proportional to the mixing time of the chain. Our theoretical guarantees of optimality are corroborated by numerical experiments.

Neural ordinary differential equations (NODE) have been proposed as a continuous depth generalization to popular deep learning models such as Residual networks (ResNets). They provide parameter efficiency and automate the model selection process in deep learning models to some extent. However, they lack the much-required uncertainty modelling and robustness capabilities which are crucial for their use in several real-world applications such as autonomous driving and healthcare. We propose a novel and unique approach to model uncertainty in NODE by considering a distribution over the end-time $T$ of the ODE solver. The proposed approach, latent time NODE (LT-NODE), treats $T$ as a latent variable and apply Bayesian learning to obtain a posterior distribution over $T$ from the data. In particular, we use variational inference to learn an approximate posterior and the model parameters. Prediction is done by considering the NODE representations from different samples of the posterior and can be done efficiently using a single forward pass. As $T$ implicitly defines the depth of a NODE, posterior distribution over $T$ would also help in model selection in NODE. We also propose, adaptive latent time NODE (ALT-NODE), which allow each data point to have a distinct posterior distribution over end-times. ALT-NODE uses amortized variational inference to learn an approximate posterior using inference networks. We demonstrate the effectiveness of the proposed approaches in modelling uncertainty and robustness through experiments on synthetic and several real-world image classification data.

The last decades have seen significant advancements in non-invasive neuroimaging technologies that have been increasingly adopted to examine human brain development. However, these improvements have not necessarily been followed by more sophisticated data analysis measures that are able to explain the mechanisms underlying functional brain development. For example, the shift from univariate (single area in the brain) to multivariate (multiple areas in brain) analysis paradigms is of significance as it allows investigations into the interactions between different brain regions. However, despite the potential of multivariate analysis to shed light on the interactions between developing brain regions, artificial intelligence (AI) techniques applied render the analysis non-explainable. The purpose of this paper is to understand the extent to which current state-of-the-art AI techniques can inform functional brain development. In addition, a review of which AI techniques are more likely to explain their learning based on the processes of brain development as defined by developmental cognitive neuroscience (DCN) frameworks is also undertaken. This work also proposes that eXplainable AI (XAI) may provide viable methods to investigate functional brain development as hypothesised by DCN frameworks.

Variational inference is a powerful paradigm for approximate Bayesian inference with a number of appealing properties, including support for model learning and data subsampling. By contrast MCMC methods like Hamiltonian Monte Carlo do not share these properties but remain attractive since, contrary to parametric methods, MCMC is asymptotically unbiased. For these reasons researchers have sought to combine the strengths of both classes of algorithms, with recent approaches coming closer to realizing this vision in practice. However, supporting data subsampling in these hybrid methods can be a challenge, a shortcoming that we address by introducing a surrogate likelihood that can be learned jointly with other variational parameters. We argue theoretically that the resulting algorithm permits the user to make an intuitive trade-off between inference fidelity and computational cost. In an extensive empirical comparison we show that our method performs well in practice and that it is well-suited for black-box inference in probabilistic programming frameworks.

Today, we're excited to announce an early beta release of Bean Machine, a PyTorch-based probabilistic programming system that makes it easy to represent and to learn about uncertainties in the machine learning models that we work with every day. Bean Machine enables you to develop domain-specific probabilistic models, and automatically learn about unobserved properties of the model with automatic, uncertainty-aware learning algorithms. Though powerful, probabilistic modeling does take some getting used to. If this is your first exposure to the topic, we welcome you to check out a short overview of the concept in the Fabulous Adventures in Coding blog. We on the Bean Machine development team believe that the usability of a system forms the bedrock for its success, and we've taken care to center Bean Machine's design around a declarative philosophy within the PyTorch ecosystem.

Herding is a deterministic algorithm used to generate data points that can be regarded as random samples satisfying input moment conditions. The algorithm is based on the complex behavior of a high-dimensional dynamical system and is inspired by the maximum entropy principle of statistical inference. In this paper, we propose an extension of the herding algorithm, called entropic herding, which generates a sequence of distributions instead of points. Entropic herding is derived as the optimization of the target function obtained from the maximum entropy principle. Using the proposed entropic herding algorithm as a framework, we discuss a closer connection between herding and the maximum entropy principle. Specifically, we interpret the original herding algorithm as a tractable version of entropic herding, the ideal output distribution of which is mathematically represented. We further discuss how the complex behavior of the herding algorithm contributes to optimization. We argue that the proposed entropic herding algorithm extends the application of herding to probabilistic modeling. In contrast to original herding, entropic herding can generate a smooth distribution such that both efficient probability density calculation and sample generation become possible. To demonstrate the viability of these arguments in this study, numerical experiments were conducted, including a comparison with other conventional methods, on both synthetic and real data.

A Bayesian Network is a directed acyclic graph (DAG) on a set of $n$ random variables (identified with the vertices); a Bayesian Network Distribution (BND) is a probability distribution on the rv's that is Markovian on the graph. A finite mixture of such models is the projection on these variables of a BND on the larger graph which has an additional "hidden" (or "latent") random variable $U$, ranging in $\{1,\ldots,k\}$, and a directed edge from $U$ to every other vertex. Models of this type are fundamental to research in Causal Inference, where $U$ models a confounding effect. One extremely special case has been of longstanding interest in the theory literature: the empty graph. Such a distribution is simply a mixture of $k$ product distributions. A longstanding problem has been, given the joint distribution of a mixture of $k$ product distributions, to identify each of the product distributions, and their mixture weights. Our results are: (1) We improve the sample complexity (and runtime) for identifying mixtures of $k$ product distributions from $\exp(O(k^2))$ to $\exp(O(k \log k))$. This is almost best possible in view of a known $\exp(\Omega(k))$ lower bound. (2) We give the first algorithm for the case of non-empty graphs. The complexity for a graph of maximum degree $\Delta$ is $\exp(O(k(\Delta^2 + \log k)))$. (The above complexities are approximate and suppress dependence on secondary parameters.)

Currently, it is hard to reap the benefits of deep learning for Bayesian methods, which allow the explicit specification of prior knowledge and accurately capture model uncertainty. We present Prior-Data Fitted Networks (PFNs). PFNs leverage large-scale machine learning techniques to approximate a large set of posteriors. The only requirement for PFNs to work is the ability to sample from a prior distribution over supervised learning tasks (or functions). Our method restates the objective of posterior approximation as a supervised classification problem with a set-valued input: it repeatedly draws a task (or function) from the prior, draws a set of data points and their labels from it, masks one of the labels and learns to make probabilistic predictions for it based on the set-valued input of the rest of the data points. Presented with a set of samples from a new supervised learning task as input, PFNs make probabilistic predictions for arbitrary other data points in a single forward propagation, having learned to approximate Bayesian inference. We demonstrate that PFNs can near-perfectly mimic Gaussian processes and also enable efficient Bayesian inference for intractable problems, with over 200-fold speedups in multiple setups compared to current methods. We obtain strong results in very diverse areas such as Gaussian process regression, Bayesian neural networks, classification for small tabular data sets, and few-shot image classification, demonstrating the generality of PFNs. Code and trained PFNs are released at https://github.com/automl/TransformersCanDoBayesianInference.

We are interested in privatizing an approximate posterior inference algorithm called Expectation Propagation (EP). EP approximates the posterior by iteratively refining approximations to the local likelihoods, and is known to provide better posterior uncertainties than those by variational inference (VI). However, using EP for large-scale datasets imposes a challenge in terms of memory requirements as it needs to maintain each of the local approximates in memory. To overcome this problem, stochastic expectation propagation (SEP) was proposed, which only considers a unique local factor that captures the average effect of each likelihood term to the posterior and refines it in a way analogous to EP. In terms of privacy, SEP is more tractable than EP because at each refining step of a factor, the remaining factors are fixed to the same value and do not depend on other datapoints as in EP, which makes the sensitivity analysis tractable. We provide a theoretical analysis of the privacy-accuracy trade-off in the posterior estimates under differentially private stochastic expectation propagation (DP-SEP). Furthermore, we demonstrate the performance of our DP-SEP algorithm evaluated on both synthetic and real-world datasets in terms of the quality of posterior estimates at different levels of guaranteed privacy.

With the development of AIoT, data-driven attack detection methods for cyber-physical systems (CPSs) have attracted lots of attention. However, existing methods usually adopt tractable distributions to approximate data distributions, which are not suitable for complex systems. Besides, the correlation of the data in different channels does not attract sufficient attention. To address these issues, we use energy-based generative models, which are less restrictive on functional forms of the data distribution. In addition, graph neural networks are used to explicitly model the correlation of the data in different channels. In the end, we propose TFDPM, a general framework for attack detection tasks in CPSs. It simultaneously extracts temporal pattern and feature pattern given the historical data. Then extract features are sent to a conditional diffusion probabilistic model. Predicted values can be obtained with the conditional generative network and attacks are detected based on the difference between predicted values and observed values. In addition, to realize real-time detection, a conditional noise scheduling network is proposed to accelerate the prediction process. Experimental results show that TFDPM outperforms existing state-of-the-art attack detection methods. The noise scheduling network increases the detection speed by three times.