The steep computational cost of diffusion models at inference hinders their use as fast physics emulators. In the context of image and video generation, this computational drawback has been addressed by generating in the latent space of an autoencoder instead of the pixel space. In this work, we investigate whether a similar strategy can be effectively applied to the emulation of dynamical systems and at what cost. We find that the accuracy of latent-space emulation is surprisingly robust to a wide range of compression rates (up to 1000x). We also show that diffusion-based emulators are consistently more accurate than non-generative counterparts and compensate for uncertainty in their predictions with greater diversity. Finally, we cover practical design choices, spanning from architectures to optimizers, that we found critical to train latent-space emulators.

Diffusion models recently proved to be remarkable priors for Bayesian inverse problems. However, training these models typically requires access to large amounts of clean data, which could prove difficult in some settings. In this work, we present a novel method based on the expectation-maximization algorithm for training diffusion models from incomplete and noisy observations only. Unlike previous works, our method leads to proper diffusion models, which is crucial for downstream tasks. As part of our method, we propose and motivate an improved posterior sampling scheme for unconditional diffusion models. We present empirical evidence supporting the effectiveness of our method.

Bayesian inference allows expressing the uncertainty of posterior belief under a probabilistic model given prior information and the likelihood of the evidence. Predominantly, the likelihood function is only implicitly established by a simulator posing the need for simulation-based inference (SBI). However, the existing algorithms can yield overconfident posteriors (Hermans *et al.*, 2022) defeating the whole purpose of credibility if the uncertainty quantification is inaccurate. We propose to include a calibration term directly into the training objective of the neural model in selected amortized SBI techniques. By introducing a relaxation of the classical formulation of calibration error we enable end-to-end backpropagation. The proposed method is not tied to any particular neural model and brings moderate computational overhead compared to the profits it introduces. It is directly applicable to existing computational pipelines allowing reliable black-box posterior inference. We empirically show on six benchmark problems that the proposed method achieves competitive or better results in terms of coverage and expected posterior density than the previously existing approaches.

Data assimilation, in its most comprehensive form, addresses the Bayesian inverse problem of identifying plausible state trajectories that explain noisy or incomplete observations of stochastic dynamical systems. Various approaches have been proposed to solve this problem, including particle-based and variational methods. However, most algorithms depend on the transition dynamics for inference, which becomes intractable for long time horizons or for high-dimensional systems with complex dynamics, such as oceans or atmospheres. In this work, we introduce score-based data assimilation for trajectory inference. We learn a score-based generative model of state trajectories based on the key insight that the score of an arbitrarily long trajectory can be decomposed into a series of scores over short segments. After training, inference is carried out using the score model, in a non-autoregressive manner by generating all states simultaneously. Quite distinctively, we decouple the observation model from the training procedure and use it only at inference to guide the generative process, which enables a wide range of zero-shot observation scenarios. We present theoretical and empirical evidence supporting the effectiveness of our method.

Modern approaches for simulation-based inference build upon deep learning surrogates to enable approximate Bayesian inference with computer simulators. In practice, the estimated posteriors' computational faithfulness is, however, rarely guaranteed. For example, Hermans et al., 2021 have shown that current simulation-based inference algorithms can produce posteriors that are overconfident, hence risking false inferences. In this work, we introduce Balanced Neural Ratio Estimation (BNRE), a variation of the NRE algorithm designed to produce posterior approximations that tend to be more conservative, hence improving their reliability, while sharing the same Bayes optimal solution. We achieve this by enforcing a balancing condition that increases the quantified uncertainty in low simulation budget regimes while still converging to the exact posterior as the budget increases. We provide theoretical arguments showing that BNRE tends to produce posterior surrogates that are more conservative than NRE's. We evaluate BNRE on a wide variety of tasks and show that it produces conservative posterior surrogates on all tested benchmarks and simulation budgets. Finally, we emphasize that BNRE is straightforward to implement over NRE and does not introduce any computational overhead.

We revisit g-modeling empirical Bayes in the absence of a tractable likelihood function, as is typical in scientific domains relying on computer simulations. We investigate how the empirical Bayesian can make use of neural density estimators first to use all noise-corrupted observations to estimate a prior or source distribution over uncorrupted samples, and then to perform single-observation posterior inference using the fitted source distribution. We propose an approach based on the direct maximization of the log-marginal likelihood of the observations, examining both biased and de-biased estimators, and comparing to variational approaches. We find that, up to symmetries, a neural empirical Bayes approach recovers ground truth source distributions. With the learned source distribution in hand, we show the applicability to likelihood-free inference and examine the quality of the resulting posterior estimates. Finally, we demonstrate the applicability of Neural Empirical Bayes on an inverse problem from collider physics.

Normalizing flows model complex probability distributions by combining a base distribution with a series of bijective neural networks. State-of-the-art architectures rely on coupling and autoregressive transformations to lift up invertible functions from scalars to vectors. In this work, we revisit these transformations as probabilistic graphical models, showing they reduce to Bayesian networks with a pre-defined topology and a learnable density at each node. From this new perspective, we propose the graphical normalizing flow, a new invertible transformation with either a prescribed or a learnable graphical structure. This model provides a promising way to inject domain knowledge into normalizing flows while preserving both the interpretability of Bayesian networks and the representation capacity of normalizing flows. We show that graphical conditioners discover relevant graph structure when we cannot hypothesize it. In addition, we analyze the effect of $\ell_1$-penalization on the recovered structure and on the quality of the resulting density estimation. Finally, we show that graphical conditioners lead to competitive white box density estimators. Our implementation is available at . Cite this Paper BibTeX @InProceedings{pmlr-v130-wehenkel21a, title = { Graphical Normalizing Flows }, author = {Wehenkel, Antoine and Louppe, Gilles}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {37--45}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/wehenkel21a/wehenkel21a.pdf}, url = {https://proceedings.mlr.press/v130/wehenkel21a.html}, abstract = { Normalizing flows model complex probability distributions by combining a base distribution with a series of bijective neural networks. State-of-the-art architectures rely on coupling and autoregressive transformations to lift up invertible functions from scalars to vectors. In this work, we revisit these transformations as probabilistic graphical models, showing they reduce to Bayesian networks with a pre-defined topology and a learnable density at each node. From this new perspective, we propose the graphical normalizing flow, a new invertible transformation with either a prescribed or a learnable graphical structure. This model provides a promising way to inject domain knowledge into normalizing flows while preserving both the interpretability of Bayesian networks and the representation capacity of normalizing flows. We show that graphical conditioners discover relevant graph structure when we cannot hypothesize it. In addition, we analyze the effect of $\ell_1$-penalization on the recovered structure and on the quality of the resulting density estimation. Finally, we show that graphical conditioners lead to competitive white box density estimators. Our implementation is available at . } } Copy to Clipboard Download Endnote %0 Conference Paper %T Graphical Normalizing Flows %A Antoine Wehenkel %A Gilles Louppe %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-wehenkel21a %I PMLR %P 37--45 %U https://proceedings.mlr.press/v130/wehenkel21a.html %V 130 %X Normalizing flows model complex probability distributions by combining a base distribution with a series of bijective neural networks. State-of-the-art architectures rely on coupling and autoregressive transformations to lift up invertible functions from scalars to vectors. In this work, we revisit these transformations as probabilistic graphical models, showing they reduce to Bayesian networks with a pre-defined topology and a learnable density at each node. From this new perspective, we propose the graphical normalizing flow, a new invertible transformation with either a prescribed or a learnable graphical structure. This model provides a promising way to inject domain knowledge into normalizing flows while preserving both the interpretability of Bayesian networks and the representation capacity of normalizing flows. We show that graphical conditioners discover relevant graph structure when we cannot hypothesize it. In addition, we analyze the effect of $\ell_1$-penalization on the recovered structure and on the quality of the resulting density estimation. Finally, we show that graphical conditioners lead to competitive white box density estimators. Our implementation is available at . Copy to Clipboard Download APA Wehenkel, A. & Louppe, G.. (2021). Graphical Normalizing Flows . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:37-45 Available from https://proceedings.mlr.press/v130/wehenkel21a.html. Copy to Clipboard Download Related Material Download PDF Supplementary PDF

Parametric stochastic simulators are ubiquitous in science, often featuring high-dimensional input parameters and/or an intractable likelihood. Performing Bayesian parameter inference in this context can be challenging. We present a neural simulation-based inference algorithm which simultaneously offers simulation efficiency and fast empirical posterior testability, which is unique among modern algorithms. Our approach is simulation efficient by simultaneously estimating low-dimensional marginal posteriors instead of the joint posterior and by proposing simulations targeted to an observation of interest via a prior suitably truncated by an indicator function. Furthermore, by estimating a locally amortized posterior our algorithm enables efficient empirical tests of the robustness of the inference results. Since scientists cannot access the ground truth, these tests are necessary for trusting inference in real-world applications. We perform experiments on a marginalized version of the simulation-based inference benchmark and two complex and narrow posteriors, highlighting the simulator efficiency of our algorithm as well as the quality of the estimated marginal posteriors.

Inferring the parameters of a stochastic model based on experimental observations is central to the scientific method. A particularly challenging setting is when the model is strongly indeterminate, i.e. when distinct sets of parameters yield identical observations. This arises in many practical situations, such as when inferring the distance and power of a radio source (is the source close and weak or far and strong?) or when estimating the amplifier gain and underlying brain activity of an electrophysiological experiment. In this work, we present hierarchical neural posterior estimation (HNPE), a novel method for cracking such indeterminacy by exploiting additional information conveyed by an auxiliary set of observations sharing global parameters. Our method extends recent developments in simulation-based inference (SBI) based on normalizing flows to Bayesian hierarchical models. We validate quantitatively our proposal on a motivating example amenable to analytical solutions and then apply it to invert a well known non-linear model from computational neuroscience, using both simulated and real EEG data.

Random forests have been widely used for their ability to provide so-called importance measures, which give insight at a global (per dataset) level on the relevance of input variables to predict a certain output. On the other hand, methods based on Shapley values have been introduced to refine the analysis of feature relevance in tree-based models to a local (per instance) level. In this context, we first show that the global Mean Decrease of Impurity (MDI) variable importance scores correspond to Shapley values under some conditions. Then, we derive a local MDI importance measure of variable relevance, which has a very natural connection with the global MDI measure and can be related to a new notion of local feature relevance. We further link local MDI importances with Shapley values and discuss them in the light of related measures from the literature. The measures are illustrated through experiments on several classification and regression problems.

Posterior inference with an intractable likelihood is becoming an increasingly common task in scientific domains which rely on sophisticated computer simulations. Typically, these forward models do not admit tractable densities forcing practitioners to rely on approximations. This work introduces a novel approach to address the intractability of the likelihood and the marginal model. We achieve this by learning a flexible amortized estimator which approximates the likelihood-to-evidence ratio. We demonstrate that the learned ratio estimator can be embedded in \textsc{mcmc} samplers to approximate likelihood-ratios between consecutive states in the Markov chain, allowing us to draw samples from the intractable posterior. Techniques are presented to improve the numerical stability and to measure the quality of an approximation. The accuracy of our approach is demonstrated on a variety of benchmarks against well-established techniques. Scientific applications in physics show its applicability.

We present a novel approach for learning an approximation of the optimal state-action value function (Q) in model-free Deep Reinforcement Learning (DRL). We propose to learn this approximation while simultaneously learning an approximation of the state-value function (V). We introduce two new DRL algorithms, called DQV-Learning and DQV-Max Learning, which follow this specific learning dynamic. In short, both algorithms use two neural networks for separately learning the V function and the Q function. We validate the effectiveness of this training scheme by thoroughly comparing our algorithms to DRL methods which only learn an approximation of the Q function, namely DQN and DDQN. Our results show that DQV and DQV-Max present several important benefits: they converge significantly faster, can achieve super-human performance on DRL testbeds on which DQN and DDQN failed to do so, and suffer less from the overestimation bias of the Q function.

Complex computer simulators are increasingly used across fields of science as generative models tying parameters of an underlying theory to experimental observations. Inference in this setup is often difficult, as simulators rarely admit a tractable density or likelihood function. We introduce Adversarial Variational Optimization (AVO), a likelihood-free inference algorithm for fitting a non-differentiable generative model incorporating ideas from generative adversarial networks, variational optimization and empirical Bayes. We adapt the training procedure of generative adversarial networks by replacing the differentiable generative network with a domain-specific simulator. We solve the resulting non-differentiable minimax problem by minimizing variational upper bounds of the two adversarial objectives. Effectively, the procedure results in learning a proposal distribution over simulator parameters, such that the JS divergence between the marginal distribution of the synthetic data and the empirical distribution of observed data is minimized. We evaluate and compare the method with simulators producing both discrete and continuous data.

We present a novel probabilistic programming framework that couples directly to existing large-scale simulators through a cross-platform probabilistic execution protocol, which allows general-purpose inference engines to record and control random number draws within simulators in a language-agnostic way. The execution of existing simulators as probabilistic programs enables highly interpretable posterior inference in the structured model defined by the simulator code base. We demonstrate the technique in particle physics, on a scientifically accurate simulation of the tau lepton decay, which is a key ingredient in establishing the properties of the Higgs boson. Inference efficiency is achieved via inference compilation where a deep recurrent neural network is trained to parameterize proposal distributions and control the stochastic simulator in a sequential importance sampling scheme, at a fraction of the computational cost of a Markov chain Monte Carlo baseline.

Monotonic neural networks have recently been proposed as a way to define invertible transformations. These transformations can be combined into powerful autoregressive flows that have been shown to be universal approximators of continuous probability distributions. Architectures that ensure monotonicity typically enforce constraints on weights and activation functions, which enables invertibility but leads to a cap on the expressiveness of the resulting transformations. In this work, we propose the Unconstrained Monotonic Neural Network (UMNN) architecture based on the insight that a function is monotonic as long as its derivative is strictly positive. In particular, this latter condition can be enforced with a free-form neural network whose only constraint is the positiveness of its output. We evaluate our new invertible building block within a new autoregressive flow (UMNN-MAF) and demonstrate its effectiveness on density estimation experiments. We also illustrate the ability of UMNNs to improve variational inference.

Probabilistic programming languages (PPLs) are receiving widespread attention for performing Bayesian inference in complex generative models. However, applications to science remain limited because of the impracticability of rewriting complex scientific simulators in a PPL, the computational cost of inference, and the lack of scalable implementations. To address these, we present a novel PPL framework that couples directly to existing scientific simulators through a cross-platform probabilistic execution protocol and provides Markov chain Monte Carlo (MCMC) and deep-learning-based inference compilation (IC) engines for tractable inference. To guide IC inference, we perform distributed training of a dynamic 3DCNN-LSTM architecture with a PyTorch-MPI-based framework on 1,024 32-core CPU nodes of the Cori supercomputer with a global mini-batch size of 128k: achieving a performance of 450 Tflop/s through enhancements to PyTorch. We demonstrate a Large Hadron Collider (LHC) use-case with the C++ Sherpa simulator and achieve the largest-scale posterior inference in a Turing-complete PPL.

Dealing with datasets of very high dimension is a major challenge in machine learning. In this paper, we consider the problem of feature selection in applications where the memory is not large enough to contain all features. In this setting, we propose a novel tree-based feature selection approach that builds a sequence of randomized trees on small subsamples of variables mixing both variables already identified as relevant by previous models and variables randomly selected among the other variables. As our main contribution, we provide an in-depth theoretical analysis of this method in infinite sample setting. In particular, we study its soundness with respect to common definitions of feature relevance and its convergence speed under various variable dependance scenarios. We also provide some preliminary empirical results highlighting the potential of the approach.

Several techniques for domain adaptation have been proposed to account for differences in the distribution of the data used for training and testing. The majority of this work focuses on a binary domain label. Similar problems occur in a scientific context where there may be a continuous family of plausible data generation processes associated to the presence of systematic uncertainties. Robust inference is possible if it is based on a pivot -- a quantity whose distribution does not depend on the unknown values of the nuisance parameters that parametrize this family of data generation processes. In this work, we introduce and derive theoretical results for a training procedure based on adversarial networks for enforcing the pivotal property (or, equivalently, fairness with respect to continuous attributes) on a predictive model. The method includes a hyperparameter to control the trade-off between accuracy and robustness. We demonstrate the effectiveness of this approach with a toy example and examples from particle physics.

MOTIVATION: Collaborative analysis of massive imaging datasets is essential to enable scientific discoveries. RESULTS: We developed Cytomine to foster active and distributed collaboration of multidisciplinary teams for large-scale image-based studies. It uses web development methodologies and machine learning in order to readily organize, explore, share and analyze (semantically and quantitatively) multi-gigapixel imaging data over the internet. We illustrate how it has been used in several biomedical applications. AVAILABILITY AND IMPLEMENTATION: Cytomine (http://www.cytomine.be/) is freely available under an open-source license from http://github.com/cytomine/ A documentation wiki (http://doc.cytomine.be) and a demo server (http://demo.cytomine.be) are also available. CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Despite growing interest and practical use in various scientific areas, variable importances derived from tree-based ensemble methods are not well understood from a theoretical point of view. In this work we characterize the Mean Decrease Impurity (MDI) variable importances as measured by an ensemble of totally randomized trees in asymptotic sample and ensemble size conditions. We derive a three-level decomposition of the information jointly provided by all input variables about the output in terms of i) the MDI importance of each input variable, ii) the degree of interaction of a given input variable with the other input variables, iii) the different interaction terms of a given degree. We then show that this MDI importance of a variable is equal to zero if and only if the variable is irrelevant and that the MDI importance of a relevant variable is invariant with respect to the removal or the addition of irrelevant variables. We illustrate these properties on a simple example and discuss how they may change in the case of non-totally randomized trees such as Random Forests and Extra-Trees.