Danilo Jimenez Rezende

Nan Rosemary Ke, Silvia Chiappa, Jane Wang et al. · deepmind, mila

The fundamental challenge in causal induction is to infer the underlying graph structure given observational and/or interventional data. Most existing causal induction algorithms operate by generating candidate graphs and evaluating them using either score-based methods (including continuous optimization) or independence tests. In our work, we instead treat the inference process as a black box and design a neural network architecture that learns the mapping from both observational and interventional data to graph structures via supervised training on synthetic graphs. The learned model generalizes to new synthetic graphs, is robust to train-test distribution shifts, and achieves state-of-the-art performance on naturalistic graphs for low sample complexity.

Danny P. Sawyer, Nan Rosemary Ke, Hubert Soyer et al. · deepmind

Foundation models excel at single-turn reasoning but struggle with multi-turn exploration in dynamic environments, a requirement for many real-world challenges. We evaluated these models on their ability to learn from experience, adapt, and gather information. First, in "Feature World," a simple setting for testing information gathering, models performed near-optimally. However, to test more complex, multi-trial learning, we implemented a text-based version of the "Alchemy" environment, a benchmark for meta-learning. Here, agents must deduce a latent causal structure by integrating information across many trials. In this setting, recent foundation models initially failed to improve their performance over time. Crucially, we found that prompting the models to summarize their observations at regular intervals enabled an emergent meta-learning process. This allowed them to improve across trials and even adaptively re-learn when the environment's rules changed unexpectedly. While most models handled the simple task, Alchemy revealed stark differences in robustness: Gemini 2.5 performed best, followed by Claude 3.7, while ChatGPT-4o and o4-mini struggled. This underscores Alchemy's value as a benchmark. Our findings demonstrate that the biggest challenge for foundation models is not selecting informative actions in the moment, but integrating knowledge through adaptive strategies over time. Encouragingly, there appears to be no intrinsic barrier to future models mastering these abilities.

Michael S. Albergo, Denis Boyda, Daniel C. Hackett et al.

This notebook tutorial demonstrates a method for sampling Boltzmann distributions of lattice field theories using a class of machine learning models known as normalizing flows. The ideas and approaches proposed in arXiv:1904.12072, arXiv:2002.02428, and arXiv:2003.06413 are reviewed and a concrete implementation of the framework is presented. We apply this framework to a lattice scalar field theory and to U(1) gauge theory, explicitly encoding gauge symmetries in the flow-based approach to the latter. This presentation is intended to be interactive and working with the attached Jupyter notebook is recommended.

Denis Boyda, Gurtej Kanwar, Sébastien Racanière et al.

We develop a flow-based sampling algorithm for $SU(N)$ lattice gauge theories that is gauge-invariant by construction. Our key contribution is constructing a class of flows on an $SU(N)$ variable (or on a $U(N)$ variable by a simple alternative) that respect matrix conjugation symmetry. We apply this technique to sample distributions of single $SU(N)$ variables and to construct flow-based samplers for $SU(2)$ and $SU(3)$ lattice gauge theory in two dimensions.

Karen Ullrich, Fabio Viola, Danilo Jimenez Rezende

Reliably transmitting messages despite information loss due to a noisy channel is a core problem of information theory. One of the most important aspects of real world communication, e.g. via wifi, is that it may happen at varying levels of information transfer. The bandwidth-limited channel models this phenomenon. In this study we consider learning coding with the bandwidth-limited channel (BWLC). Recently, neural communication models such as variational autoencoders have been studied for the task of source compression. We build upon this work by studying neural communication systems with the BWLC. Specifically,we find three modelling choices that are relevant under expected information loss. First, instead of separating the sub-tasks of compression (source coding) and error correction (channel coding), we propose to model both jointly. Framing the problem as a variational learning problem, we conclude that joint systems outperform their separate counterparts when coding is performed by flexible learnable function approximators such as neural networks. To facilitate learning, we introduce a differentiable and computationally efficient version of the bandwidth-limited channel. Second, we propose a design to model missing information with a prior, and incorporate this into the channel model. Finally, sampling from the joint model is improved by introducing auxiliary latent variables in the decoder. Experimental results justify the validity of our design decisions through improved distortion and FID scores.

Gurtej Kanwar, Michael S. Albergo, Denis Boyda et al.

We define a class of machine-learned flow-based sampling algorithms for lattice gauge theories that are gauge-invariant by construction. We demonstrate the application of this framework to U(1) gauge theory in two spacetime dimensions, and find that near critical points in parameter space the approach is orders of magnitude more efficient at sampling topological quantities than more traditional sampling procedures such as Hybrid Monte Carlo and Heat Bath.

Peter Wirnsberger, Andrew J. Ballard, George Papamakarios et al.

Free energy perturbation (FEP) was proposed by Zwanzig more than six decades ago as a method to estimate free energy differences, and has since inspired a huge body of related methods that use it as an integral building block. Being an importance sampling based estimator, however, FEP suffers from a severe limitation: the requirement of sufficient overlap between distributions. One strategy to mitigate this problem, called Targeted Free Energy Perturbation, uses a high-dimensional mapping in configuration space to increase overlap of the underlying distributions. Despite its potential, this method has attracted only limited attention due to the formidable challenge of formulating a tractable mapping. Here, we cast Targeted FEP as a machine learning problem in which the mapping is parameterized as a neural network that is optimized so as to increase overlap. We develop a new model architecture that respects permutational and periodic symmetries often encountered in atomistic simulations and test our method on a fully-periodic solvation system. We demonstrate that our method leads to a substantial variance reduction in free energy estimates when compared against baselines, without requiring any additional data.

Danilo Jimenez Rezende, George Papamakarios, Sébastien Racanière et al.

Normalizing flows are a powerful tool for building expressive distributions in high dimensions. So far, most of the literature has concentrated on learning flows on Euclidean spaces. Some problems however, such as those involving angles, are defined on spaces with more complex geometries, such as tori or spheres. In this paper, we propose and compare expressive and numerically stable flows on such spaces. Our flows are built recursively on the dimension of the space, starting from flows on circles, closed intervals or spheres.

George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende et al.

Normalizing flows provide a general mechanism for defining expressive probability distributions, only requiring the specification of a (usually simple) base distribution and a series of bijective transformations. There has been much recent work on normalizing flows, ranging from improving their expressive power to expanding their application. We believe the field has now matured and is in need of a unified perspective. In this review, we attempt to provide such a perspective by describing flows through the lens of probabilistic modeling and inference. We place special emphasis on the fundamental principles of flow design, and discuss foundational topics such as expressive power and computational trade-offs. We also broaden the conceptual framing of flows by relating them to more general probability transformations. Lastly, we summarize the use of flows for tasks such as generative modeling, approximate inference, and supervised learning.

Slava Voloshynovskiy, Mouad Kondah, Shideh Rezaeifar et al.

Information bottleneck (IB) principle [1] has become an important element in information-theoretic analysis of deep models. Many state-of-the-art generative models of both Variational Autoencoder (VAE) [2; 3] and Generative Adversarial Networks (GAN) [4] families use various bounds on mutual information terms to introduce certain regularization constraints [5; 6; 7; 8; 9; 10]. Accordingly, the main difference between these models consists in add regularization constraints and targeted objectives. In this work, we will consider the IB framework for three classes of models that include supervised, unsupervised and adversarial generative models. We will apply a variational decomposition leading a common structure and allowing easily establish connections between these models and analyze underlying assumptions. Based on these results, we focus our analysis on unsupervised setup and reconsider the VAE family. In particular, we present a new interpretation of VAE family based on the IB framework using a direct decomposition of mutual information terms and show some interesting connections to existing methods such as VAE [2; 3], beta-VAE [11], AAE [12], InfoVAE [5] and VAE/GAN [13]. Instead of adding regularization constraints to an evidence lower bound (ELBO) [2; 3], which itself is a lower bound, we show that many known methods can be considered as a product of variational decomposition of mutual information terms in the IB framework. The proposed decomposition might also contribute to the interpretability of generative models of both VAE and GAN families and create a new insights to a generative compression [14; 15; 16; 17]. It can also be of interest for the analysis of novelty detection based on one-class classifiers [18] with the IB based discriminators.

Peter Toth, Danilo Jimenez Rezende, Andrew Jaegle et al.

The Hamiltonian formalism plays a central role in classical and quantum physics. Hamiltonians are the main tool for modelling the continuous time evolution of systems with conserved quantities, and they come equipped with many useful properties, like time reversibility and smooth interpolation in time. These properties are important for many machine learning problems - from sequence prediction to reinforcement learning and density modelling - but are not typically provided out of the box by standard tools such as recurrent neural networks. In this paper, we introduce the Hamiltonian Generative Network (HGN), the first approach capable of consistently learning Hamiltonian dynamics from high-dimensional observations (such as images) without restrictive domain assumptions. Once trained, we can use HGN to sample new trajectories, perform rollouts both forward and backward in time and even speed up or slow down the learned dynamics. We demonstrate how a simple modification of the network architecture turns HGN into a powerful normalising flow model, called Neural Hamiltonian Flow (NHF), that uses Hamiltonian dynamics to model expressive densities. We hope that our work serves as a first practical demonstration of the value that the Hamiltonian formalism can bring to deep learning.

Danilo Jimenez Rezende, Sébastien Racanière, Irina Higgins et al.

This paper introduces equivariant hamiltonian flows, a method for learning expressive densities that are invariant with respect to a known Lie-algebra of local symmetry transformations while providing an equivariant representation of the data. We provide proof of principle demonstrations of how such flows can be learnt, as well as how the addition of symmetry invariance constraints can improve data efficiency and generalisation. Finally, we make connections to disentangled representation learning and show how this work relates to a recently proposed definition.

Karol Gregor, Danilo Jimenez Rezende, Frederic Besse et al.

When agents interact with a complex environment, they must form and maintain beliefs about the relevant aspects of that environment. We propose a way to efficiently train expressive generative models in complex environments. We show that a predictive algorithm with an expressive generative model can form stable belief-states in visually rich and dynamic 3D environments. More precisely, we show that the learned representation captures the layout of the environment as well as the position and orientation of the agent. Our experiments show that the model substantially improves data-efficiency on a number of reinforcement learning (RL) tasks compared with strong model-free baseline agents. We find that predicting multiple steps into the future (overshooting), in combination with an expressive generative model, is critical for stable representations to emerge. In practice, using expressive generative models in RL is computationally expensive and we propose a scheme to reduce this computational burden, allowing us to build agents that are competitive with model-free baselines.

Simon A. A. Kohl, Bernardino Romera-Paredes, Klaus H. Maier-Hein et al.

Medical imaging only indirectly measures the molecular identity of the tissue within each voxel, which often produces only ambiguous image evidence for target measures of interest, like semantic segmentation. This diversity and the variations of plausible interpretations are often specific to given image regions and may thus manifest on various scales, spanning all the way from the pixel to the image level. In order to learn a flexible distribution that can account for multiple scales of variations, we propose the Hierarchical Probabilistic U-Net, a segmentation network with a conditional variational auto-encoder (cVAE) that uses a hierarchical latent space decomposition. We show that this model formulation enables sampling and reconstruction of segmenations with high fidelity, i.e. with finely resolved detail, while providing the flexibility to learn complex structured distributions across scales. We demonstrate these abilities on the task of segmenting ambiguous medical scans as well as on instance segmentation of neurobiological and natural images. Our model automatically separates independent factors across scales, an inductive bias that we deem beneficial in structured output prediction tasks beyond segmentation.

Danilo Jimenez Rezende, Fabio Viola

In spite of remarkable progress in deep latent variable generative modeling, training still remains a challenge due to a combination of optimization and generalization issues. In practice, a combination of heuristic algorithms (such as hand-crafted annealing of KL-terms) is often used in order to achieve the desired results, but such solutions are not robust to changes in model architecture or dataset. The best settings can often vary dramatically from one problem to another, which requires doing expensive parameter sweeps for each new case. Here we develop on the idea of training VAEs with additional constraints as a way to control their behaviour. We first present a detailed theoretical analysis of constrained VAEs, expanding our understanding of how these models work. We then introduce and analyze a practical algorithm termed Generalized ELBO with Constrained Optimization, GECO. The main advantage of GECO for the machine learning practitioner is a more intuitive, yet principled, process of tuning the loss. This involves defining of a set of constraints, which typically have an explicit relation to the desired model performance, in contrast to tweaking abstract hyper-parameters which implicitly affect the model behavior. Encouraging experimental results in several standard datasets indicate that GECO is a very robust and effective tool to balance reconstruction and compression constraints.

Simon A. A. Kohl, Bernardino Romera-Paredes, Clemens Meyer et al.

Many real-world vision problems suffer from inherent ambiguities. In clinical applications for example, it might not be clear from a CT scan alone which particular region is cancer tissue. Therefore a group of graders typically produces a set of diverse but plausible segmentations. We consider the task of learning a distribution over segmentations given an input. To this end we propose a generative segmentation model based on a combination of a U-Net with a conditional variational autoencoder that is capable of efficiently producing an unlimited number of plausible hypotheses. We show on a lung abnormalities segmentation task and on a Cityscapes segmentation task that our model reproduces the possible segmentation variants as well as the frequencies with which they occur, doing so significantly better than published approaches. These models could have a high impact in real-world applications, such as being used as clinical decision-making algorithms accounting for multiple plausible semantic segmentation hypotheses to provide possible diagnoses and recommend further actions to resolve the present ambiguities.

Marco Fraccaro, Danilo Jimenez Rezende, Yori Zwols et al.

In model-based reinforcement learning, generative and temporal models of environments can be leveraged to boost agent performance, either by tuning the agent's representations during training or via use as part of an explicit planning mechanism. However, their application in practice has been limited to simplistic environments, due to the difficulty of training such models in larger, potentially partially-observed and 3D environments. In this work we introduce a novel action-conditioned generative model of such challenging environments. The model features a non-parametric spatial memory system in which we store learned, disentangled representations of the environment. Low-dimensional spatial updates are computed using a state-space model that makes use of knowledge on the prior dynamics of the moving agent, and high-dimensional visual observations are modelled with a Variational Auto-Encoder. The result is a scalable architecture capable of performing coherent predictions over hundreds of time steps across a range of partially observed 2D and 3D environments.

Théophane Weber, Sébastien Racanière, David P. Reichert et al.

We introduce Imagination-Augmented Agents (I2As), a novel architecture for deep reinforcement learning combining model-free and model-based aspects. In contrast to most existing model-based reinforcement learning and planning methods, which prescribe how a model should be used to arrive at a policy, I2As learn to interpret predictions from a learned environment model to construct implicit plans in arbitrary ways, by using the predictions as additional context in deep policy networks. I2As show improved data efficiency, performance, and robustness to model misspecification compared to several baselines.

Karol Gregor, Danilo Jimenez Rezende, Daan Wierstra

In this paper we introduce a new unsupervised reinforcement learning method for discovering the set of intrinsic options available to an agent. This set is learned by maximizing the number of different states an agent can reliably reach, as measured by the mutual information between the set of options and option termination states. To this end, we instantiate two policy gradient based algorithms, one that creates an explicit embedding space of options and one that represents options implicitly. The algorithms also provide an explicit measure of empowerment in a given state that can be used by an empowerment maximizing agent. The algorithm scales well with function approximation and we demonstrate the applicability of the algorithm on a range of tasks.

Danilo Jimenez Rezende, S. M. Ali Eslami, Shakir Mohamed et al.

A key goal of computer vision is to recover the underlying 3D structure from 2D observations of the world. In this paper we learn strong deep generative models of 3D structures, and recover these structures from 3D and 2D images via probabilistic inference. We demonstrate high-quality samples and report log-likelihoods on several datasets, including ShapeNet [2], and establish the first benchmarks in the literature. We also show how these models and their inference networks can be trained end-to-end from 2D images. This demonstrates for the first time the feasibility of learning to infer 3D representations of the world in a purely unsupervised manner.

Karol Gregor, Frederic Besse, Danilo Jimenez Rezende et al.

We introduce a simple recurrent variational auto-encoder architecture that significantly improves image modeling. The system represents the state-of-the-art in latent variable models for both the ImageNet and Omniglot datasets. We show that it naturally separates global conceptual information from lower level details, thus addressing one of the fundamentally desired properties of unsupervised learning. Furthermore, the possibility of restricting ourselves to storing only global information about an image allows us to achieve high quality 'conceptual compression'.

Danilo Jimenez Rezende, Shakir Mohamed, Ivo Danihelka et al.

Humans have an impressive ability to reason about new concepts and experiences from just a single example. In particular, humans have an ability for one-shot generalization: an ability to encounter a new concept, understand its structure, and then be able to generate compelling alternative variations of the concept. We develop machine learning systems with this important capacity by developing new deep generative models, models that combine the representational power of deep learning with the inferential power of Bayesian reasoning. We develop a class of sequential generative models that are built on the principles of feedback and attention. These two characteristics lead to generative models that are among the state-of-the art in density estimation and image generation. We demonstrate the one-shot generalization ability of our models using three tasks: unconditional sampling, generating new exemplars of a given concept, and generating new exemplars of a family of concepts. In all cases our models are able to generate compelling and diverse samples---having seen new examples just once---providing an important class of general-purpose models for one-shot machine learning.

Shakir Mohamed, Danilo Jimenez Rezende

The mutual information is a core statistical quantity that has applications in all areas of machine learning, whether this is in training of density models over multiple data modalities, in maximising the efficiency of noisy transmission channels, or when learning behaviour policies for exploration by artificial agents. Most learning algorithms that involve optimisation of the mutual information rely on the Blahut-Arimoto algorithm --- an enumerative algorithm with exponential complexity that is not suitable for modern machine learning applications. This paper provides a new approach for scalable optimisation of the mutual information by merging techniques from variational inference and deep learning. We develop our approach by focusing on the problem of intrinsically-motivated learning, where the mutual information forms the definition of a well-known internal drive known as empowerment. Using a variational lower bound on the mutual information, combined with convolutional networks for handling visual input streams, we develop a stochastic optimisation algorithm that allows for scalable information maximisation and empowerment-based reasoning directly from pixels to actions.

Danilo Jimenez Rezende, Shakir Mohamed

The choice of approximate posterior distribution is one of the core problems in variational inference. Most applications of variational inference employ simple families of posterior approximations in order to allow for efficient inference, focusing on mean-field or other simple structured approximations. This restriction has a significant impact on the quality of inferences made using variational methods. We introduce a new approach for specifying flexible, arbitrarily complex and scalable approximate posterior distributions. Our approximations are distributions constructed through a normalizing flow, whereby a simple initial density is transformed into a more complex one by applying a sequence of invertible transformations until a desired level of complexity is attained. We use this view of normalizing flows to develop categories of finite and infinitesimal flows and provide a unified view of approaches for constructing rich posterior approximations. We demonstrate that the theoretical advantages of having posteriors that better match the true posterior, combined with the scalability of amortized variational approaches, provides a clear improvement in performance and applicability of variational inference.

Karol Gregor, Ivo Danihelka, Alex Graves et al.

This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural network architecture for image generation. DRAW networks combine a novel spatial attention mechanism that mimics the foveation of the human eye, with a sequential variational auto-encoding framework that allows for the iterative construction of complex images. The system substantially improves on the state of the art for generative models on MNIST, and, when trained on the Street View House Numbers dataset, it generates images that cannot be distinguished from real data with the naked eye.

Danilo Jimenez Rezende, Shakir Mohamed, Daan Wierstra

We marry ideas from deep neural networks and approximate Bayesian inference to derive a generalised class of deep, directed generative models, endowed with a new algorithm for scalable inference and learning. Our algorithm introduces a recognition model to represent approximate posterior distributions, and that acts as a stochastic encoder of the data. We develop stochastic back-propagation -- rules for back-propagation through stochastic variables -- and use this to develop an algorithm that allows for joint optimisation of the parameters of both the generative and recognition model. We demonstrate on several real-world data sets that the model generates realistic samples, provides accurate imputations of missing data and is a useful tool for high-dimensional data visualisation.