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Probabilistic Machine Learning in the Age of Deep Learning: New Perspectives for Gaussian Processes, Bayesian Optimization and Beyond (PhD Thesis)

Fri, 01 Sep 2023 00:00:00 +0000

The full text is available as a single PDF file

You can also find a list of contents and PDFs corresponding to each individual chapter below:

Chapter 1: Introduction
Chapter 2: Background
Chapter 3: Orthogonally-Decoupled Sparse Gaussian Processes with Spherical Neural Network Activation Features
Chapter 4: Cycle-Consistent Generative Adversarial Networks as a Bayesian Approximation
Chapter 5: Bayesian Optimisation by Classification with Deep Learning and Beyond
Chapter 6: Conclusion
Appendix A: Numerical Methods for Improved Decoupled Sampling of Gaussian Processes
Bibliography

Please find Chapter 1: Introduction reproduced in full below:

Introduction

Artificial intelligence (AI) stands poised to be among the most disruptive technologies of our era. The breakneck pace of recent AI advancements has been spearheaded by machine learning (ML), particularly the resurgence of deep learning. Deep learning is as old as the first general-purpose electronic computer; with roots tracing back to the 1940s and ’50s ( ; ), the revival of deep learning, beginning in the early 2010s, was catalysed by a series of breakthroughs that shattered previously perceived limitations and captivated the collective imagination. These breakthroughs span various domains, including computer vision ( ; ; ; ), speech recognition ( ; ), natural language processing ( ; ), protein folding ( ), generative art and artificial creativity ( ; ; ; ), as well as reinforcement learning for robotics control ( ; ) and achieving superhuman-level gameplay ( ; ).

Nevertheless, it is crucial to view these developments as means to an ultimate end rather than an end in themselves. Arguably, the true pinnacle of AI’s capabilities lies in optimal decision-making, whether that entails offering analyses and insights to aid humans in making better decisions or completely automating the decision-making process altogether. Practically any task directed towards a well-defined objective can be boiled down to a cascade of decisions. At a fundamental level, operating a vehicle involves a continuous stream of decisions involving accelerating, braking, and turning. Financial trading revolves around decisions to buy, sell, or hold various assets. Even complex engineering tasks, such as designing an aerofoil, involve a sequence of decisions about adjusting design variables to achieve desirable aerodynamic characteristics.

Yet, the intricacies of decision-making surpass what any single advancement in deep learning can address. While convolutional neural networks (CNNs) can facilitate object detection tasks in autonomous vehicles, recurrent neural networks (RNNs) can aid in forecasting market dynamics for systematic trading, and physics-informed NNs can assist in predicting aerodynamic effects, it remains the case that no target or quantity of interest can be entirely known or predictable (indeed, if they were, the pursuit of predictive modelling and ML would be superfluous). Instead, predictions often prove unreliable, or at best, uncertain, due to the limitations of our knowledge and the complexity and variability inherent in the underlying real-world processes. The impressive power of deep learning models often overshadows their ignorance of the limits of their own knowledge and the extent of uncertainty in their predictions. When these predictions are integrated into a sequential decision-making framework, such uncertainty can amplify, compound, and lead to catastrophic consequences. In the context of aeronautical engineering, this could result in inefficient designs; in quantitative finance, it can lead to devastating capital losses; and in autonomous driving, it can even cost lives.

Probabilistic Machine Learning

Grounded in the laws of probability and Bayesian statistics ( ; ), probabilistic ML provides a consistent framework for systematically reasoning about the unknown. The probabilistic approach to ML acknowledges that the real world is fraught with uncertainty and embraces this uncertainty as an inherent part of decision-making. Unlike traditional methods, including those of deep learning, it recognises model predictions not as absolute truths that can be represented as single point estimates produced from a deterministic mapping, but as full probability distributions that capture the potential outcomes of a random variable as it propagates through some underlying data-generating process. In a probabilistic model, all quantities are treated as random variables governed by probability distributions – the data are treated as observed variables, which are influenced by some underlying hidden variables, e.g., the model parameters. A prior distribution is used to express reasonable values for these hidden variables and to eliminate implausible ones. The relationship between observed and hidden variables is described using the likelihood, and the process of Bayesian inference amounts to calculating, using basic laws of probability, a posterior distribution over the hidden factors conditioned on the observed data, which can be seen as a refinement of the prior beliefs in light of new evidence. While the posterior distribution can be useful in and of itself, its primary role lies in facilitating subsequent prediction and decision-making by providing full probability distributions over predicted outcomes. This capability allows the decision-maker to assess the range of possible scenarios and their associated probabilities, enabling a more nuanced understanding of uncertainty and risk, which is indispensable in complex, dynamic environments where the repercussions of incorrect decisions can be severe. In essence, probabilistic ML equips autonomous decision-making systems with a probabilistic worldview, enabling them to navigate ambiguity and make sound decisions in the face of imperfect information.

Probabilistic ML vs. Deep Learning

While deep learning has dominated recent AI advances, probabilistic ML remains as important as ever and continues to offer valuable tools for addressing AI challenges that can not be fully resolved by deep learning alone. Although both approaches can be combined to create hybrid methods that leverage their respective strengths, some defining characteristics have traditionally set deep learning apart from probabilistic ML. Perhaps most notably, probabilistic ML approaches can achieve remarkable predictive performance even when data is scarce. In contrast, deep learning models tend to be data-intensive by nature, often demanding datasets of a scale proportional to their size (i.e., their parameter count) ( ), which has seen explosive growth in recent years ( ; ; ; ; ). With that being said, inference in many probabilistic models poses computational problems that are difficult to scale. On the other hand, deep learning approaches have excelled in scalability, a key factor contributing to their widespread success. This scalability is bolstered by their compatibility with various speed-enhancing mechanisms such as stochastic optimisation, specialised hardware accelerators (GPUs and TPUs), as well as distributed and/or cloud-based computing infrastructure. To bridge this gap, substantial research effort has been devoted to enabling probabilistic ML to benefit from these advantages through optimisation-based approximations to Bayesian inference ( ).

Moreover, as mentioned earlier, these paradigms are by no means mutually exclusive. Indeed, it is often possible to directly extend existing models with a Bayesian treatment of their parameters, adding a layer of probabilistic reasoning to the model, and allowing it to not only make predictions but also estimate the uncertainty associated with those predictions. An excellent example is the BNN, which treats the weights as hidden variables and leverages posterior inference to provide predictions while estimating associated uncertainties, delivering a more robust and principled approach to deep learning ( ; ; ).

The Bayesian formalism naturally gives rise to many popular methods and paradigms, often in the form of point estimates or other kinds of approximations. The quintessential example of this is found in linear regression, in particular, in ridge and lasso regression ( ), which correspond variously to maximum a posteriori (MAP) estimates in Bayesian linear regression (BLR) models with prior distributions possessing different sparsity-inducing characteristics ( ) – more broadly, mitigations against over-fitting tend to arise organically in Bayesian methods, which is why they are frequently characterised as being fundamentally more robust against over-fitting ( ). Likewise, the once à la mode support vector machines (SVMs) can be seen as MAP estimates for a class of nonparametric Bayesian models ( ), dropout ( ) in NNs can be seen as a variational approximation to exact inference in BNNs ( ), and unsupervised learning methods such as factor analysis (FA) ( ) and principal component analysis (PCA) ( ) are instances of a class of LVMs ( ; ) known as linear-Gaussian factor models ( ), to name just a few examples. Time and again, classical approaches have not only benefitted from being viewed through the Bayesian perspective but have also been enriched and redefined by the depth of insights this framework provides.

Thesis Goals

The over-arching goal of this thesis is to continue advancing the integration and cross-pollination between deep learning and probabilistic ML. We aim to further the interplay between these two fields, both by incorporating probabilistic interpretations and uncertainty quantification into popular deep learning frameworks, and by leveraging the representational power of deep NNs to improve established Bayesian methods. This dual-pronged approach provides fresh perspectives and taps the complementary strengths of both paradigms, advancing the foundations of AI and facilitating the development of more capable and dependable decision support frameworks. Ultimately, we strive to unlock the potential of deep learning within high-impact probabilistic ML methodologies, and to lend useful Bayesian perspectives on current deep learning techniques.

Gaussian Process Models

Arguably, no family of probabilistic models embodies the ethos of probabilistic ML and illustrates its nuances and parallels with deep learning quite like the GP. Accordingly, they shall occupy a prominent place in our thesis. In particular, GPs stand out as the ideal choice when dealing with limited data, offer the flexibility to encode prior beliefs through the covariance function, and provide predictive uncertainty estimates with a fine calibration that is second to none. Conversely, they are challenging to scale to large datasets, a limitation that has spurred extensive research and development efforts. Furthermore, in contrast to deep learning models, which are often lauded for their ability to automatically uncover valuable patterns and features in data, GPs have at times been dismissed as unsophisticated smoothing mechanisms ( ). Despite these apparent disparities, GPs are intricately connected to NNs in numerous ways. Among these, one of the most classical and well-known relationships is the convergence of single-layer NNs with randomly initialised weights toward GPs in the infinite-width limit ( ). Similar links have also been identified between GPs and infinitely wide deep NNs ( ; ).

In an effort to elevate the representational capabilities of GPs to a level comparable with deep NNs, DGPs ( ) stack together multiple layers of GPs. Additional efforts to construct efficient sparse GP approximations have leveraged the advantageous properties of computations on the hypersphere ( ), which has led to deep GP (DGP) models in which the propagation of posterior predictive means is equivalent to a forward pass through a deep neural network (NN) ( ; ). Notably, as a side effect, this model effectively provides uncertainty estimates for deep NN through its predictive variance. Among the contributions of our thesis is the further development of this framework, integrating cutting-edge techniques ( ; ) to address some of its practical limitations, thereby narrowing the performance gap between GPs and deep NNs.

Probabilistic models, serving a crucial role as decision support tools, routinely aid scientific discovery in fields such as physics and astronomy, guiding advancements in areas of medicine and healthcare encompassing bioinformatics, epidemiology, and medical diagnosis. Beyond that, these models have wide-ranging applications in economics, econometrics, and the social sciences. Moreover, they are indispensable in various engineering disciplines, such as robotics and environmental engineering. Among the many probabilistic models, GPs stand out as a powerful driving force behind a number of important sequential decision-making frameworks, including active learning ( ) and reinforcement learning ( ), and the broader area of probabilistic numerics at large ( ). Notably, Bayesian optimisation (BO) ( ; ; ) is one major area that relies heavily on GPs and will feature extensively in our thesis.

Bayesian optimisation

BO is a powerful methodology dedicated to the global optimisation of complex and resource-intensive objective functions. In contrast to classical optimisation methods, BO excels even when dealing with functions that lack strong assumptions or guarantees. These functions may not be convex, possess no gradients, lack a well-defined mathematical form, and observable only indirectly through noisy measurements.

At its core, BO is a sequential decision-making algorithm.

It relies on observations from past function evaluations to determine the next candidate location for evaluation in pursuit of optimal solutions. BO leverages a probabilistic model, often a GP, to represent its knowledge and beliefs about the unknown function. This model is continuously updated with the acquisition of each new observation, enabling the algorithm to adapt its behaviour and make sound decisions based on the evolving information.

BO effectively manages uncertainty inherent in such sequential decision-making processes by making use of the probabilistic model to the fullest, harnessing the entire predictive distribution, particularly, the predictive uncertainty, to select promising candidate solutions that bring the most value to the optimisation process. This generally consists not merely of those most likely to optimise the objective function (i.e., exploiting that which is known), but also those likely to reveal the most knowledge and information about the function itself (i.e., exploring that which remains unknown).

This pronounced emphasis on well-calibrated uncertainty distinguishes BO as one of the standout “killer apps” for GPs and a jewel in the crown of probabilistic ML applications. In practice, BO has proven instrumental across science, engineering, and industry, where efficiency and cost-effectiveness are paramount. Its applications include protein engineering ( ; ), material discovery ( ), experimental physics (e.g., experiments involving ultra-cold atoms ( ) and free-electron lasers ( )), environmental monitoring (sensor placement) ( ; ), and the design of aerodynamic aerofoils ( ; ), integrated circuits ( ; ), broadband high-efficiency power amplifiers ( ), and fast-charging protocols for lithium-ion batteries ( ). Notably, it has played a crucial role in automating the hyperparameter tuning of various ML models ( ; ), especially deep learning models, thus representing yet another way in which probabilistic ML has contributed to the advancement of deep learning.

However, GPs are not universally suitable for all BO problem scenarios. They are most effective when dealing with smooth, stationary functions with homoscedastic noise and a relatively modest input dimensionality. Additionally, GPs are easiest to work with for functions with a single output and purely continuous inputs. While a surprisingly wide array of real-world challenges satisfy these conditions, many high-impact problems, such as gene and protein design, which involves sequential inputs ( ; ; ; ; ); NAS, which involves structured inputs with intricate conditional dependencies; and automotive safety engineering, which involve numerous constraints and multiple objectives, clearly fall outside of this scope. This is not to say that GPs cannot be extended to such challenging scenarios. However, such extensions almost always come at a cost. Consequently, it makes sense to appeal to alternative modelling paradigms more naturally suited to specific tasks, e.g., employing random forests (RFs) to handle discrete and structured inputs, or deep NNs for capturing nonstationary behaviour and dealing with multiple objectives. A major contribution of this thesis is the introduction of a new formulation of BO that seamlessly accommodates virtually any modelling paradigm, including deep learning, without any compromise.

Thesis Overview

The core contributions of our thesis are summarised as follows:

We improve upon the framework for sparse hyperspherical GP approximations that employ nonlinear activations as inter-domain inducing features. This framework serves as a bridge between GPs and NNs, with posterior predictive mean taking the form of single-layer feedforward NNs. Our thesis examines some practical issues associated with this approach and proposes an extension that takes advantage of the orthogonal decoupling of GPs to mitigate these limitations. In particular, we introduce spherical inter-domain features to construct more flexible data-dependent basis functions for both the principal and orthogonal components of the GP approximation. We demonstrate that incorporating orthogonal inducing variables under this framework not only alleviates these shortcomings but also offers superior scalability compared to alternative strategies.
We provide a probabilistic perspective on cycle-consistent adversarial networks (CYCLEGANs), a cutting-edge deep generative model for style transfer and image-to-image translation. Specifically, we frame the problem of learning cross-domain correspondences without paired data as Bayesian inference in a latent variable model (LVM), in which the goal is to uncover the hidden representations of entities from one domain as entities in another. First, we introduce implicit LVMs, which allow flexible prior specification over latent representations as implicit distributions. Next, we develop a new variational inference (VI) framework that minimises a symmetrised statistical divergence between the variational and true joint distributions. Finally, we show that CYCLEGANs emerge as a closely-related variant of our framework, providing a useful interpretation as a Bayesian approximation.
We introduce a model-agnostic formulation of BO based on classification. Building on the established links between class-probability estimation (CPE), density-ratio estimation (DRE), and the improvement-based acquisition functions, we reformulate the acquisition function as a binary classifier over candidate solutions. This approach eliminates the need for an explicit probabilistic model of the objective function and casts aside the limitations of tractability constraints. As a result, our model-agnostic BO approach substantially broadens its applicability across diverse problem scenarios, accommodating flexible and scalable modelling paradigms such as deep learning without necessitating approximations or sacrificing expressive and representational capacity.

Accordingly, our thesis is organised as follows:

Chapter 2 (Background) lays the necessary groundwork for our thesis. We begin by outlining the fundamental principles of probability and Bayesian statistics, which form the basis of probabilistic ML. Additionally, we introduce the widely-adopted method of approximate Bayesian inference known as VI. Our discussion underscores the central role played by statistical divergences, prompting us to delve into a larger family of divergences and motivating our discussion of DRE. With a solid foundation in place, we shift our focus to GPs, providing an introductory overview and highlighting the most commonly-used sparse approximations. Finally, we conclude this background chapter by introducing the basic concepts behind BO.
Chapter 3 (Orthogonally-Decoupled Sparse GPs with Spherical Inducing Features) examines orthogonally-decoupled sparse GPs with spherical NN activation features, as summarised in the corresponding item above.
Chapter 4 (Cycle-Consistent Adversarial Learning as Bayesian Inference) examines from the perspective of approximate Bayesian inference, as summarised in the corresponding item above.
Chapter 5 (Bayesian Optimization by Density-Ratio Estimation) examines our model-agnostic approach to BO based on binary classification and DRE, as summarised in the corresponding item above.
Chapter 6 (Conclusion) brings this thesis to a close by reflecting on our main contributions and situating them in the broader landscape of probabilistic methods in ML. Finally, we conclude by presenting our outlook on the avenues for future research and development in this rapidly evolving field.

References

Anil, R., Dai, A. M., Firat, O., Johnson, M., Lepikhin, D., Passos, A., Shakeri, S., Taropa, E., Bailey, P., Chen, Z., et al. (2023). Palm 2 technical report. arXiv Preprint arXiv:2305.10403.

Attia, P. M., Grover, A., Jin, N., Severson, K. A., Markov, T. M., Liao, Y.-H., Chen, M. H., Cheong, B., Perkins, N., Yang, Z., et al. (2020). Closed-loop optimization of fast-charging protocols for batteries with machine learning. Nature, 578(7795), 397–402.

Bartholomew, D. J., Knott, M., & Moustaki, I. (2011). Latent variable models and factor analysis: A unified approach. John Wiley & Sons.

Bayes, T. (1763). LII. An essay towards solving a problem in the doctrine of chances. By the late rev. Mr. Bayes, FRS communicated by mr. Price, in a letter to john canton, AMFR s. Philosophical Transactions of the Royal Society of London, 53, 370–418.

Blundell, C., Cornebise, J., Kavukcuoglu, K., & Wierstra, D. (2015). Weight uncertainty in neural network. International Conference on Machine Learning, 1613–1622.

Brochu, E., Cora, V. M., & De Freitas, N. (2010). A tutorial on bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. arXiv Preprint arXiv:1012.2599.

Brown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J. D., Dhariwal, P., Neelakantan, A., Shyam, P., Sastry, G., Askell, A., et al. (2020). Language models are few-shot learners. Advances in Neural Information Processing Systems, 33, 1877–1901.

Chen, P., Merrick, B. M., & Brazil, T. J. (2015). Bayesian optimization for broadband high-efficiency power amplifier designs. IEEE Transactions on Microwave Theory and Techniques, 63(12), 4263–4272.

Damianou, A., & Lawrence, N. D. (2013). Deep gaussian processes. Artificial Intelligence and Statistics, 207–215.

Deisenroth, M., & Rasmussen, C. E. (2011). PILCO: A model-based and data-efficient approach to policy search. Proceedings of the 28th International Conference on Machine Learning (ICML-11), 465–472.

Duris, J., Kennedy, D., Hanuka, A., Shtalenkova, J., Edelen, A., Baxevanis, P., Egger, A., Cope, T., McIntire, M., Ermon, S., et al. (2020). Bayesian optimization of a free-electron laser. Physical Review Letters, 124(12), 124801.

Dutordoir, V., Durrande, N., & Hensman, J. (2020). Sparse Gaussian processes with spherical harmonic features. International Conference on Machine Learning, 2793–2802.

Dutordoir, V., Hensman, J., Wilk, M. van der, Ek, C. H., Ghahramani, Z., & Durrande, N. (2021). Deep neural networks as point estimates for deep Gaussian processes. Advances in Neural Information Processing Systems, 34.

Forrester, A. I., & Keane, A. J. (2009). Recent advances in surrogate-based optimization. Progress in Aerospace Sciences, 45(1-3), 50–79.

Gal, Y., & Ghahramani, Z. (2016). Dropout as a bayesian approximation: Representing model uncertainty in deep learning. International Conference on Machine Learning, 1050–1059.

Garnett, R. (2023). Bayesian Optimization. Cambridge University Press.

Garnett, R., Osborne, M. A., & Roberts, S. J. (2010). Bayesian optimization for sensor set selection. Proceedings of the 9th ACM/IEEE International Conference on Information Processing in Sensor Networks, 209–219.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian data analysis. CRC press.

Girshick, R., Donahue, J., Darrell, T., & Malik, J. (2014). Rich feature hierarchies for accurate object detection and semantic segmentation. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 580–587.

Gonzalez, J., Longworth, J., James, D. C., & Lawrence, N. D. (2015). Bayesian optimization for synthetic gene design. arXiv Preprint arXiv:1505.01627.

Goodfellow, I. J., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., & Bengio, Y. (2014). Generative adversarial networks. arXiv Preprint arXiv:1406.2661.

Graves, A., Mohamed, A., & Hinton, G. (2013). Speech recognition with deep recurrent neural networks. 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, 6645–6649.

Hennig, P., Osborne, M. A., & Kersting, H. P. (2022). Probabilistic numerics. Cambridge University Press.

Hie, B. L., & Yang, K. K. (2022). Adaptive machine learning for protein engineering. Current Opinion in Structural Biology, 72, 145–152.

Hinton, G., Deng, L., Yu, D., Dahl, G. E., Mohamed, A., Jaitly, N., Senior, A., Vanhoucke, V., Nguyen, P., Sainath, T. N., et al. (2012). Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups. IEEE Signal Processing Magazine, 29(6), 82–97.

Ho, J., Jain, A., & Abbeel, P. (2020). Denoising diffusion probabilistic models. Advances in Neural Information Processing Systems, 33, 6840–6851.

Hoffmann, J., Borgeaud, S., Mensch, A., Buchatskaya, E., Cai, T., Rutherford, E., Casas, D. de L., Hendricks, L. A., Welbl, J., Clark, A., et al. (2022). Training compute-optimal large language models. arXiv Preprint arXiv:2203.15556.

Houlsby, N., Huszár, F., Ghahramani, Z., & Lengyel, M. (2011). Bayesian active learning for classification and preference learning. arXiv Preprint arXiv:1112.5745.

Jordan, M. I., Ghahramani, Z., Jaakkola, T. S., & Saul, L. K. (1998). An introduction to variational methods for graphical models. Learning in Graphical Models, 105–161.

Jumper, J., Evans, R., Pritzel, A., Green, T., Figurnov, M., Ronneberger, O., Tunyasuvunakool, K., Bates, R., Žídek, A., Potapenko, A., et al. (2021). Highly accurate protein structure prediction with AlphaFold. Nature, 596(7873), 583–589.

Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2012). Imagenet classification with deep convolutional neural networks. Advances in Neural Information Processing Systems, 25.

Lam, R., Poloczek, M., Frazier, P., & Willcox, K. E. (2018). Advances in bayesian optimization with applications in aerospace engineering. 2018 AIAA Non-Deterministic Approaches Conference, 1656.

Laplace, P. S. (1814). Théorie analytique des probabilités. Courcier.

Lee, J., Bahri, Y., Novak, R., Schoenholz, S. S., Pennington, J., & Sohl-Dickstein, J. (2017). Deep neural networks as gaussian processes. arXiv Preprint arXiv:1711.00165.

Lillicrap, T. P., Hunt, J. J., Pritzel, A., Heess, N., Erez, T., Tassa, Y., Silver, D., & Wierstra, D. (2015). Continuous control with deep reinforcement learning. arXiv Preprint arXiv:1509.02971.

Lyu, W., Xue, P., Yang, F., Yan, C., Hong, Z., Zeng, X., & Zhou, D. (2017). An efficient bayesian optimization approach for automated optimization of analog circuits. IEEE Transactions on Circuits and Systems I: Regular Papers, 65(6), 1954–1967.

MacKay, D. J. (1992). A practical bayesian framework for backpropagation networks. Neural Computation, 4(3), 448–472.

MacKay, D. J. (2003). Information theory, inference and learning algorithms. Cambridge university press.

Marchant, R., & Ramos, F. (2012). Bayesian optimisation for intelligent environmental monitoring. 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2242–2249.

Matthews, A. G. de G., Rowland, M., Hron, J., Turner, R. E., & Ghahramani, Z. (2018). Gaussian process behaviour in wide deep neural networks. arXiv Preprint arXiv:1804.11271.

McCulloch, W. S., & Pitts, W. (1943). A logical calculus of the ideas immanent in nervous activity. The Bulletin of Mathematical Biophysics, 5, 115–133.

Mnih, V., Kavukcuoglu, K., Silver, D., Graves, A., Antonoglou, I., Wierstra, D., & Riedmiller, M. (2013). Playing atari with deep reinforcement learning. arXiv Preprint arXiv:1312.5602.

Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., Graves, A., Riedmiller, M., Fidjeland, A. K., Ostrovski, G., et al. (2015). Human-level control through deep reinforcement learning. Nature, 518(7540), 529–533.

Moss, H. B., Beck, D., González, J., Leslie, D. S., & Rayson, P. (2020). BOSS: Bayesian optimization over string spaces. arXiv Preprint arXiv:2010.00979.

Neal, R. M. (1995). BAYESIAN LEARNING FOR NEURAL NETWORKS

\[PhD thesis\]

. University of Toronto.

OpenAI, R. (2023). GPT-4 technical report. arXiv, 2303–08774.

Opper, M., & Winther, O. (2000). Gaussian processes and SVM: Mean field results and leave-one-out.

Pearson, K. (1901). LIII. On lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11), 559–572.

Rae, J. W., Borgeaud, S., Cai, T., Millican, K., Hoffmann, J., Song, F., Aslanides, J., Henderson, S., Ring, R., Young, S., et al. (2021). Scaling language models: Methods, analysis & insights from training gopher. arXiv Preprint arXiv:2112.11446.

Ramesh, A., Dhariwal, P., Nichol, A., Chu, C., & Chen, M. (2022). Hierarchical text-conditional image generation with clip latents. arXiv Preprint arXiv:2204.06125, 1(2), 3.

Rasmussen, C. E., & Williams, C. K. I. (2005). Gaussian Processes for Machine Learning. The MIT Press.

Redmon, J., Divvala, S., Girshick, R., & Farhadi, A. (2016). You only look once: Unified, real-time object detection. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 779–788.

Rombach, R., Blattmann, A., Lorenz, D., Esser, P., & Ommer, B. (2022). High-resolution image synthesis with latent diffusion models. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 10684–10695.

Romero, P. A., Krause, A., & Arnold, F. H. (2013). Navigating the protein fitness landscape with gaussian processes. Proceedings of the National Academy of Sciences, 110(3), E193–E201.

Ronneberger, O., Fischer, P., & Brox, T. (2015). U-net: Convolutional networks for biomedical image segmentation. Medical Image Computing and Computer-Assisted Intervention–MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceedings, Part III 18, 234–241.

Rosenblatt, F. (1958). The perceptron: A probabilistic model for information storage and organization in the brain. Psychological Review, 65(6), 386.

Roweis, S., & Ghahramani, Z. (1999). A unifying review of linear gaussian models. Neural Computation, 11(2), 305–345.

Salimbeni, H., Cheng, C.-A., Boots, B., & Deisenroth, M. (2018). Orthogonally decoupled variational Gaussian processes. Advances in Neural Information Processing Systems, 31.

Seko, A., Togo, A., Hayashi, H., Tsuda, K., Chaput, L., & Tanaka, I. (2015). Prediction of low-thermal-conductivity compounds with first-principles anharmonic lattice-dynamics calculations and bayesian optimization. Physical Review Letters, 115(20), 205901.

Shahriari, B., Swersky, K., Wang, Z., Adams, R. P., & De Freitas, N. (2015). Taking the human out of the loop: A review of bayesian optimization. Proceedings of the IEEE, 104(1), 148–175.

Shi, J., Titsias, M., & Mnih, A. (2020). Sparse orthogonal variational inference for Gaussian processes. International Conference on Artificial Intelligence and Statistics, 1932–1942.

Shoeybi, M., Patwary, M., Puri, R., LeGresley, P., Casper, J., & Catanzaro, B. (2019). Megatron-lm: Training multi-billion parameter language models using model parallelism. arXiv Preprint arXiv:1909.08053.

Silver, D., Huang, A., Maddison, C. J., Guez, A., Sifre, L., Van Den Driessche, G., Schrittwieser, J., Antonoglou, I., Panneershelvam, V., Lanctot, M., et al. (2016). Mastering the game of go with deep neural networks and tree search. Nature, 529(7587), 484–489.

Snoek, J., Larochelle, H., & Adams, R. P. (2012). Practical Bayesian optimization of machine learning algorithms. Advances in Neural Information Processing Systems, 25, 2951–2959.

Spearman, C. (1904). " general intelligence," objectively determined and measured. The American Journal of Psychology, 15(2), 201–292.

Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. (2014). Dropout: A simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 15(1), 1929–1958.

Sun, S., Shi, J., & Grosse, R. B. (2020). Neural networks as inter-domain inducing points. Third Symposium on Advances in Approximate Bayesian Inference.

Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology, 58(1), 267–288.

Tipping, M. E., & Bishop, C. M. (1999). Probabilistic principal component analysis. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(3), 611–622.

Torun, H. M., Swaminathan, M., Davis, A. K., & Bellaredj, M. L. F. (2018). A global bayesian optimization algorithm and its application to integrated system design. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 26(4), 792–802.

Touvron, H., Martin, L., Stone, K., Albert, P., Almahairi, A., Babaei, Y., Bashlykov, N., Batra, S., Bhargava, P., Bhosale, S., et al. (2023). Llama 2: Open foundation and fine-tuned chat models. arXiv Preprint arXiv:2307.09288.

Turner, R., Eriksson, D., McCourt, M., Kiili, J., Laaksonen, E., Xu, Z., & Guyon, I. (2021). Bayesian optimization is superior to random search for machine learning hyperparameter tuning: Analysis of the black-box optimization challenge 2020. NeurIPS 2020 Competition and Demonstration Track, 3–26.

Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, Ł., & Polosukhin, I. (2017). Attention is all you need. Advances in Neural Information Processing Systems, 30.

Wigley, P. B., Everitt, P. J., Hengel, A. van den, Bastian, J. W., Sooriyabandara, M. A., McDonald, G. D., Hardman, K. S., Quinlivan, C. D., Manju, P., Kuhn, C. C., et al. (2016). Fast machine-learning online optimization of ultra-cold-atom experiments. Scientific Reports, 6(1), 25890.

Yang, K. K., Wu, Z., & Arnold, F. H. (2019). Machine-learning-guided directed evolution for protein engineering. Nature Methods, 16(8), 687–694.

GPflux

Wed, 01 Sep 2021 00:00:00 +0000

is a TensorFlow/Keras framework for Deep , developed at Secondmind Labs. It builds on and exposes Deep GP layers as familiar Keras building blocks, making it easier to compose deep .

Contributed during my doctoral student researcher appointment at Secondmind Labs, alongside Vincent Dutordoir, ST John, and other members of the lab.

AutoGluon

Sun, 01 Sep 2019 00:00:00 +0000

is an open-source toolkit from AWS that automates ML for tabular, image, and text data. During my AWS Berlin internship I was a core developer of the -based searcher module — described in and later forming the basis of .

Aboleth

Thu, 01 Jun 2017 00:00:00 +0000

is a minimalistic TensorFlow framework for scalable and approximation, focused on techniques. Built at CSIRO Data61 with Daniel Steinberg and Lachlan McCalman.

A Tutorial on Variational Autoencoders with a Concise Keras Implementation

Wed, 20 Apr 2016 00:00:00 +0000

is awesome. It is a very well-designed library that clearly abides by its of modularity and extensibility, enabling us to easily assemble powerful, complex models from primitive building blocks. This has been demonstrated in numerous blog posts and tutorials, in particular, the excellent tutorial on . As the name suggests, that tutorial provides examples of how to implement various kinds of autoencoders in Keras, including the variational autoencoder (VAE)¹.

Like all autoencoders, the variational autoencoder is primarily used for unsupervised learning of hidden representations. However, they are fundamentally different to your usual neural network-based autoencoder in that they approach the problem from a probabilistic perspective. They specify a joint distribution over the observed and latent variables, and approximate the intractable posterior conditional density over latent variables with variational inference, using an inference network ² ³ (or more classically, a recognition model ⁴) to amortize the cost of inference.

While the examples in the aforementioned tutorial do well to showcase the versatility of Keras on a wide range of autoencoder model architectures, doesn’t properly take advantage of Keras’ modular design, making it difficult to generalize and extend in important ways. As we will see, it relies on implementing custom layers and constructs that are restricted to a specific instance of variational autoencoders. This is a shame because when combined, Keras’ building blocks are powerful enough to encapsulate most variants of the variational autoencoder and more generally, recognition-generative model combinations for which the generative model belongs to a large family of deep latent Gaussian models (DLGMs)⁵.

The goal of this post is to propose a clean and elegant alternative implementation that takes better advantage of Keras’ modular design. It is not intended as tutorial on variational autoencoders ⁶. Rather, we study variational autoencoders as a special case of variational inference in deep latent Gaussian models using inference networks, and demonstrate how we can use Keras to implement them in a modular fashion such that they can be easily adapted to approximate inference in tasks beyond unsupervised learning, and with complicated (non-Gaussian) likelihoods.

This first post will lay the groundwork for a series of future posts that explore ways to extend this basic modular framework to implement the cutting-edge methods proposed in the latest research, such as the normalizing flows for building richer posterior approximations ⁷, importance weighted autoencoders ⁸, the Gumbel-softmax trick for inference in discrete latent variables ⁹, and even the most recent GAN-based density-ratio estimation techniques for likelihood-free inference ¹⁰ ¹¹.

Model specification

First, it is important to understand that the variational autoencoder . Rather, the generative model is a component of the variational autoencoder and is, in general, a deep latent Gaussian model. In particular, let $\mathbf{x}$ be a local observed variable and $\mathbf{z}$ its corresponding local latent variable, with joint distribution

$$ p_{\theta}(\mathbf{x}, \mathbf{z}) = p_{\theta}(\mathbf{x} | \mathbf{z}) p(\mathbf{z}). $$

In Bayesian modelling, we assume the distribution of observed variables to be governed by the latent variables. Latent variables are drawn from a prior density $p(\mathbf{z})$ and related to the observations through the likelihood $p_{\theta}(\mathbf{x} | \mathbf{z})$. Deep latent Gaussian models (DLGMs) are a general class of models where the observed variable is governed by a hierarchy of latent variables, and the latent variables at each level of the hierarchy are Gaussian a priori ⁵.

In a typical instance of the variational autoencoder, we have only a single layer of latent variables with a Normal prior distribution,

$$ p(\mathbf{z}) = \mathcal{N}(\mathbf{0}, \mathbf{I}). $$

Now, each local latent variable is related to its corresponding observation through the likelihood $p_{\theta}(\mathbf{x} | \mathbf{z})$, which can be viewed as a probabilistic decoder. Given a hidden lower-dimensional representation (or “code”) $\mathbf{z}$, it “decodes” it into a distribution over the observation $\mathbf{x}$.

Decoder

In this example, we define $p_{\theta}(\mathbf{x} | \mathbf{z})$ to be a multivariate Bernoulli whose probabilities are computed from $\mathbf{z}$ using a fully-connected neural network with a single hidden layer,

$$ \begin{align*} p_{\theta}(\mathbf{x} | \mathbf{z}) & = \mathrm{Bern}( \sigma( \mathbf{W}_2 \mathbf{h} + \mathbf{b}_2 ) ), \newline \mathbf{h} & = h(\mathbf{W}_1 \mathbf{z} + \mathbf{b}_1), \end{align*} $$

where $\sigma$ is the logistic sigmoid function, $h$ is some non-linearity, and the model parameters $\theta = \{ \mathbf{W}_1, \mathbf{W}_2, \mathbf{b}_1, \mathbf{b}_2 \}$ consist of the weights and biases of this neural network.

It is straightforward to implement this in Keras with the :

decoder = Sequential([
 Dense(intermediate_dim, input_dim=latent_dim, activation='relu'),
 Dense(original_dim, activation='sigmoid')
])

You can view a summary of the model parameters $\theta$ by calling decoder.summary(). Additionally, you can produce a high-level diagram of the network architecture, and optionally the input and output shapes of each layer using from the keras.utils.vis_utils module. Although our architecture is about as simple as it gets, it is included in the figure below as an example of what the diagrams look like.

Note that by fixing $\mathbf{W}_1$, $\mathbf{b}_1$ and $h$ to be the identity matrix, the zero vector, and the identity function, respectively (or equivalently dropping the first Dense layer in the snippet above altogether), we recover logistic factor analysis. With similarly minor modifications, we can recover other members from the family of DLGMs, which include non-linear factor analysis, non-linear Gaussian belief networks, sigmoid belief networks, and many others ⁵.

Having specified how the probabilities are computed, we can now define the negative log likelihood of a Bernoulli $- \log p_{\theta}(\mathbf{x}|\mathbf{z})$, which is in fact equivalent to the :

def nll(y_true, y_pred):
 """ Negative log likelihood (Bernoulli). """

 # keras.losses.binary_crossentropy gives the mean
 # over the last axis. we require the sum
 return K.sum(K.binary_crossentropy(y_true, y_pred), axis=-1)

As we discuss later, this will not be the loss we ultimately minimize, but will constitute the data-fitting term of our final loss.

Note this is a valid definition of a , which is required to compile and optimize a model. It is a symbolic function that returns a scalar for each data-point in y_true and y_pred. In our example, y_pred will be the output of our decoder network, which are the predicted probabilities, and y_true will be the true probabilities.

Side note: Using TensorFlow Distributions in loss

If you are using the TensorFlow backend, you can directly use the (negative) log probability of Bernoulli from TensorFlow Distributions as a Keras loss, as I demonstrate in my post on .

Specifically we can define the loss as,

def nll(y_true, y_pred):
 """ Negative log likelihood (Bernoulli). """

 lh = K.tf.distributions.Bernoulli(probs=y_pred)

 return - K.sum(lh.log_prob(y_true), axis=-1)

This is exactly equivalent to the previous definition, but does not call K.binary_crossentropy directly.

Inference

Having specified the generative process, we would now like to perform inference on the latent variables and model parameters $\mathbf{z}$ and $\theta$, respectively. In particular, our goal is to compute the posterior $p_{\theta}(\mathbf{z} | \mathbf{x})$, the conditional density of the latent variable $\mathbf{z}$ given observed variable $\mathbf{x}$. Additionally, we wish to optimize the model parameters $\theta$ with respect to the marginal likelihood $p_{\theta}(\mathbf{x})$. Both depend on the marginal likelihood, whose calculation requires marginalizing out the latent variables $\mathbf{z}$. In general, this is computational intractable, requiring exponential time to compute, or it is analytically intractable and cannot be evaluated in closed-form. In our case, we suffer from the latter intractability, since our prior is Gaussian non-conjugate to the Bernoulli likelihood.

To circumvent this intractability we turn to variational inference, which formulates inference as an optimization problem. It seeks an approximate posterior $q_{\phi}(\mathbf{z} | \mathbf{x})$ closest in Kullback-Leibler (KL) divergence to the true posterior. More precisely, the approximate posterior is parameterized by variational parameters $\phi$, and we seek a setting of these parameters that minimizes the aforementioned KL divergence,

$$ \phi^* = \mathrm{argmin}_{\phi} \mathrm{KL} [q_{\phi}(\mathbf{z} | \mathbf{x}) || p_{\theta}(\mathbf{z} | \mathbf{x}) ] $$

With the luck we’ve had so far, it shouldn’t come as a surprise anymore that this too is intractable. It also depends on the log marginal likelihood, whose intractability is the reason we appealed to approximate inference in the first place. Instead, we maximize an alternative objective function, the evidence lower bound (ELBO), which is expressed as

$$ \begin{align*} \mathrm{ELBO}(q) & = \mathbb{E}_{q_{\phi}(\mathbf{z} | \mathbf{x})} [ \log p_{\theta}(\mathbf{x} | \mathbf{z}) + \log p(\mathbf{z}) - \log q_{\phi}(\mathbf{z} | \mathbf{x}) ] \newline & = \mathbb{E}_{q_{\phi}(\mathbf{z} | \mathbf{x})} [ \log p_{\theta}(\mathbf{x} | \mathbf{z}) ] -\mathrm{KL} [ q_{\phi}(\mathbf{z} | \mathbf{x}) || p(\mathbf{z}) ]. \end{align*} $$

Importantly, the ELBO is a lower bound to the log marginal likelihood. Therefore, maximizing it with respect to the model parameters $\theta$ approximately maximizes the log marginal likelihood. Additionally, maximizing it with respect to variational parameters $\phi$ can be shown to minimize $\mathrm{KL} [q_{\phi}(\mathbf{z} | \mathbf{x}) || p_{\theta}(\mathbf{z} | \mathbf{x}) ]$. Also, it turns out that the KL divergence determines the tightness of the lower bound, where we have equality iff the KL divergence is zero, which happens iff $q_{\phi}(\mathbf{z} | \mathbf{x}) = p_{\theta}(\mathbf{z} | \mathbf{x})$. Hence, simultaneously maximizing it with respect to $\theta$ and $\phi$ gets us two birds with one stone.

Next we discuss the form of the approximate posterior $q_{\phi}(\mathbf{z} | \mathbf{x})$, which can be viewed as a probabilistic encoder. Its role is opposite to that of the decoder. Given an observation $\mathbf{x}$, it “encodes” it into a distribution over its hidden lower-dimensional representations.

Encoder

For each local observed variable $\mathbf{x}_n$, we wish to approximate the true posterior distribution $p(\mathbf{z}_n|\mathbf{x}_n)$ over its corresponding local latent variables $\mathbf{z}_n$. A common approach is to approximate it using a variational distribution $q_{\lambda_n}(\mathbf{z}_n)$, specified as a diagonal Gaussian, where the local variational parameters $\lambda_n = \{ \boldsymbol{\mu}_n, \boldsymbol{\sigma}_n \}$ are the mean and standard deviation of this approximating distribution,

$$ q_{\lambda_n}(\mathbf{z}_n) = \mathcal{N}( \mathbf{z}_n | \boldsymbol{\mu}_n, \mathrm{diag}(\boldsymbol{\sigma}_n^2) ). $$

This approach has a number of shortcomings. First, the number of local variational parameters we need to optimize grows with the size of the dataset. Second, a new set of local variational parameters need to be optimized for new unseen test points. This is not to mention the strong factorization assumption we make by specifying diagonal Gaussian distributions as the family of approximations. The last is still an active area of research, and the first two can be addressed by introducing a further approximation using an inference network.

Inference network

We amortize the cost of inference by introducing an inference network which approximates the local variational parameters $\lambda_n$ for a given local observed variable $\textbf{x}_n$. For our approximating distribution in particular, given $\textbf{x}_n$ the inference network yields two vector-valued outputs $\boldsymbol{\mu}_{\phi}(\textbf{x}_n)$ and $\boldsymbol{\sigma}_{\phi}(\textbf{x}_n)$, which we use to approximate its local variational parameters $\boldsymbol{\mu}_n$ and $\boldsymbol{\sigma}_n$, respectively. Our approximate posterior distribution now becomes $$ q_{\phi}(\mathbf{z}_n | \mathbf{x}_n)

\mathcal{N}(\mathbf{z}n | \boldsymbol{\mu}{\phi}(\mathbf{x}n), \mathrm{diag}(\boldsymbol{\sigma}{\phi}^2(\mathbf{x}_n)) ). $$ Instead of learning local variational parameters $\lambda_n$ for each data-point, we now learn a fixed number of global variational parameters $\phi$ which constitute the parameters (i.e. weights) of the inference network. Moreover, this approximation allows statistical strength to be shared across observed data-points and also generalize to unseen test points.

We specify the mean $\boldsymbol{\mu}_{\phi}(\mathbf{x})$ and log variance $\log \boldsymbol{\sigma}_{\phi}^2(\mathbf{x})$ of this distribution as the output of an inference network. For this post, we keep the architecture of the network simple, with only a single hidden layer and two fully-connected output layers. Again, this is simple to define in Keras:

# input layer
x = Input(shape=(original_dim,))

# hidden layer
h = Dense(intermediate_dim, activation='relu')(x)

# output layer for mean and log variance
z_mu = Dense(latent_dim)(h)
z_log_var = Dense(latent_dim)(h)

Since this network has multiple outputs, we couldn’t use the Sequential model API as we did for the decoder. Instead, we will resort to the more powerful , which allows us to implement complex models with shared layers, multiple inputs, multiple outputs, and so on.

Note that we output the log variance instead of the standard deviation because this is not only more convenient to work with, but also helps with numerical stability. However, we still require the standard deviation later. To recover it, we simply implement the appropriate transformation and encapsulate it in a .

# normalize log variance to std dev
z_sigma = Lambda(lambda t: K.exp(.5*t))(z_log_var)

Before moving on, we give a few words on nomenclature and context. In the prelude and title of this section, we characterized the approximate posterior distribution with an inference network as a probabilistic encoder (analogously to its counterpart, the probabilistic decoder). Although this is an accurate interpretation, it is a limited one. Classically, inference networks are known as recognition models, and have now been used for decades in a wide variety of probabilistic methods. When composed end-to-end, the recognition-generative model combination can be seen as having an autoencoder structure. Indeed, this structure contains the variational autoencoder as a special case, and also the now less fashionable Helmholtz machine ⁴. Even more generally, this recognition-generative model combination constitutes a widely-applicable approach currently known as amortized variational inference, which can be used to perform approximate inference in models that lie beyond even the large class of deep latent Gaussian models.

Having specified all the ingredients necessary to carry out variational inference (namely, the prior, likelihood and approximate posterior), we next focus on finalizing the definition of the (negative) ELBO as our loss function in Keras. As written earlier, the ELBO can be decomposed into two terms, $\mathbb{E}_{q_{\phi}(\mathbf{z} | \mathbf{x})} [ \log p_{\theta}(\mathbf{x} | \mathbf{z}) ]$ the expected log likelihood (ELL) over $q_{\phi}(\mathbf{z} | \mathbf{x})$, and $- \mathrm{KL} [q_{\phi}(\mathbf{z} | \mathbf{x}) || p(\mathbf{z}) ]$ the negative KL divergence between prior $p(\mathbf{z})$ and approximate posterior $q_{\phi}(\mathbf{z} | \mathbf{x})$. We first turn our attention to the KL divergence term.

KL Divergence

Intuitively, maximizing the negative KL divergence term encourages approximate posterior densities that place its mass on configurations of the latent variables which are closest to the prior. Effectively, this regularizes the complexity of latent space. Now, since both the prior $p(\mathbf{z})$ and approximate posterior $q_{\phi}(\mathbf{z} | \mathbf{x})$ are Gaussian, the KL divergence can actually be calculated with the closed-form expression,

$$ \mathrm{KL} [ q_{\phi}(\mathbf{z} | \mathbf{x}) || p(\mathbf{z}) ] = - \frac{1}{2} \sum_{k=1}^K \{ 1 + \log \sigma_k^2 - \mu_k^2 - \sigma_k^2 \} $$

where $\mu_k$ and $\sigma_k$ are the $k$-th components of output vectors $\mu_{\phi}(\mathbf{x})$ and $\sigma_{\phi}(\mathbf{x})$, respectively. This is not too difficult to derive, and I would recommend verifying this as an exercise. You can also find a derivation in the appendix of Kingma and Welling’s (2014) paper ¹.

Recall that earlier, we defined the expected log likelihood term of the ELBO as a Keras loss. We were able to do this since the log likelihood is a function of the network’s final output (the predicted probabilities), so it maps nicely to a Keras loss. Unfortunately, the same does not apply for the KL divergence term, which is a function of the network’s intermediate layer outputs, the mean mu and log variance log_var.

We define an auxiliary which takes mu and log_var as input and simply returns them as output without modification. We do however explicitly introduce the of calculating the KL divergence and adding it to a collection of losses, by calling the method add_loss ¹².

class KLDivergenceLayer(Layer):

 """ Identity transform layer that adds KL divergence
 to the final model loss.
 """

 def __init__(self, *args, **kwargs):
 self.is_placeholder = True
 super(KLDivergenceLayer, self).__init__(*args, **kwargs)

 def call(self, inputs):

 mu, log_var = inputs

 kl_batch = - .5 * K.sum(1 + log_var -
 K.square(mu) -
 K.exp(log_var), axis=-1)

 self.add_loss(K.mean(kl_batch), inputs=inputs)

 return inputs

Next we feed z_mu and z_log_var through this layer (this needs to take place before feeding z_log_var through the Lambda layer to recover z_sigma).

z_mu, z_log_var = KLDivergenceLayer()([z_mu, z_log_var])

Now when the Keras model is finally compiled, the collection of losses will be aggregated and added to the specified Keras loss function to form the loss we ultimately minimize. If we specify the loss as the negative log-likelihood we defined earlier (nll), we recover the negative ELBO as the final loss we minimize, as intended.

Side note: Alternative divergences

A key benefit of encapsulating the divergence in an auxiliary layer is that we can easily implement and swap in other divergences, such as the $\chi$-divergence or the $\alpha$-divergence. Using alternative divergences for variational inference is an active research topic ¹³ ¹⁴.

Side note: Implicit models and adversarial learning

Additionally, we could also extend the divergence layer to use an auxiliary density ratio estimator function, instead of evaluating the KL divergence in the analytical form above. This relaxes the requirement on approximate posterior $q_{\phi}(\mathbf{z}|\mathbf{x})$ (and incidentally also prior $p(\mathbf{z})$) to yield tractable densities, at the cost of maximizing a cruder estimate of the ELBO. This is known as Adversarial Variational Bayes¹⁰, and is an important line of recent research that, when taken to its logcal conclusion, can extend the applicability of variational inference to arbitrarily expressive implicit probabilistic models with intractable likelihoods¹¹.

Reparameterization using Merge Layers

To perform gradient-based optimization of ELBO with respect to model parameters $\theta$ and variational parameters $\phi$, we require its gradients with respect to these parameters, which is generally intractable. Currently, the dominant approach for circumventing this is by Monte Carlo (MC) estimation of the gradients. The basic idea is to write the gradient of the ELBO as an expectation of the gradient, approximate it with MC estimates, then perform stochastic gradient descent with the repeated MC gradient estimates.

There exist a number of estimators based on different variance reduction techniques. However, MC gradient estimates based on the reparameterization trick, known as the reparameterization gradients, have be shown to have the lowest variance among competing estimators for continuous latent variables⁵. The reparameterization trick is a straightforward change of variables that expresses the random variable $\mathbf{z} \sim q_{\phi}(\mathbf{z} | \mathbf{x})$ as a deterministic transformation $g_{\phi}$ of another random variable $\boldsymbol{\epsilon}$ and input $\mathbf{x}$, with parameters $\phi$,

$$ z = g_{\phi}(\mathbf{x}, \boldsymbol{\epsilon}), \quad \boldsymbol{\epsilon} \sim p(\boldsymbol{\epsilon}). $$

Note that $p(\boldsymbol{\epsilon})$ is simpler base distribution which is parameter-free and independent of $\mathbf{x}$ or $\phi$. To prevent clutter, we write the ELBO as an expectation of the function $f(\mathbf{x}, \mathbf{z}) = \log p_{\theta}(\mathbf{x} , \mathbf{z}) - \log q_{\phi}(\mathbf{z} | \mathbf{x})$ over distribution $q_{\phi}(\mathbf{z} | \mathbf{x})$. Now, for any function $f(\mathbf{x}, \mathbf{z})$, taking the gradient of the expectation with respect to $\phi$, and substituting all occurrences of $\mathbf{z}$ with $g_{\phi}(\mathbf{x}, \boldsymbol{\epsilon})$, we have

$$ \begin{align*} \nabla_{\phi} \mathbb{E}_{q_{\phi}(\mathbf{z} | \mathbf{x})} [ f(\mathbf{x}, \mathbf{z}) ] & = \nabla_{\phi} \mathbb{E}_{p(\boldsymbol{\epsilon})} [ f(\mathbf{x}, g_{\phi}(\mathbf{x}, \boldsymbol{\epsilon})) ] \newline & = \mathbb{E}_{p(\mathbf{\epsilon})} [ \nabla_{\phi} f(\mathbf{x}, g_{\phi}(\mathbf{x}, \boldsymbol{\epsilon})) ]. \end{align*} $$

In other words, this simple reparameterization allows the gradient and the expectation to commute, thereby allowing us to compute unbiased stochastic estimates of the ELBO gradients by drawing noise samples $\boldsymbol{\epsilon}$ from $p(\boldsymbol{\epsilon})$.

To recover the diagonal Gaussian approximation we specified earlier $q_{\phi}(\mathbf{z}_n | \mathbf{x}_n) = \mathcal{N}(\mathbf{z}_n | \boldsymbol{\mu}_{\phi}(\mathbf{x}_n), \mathrm{diag}(\boldsymbol{\sigma}_{\phi}^2(\mathbf{x}_n)))$, we draw noise from the Normal base distribution, and specify a simple location-scale transformation

$$ \mathbf{z} = g_{\phi}(\mathbf{x}, \boldsymbol{\epsilon}) = \mu_{\phi}(\mathbf{x}) + \sigma_{\phi}(\mathbf{x}) \odot \boldsymbol{\epsilon}, \quad \boldsymbol{\epsilon} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}), $$

where $\mu_{\phi}(\mathbf{x})$ and $\sigma_{\phi}(\mathbf{x})$ are the outputs of the inference network defined earlier with parameters $\phi$, and $\odot$ denotes the elementwise product. In Keras, we explicitly make the noise vector an input to the model by defining an Input layer for it. We then implement the above location-scale transformation using , namely Add and Multiply.

eps = Input(shape=(latent_dim,))

z_eps = Multiply()([z_sigma, eps])
z = Add()([z_mu, z_eps])

Side note: Monte Carlo sample size

Note both the inputs for observed variables and noise (x and eps) need to be specified explicitly as inputs to our final model. Furthermore, the size of their first dimension (i.e. batch size) are required to be the same. This corresponds to using a exactly one Monte Carlo sample to approximate the expected log likelihood, drawing a single sample $\mathbf{z}_n$ from $q_{\phi}(\mathbf{z}_n | \mathbf{x}_n)$ for each data-point $\mathbf{x}_n$ in the batch. Although you might find an MC sample size of 1 surprisingly small, it is actually adequate for a sufficiently large batch size (~100) ¹. In a , I demonstrate how to extend our approach to support larger MC sample sizes using just a few minor tweaks. This extension is crucial for implementing the importance weighted autoencoder ⁸.

Now, since the noise input is drawn from the Normal distribution, we can save from having to feed in values for this input from outside the computation graph by binding a tensor to this Input layer. Specifically, we bind a tensor created using K.random_normal with the required shape,

eps = Input(tensor=K.random_normal(shape=(K.shape(x)[0], latent_dim)))

While eps still needs to be explicitly specified as an input to compile the model, values for this input will no longer be expected by methods such as fit, predict. Instead, samples from this distribution will be lazily generated inside the computation graph when required. See my notes on for more details.

In the , all of this logic is encapsulated in a single Lambda layer, which simultaneously draws samples from a hard-coded base distribution and also performs the location-scale transformation. In contrast, this approach achieves a good level of and . By decoupling the random noise vector from the layer’s internal logic and explicitly making it a model input, we emphasize the fact that all sources of stochasticity emanate from this input. It thereby becomes clear that a random sample drawn from a particular approximating distribution is obtained by feeding this source of stochasticity through a number of successive deterministic transformations.

Side notes: Gumbel-softmax trick for discrete latent variables

As an example, we could provide samples drawn from the Uniform distribution as noise input. By applying a number of deterministic transformations that constitute the Gumbel-softmax reparameterization trick ⁹, we are able to obtain samples from the Categorical distribution. This allows us to perform approximate inference on discrete latent variables, and can be implemented in this framework by adding a dozen or so lines of code!

Putting it all together

So far, we’ve dissected the variational autoencoder into modular components and discussed the role and implementation of each one at some length. Now let’s compose these components together end-to-end to form the final autoencoder architecture.

x = Input(shape=(original_dim,))
h = Dense(intermediate_dim, activation='relu')(x)

z_mu = Dense(latent_dim)(h)
z_log_var = Dense(latent_dim)(h)

z_mu, z_log_var = KLDivergenceLayer()([z_mu, z_log_var])
z_sigma = Lambda(lambda t: K.exp(.5*t))(z_log_var)

eps = Input(tensor=K.random_normal(shape=(K.shape(x)[0], latent_dim)))
z_eps = Multiply()([z_sigma, eps])
z = Add()([z_mu, z_eps])

decoder = Sequential([
 Dense(intermediate_dim, input_dim=latent_dim, activation='relu'),
 Dense(original_dim, activation='sigmoid')
])

x_pred = decoder(z)

It’s surprisingly concise, taking up around 20 lines of code. The diagram of the full model architecture is visualized below.

Finally, we specify and compile the model, using the negative log likelihood nll defined earlier as the loss.

vae = Model(inputs=[x, eps], outputs=x_pred)
vae.compile(optimizer='rmsprop', loss=nll)

Model fitting

Dataset: MNIST digits

(x_train, y_train), (x_test, y_test) = mnist.load_data()
x_train = x_train.reshape(-1, original_dim) / 255.
x_test = x_test.reshape(-1, original_dim) / 255.

vae.fit(x_train,
 x_train,
 shuffle=True,
 epochs=epochs,
 batch_size=batch_size,
 validation_data=(x_test, x_test))

Loss (NELBO) convergence

pd.DataFrame(hist.history).plot(ax=ax)

Model evaluation

encoder = Model(x, z_mu)

# display a 2D plot of the digit classes in the latent space
z_test = encoder.predict(x_test, batch_size=batch_size)
plt.figure(figsize=(6, 6))
plt.scatter(z_test[:, 0], z_test[:, 1], c=y_test,
 alpha=.4, s=3**2, cmap='viridis')
plt.colorbar()
plt.show()

# display a 2D manifold of the digits
n = 15 # figure with 15x15 digits
digit_size = 28

# linearly spaced coordinates on the unit square were transformed
# through the inverse CDF (ppf) of the Gaussian to produce values
# of the latent variables z, since the prior of the latent space
# is Gaussian

z1 = norm.ppf(np.linspace(0.01, 0.99, n))
z2 = norm.ppf(np.linspace(0.01, 0.99, n))
z_grid = np.dstack(np.meshgrid(z1, z2))

x_pred_grid = decoder.predict(z_grid.reshape(n*n, latent_dim)) \
 .reshape(n, n, digit_size, digit_size)

plt.figure(figsize=(10, 10))
plt.imshow(np.block(list(map(list, x_pred_grid))), cmap='gray')
plt.show()

Recap

In this post, we covered the basics of amortized variational inference, looking at variational autoencoders as a specific example. In particular, we

Implemented the decoder and encoder using the and respectively.
Augmented the final loss with the KL divergence term by writing an auxiliary .
Worked with the log variance for numerical stability, and used a to transform it to the standard deviation when necessary.
Explicitly made the noise an Input layer, and implemented the reparameterization trick using .
, so random samples are generated within the computation graph.

What’s next

Next, we will extend the divergence layer to use an auxiliary density ratio estimator function, instead of evaluating the KL divergence in the analytical form above. This relaxes the requirement on approximate posterior $q_{\phi}(\mathbf{z}|\mathbf{x})$ (and incidentally also prior $p(\mathbf{z})$) to yield tractable densities, at the cost of maximizing a cruder estimate of the ELBO. This is known as Adversarial Variational Bayes¹⁰, and is an important line of recent research that, when taken to its logcal conclusion, can extend the applicability of variational inference to arbitrarily expressive implicit probabilistic models with intractable likelihoods¹¹.

Cite as:

@article{tiao2017vae,
 title = "{A} {T}utorial on {V}ariational {A}utoencoders with a {C}oncise {K}eras {I}mplementation",
 author = "Tiao, Louis C",
 journal = "tiao.io",
 year = "2017",
 url = "https://tiao.io/post/tutorial-on-variational-autoencoders-with-a-concise-keras-implementation/"
}

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Links & Resources

Below, you can find:

The used to generate the diagrams and plots in this post.
The above snippets combined in a single executable Python file:

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm

from keras import backend as K

from keras.layers import Input, Dense, Lambda, Layer, Add, Multiply
from keras.models import Model, Sequential
from keras.datasets import mnist


original_dim = 784
intermediate_dim = 256
latent_dim = 2
batch_size = 100
epochs = 50
epsilon_std = 1.0


def nll(y_true, y_pred):
 """ Negative log likelihood (Bernoulli). """

 # keras.losses.binary_crossentropy gives the mean
 # over the last axis. we require the sum
 return K.sum(K.binary_crossentropy(y_true, y_pred), axis=-1)


class KLDivergenceLayer(Layer):

 """ Identity transform layer that adds KL divergence
 to the final model loss.
 """

 def __init__(self, *args, **kwargs):
 self.is_placeholder = True
 super(KLDivergenceLayer, self).__init__(*args, **kwargs)

 def call(self, inputs):

 mu, log_var = inputs

 kl_batch = - .5 * K.sum(1 + log_var -
 K.square(mu) -
 K.exp(log_var), axis=-1)

 self.add_loss(K.mean(kl_batch), inputs=inputs)

 return inputs


decoder = Sequential([
 Dense(intermediate_dim, input_dim=latent_dim, activation='relu'),
 Dense(original_dim, activation='sigmoid')
])

x = Input(shape=(original_dim,))
h = Dense(intermediate_dim, activation='relu')(x)

z_mu = Dense(latent_dim)(h)
z_log_var = Dense(latent_dim)(h)

z_mu, z_log_var = KLDivergenceLayer()([z_mu, z_log_var])
z_sigma = Lambda(lambda t: K.exp(.5*t))(z_log_var)

eps = Input(tensor=K.random_normal(stddev=epsilon_std,
 shape=(K.shape(x)[0], latent_dim)))
z_eps = Multiply()([z_sigma, eps])
z = Add()([z_mu, z_eps])

x_pred = decoder(z)

vae = Model(inputs=[x, eps], outputs=x_pred)
vae.compile(optimizer='rmsprop', loss=nll)

# train the VAE on MNIST digits
(x_train, y_train), (x_test, y_test) = mnist.load_data()
x_train = x_train.reshape(-1, original_dim) / 255.
x_test = x_test.reshape(-1, original_dim) / 255.

vae.fit(x_train,
 x_train,
 shuffle=True,
 epochs=epochs,
 batch_size=batch_size,
 validation_data=(x_test, x_test))

encoder = Model(x, z_mu)

# display a 2D plot of the digit classes in the latent space
z_test = encoder.predict(x_test, batch_size=batch_size)
plt.figure(figsize=(6, 6))
plt.scatter(z_test[:, 0], z_test[:, 1], c=y_test,
 alpha=.4, s=3**2, cmap='viridis')
plt.colorbar()
plt.show()

# display a 2D manifold of the digits
n = 15 # figure with 15x15 digits
digit_size = 28

# linearly spaced coordinates on the unit square were transformed
# through the inverse CDF (ppf) of the Gaussian to produce values
# of the latent variables z, since the prior of the latent space
# is Gaussian
u_grid = np.dstack(np.meshgrid(np.linspace(0.05, 0.95, n),
 np.linspace(0.05, 0.95, n)))
z_grid = norm.ppf(u_grid)
x_decoded = decoder.predict(z_grid.reshape(n*n, 2))
x_decoded = x_decoded.reshape(n, n, digit_size, digit_size)

plt.figure(figsize=(10, 10))
plt.imshow(np.block(list(map(list, x_decoded))), cmap='gray')
plt.show()

D. P. Kingma and M. Welling, “Auto-Encoding Variational Bayes,” in Proceedings of the 2nd International Conference on Learning Representations (ICLR), 2014. ↩︎ ↩︎ ↩︎
↩︎
Section “Recognition models and amortised inference” in ↩︎
Dayan, P., Hinton, G. E., Neal, R. M., & Zemel, R. S. (1995). The Helmholtz machine. Neural Computation, 7(5), 889–904. ↩︎ ↩︎
Rezende, D. J., Mohamed, S., & Wierstra, D. (2014). “Stochastic backpropagation and approximate inference in deep generative models,” in Proceedings of The 31st International Conference on Machine Learning, 2014, (Vol. 32, pp. 1278–1286). Bejing, China: PMLR. ↩︎ ↩︎ ↩︎ ↩︎
For a complete treatment of variational autoencoders, and variational inference in general, I highly recommend:
- Jaan Altosaar’s blog post,
- Diederik P. Kingma’s PhD Thesis, .
↩︎
D. Rezende and S. Mohamed, “Variational Inference with Normalizing Flows,” in Proceedings of the 32nd International Conference on Machine Learning, 2015, vol. 37, pp. 1530–1538. ↩︎
Y. Burda, R. Grosse, and R. Salakhutdinov, “Importance Weighted Autoencoders,” in Proceedings of the 3rd International Conference on Learning Representations (ICLR), 2015. ↩︎ ↩︎
E. Jang, S. Gu, and B. Poole, “Categorical Reparameterization with Gumbel-Softmax,” Nov. 2016. in Proceedings of the 5th International Conference on Learning Representations (ICLR), 2017. ↩︎ ↩︎
L. Mescheder, S. Nowozin, and A. Geiger, “Adversarial Variational Bayes: Unifying Variational Autoencoders and Generative Adversarial Networks,” in Proceedings of the 34th International Conference on Machine Learning, 2017, vol. 70, pp. 2391–2400. ↩︎ ↩︎ ↩︎
D. Tran, R. Ranganath, and D. Blei, “Hierarchical Implicit Models and Likelihood-Free Variational Inference,” in Advances in Neural Information Processing Systems 30, 2017. ↩︎ ↩︎ ↩︎
To support sample weighting (fined-tuning how much each data-point contributes to the loss), Keras losses are expected returns a scalar for each data-point in the batch. In contrast, losses appended with the add_loss method don’t support this, and are expected to be a single scalar. Hence, we calculate the KL divergence for all data-points in the batch and take the mean before passing it to add_loss. ↩︎
Y. Li and R. E. Turner, “Rényi Divergence Variational Inference,” in Advances in Neural Information Processing Systems 29, 2016. ↩︎
A. B. Dieng, D. Tran, R. Ranganath, J. Paisley, and D. Blei, “Variational Inference via chi Upper Bound Minimization,” in Advances in Neural Information Processing Systems 30, 2017. ↩︎