Bayesian Reasoning and Deep Learning Shakir Mohamed

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1 Bayesian Reasoning and Deep Learning Shakir Mohamed DeepMind 9 October 2015

2 Abstract Deep learning and Bayesian machine learning are currently two of the most active areas of machine learning research. Deep learning provides a powerful class of models and an easy framework for learning that now provides state-ofthe-art methods for applications ranging from image classification to speech recognition. Bayesian reasoning provides a powerful approach for information integration, inference and decision making that has established it as the key tool for data-efficient learning, uncertainty quantification and robust model composition that is widely used in applications ranging from information retrieval to large-scale ranking. Each of these research areas has shortcomings that can be effectively addressed by the other, pointing towards a needed convergence of these two areas of machine learning; the complementary aspects of these two research areas is the focus of this talk. Using the tools of auto-encoders and latent variable models, we shall discuss some of the ways in which our machine learning practice is enhanced by combining deep learning with Bayesian reasoning. This is an essential, and ongoing, convergence that will only continue to accelerate and provides some of the most exciting prospects, some of which we shall discuss, for contemporary machine learning research. Bayesian Reasoning and Deep Learning 2

3 Deep Learning Bayesian Reasoning Better ML Bayesian Reasoning and Deep Learning 3

4 Deep Learning A framework for constructing flexible models + Rich non-linear models for classification and sequence prediction. + Scalable learning using stochastic approximations and conceptually simple. + Easily composable with other gradientbased methods - Only point estimates - Hard to score models, do model selection and complexity penalisation. Bayesian Reasoning and Deep Learning 4

Bayesian Reasoning A framework for inference and decision making + Unified framework for model building, inference, prediction and decision making + Explicit accounting for uncertainty and

5 Bayesian Reasoning A framework for inference and decision making + Unified framework for model building, inference, prediction and decision making + Explicit accounting for uncertainty and variability of outcomes + Robust to overfitting; tools for model selection and composition. - Mainly conjugate and linear models - Potentially intractable inference leading to expensive computation or long simulation times. Bayesian Reasoning and Deep Learning 5

6 Two Streams of Machine Learning + Rich non-linear models for classification and sequence prediction. + Scalable learning using stochastic approximation and conceptually simple. + Easily composable with other gradient-based methods - Only point estimates Deep Learning - Hard to score models, do selection and complexity penalisation. Bayesian Reasoning - Mainly conjugate and linear models - Potentially intractable inference, computationally expensive or long simulation time. + Unified framework for model building, inference, prediction and decision making + Explicit accounting for uncertainty and variability of outcomes + Robust to overfitting; tools for model selection and composition. Bayesian Reasoning and Deep Learning 6

7 Outline Bayesian Reasoning + Deep Learning Complementary strengths that we should expect to be successfully combined. 1 2 Why is this a good idea? Review of deep learning Limitations of maximum likelihood and MAP estimation How can we achieve this convergence? Case study using auto-encoders and latent variable models Approximate Bayesian inference 3 What else can we do? Semi-supervised learning, classification, better inference and more. Bayesian Reasoning and Deep Learning 7

A (Statistical) Review of Deep Learning Generalised Linear Regression = w > x + b p(y x) =p(y g( ); ) The basic function can be any linear function, e.g., affine, convolution. g(.) is an inverse link function that we ll refer to as an activation function.

8 A (Statistical) Review of Deep Learning Generalised Linear Regression = w > x + b p(y x) =p(y g( ); ) The basic function can be any linear function, e.g., affine, convolution. g(.) is an inverse link function that we ll refer to as an activation function. generalised regression. Target Regression Link Inv link Activation Real Linear Identity Identity Binary Logistic Logit log µ 1-µ Sigmoid 1 Sigmoid 1+exp(- ) Binary Probit Inv Gauss Gauss CDF Probit CDF -1 (µ) ( ) Binary Gumbel Compl. Gumbel CDF log-log log(-log(µ)) e -e-x Binary Logistic Hyperbolic Tangent tanh( ) Tanh Categorical Multinomial Multin. Logit Softmax P i j j Counts Poisson log(µ) p exp( ) Counts Poisson (µ) 2 Non-neg. Gamma Reciprocal 1 1 µ Sparse Tobit max max(0; ) ReLU Ordered Ordinal Cum. Logit ( k - ) Maximum likelihood estimation Optimise the negative log-likelihood L = log p(y g( ); ) Bayesian Reasoning and Deep Learning 8

A (Statistical) Review of Deep Learning Recursive Generalised Linear Regression Recursively compose the basic linear functions. Gives a deep neural network. E[y] =h L.

9 A (Statistical) Review of Deep Learning Recursive Generalised Linear Regression Recursively compose the basic linear functions. Gives a deep neural network. E[y] =h L... h l h 0 (x) A general framework for building non-linear, parametric models Problem: Overfitting of MLE leading to limited generalisation. Bayesian Reasoning and Deep Learning 9

10 A (Statistical) Review of Deep Learning Regularisation Strategies for Deep Networks Regularisation is essential to overcome the limitations of maximum likelihood estimation. Regularisation, penalised regression, shrinkage. A wide range of available regularisation techniques: Large data sets Input noise/jittering and data augmentation/expansion. L2 /L1 regularisation (Weight decay, Gaussian prior) Binary or Gaussian Dropout Batch normalisation More robust loss function using MAP estimation instead. Bayesian Reasoning and Deep Learning 10

Sensitivities affect gradients and can make learning hard Invariant MAP estimators and exploiting natural gradients, trust

11 More Robust Learning MAP estimators and limitations Power of MAP estimators is that they provide some robustness to overfitting. Creates sensitivities to parameterisation. 1. Sensitivities affect gradients and can make learning hard Invariant MAP estimators and exploiting natural gradients, trust region methods and other improved optimisation. 2. Still no way to measure confidence of our model. Can generate frequentist confidence intervals and bootstrap estimates. Bayesian Reasoning and Deep Learning 11

12 Towards Bayesian Reasoning Proposed solutions have not fully dealt with the underlying issues. Issues arise as a consequence of: Reasoning only about the most likely solution and Not maintaining knowledge of the underlying variability (and averaging over this). Given this powerful model class and invaluable tools for regularisation and optimisation, let us develop a Pragmatic Bayesian Approach for Probabilistic Reasoning in Deep Networks. Bayesian reasoning over some, but not all parts of our models (yet). Bayesian Reasoning and Deep Learning 12

13 Outline Bayesian Reasoning + Deep Learning Complementary strengths that we should expect to be successfully combined. 1 Why is this a good idea? Review of deep learning Limitations of maximum likelihood and MAP estimation 2 How can we achieve this convergence? Case study using auto-encoders and latent variable models Approximate Bayesian inference 3 What else can we do? Semi-supervised learning, classification, better inference and more. Bayesian Reasoning and Deep Learning 13

14 Dimensionality Reduction and Auto-encoders Unsupervised learning and auto-encoders A generic tool for dimensionality reduction and feature extraction. Minimise reconstruction error using an encoder and a decoder. + Non-linear dimensionality reduction using deep networks for encoder and decoder. + Easy to implement as a single computational graph and train using SGD z Decoder g(.) y* = g(z) z = f(y) Encoder f(.) Data y - No natural handling of missing data L = log p(y g(z)) - No representation of variability of the representation space. L = ky g(f(y))k 2 2 Bayesian Reasoning and Deep Learning 14

15 Dimensionality Reduction and Auto-encoders z z = f(y) Some questions about auto-encoders: What is the model we are interested in? Why use an encoder? How do we regularise? Decoder g(.) Encoder f(.) y* = g(z) Data y Best to be explicit about the: Probabilistic model of interest and Mechanism we use for inference. Bayesian Reasoning and Deep Learning 15

16 Density Estimation and Latent Variable Models Latent variable models: Generic and flexible model class for density estimation. Specifies a generative process that gives rise to the data. Latent Gaussian Models: Probabilistic PCA, Factor analysis (FA), Bayesian Exponential Family PCA (BXPCA). BXPCA Latent Variable z N (z µ, ) Observation Model μ z Σ = Wz + b W y Expon(y ) Exponential fam natural parameters η. y n = 1,, N Use our knowledge of deep learning to design even richer models. Bayesian Reasoning and Deep Learning 16

17 Deep Generative Models Rich extension of previous model using deep neural networks. E.g., non-linear factor analysis, non-linear Gaussian belief networks, deep latent Gaussian models (DLGM). μ z 2 Σ DLGM Latent Variables (Stochastic layers) z l N (z l f l (z l+1 ), l ) f l (z) = (Wh(z)+b) h 4 h 3 W 1 Deterministic layers z 1 h i (x) = (Ax + c) h 2 Observation Model = Wh 1 + b h 1 W y Expon(y ) y Can also use non-exponential family. n = 1,, N Bayesian Reasoning and Deep Learning 17

18 Deep Latent Gaussian Models 1. Explain this data Our inferential tasks are: p(z y, W) / p(y z, W)p(z) μ Σ z 1 p(y y) = 2. Make predictions: Z p(y z, W)p(z y, W)dz h 2 h 1 W 3. Choose the best model Z p(y W) = p(y z, W)p(z)dz y n = 1,, N Bayesian Reasoning and Deep Learning 18

19 Variational Inference Use tools from approximate inference to handle intractable integrals. True posterior KL[q(z y)kp(z y)] Approximation class Reconstruction cost: Expected log-likelihood measures how well samples from q(z) are able to explain the data y. q (z) Penalty: Explanation of the data q(z) doesn t deviate too far from your beliefs p(z) - Okham s razor. Reconstruction F(y, q) =E q(z) [log p(y z)] Penalty KL[q(z)kp(z)] Penalty is derived from your model and does not need to be designed. Bayesian Reasoning and Deep Learning 19

20 Amortised Variational Inference z ~ q(z y) F(y, q) =E q(z) [log p(y z)] KL[q(z)kp(z)] Approx. Posterior Reconstruction Penalty Approximate posterior distribution q(z): Best match to true posterior p(z y), one of the unknown inferential quantities of interest to us. Inference/ Encoder q(z y) Inference network: q is an encoder or inverse model. Parameters of q are now a set of global parameters used for inference of all data points - test and train. Amortise (spread) the cost of inference over all data. Data y Encoders provide an efficient mechanism for amortised posterior inference Bayesian Reasoning and Deep Learning 20

21 Auto-encoders and Inference in DGMs F(y, q) =E q(z) [log p(y z)] KL[q(z)kp(z)] z z ~ q(z y) Approx. Posterior Reconstruction Penalty Model (Decoder): likelihood p(y z). Inference (Encoder): variational distribution q(z y) Model p(y z) Inference Network q(z y) Stochastic encoder-decoder systems implement variational inference. y ~ p(y z) Data y Specific combination of variational inference in latent variable models using inference networks Variational Auto-encoder But don t forget what your model is, and what inference you use. Bayesian Reasoning and Deep Learning 21

22 What Have we Gained + Transformed an auto-encoders into more interesting deep generative models. + Rich new class of density estimators built with non-linear models. + Used a principled approach for deriving loss functions that automatically include appropriate penalty functions. + Explained how an encoder enters into our models and why this is a good idea. + Able to answer all our desired inferential questions. + Knowledge of the uncertainty associated with our latent variables. F(y, q) =E q(z) [log p(y z)] z Model p(y z) y ~ p(y z) KL[q(z)kp(z)] z ~ q(z y) Inference Network q(z y) Data y Bayesian Reasoning and Deep Learning 22

23 What Have we Gained F(y, q) =E q(z) [log p(y z)] KL[q(z)kp(z)] + Able to score our models and do model selection using the free energy. z Model p(y z) y ~ p(y z) z ~ q(z y) Inference Network q(z y) Data y + Can impute missing data under any missingness assumption + Can still combine with natural gradient and improved optimisation tools. + Easy implementation - have a single computational graph and simple Monte Carlo gradient estimators. + Computational complexity the same as any large-scale deep learning system. A true marriage of Bayesian Reasoning and Deep Learning Bayesian Reasoning and Deep Learning 23

24 Data Visualisation MNIST Handwritten digits x28 DLGM Samples from 2D latent model Labels in 2D latent space Bayesian Reasoning and Deep Learning 24

25 Visualising MNIST in 3D DLGM Bayesian Reasoning and Deep Learning 25

26 Data Simulation DLGM Data Samples Bayesian Reasoning and Deep Learning 26

27 Missing Data Imputation Original Data unobserved pixels Inferred Image 10% observed DLGM 50% observed Bayesian Reasoning and Deep Learning 27

28 Outline Bayesian Reasoning + Deep Learning Complementary strengths that we should expect to be successfully combined. 1 Why is this a good idea? Review of deep learning Limitations of maximum likelihood and MAP estimation 2 How can we achieve this convergence? Auto-encoders and latent variable models Approximate and variational inference 3 What else can we do? Semi-supervised learning, recurrent networks, classification, better inference and more. Bayesian Reasoning and Deep Learning 28

29 Semi-supervised Learning Can extend the marriage of Bayesian reasoning and deep learning to the problem of semi-supervised classification. Semi-supervised DLGM π y μ x z Σ W n = 1,, N Bayesian Reasoning and Deep Learning 29

30 Analogical Reasoning Semi-supervised DLGM Bayesian Reasoning and Deep Learning 30

Generative Models with Attention Figure 7. MNIST generation sequences for DRAW without atwe can also combine other tools from deep learning to design tention.

Generated MNIST images with two digits. attention it constructs the digit by tracing the lines nt Neural Network For with Image Generation much like a person with a pen. DRAW ts scenes d by the.

MNIST Generation with Two Digits dec decoder P (x z1:t ) decoder ht motivation The main for using 1 RNN RNNan attention-based generative model is that large images can be built up iteratively, z zt+1

To test (generative model) sample this capability sample sample in a controlled fashion, we trained DRAW encoding two 28 x, 28 zmnist images choq(z x) to generate Q(ztimages x, z1:t with Q(z 1) t+1

31 Generative Models with Attention Figure 7. MNIST generation sequences for DRAW without atwe can also combine other tools from deep learning to design tention. Notice how the network first generates a very blurry imthat is subsequently refined.generative models: recurrent networks even age more powerful and attention. Figure 8. Generated MNIST images with two digits. attention it constructs the digit by tracing the lines nt Neural Network For with Image Generation much like a person with a pen. DRAW ts scenes d by the. y step is ne while ew years by a seby a sin& Hinton, to, 2014; 014; Serequential h can be s such as model in possible nse it re- P (x z) decoder FNN ct 1 write ct write... ct 4.3. MNIST Generation with Two Digits dec decoder P (x z1:t ) decoder ht motivation The main for using 1 RNN RNNan attention-based generative model is that large images can be built up iteratively, z zt+1 zt decoding by adding to a small part of the image at a time. To test (generative model) sample this capability sample sample in a controlled fashion, we trained DRAW encoding two 28 x, 28 zmnist images choq(z x) to generate Q(ztimages x, z1:t with Q(z 1) t+1 1:t ) (inference) sen at random and placed at random locations in a encoderin casesencoder black background. where the two digits overlap, henc t 1 RNN RNNtogether at each point and encoder the pixel intensities were added FNN clipped to be noread greater thanread one. Examples of generated data are shown in Fig. 8. The network typically generates x the other, suggesting x x one digit and then an ability to recreate composite scenes from simple pieces. Figure 2. Left: Conventional Auto-Encoder. Dur4.4. Street View House Variational Number Generation ing generation, a sample z is drawn from a prior P (z) and passedfigure 9. Generated SVHN images. The rightmost column MNIST digits are very simplistic in terms of visual strucshows the training images closest (in L2 distance) to the generthrough the feedforward decoder network to compute the probature, and we were keen to see how well DRAW performed ated images beside them. Note that the two columns are visually bility of the input P (x z) given the sample. During inference the on natural images. Our first natural image generation exsimilar, but the numbers are generally different. input x is periment passed to thetheencoder network, producing an approxused multi-digit Street View House Numbers datasetq(z x) (Netzer etover al., 2011). used the same preprocessimate posterior latentwe variables. During training, z ing as (Goodfellow et al., 2013), a Bayesian Reasoning and Deep Learning is sampled from Q(z x) and then usedyielding to compute thehouse total de-highly realistic, as shown in Figs. 9 and 10. Fig. 11 reveals

32 Uncertainty on Model Parameters We can also combine other tools from deep learning to design even more powerful generative models: recurrent networks and attention. Bayesian Neural Networks x h 1 y n = 1,, N W 1 h 2 W 2 W 3 Y H 1 H 2 H 3 1 X 1 Bayesian Reasoning and Deep Learning 32

33 In Review Deep learning as a framework for building highly flexible non-linear parametric models, but regularisation and accounting for uncertainty and lack of knowledge is still needed. Bayesian reasoning as a general framework for inference that allows us to account for uncertainty and a principled approach for regularisation and model scoring. z z ~ q(z y) Combined Bayesian reasoning with auto-encoders and showed just how much can be gained by a marriage of these two streams of machine learning research. Model p(y z) y ~ p(y z) Inference Network q(z y) Data y Bayesian Reasoning and Deep Learning 33

34 Thanks to many people: Danilo Rezende, Ivo Danihelka, Karol Gregor, Charles Blundell, Theophane Weber, Andriy Mnih, Daan Wierstra (Google DeepMind), Durk Kingma, Max Welling (U. Amsterdam) Thank You. Bayesian Reasoning and Deep Learning 34

35 Some References Probabilistic Deep Learning Rezende, Danilo Jimenez, Shakir Mohamed, and Daan Wierstra. "Stochastic backpropagation and approximate inference in deep generative models." ICML (2014). Kingma, Diederik P., and Max Welling. "Auto-encoding variational Bayes." ICLR Mnih, Andriy, and Karol Gregor. "Neural variational inference and learning in belief networks." ICML (2014). Gregor, Karol, et al. "Deep autoregressive networks." ICML (2014). Kingma, D. P., Mohamed, S., Rezende, D. J., & Welling, M. (2014). Semi-supervised learning with deep generative models. NIPS (pp ). Gregor, K., Danihelka, I., Graves, A., & Wierstra, D. (2015). DRAW: A recurrent neural network for image generation. arxiv preprint arxiv: Rezende, D. J., & Mohamed, S. (2015). Variational Inference with Normalizing Flows. arxiv preprint arxiv: Blundell, C., Cornebise, J., Kavukcuoglu, K., & Wierstra, D. (2015). Weight Uncertainty in Neural Networks. arxiv preprint arxiv: Hernández-Lobato, J. M., & Adams, R. P. (2015). Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks. arxiv preprint arxiv: Gal, Y., & Ghahramani, Z. (2015). Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. arxiv preprint arxiv: Bayesian Reasoning and Deep Learning 35

36 What is a Variational Method? Variational Principle General family of methods for approximating complicated densities by a simpler class of densities. KL[q(z y)kp(z y)] Approximation class True posterior q (z) Deterministic approximation procedures with bounds on probabilities of interest. Fit the variational parameters. Bayesian Reasoning and Deep Learning 36

37 From IS to Variational Inference Integral problem Proposal Importance Weight Jensen s inequality Z log Z p(x)g(x)dx p(x) log g(x)dx log p(y) = log log p(y) = log Z log p(y) = log p(y z) p(z) q(z) q(z)dz Z log p(y) q(z) log p(y z) p(z) dz q(z) = Z Z Z q(z) log p(y z) p(y z)p(z)dz p(y z)p(z) q(z) q(z) dz Z q(z) log q(z) p(z) Variational lower bound = E q(z) [log p(y z)] KL[q(z)kp(z)] Bayesian Reasoning and Deep Learning 37

38 Minimum Description Length (MDL) F(y, q) =E q(z) [log p(y z)] KL[q(z)kp(z)] Stochastic encoder Data code-length Hypothesis code Stochastic encoder-decoder systems implement variational inference. Regularity in our data that can be explained with latent variables, implies that the data is compressible. MDL: inference seen as a problem of compression we must find the ideal shortest message of our data y: marginal likelihood. Must introduce an approximation to the ideal message. Encoder: variational distribution q(z y), Decoder: likelihood p(y z). z Decoder p(y z) y ~ p(y z) z ~ q(z y) Encoder q(z y) Data y Bayesian Reasoning and Deep Learning 38

39 Denoising Auto-encoders (DAE) F(y, q) =E q(z) [log p(y z)] (z,y) Stochastic encoder Reconstruction Penalty Stochastic encoder-decoder systems implement variational inference. DAE: A mechanism for finding representations or features of data (i.e. latent variable explanations). Encoder: variational distribution q(z y), Decoder: likelihood p(y z). z Decoder p(y z) z ~ q(z y) Encoder q(z y) The variational approach requires you to be explicit about your assumptions. Penalty is derived from your model and does not need to be designed. y ~ p(y z) Data y Bayesian Reasoning and Deep Learning 39

40 Amortising the Cost of Inference Repeat: E-step For i = 1, N n /r E q (z) [log p (y n z n )] r KL[q(z n )kp(z n )] Instead of solving this optimisation for every data point n, we can instead use a model. M-step / 1 N X r log p (y n z n ) n z Model p(y z) y ~ p(y z) z ~ q(z y) Inference Network q(z y) Data y Inference network: q is an encoder or inverse model. Parameters of q are now a set of global parameters used for inference of all data points - test and train. Share the cost of inference (amortise) over all data. Combines easily with mini-batches and Monte Carlo expectations. Can jointly optimise variational and model parameters: no need for alternating optimisation. Bayesian Reasoning and Deep Learning 40

41 Implementing your Variational Algorithm Avoid deriving pages of gradient updates for variational inference. Variational inference turns integration into optimisation: Automated Tools: Differentiation: Theano, Torch7, Stan Message passing: infer.net Stochastic gradient descent and other preconditioned optimisation. Same code can run on both GPUs or on distributed clusters. Probabilistic models are modular, can easily be combined. E q [( log p(y z) + log q(z) log p(z)] Prior p(z) log p(z) Model p(x z) log p(x z) Forward pass z H[q(z)] Inference q(z x) Data x Backward pass Prior p(z) r Model p(x z) Inference q(z x) Ideally want probabilistic programming using variational inference. r r Bayesian Reasoning and Deep Learning 41

42 Stochastic Backpropagation A Monte Carlo method that works with continuous latent variables. Original problem r E q(z) [f(z)] Reparameterisation z N (µ, 2 ) z = µ + N (0, 1) Backpropagation with Monte Carlo r E N (0,1) [f(µ + )] E N (0,1) [r ={µ, } f(µ + )] Can use any likelihood function, avoids the need for additional lower bounds. Low-variance, unbiased estimator of the gradient. Can use just one sample from the base distribution. Possible for many distributions with location-scale or other known transformations, such as the CDF. Bayesian Reasoning and Deep Learning 42

43 Monte Carlo Control Variate Estimators More general Monte Carlo approach that can be used with both discrete or continuous latent variables. Property of the score function: r log q (z x) = r q (z x) q (z x) Original problem Score ratio r E q (z) [log p (y z)] E q (z) [log p (y z)r log q(z y)] MCCV Estimate E q (z) [(log p (y z) c)r log q(z y)] c is known as a control variate and is used to control the variance of the estimator. Bayesian Reasoning and Deep Learning 43

Lecture 1: Machine Learning Basics

1/69 Lecture 1: Machine Learning Basics Ali Harakeh University of Waterloo WAVE Lab ali.harakeh@uwaterloo.ca May 1, 2017 2/69 Overview 1 Learning Algorithms 2 Capacity, Overfitting, and Underfitting 3