Variational AutoEncoder: An Introduction and Recent Perspectives

Transcription

1 Metode de optimizare Riemanniene pentru învățare profundă Proiect cofinanțat din Fondul European de Dezvoltare Regională prin Programul Operațional Competitivitate Variational AutoEncoder: An Introduction and Recent Perspectives Luigi Malagò, Alexandra Peste, and Septimia Sarbu Romanian Institute of Science and Technology Politecnico di Milano 23 February 2018

2 Outline 1 Plan of the presentation 2 General View of Variational Autoencoders Introduction Research Directions 3 Work-in-progress Using Gaussian Graphical Models Geometry of the Latent Space 4 Future Work L.Malagò, A.Peşte (RIST) Variational Autoencoders 2/36

3 Plan of the presentation Overview of variational inference in deep learning: general algorithm, research directions and fast review of the existing literature Present some of the work we have done so far: the use of graphical models for introducing correlations between the latent variables analysis of the geometry of the latent space Present some ideas and questions that we have been thinking about, along with possible research directions L.Malagò, A.Peşte (RIST) Variational Autoencoders 3/36

4 Outline 1 Plan of the presentation 2 General View of Variational Autoencoders Introduction Research Directions 3 Work-in-progress Using Gaussian Graphical Models Geometry of the Latent Space 4 Future Work L.Malagò, A.Peşte (RIST) Variational Autoencoders 4/36

5 Generative Models in Deep Learning Real images Generated images [12] L.Malagò, A.Peşte (RIST) Variational Autoencoders 5/36

6 Notations for Bayesian Inference X, Z multivariate random variables, Z continuous, with probability density functions (pdf) p (x) andp (z) respectively,withparameters ; p (z) isthe prior and p (x) themarginal; p (x, z): pdf of the joint random variable (X, Z), with parameters ; p (x z), p (z x): pdfs of the random variables X Z = z and Z X = x; p (z x) istheposterior. L.Malagò, A.Peşte (RIST) Variational Autoencoders 6/36

7 General Setting Formulation of the problem: The continuous latent r.v. Z generates Z X, through f ( ) a di erentiable function, such that p (x z)p (z)dz is intractable. The goal is inference, i.e.,findingp (z x). Variational inference [1] approximates the true posterior p (z x) withq (z x), by minimizing the Kullback-Leibler divergence KL(q (z x) k p (z x)). z q (z x) f ( ) x Approach to the solution: maximizing a lower bound of the log likelihood. L.Malagò, A.Peşte (RIST) Variational Autoencoders 7/36

8 General Setting Formulation of the problem: The continuous latent r.v. Z generates Z X, through f ( ) a di erentiable function, such that p (x z)p (z)dz is intractable. The goal is inference, i.e.,findingp (z x). Variational inference [1] approximates the true posterior p (z x) withq (z x), by minimizing the Kullback-Leibler divergence KL(q (z x) k p (z x)). z q (z x) f ( ) x Approach to the solution: maximizing a lower bound of the log likelihood. L.Malagò, A.Peşte (RIST) Variational Autoencoders 7/36

9 General Setting Formulation of the problem: The continuous latent r.v. Z generates Z X, through f ( ) a di erentiable function, such that p (x z)p (z)dz is intractable. The goal is inference, i.e.,findingp (z x). Variational inference [1] approximates the true posterior p (z x) withq (z x), by minimizing the Kullback-Leibler divergence KL(q (z x) k p (z x)). z q (z x) f ( ) x Approach to the solution: maximizing a lower bound of the log likelihood. L.Malagò, A.Peşte (RIST) Variational Autoencoders 7/36

10 Variational Inference I Deriving the lower bound: Z ln p (x) =ln q (z x) p (x, z) q (z x) dz Evidence lower bound: L(, Z q (z x)ln p (x, z) dz (Jensen s inequality) q (z x) apple ; x) :=E q (z x) ln p (x, z) apple ln p (x) q (z x) Minimizing KL () maximizing the lower-bound: apple E q (z x) ln p (x, z) =lnp (x) KL(q (z x) k p (z x)) q (z x) The maximum of the lower-bound is the log-likelihood, and it is obtained when KL(q (z x) k p (z x)) = 0. Thus, the problems are equivalent. L.Malagò, A.Peşte (RIST) Variational Autoencoders 8/36

11 Variational Inference I Deriving the lower bound: Z ln p (x) =ln q (z x) p (x, z) q (z x) dz Evidence lower bound: L(, Z q (z x)ln p (x, z) dz (Jensen s inequality) q (z x) apple ; x) :=E q (z x) ln p (x, z) apple ln p (x) q (z x) Minimizing KL () maximizing the lower-bound: apple E q (z x) ln p (x, z) =lnp (x) KL(q (z x) k p (z x)) q (z x) The maximum of the lower-bound is the log-likelihood, and it is obtained when KL(q (z x) k p (z x)) = 0. Thus, the problems are equivalent. L.Malagò, A.Peşte (RIST) Variational Autoencoders 8/36

12 Variational Inference II Optimizing the lower bound maximizes the log likelihood. The distribution of X can be approximated with importance sampling: ln p (x) ln 1 S SX i=1 p (x z (i) )p (z (i) ) q (z (i) x) where z (i) q ( x). Fixing the family of distributions for the r.v., e.g. we assume they are Gaussians, we move from variational calculus to regular optimization of the parameters. The problem becomes: max E q (z x) [ln p (x, z), ln q (z x)] L.Malagò, A.Peşte (RIST) Variational Autoencoders 9/36

13 Variational Inference II Optimizing the lower bound maximizes the log likelihood. The distribution of X can be approximated with importance sampling: ln p (x) ln 1 S SX i=1 p (x z (i) )p (z (i) ) q (z (i) x) where z (i) q ( x). Fixing the family of distributions for the r.v., e.g. we assume they are Gaussians, we move from variational calculus to regular optimization of the parameters. The problem becomes: max E q (z x) [ln p (x, z), ln q (z x)] L.Malagò, A.Peşte (RIST) Variational Autoencoders 9/36

14 Variational Autoencoders Variational Autoencoders ([6], [11]) tackle the problem of variational inference in the context of neural networks. Theparameters and of q (z x) andp (x z) are learned through two di erent neural networks: encoder and decoder. sampling layer Encoder Network Decoder Network L.Malagò, A.Peşte (RIST) Variational Autoencoders 10 / 36

15 Applications Encode: learn a lower dimensional representation of the dataset, by sampling from q ( x). The dimension of the latent variable Z is assumed to be much smaller than the dimension of the dataset. Generate from noise examples that resemble the ones seen during training. The prior p (z) on the latent variable is assumed Gaussian N (0, I) and samples are fed through the network to output the conditional probabilities p (x z). L.Malagò, A.Peşte (RIST) Variational Autoencoders 11 / 36

16 Applications Encode: learn a lower dimensional representation of the dataset, by sampling from q ( x). The dimension of the latent variable Z is assumed to be much smaller than the dimension of the dataset. Generate from noise examples that resemble the ones seen during training. The prior p (z) on the latent variable is assumed Gaussian N (0, I) and samples are fed through the network to output the conditional probabilities p (x z). L.Malagò, A.Peşte (RIST) Variational Autoencoders 11 / 36

17 Details of the Algorithm Encoder: q (z x) - Gaussian N (µ, D) with diagonal covariance; - the set of parameters of the encoder Decoder: p (x z) - Gaussian with diagonal covariance (continuous data) or Bernoulli vector (discrete data); - the set of parameters of the decoder For a data point x, rewrite the lower bound L(, ; x) L(, ; x) = E q (z x) [ln p (x z)] KL(q (z x) p (z)) Reconstruction error Regularization Cost function to be optimized: 1 N NX L(, n=1 ; x n ), from dataset X = {x n } n=1,n L.Malagò, A.Peşte (RIST) Variational Autoencoders 12 / 36

18 Details of the Algorithm Encoder: q (z x) - Gaussian N (µ, D) with diagonal covariance; - the set of parameters of the encoder Decoder: p (x z) - Gaussian with diagonal covariance (continuous data) or Bernoulli vector (discrete data); - the set of parameters of the decoder For a data point x, rewrite the lower bound L(, ; x) L(, ; x) = E q (z x) [ln p (x z)] KL(q (z x) p (z)) Reconstruction error Regularization Cost function to be optimized: 1 N NX L(, n=1 ; x n ), from dataset X = {x n } n=1,n L.Malagò, A.Peşte (RIST) Variational Autoencoders 12 / 36

19 Backpropagating through Stochastic Layers Training neural networks requires computing the gradient of the cost function, using backpropagation Di culty when computing r E q (z x) [ln p (x z)] - Monte Carlo estimation of the gradient has high variance The reparameterization trick: find g ( ) di erentiable transformation and random variable with pdf p( ), such that Z = g ( ). E q (z x) [ln p (x z)] = E p( ) [ln p (x g ( ))] r E q (z x) [ln p (x z)] = E p( ) [r ln p (x g ( )] Example for X N(µ, ), with = LL T Cholesky decomposition: X = µ + L,with N(0, I). L.Malagò, A.Peşte (RIST) Variational Autoencoders 13 / 36

20 Backpropagating through Stochastic Layers Training neural networks requires computing the gradient of the cost function, using backpropagation Di culty when computing r E q (z x) [ln p (x z)] - Monte Carlo estimation of the gradient has high variance The reparameterization trick: find g ( ) di erentiable transformation and random variable with pdf p( ), such that Z = g ( ). E q (z x) [ln p (x z)] = E p( ) [ln p (x g ( ))] r E q (z x) [ln p (x z)] = E p( ) [r ln p (x g ( )] Example for X N(µ, ), with = LL T Cholesky decomposition: X = µ + L,with N(0, I). L.Malagò, A.Peşte (RIST) Variational Autoencoders 13 / 36

21 Limitations and Challenges Limitations The conditional independence assumption on the latent variables given the observations limits the expressive power of the approximate posterior Limitation on the number of active latent variables when using a hierarchy of stochastic layers [13] Challenges Di culty when training on text data: empiricalobservationthatthelearned latent representation is not meaningful [2] How to improve the quality of the generated samples, incaseofadatasetof images? How can we find a better correlation between the images generated and the maximization of the lower bound? How to estimate the tightness of the bound? L.Malagò, A.Peşte (RIST) Variational Autoencoders 14 / 36

22 Limitations and Challenges Limitations The conditional independence assumption on the latent variables given the observations limits the expressive power of the approximate posterior Limitation on the number of active latent variables when using a hierarchy of stochastic layers [13] Challenges Di culty when training on text data: empiricalobservationthatthelearned latent representation is not meaningful [2] How to improve the quality of the generated samples, incaseofadatasetof images? How can we find a better correlation between the images generated and the maximization of the lower bound? How to estimate the tightness of the bound? L.Malagò, A.Peşte (RIST) Variational Autoencoders 14 / 36

23 Research Directions More complex representations for q (z x), bytransformingasimple distribution through invertible di erentiable functions, as in [10] and [5] Increased complexity of the graphical models, e.g. a hierarchy of latent variables or auxiliary variables as in [13] and [9] Designing tighter bounds: importance weighting estimates of the log-likelihood [3] apple L K (, ; x) =E z 1,z 2,...z K q (z x) log 1 KX p (x, z k ) K q (z k x) minimizing di erent divergences (Renyi [8], -divergence [4]) Overcoming the challenge of training VAE on text data [2] k=1 L.Malagò, A.Peşte (RIST) Variational Autoencoders 15 / 36

24 Research Directions More complex representations for q (z x), bytransformingasimple distribution through invertible di erentiable functions, as in [10] and [5] Increased complexity of the graphical models, e.g. a hierarchy of latent variables or auxiliary variables as in [13] and [9] Designing tighter bounds: importance weighting estimates of the log-likelihood [3] apple L K (, ; x) =E z 1,z 2,...z K q (z x) log 1 KX p (x, z k ) K q (z k x) minimizing di erent divergences (Renyi [8], -divergence [4]) Overcoming the challenge of training VAE on text data [2] k=1 L.Malagò, A.Peşte (RIST) Variational Autoencoders 15 / 36

25 Research Directions More complex representations for q (z x), bytransformingasimple distribution through invertible di erentiable functions, as in [10] and [5] Increased complexity of the graphical models, e.g. a hierarchy of latent variables or auxiliary variables as in [13] and [9] Designing tighter bounds: importance weighting estimates of the log-likelihood [3] apple L K (, ; x) =E z 1,z 2,...z K q (z x) log 1 KX p (x, z k ) K q (z k x) minimizing di erent divergences (Renyi [8], -divergence [4]) Overcoming the challenge of training VAE on text data [2] k=1 L.Malagò, A.Peşte (RIST) Variational Autoencoders 15 / 36

26 Research Directions More complex representations for q (z x), bytransformingasimple distribution through invertible di erentiable functions, as in [10] and [5] Increased complexity of the graphical models, e.g. a hierarchy of latent variables or auxiliary variables as in [13] and [9] Designing tighter bounds: importance weighting estimates of the log-likelihood [3] apple L K (, ; x) =E z 1,z 2,...z K q (z x) log 1 KX p (x, z k ) K q (z k x) minimizing di erent divergences (Renyi [8], -divergence [4]) Overcoming the challenge of training VAE on text data [2] k=1 L.Malagò, A.Peşte (RIST) Variational Autoencoders 15 / 36

27 Outline 1 Plan of the presentation 2 General View of Variational Autoencoders Introduction Research Directions 3 Work-in-progress Using Gaussian Graphical Models Geometry of the Latent Space 4 Future Work L.Malagò, A.Peşte (RIST) Variational Autoencoders 16 / 36

28 Gaussian Graphical Models for VAE Gaussian Graphical Models [7] introduce correlations in the latent variables. Chain and 2D grid topologies =) sparse precision matrix P = 1,withthe number of non-zero components linear in the dimension of the latent variable The encoder network outputs the mean µ and the Cholesky factor L of the precision matrix. L will have a special sparse structure and will ensure the positive definiteness of. To sample from N (µ, ): solve linear system L T =, where N(0, I), and output z = µ +. Sampling from N (µ, ) and computing KL(N (µ, ) kn(0, I)) can be done in linear time =) introduce expressiveness without extra computational complexity. L.Malagò, A.Peşte (RIST) Variational Autoencoders 17 / 36

29 Gaussian Graphical Models for VAE Gaussian Graphical Models [7] introduce correlations in the latent variables. Chain and 2D grid topologies =) sparse precision matrix P = 1,withthe number of non-zero components linear in the dimension of the latent variable The encoder network outputs the mean µ and the Cholesky factor L of the precision matrix. L will have a special sparse structure and will ensure the positive definiteness of. To sample from N (µ, ): solve linear system L T =, where N(0, I), and output z = µ +. Sampling from N (µ, ) and computing KL(N (µ, ) kn(0, I)) can be done in linear time =) introduce expressiveness without extra computational complexity. L.Malagò, A.Peşte (RIST) Variational Autoencoders 17 / 36

30 Chain Topology z 1 z 2... z k Chain Model 0 P = k 1 k 1 C A The precision matrix P is tridiagonal; the Cholesky factor of such a matrix is lower-bidiagonal. L.Malagò, A.Peşte (RIST) Variational Autoencoders 18 / 36

31 Grid Topology z 1 z 2 z 3 z 1 z 2 z 3 z 4 z 5 z 6 z 4 z 5 z 6 z 7 z 8 z 9 z 7 z 8 z 9 Regular Grid Extended Grid We use in our experiments the extended grid, which corresponds to a block tridiagonal precision matrix P; we assume the Cholesky factor has a lower-block-bidiagonal structure. L.Malagò, A.Peşte (RIST) Variational Autoencoders 19 / 36

32 Motivation for Next Research Direction The purpose was to approximate the posterior with more complex distributions. Although the results show a slight improvement, they do not motivate the future use of these models. A more comprehensive analysis should be made to understand the geometry of the latent space. L.Malagò, A.Peşte (RIST) Variational Autoencoders 20 / 36

33 Analysis of the Representations in the Latent Space Experiments on MNIST dataset to understand the representation of the images in the learned latent space. Principal Components Analysis of the latent means will give us insights about which components are relevant for the representation. Claim: components with a low variation along the dataset are the ones not meaningful. PCA eigenvalues of the posterior samples are very close to 1 =) the KL minimization forces some components to be N (0, I). L.Malagò, A.Peşte (RIST) Variational Autoencoders 21 / 36

34 PCA Analysis 1/2 k = 20, RELU k = 30, RELU 1/5

35 PCA Analysis 2/2 k = 30, RELU k = 30, ELU 2/5

36 PCA Plots L.Malagò, A.Peşte (RIST) Variational Autoencoders 22 / 36

37 Interpretation of the Plot When training a VAE with latent size 20 on MNIST, only around 15 of the latent variables are relevant for the representation. The number remains constant when training with a larger latent size. This is a consequence of the KL regularization term in the cost function, which forces some components to be Gaussian noise. Is this number a particularity of the dataset? What is the impact on this number when using more complicated network architectures? Would we observe the same behavior with other bounds derived from di erent divergences (e.g. Rényi)? L.Malagò, A.Peşte (RIST) Variational Autoencoders 23 / 36

38 Interpretation of the Plot When training a VAE with latent size 20 on MNIST, only around 15 of the latent variables are relevant for the representation. The number remains constant when training with a larger latent size. This is a consequence of the KL regularization term in the cost function, which forces some components to be Gaussian noise. Is this number a particularity of the dataset? What is the impact on this number when using more complicated network architectures? Would we observe the same behavior with other bounds derived from di erent divergences (e.g. Rényi)? L.Malagò, A.Peşte (RIST) Variational Autoencoders 23 / 36

39 Correlations Plot L.Malagò, A.Peşte (RIST) Variational Autoencoders 24 / 36

40 Interpretation of the Plot With the previous plot we want to better understand the distribution of the latent means vector across the dataset to identify the inactive components. Distribution of (µ i,µ j ), samples corresponding to the points in the dataset. Inactive components are close to 0 and remain constant along the data set. L.Malagò, A.Peşte (RIST) Variational Autoencoders 25 / 36

41 Generated images Images generated by training VAE on MNIST, with the encoder and decoder feed-forward neural networks with two hidden layers: Samples from MNIST dataset Generated after 100 epochs Generated after 1000 epochs L.Malagò, A.Peşte (RIST) Variational Autoencoders 26 / 36

42 Linear Separability in the Latent Space VAE is trained on MNIST with 2 latent variables. The plot represents the means of the posterior for each point in the dataset, colored by corresponding class. Linear separability of the classes in the space of latent representations Sampling in the latent space from the empty regions =) images that are not digits Linear interpolation property =) continuous deformation in the latent space between two di erent images. L.Malagò, A.Peşte (RIST) Variational Autoencoders 27 / 36

43 Classification Performance 4/5

44 Outline 1 Plan of the presentation 2 General View of Variational Autoencoders Introduction Research Directions 3 Work-in-progress Using Gaussian Graphical Models Geometry of the Latent Space 4 Future Work L.Malagò, A.Peşte (RIST) Variational Autoencoders 28 / 36

45 Short-Term Linear separability of the dataset in the space of multi-dimensional latent representations Use skew distributions to model the posterior Study the behavior of the latent relevant components in the case of more complex posteriors, like the ones presented in [10] and [5] L.Malagò, A.Peşte (RIST) Variational Autoencoders 29 / 36

46 Medium-Term Bounds derived from di erent divergences (e.g. Rényi, -divergence) impact of the parameter on the tightness of the bounds relevant components in the latent space and see how their number changes Geometric methods for training VAE the use of natural gradient study the geometry of the latent space use Riemannian optimization methods that exploit some properties of the space of the latent variables Extend the study to di erent types of generative models, e.g. Generative Adversarial Networks (GANs), Restricted Boltzmann Machines (RBMs). L.Malagò, A.Peşte (RIST) Variational Autoencoders 30 / 36

47 References I [1] David M Blei, Alp Kucukelbir, and Jon D McAuli e. Variational inference: A review for statisticians. Journal of the American Statistical Association, [2] Samuel R Bowman, Luke Vilnis, Oriol Vinyals, Andrew M Dai, Rafal Jozefowicz, and Samy Bengio. Generating sentences from a continuous space. arxiv preprint arxiv: , [3] Yuri Burda, Roger Grosse, and Ruslan Salakhutdinov. Importance weighted autoencoders. arxiv preprint arxiv: , [4] José Miguel Hernández-Lobato, Yingzhen Li, Mark Rowland, Daniel Hernández-Lobato, Thang D Bui, and Richard E Turner. Black-box -divergence minimization L.Malagò, A.Peşte (RIST) Variational Autoencoders 31 / 36

48 References II [5] Diederik P Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improving variational autoencoders with inverse autoregressive flow. In Advances In Neural Information Processing Systems, pages , [6] Diederik P Kingma and Max Welling. Auto-encoding variational bayes [7] Ste en L Lauritzen. Graphical models, volume 17. Clarendon Press, [8] Yingzhen Li and Richard E Turner. Rényi divergence variational inference. In Advances in Neural Information Processing Systems, pages , L.Malagò, A.Peşte (RIST) Variational Autoencoders 32 / 36

49 References III [9] Lars Maaløe, Casper Kaae Sønderby, Søren Kaae Sønderby, and Ole Winther. Auxiliary deep generative models. arxiv preprint arxiv: , [10] Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. arxiv preprint arxiv: , [11] Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models [12] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in Neural Information Processing Systems, pages , L.Malagò, A.Peşte (RIST) Variational Autoencoders 33 / 36

50 References IV [13] Casper Kaae Sønderby, Tapani Raiko, Lars Maaløe, Søren Kaae Sønderby, and Ole Winther. Ladder variational autoencoders. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, editors, Advances in Neural Information Processing Systems 29, pages Curran Associates, Inc., L.Malagò, A.Peşte (RIST) Variational Autoencoders 34 / 36

51 Questions? L.Malagò, A.Peşte (RIST) Variational Autoencoders 35 / 36

52 Transylvanian Machine Learning Summer School July 2018, Cluj-Napoca, Romania 5/5

53 Transylvanian Machine Learning Summer School July 2018, Cluj-Napoca, Romania Registration Deadline: 30 March 2018 More details at 5/5

54 Thank You! L.Malagò, A.Peşte (RIST) Variational Autoencoders 36 / 36