DOCSLIB.ORG

Online Optimization with Energy Based Models

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Online Optimization with Energy Based Models Online Optimization with Energy Based Models by Yilun Du B.S., Massachusetts Institute of Technology (2019) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Science in Computer Science and Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 ○c Massachusetts Institute of Technology 2020. All rights reserved. Author................................................................ Department of Electrical Engineering and Computer Science May 15, 2020 Certified by. Leslie P. Kaelbling Professor Department of Electrical Engineering and Computer Science Thesis Supervisor Certified by. Tomas Lozano-Perez Professor Department of Electrical Engineering and Computer Science Thesis Supervisor Certified by. Joshua B. Tenenbaum Professor Department of Brain and Cognitive Science Thesis Supervisor Accepted by . Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students 2 Online Optimization with Energy Based Models by Yilun Du Submitted to the Department of Electrical Engineering and Computer Science on May 15, 2020, in partial fulfillment of the requirements for the degree of Master of Science in Computer Science and Engineering Abstract This thesis examines the power of applying optimization on learned neural networks, referred to as Energy Based Models (EBMs). We first present methods that enable scalable training of EBMs, allowing an optimization procedure to generate high resolu- tion images. We simultaneously show that resultant models are robust, compositional, and are further easy to learn online. Next we showcase how this optimization pro- cedure can also be used to formulate plans in interactive environments. We further showcase how a similar procedure can be used to learn neural energy functions for proteins, enabling structural recovery through optimization. Finally, we show that by defining generation as a optimization procedure, we can combine generative models from different domains together, and apply optimization on the joint model. Weshow that this allows us to apply various logical operations on images generation, as well as learn to generate new concepts in a continual manner. Thesis Supervisor: Leslie P. Kaelbling Title: Professor Department of Electrical Engineering and Computer Science Thesis Supervisor: Tomas Lozano-Perez Title: Professor Department of Electrical Engineering and Computer Science Thesis Supervisor: Joshua B. Tenenbaum Title: Professor Department of Brain and Cognitive Science 3 4 Acknowledgments I would like to thank Leslie, Tomas, and Josh for helping me guide through the master degree process, giving insightful advice and feedback on my ideas and helping me mature as a researcher. Much of the work done in this thesis was done with Igor, who I am also thankful for both suggesting invaluable advice and guiding my development as a researcher. I am also greatly thankful for all the collaborators that helped me through this work. Finally, I would like to thank my lab-mates for providing great conversations and insights on each of the research topics I have worked on. 5 6 Contents 1 Introduction 17 2 Related Work 19 3 Learning Energy Models 21 3.1 Energy Based Models........................... 21 3.1.1 Sample Replay Buffer....................... 22 3.1.2 Regularization........................... 23 3.2 Image Generation............................. 25 3.2.1 Mode Evaluation......................... 26 3.3 Generalization............................... 27 3.3.1 Adversarial Robustness...................... 27 3.3.2 Out-of-Distribution Generalization............... 28 3.4 Online Learning.............................. 29 4 Energy Based Models for Planning 31 4.1 Planning through Online Optimization................. 31 4.1.1 Energy-Based Models and Terminology............. 31 4.1.2 Planning with Energy-Based Models.............. 32 4.1.3 Online Learning with Energy-Based Models.......... 34 4.1.4 Related Work........................... 36 4.2 Experiments................................ 37 4.2.1 Setup............................... 37 7 4.2.2 Online Model Learning...................... 38 4.2.3 Maximum Entropy Inference................... 41 4.2.4 Exploration............................ 42 5 Energy Models for Protein Structure 45 5.1 Background................................ 45 5.2 Method.................................. 46 5.2.1 Parameterization of protein conformations........... 48 5.2.2 Usage as an energy function................... 48 5.2.3 Training and loss functions.................... 48 5.2.4 Recovery of Rotamers...................... 49 5.3 Evaluation................................. 49 5.3.1 Datasets.............................. 49 5.3.2 Baselines.............................. 50 5.3.3 Evaluation............................. 50 5.3.4 Rotamer recovery results..................... 51 5.3.5 Visualizing Energies....................... 52 6 Compositionality with Energy Based Models 57 6.1 Method.................................. 57 6.1.1 Energy Based Models....................... 57 6.1.2 Logical Operators through Online Optimization........ 58 6.2 Experiments................................ 61 6.2.1 Setup............................... 61 6.2.2 Compositional Generation.................... 62 6.2.3 Continual Learning........................ 65 7 Conclusion 67 7.1 Future Directions............................. 67 8 List of Figures 1-1 A 2D example of combining EBMs through summation and the resulting sampling trajectories. ............................ 17 3-1 Table of Inception and FID scores for ImageNet32x32 and CIFAR-10. Quan- titative numbers for ImageNet32x32 from [Ostrovski et al., 2018]. (*) We use Inception Score (from original OpenAI repo) to compare with legacy models, but strongly encourage future work to compare soley with FID score, since Langevin Dynamics converges to minima that artificially inflate Inception Score. (**) conditional EBM models for 128x128 are smaller than those in SNGAN. .................................. 24 3-2 EBM image restoration on images in the test set via MCMC (online op- timization). The right column shows failure (approx. 10% objects change with ground truth initialization and 30% of objects change in salt/pepper corruption or in-painting. Bottom two rows shows worst case of change.) . 24 3-3 Comparison of image generation techniques on unconditional CIFAR-10 dataset. 24 3-4 Illustration of cross-class implicit sampling on a conditional EBM. The EBM is conditioned on a particular class but is initialized with an image from a separate class. ............................... 25 3-5 Illustration of image completions on conditional ImageNet model. Our models exhibit diversity in inpainting. ....................... 25 3-6 휖 plots under L1 and L2 attacks of conditional EBMs as compared to PGD trained models in [Madry et al., 2017] and a baseline Wide ResNet18. ... 27 9 3-7 Histogram of relative likelihoods for various datasets for Glow, PixelCNN++ and EBM models .............................. 28 4-1 Overview of online training procedure with a EBM where grey areas represent inadmissible regions. Plans from the current observation to goal state is inferred from the EBM (left). A particular plan is chosen and executed until a planned state deviates significantly from an actual state (middle). The EBM is then trained on all real transitions 휏real and all planned transitions before the deviation (in green) 휏휃, while transitions afterwards (red) are ignored. A new plan is then generated from the new location of the agent (right) ................................... 35 4-2 Illustrations of the 4 evaluated environments. Both particle and maze environ- ments have 2 degree of freedom for x, y movement. The Reacher environment has 2 degrees of freedom corresponding to torques to two motors. The Sawyer Arm environment has 7 degrees of freedoms corresponding to torques. ... 37 4-5 Qualitative image showing EBM successfully navigating finger end effector to goal position. .............................. 39 4-3 Navigation path with a central obstacle the model was not trained with. 39 4-4 Performance on Particle, Maze and Reacher environments where dynamics models are either pretrained on random transitions or learned via online interaction with the environment. Action FF: Action Feed-Forward Network. 39 4-6 Effects of varying number of planning steps to reach a goal state. Asthe number of steps of planning increases, there is a larger envelope of explored states. ................................... 41 4-7 Illustrations of two different planned trajectories from start to goal inthe Reacher environment. ........................... 41 10 4-8 Illustration of energy values of states (computed by taking the energy of a transition centered at the location) and corresponding visitation maps. While an EBM learns a probabilistic model of transitions in areas already explored, energies of unexplored regions fluctuate throughout training, leading toa natural exploration incentive. Early on in training (left), the EBM puts low energy on the upper corner, incentivizing agent exploration towards the top corner. Later on in training (right), an EBM puts low energy on the right lower corner, incentivizing agent exploration towards the bottom corner. 42 4-9 Illustration of exploration in a maze under random actions (left) as opposed to following an EBM (middle). Areas in blue in the maze environment (right) are admissible,
Load more

Recommended publications
  • Training Energy-Based Models for Time-Series Imputation
    JournalofMachineLearningResearch14(2013)2771-2797 Submitted 1/13; Revised 5/13; Published 9/13 Training Energy-Based Models for Time-Series Imputation Philemon´ Brakel [email protected] Dirk Stroobandt [email protected] Benjamin Schrauwen [email protected] Department of Electronics and Information Systems University of Ghent Sint-Pietersnieuwstraat 41 9000 Gent, Belgium Editor: Yoshua Bengio Abstract Imputing missing values in high dimensional time-series is a difficult problem. This paper presents a strategy for training energy-based graphical models for imputation directly, bypassing difficul- ties probabilistic approaches would face. The training strategy is inspired by recent work on optimization-based learning (Domke, 2012) and allows complex neural models with convolutional and recurrent structures to be trained for imputation tasks. In this work, we use this training strat- egy to derive learning rules for three substantially different neural architectures. Inference in these models is done by either truncated gradient descent or variational mean-field iterations. In our experiments, we found that the training methods outperform the Contrastive Divergence learning algorithm. Moreover, the training methods can easily handle missing values in the training data it- self during learning. We demonstrate the performance of this learning scheme and the three models we introduce on one artificial and two real-world data sets. Keywords: neural networks, energy-based models, time-series, missing values, optimization 1. Introduction Many interesting data sets in fields like meteorology, finance, and physics, are high dimensional time-series. High dimensional time-series are also used in applications like speech recognition, motion capture and handwriting recognition. To make optimal use of such data, it is often necessary to impute missing values.
    [Show full text]
  • Boltzmann Machines and Energy-Based Models
    Boltzmann machines and energy-based models Takayuki Osogami IBM Research - Tokyo [email protected] Abstract We review Boltzmann machines and energy-based models. A Boltzmann machine defines a probability distribution over binary-valued patterns. One can learn parameters of a Boltzmann machine via gradient based approaches in a way that log likelihood of data is increased. The gradi- ent and Hessian of a Boltzmann machine admit beautiful mathematical representations, although computing them is in general intractable. This intractability motivates approximate methods, in- cluding Gibbs sampler and contrastive divergence, and tractable alternatives, namely energy-based models. 1 Introduction The Boltzmann machine has received considerable attention particularly after the publication of the seminal paper by Hinton and Salakhutdinov on autoencoder with stacked restricted Boltzmann ma- chines [21], which leads to today's success of and expectation to deep learning [54, 13] as well as a wide range of applications of Boltzmann machines such as collaborative filtering [1], classification of images and documents [33], and human choice [45, 47]. The Boltzmann machine is a stochastic (generative) model that can represent a probability distribution over binary patterns and others (see Section 2). The stochastic or generative capability of the Boltzmann machine has not been fully exploited in to- day's deep learning. For further advancement of the field, it is important to understand basics of the Boltzmann machine particularly from probabilistic perspectives. In this paper, we review fundamental properties of the Boltzmann machines with particular emphasis on probabilistic representations that allow intuitive interpretations in terms of probabilities. A core of this paper is in the learning rules based on gradients or stochastic gradients for Boltzmann machines (Section 3-Section 4).
    [Show full text]
  • On Autoencoders and Score Matching for Energy Based Models
    On Autoencoders and Score Matching for Energy Based Models Kevin Swersky* [email protected] Marc'Aurelio Ranzatoy [email protected] David Buchman* [email protected] Benjamin M. Marlin* [email protected] Nando de Freitas* [email protected] *Department of Computer Science, University of British Columbia, Vancouver, BC V6T 1Z4, Canada yDepartment of Computer Science, University of Toronto, Toronto, ON M5S 2G4, Canada Abstract a long history in particular application areas including modeling natural images. We consider estimation methods for the class of continuous-data energy based mod- Recently, more sophisticated latent variable EBMs for els (EBMs). Our main result shows that es- continuous data including the PoT (Welling et al., timating the parameters of an EBM using 2003), mPoT (Ranzato et al., 2010b), mcRBM (Ran- score matching when the conditional distri- zato & Hinton, 2010), FoE (Schmidt et al., 2010) and bution over the visible units is Gaussian cor- others have become popular models for learning rep- responds to training a particular form of reg- resentations of natural images as well as other sources ularized autoencoder. We show how different of real-valued data. Such models, also called gated Gaussian EBMs lead to different autoencoder MRFs, leverage latent variables to represent higher architectures, providing deep links between order interactions between the input variables. In these two families of models. We compare the very active research area of deep learning (Hinton the score matching estimator for the mPoT et al., 2006), these models been employed as elemen- model, a particular Gaussian EBM, to several tary building blocks to construct hierarchical models other training methods on a variety of tasks that achieve very promising performance on several including image denoising and unsupervised perceptual tasks (Ranzato & Hinton, 2010; Bengio, feature extraction.
    [Show full text]
  • Joint Training of Variational Auto-Encoder and Latent Energy-Based Model
    Joint Training of Variational Auto-Encoder and Latent Energy-Based Model Tian Han1, Erik Nijkamp2, Linqi Zhou2, Bo Pang2, Song-Chun Zhu2, Ying Nian Wu2 1Department of Computer Science, Stevens Institute of Technology 2Department of Statistics, University of California, Los Angeles [email protected], {enijkamp,bopang}@ucla.edu, [email protected] {sczhu,ywu}@stat.ucla.edu Abstract learning and adversarial learning. Specifically, instead of employing a discriminator as in GANs, we recruit a latent This paper proposes a joint training method to learn both energy-based model (EBM) to mesh with VAE seamlessly the variational auto-encoder (VAE) and the latent energy- in a joint training scheme. based model (EBM). The joint training of VAE and latent The generator model in VAE is a directed model, with a EBM are based on an objective function that consists of known prior distribution on the latent vector, such as Gaus- three Kullback-Leibler divergences between three joint dis- sian white noise distribution, and a conditional distribution tributions on the latent vector and the image, and the objec- of the image given the latent vector. The advantage of such tive function is of an elegant symmetric and anti-symmetric a model is that it can generate synthesized examples by di- form of divergence triangle that seamlessly integrates vari- rect ancestral sampling. The generator model defines a joint ational and adversarial learning. In this joint training probability density of the latent vector and the image in a scheme, the latent EBM serves as a critic of the generator top-down scheme. We may call this joint density the gener- model, while the generator model and the inference model ator density.
    [Show full text]
  • Learning Latent Space Energy-Based Prior Model
    Learning Latent Space Energy-Based Prior Model Bo Pang∗1 Tian Han∗2 Erik Nijkamp∗1 Song-Chun Zhu1 Ying Nian Wu1 1University of California, Los Angeles 2Stevens Institute of Technology {bopang, enijkamp}@ucla.edu [email protected] {sczhu, ywu}@stat.ucla.edu Abstract We propose to learn energy-based model (EBM) in the latent space of a generator model, so that the EBM serves as a prior model that stands on the top-down network of the generator model. Both the latent space EBM and the top-down network can be learned jointly by maximum likelihood, which involves short-run MCMC sampling from both the prior and posterior distributions of the latent vector. Due to the low dimensionality of the latent space and the expressiveness of the top-down network, a simple EBM in latent space can capture regularities in the data effectively, and MCMC sampling in latent space is efficient and mixes well. We show that the learned model exhibits strong performances in terms of image and text generation and anomaly detection. The one-page code can be found in supplementary materials. 1 Introduction In recent years, deep generative models have achieved impressive successes in image and text generation. A particularly simple and powerful model is the generator model [35, 21], which assumes that the observed example is generated by a low-dimensional latent vector via a top-down network, and the latent vector follows a non-informative prior distribution, such as uniform or isotropic Gaussian distribution. While we can learn an expressive top-down network to map the prior distribution to the data distribution, we can also learn an informative prior model in the latent space to further improve the expressive power of the whole model.
    [Show full text]