Report

Learning Deep Energy Models Author: Jiquan Ngiam et. al., 2011 Presenter: Dibyendu Sengupta 1 The Deep Learning Problem . Output or the highest level representation: That’s the CRAZY FROG!!!!! . . Slightly higher level representation Learning and modeling the different layers is a very challenging problem Vectorized pixel intensities 2 Outline Placing DEM in context of other models Energy Based Models Description of Deep Energy Models Learning in Deep Energy Models Experiments using DEMs 3 State of the art methods • Deep Belief Network (DBN): RBMs stacked and trained in a “Greedy” manner to form DBN1 each of which models the posterior distribution of the previous layer. DBN Layers • Deep Boltzmann Machine (DBM): DBM2 has undirected connections between the layers of networks initialized by RBMs. Joint training is done on the layers. DBM Layers • Deep Energy Model (DEM): DEM consists of a feedforward NN that deterministically transforms the input and the output of the feedforward network is modeled with a stochastic hidden unit. DBM Layers 1 Hinton et al., A Fast Learning Algorithm for Deep Belief Nets, Neural Computation, 2006 2 Salakhutdinov et al., Deep Boltzmann Machines, AISTATS, 2009 4 Deep Belief Network (DBN) DBNs are graphical models which learn to extract a deep hierarchical representation of the training data by modeling observed data x and “l” hidden layers “h” as follows by the joint distribution Algorithm: 1. Train the first layer as an RBM that models the input, x as its visible layer. 2. The first layer is used as the input data for the second layer which is chosen either by mean activations of or samples of 3. Iterate for the desired number of layers, each time propagating upward either samples or mean values. 4. Fine-tune all the parameters of this deep architecture with respect to log- likelihood or with respect to a supervised training criterion. 5 Deep Boltzmann Machine (DBM) • DBM is similar to DBN: Primary contrasting feature is undirected connections in all the layers • Layerwise training algorithm is used to initialize the layers using RBMs • Joint training is performed on all the layers 6 Motivation of Deep Energy Models (DEM) • Both DBN and DBM has multiple stochastic hidden layers • Computing the conditional posterior over the stochastic hidden layers is intractable • Learning and inference is more tractable in single layer RBM but it suffers from lack of representational power • To overcome both the defects DEMs combine layers of deterministic hidden layers with a layer of stochastic hidden layer 7 Outline Placing DEM in context of other models Energy Based Models Description of Deep Energy Models Learning in Deep Energy Models Experiments using DEMs 8 Energy Based Models (EBMs) x – Visible units, h – Hidden units, Z – Partition Function, F(x) – Free Energy Independent of h We would like the configurations to be at low energy 9 General Learning Strategy for EBMs Gradient based methods on this functional formulation to learn parameters θ In general, the “positive” term is easy to compute but the “negative” term is often intractable and sampling needs to be done. Expectations are computed for both these terms to estimate their values 10 Energy based models of RBM W: weights connecting visible (v) and hidden (h) units b and c: offsets of visible and hidden units RBM representation Exploiting the structure of RBM we can obtain In particular for RBMs with binary units: 11 Outline Placing DEM in context of other models Energy Based Models Description of Deep Energy Models Learning in Deep Energy Models Experiments using DEMs 12 Sigmoid Deep Energy Model gθ(v) represents the feedforward output of the neural network gθ Similar to RBMs, an energy function defines the connections between gθ(v) and the hidden units h (assumed binary) The conditional posteriors of the hidden variables are easy to compute: Representational power of the model can be increased by adding more layers of the feedforward NN 13 Generalized Deep Energy Models Generalized Free Energy Function Sigmoid DEM with gθ as the feedforward NN Different models for H(v) enable DEMs with multiple layers of nonlinearities PoT Distribution Covariance RBM Distribution Examples of 2-layered network: First layer computes squared responses followed by a softrectification. There can also be linear combinations of models like mean-covariance RBM which is a linear 14 combination of RBM and cRBM An alternative deep version of DEM • PoT and cRBM uses shallow feedback in the energy landscape • “Stacked PoT” or SPoT is chosen as a deeper version of PoT by stacking a bunch of PoT layers • This creates a more expressive deeper model 15 Outline Placing DEM in context of other models Energy Based Models Description of Deep Energy Models Learning in Deep Energy Models Experiments using DEMs 16 Learning Parameters in DEMs • Models were trained by maximizing the log-likelihood • Stochastic gradient ascent was used to learn the parameters, θ • Obtain update rules similar to generalized Energy Based Model (EBM) updates 2nd Term: Expectation over data – can be easily computed 1st Term: Expectation over model distribution – Harder to compute and 17 often intractable and is approximated by Sampling Hybrid Monte Carlo (HMC) Sampler Hamiltonian Dynamics • Model samples obtained by simulation of physical system • Particles are subjected to potential and kinetic energies • Velocities are sampled from a univariate Gaussian to obtain state1 • State of the particles follow conservation of Hamiltonian H(s,φ) • n-steps of Leap-Frog Algorithm applied to state1 (s,φ) to obtain state2 • Acceptance is performed based on Pacc(state1, state2) Neal RM, Proabilistic inference using Markov Chain Monte Carlo Methods, Technical Report, U Toronto, 199318 Greedy Layerwise and Joint Training • First greedy layerwise training is performed by – Training successive layers to optimize data likelihood – Freeze parameters of the earlier layers – Learning objective (i.e. data likelihood) stays the same throughout training of deep model • Joint training is subsequently performed on all layers by - Unfreezing the weights - Optimizing the same objective function - Computational cost is comparable to layerwise training • Training in DEM is computationally much cheaper than DBN and DBM since only the top layer needs sampling and all intermediate layers are deterministic hidden units 19 Discriminative Deep Energy Models al: Activations of lth in gθ used to learn a linear classifier of image labels y via weights U Training: Done by hybrid generative-discriminative objective Gradient of generative cost: Can be computed by previously discussed method Gradient discriminative cost: Can be computed by considering the model to be a short-circuited feedforward NN with softmax classification 20 Outline Placing DEM in context of other models Energy Based Models Description of Deep Energy Models Learning in Deep Energy Models Experiments using DEMs 21 Experiments: Natural Images Convergence: Questionable! Remarks: • Experiments done with sigmoid and SPoT models under Annealed Importance Sampling (AIS) • M1 and M2: Training using Greedy Layerwise stacking of 1 and 2 layers respectively • M1-M2-M12: Greedy layerwise training for 2 layers followed by joint training of the two layers • Joint training results in performance improvement over pure Greedy lawerwise training but the convergence of Log-Likelihood is not evident from plots • M1-M2-M12 seem to require a significantly larger number of iterations • Adding multiple layers in SPoT significantly improves model performance 22 Experiments: Object Recognition Samples from SPoT M12 model trained on NORB dataset Remarks: • Models were trained on NORB dataset • Hybrid discriminative-generative Deep models (SPoT) performed better than the fully discriminative model • Fully discriminative model suffers from overfitting • Optimal α parameter that weighs discriminative-generative cost is obtained by cross validation on a subset of training data • Iteration counts for convergence in the models were not reported 23 Conclusions • It is often difficult to scale SPoT model to realistic datasets because of the slowness of HMC • Jointly training all layers yields significantly better models than Greedy layerwise training Filters appear Blob-like Filters appear Gabor-like Single layer Sigmoid DEM: Trained by Greedy layerwise Two layer Sigmoid DEM: Trained by Greedy and Joint Training 24 What is the best Multi-Stage Architecture for Object Recognition? Author: Kevin Jarrett et. al., 2009 Presenter: Sreeparna Mukherjee 25 Coming back to the starting problem . Output or the highest level feature extraction: That’s the CRAZY FROG!!!!! . . Feature Extraction: Stage 1 Vectorized pixel intensities Can it be done more efficiently with multiple feature extraction stages instead of just one 26 Outline Different Feature Extraction Models Multi-Stage Feature Extraction Architecture Learning Protocols Experiments 27 Existing Feature Extraction Models • There are several single-stage feature extraction systems inspired by mammalian visual cortex – Scale Invariant Feature Transform (SIFT) – Histogram of Oriented Gradients (HOG) – Geometric Blur • There are also models with two or more successive stages of feature extractions - Convolutional networks trained in supervised or unsupervised mode - Multistage systems using a non-linear MAX or HMAX models 28 Contrasts among different models The feature extraction models primarily differ in following aspects – Number of stages of feature extraction – Type of non-linearity used after filter-bank – Type of filter used – Type of classifier used 29 Questions to Address • How do the non-linearities following filter banks influence recognition accuracy? • Does unsupervised or supervised learning of filter banks improve performance over hardwired or random filters? • Is there any benefit of using a 2-stage feature extractor as opposed to single stage feature extractor? 30 Intuitions in Feature Extraction Architecture • Supervised training on a small number of labeled datasets (e.g. Caltech-101) will fail • Filters need to be carefully handpicked for good performance • Non-linearities should not be a significant factor What do you think? These intuitions are wrong – We’ll see how !!!! 31 Outline Different Feature Extraction Models Multi-Stage Feature Extraction Architecture Learning Protocols Experiments 32 General Model Architecture Output or the highest level feature extraction: That’s the CRAZY FROG!!!!! Pooling Layer: Local Averaging to remove small perturbations Non Linear Transformation Layers Filter Bank Layer Vectorized pixel intensities 33 Filter Bank Layer (FCSG) Input (x): n1 2D feature maps of size n2 × n3 xijk is each component in each feature map xi Output (y): m1 feature maps of size m2 × m3 Filter (k): kij is a filter in the filter bank of size l1 × l2 mapping xi to yj 34 Non-Linear Transformations in FCSG FCSG comprises of Convolution Filters (C), Sigmoid/tanh non-linearity (S) and gain (G) coefficients gj 35 Rectification Layer (Rabs) • This layer returns the absolute value of its input • Other rectifying non-linearities produced similar results under Rabs 36 Local Contrast Normalization Layer (N) This layer performs local – Subtractive Normalization – Divisive Normalization Subtractive Normalization Divisive Normalization wpq is a normalized Gaussian weighting window 37 Average Pooling and Subsampling Layer (PA) Averaging: This creates robustness to small distortions wpq is a uniform weighting window Subsampling: Spatial resolution is reduced by downsampling with a ratio S in both spatial directions Max Pooling and Subsampling Layer (PM) Average operation is replaced by Max operation Subsampling procedure stays the same 38 Hierarchy among the Layers Layers can be combined in various hierarchical ways to obtain different architectures FCSG – PA FCSG – Rabs – PA FCSG – Rabs– N – PA FCSG – PM A typical multistage architecture: FCSG – Rabs– N – PA 39 Outline Different Feature Extraction Models Multi-Stage Feature Extraction Architecture Learning Protocols Experiments 40 Unsupervised Training Protocols Input: X (vectorized patch or stack of patches) Dictionary: W – to be learnt Feature Vector: Z* - obtained by minimizing the energy function 41 Learning procedure: Olshausen-Field The energy function to be minimized λ – Sparsity Hyper-parameter Learning W: Done by minimizing the Loss Function LOF(W) using stochastic gradient descent Obtaining Z* from EOF via “basis pursuit” is an expensive optimization problem 42 Learning procedure: PSD Regressor function mapping X Y Loss Function • EPSD optimization is faster as it has the predictor term • Goal of algorithm is to make the regressor C(X,K,G) as close to Z as possible •After training completion Z* = C(X,K) for input X i.e. the method is fast feedforward 43 Outline Different Feature Extraction Models Multi-Stage Feature Extraction Architecture Learning Protocols Experiments 44 Experiment: Caltech 101 Dataset R and RR – Random Features and Supervised Classifier U and UU – Unsupervised Features, Supervised Classifier U+ and U+U+ - Unsupervised Feature, Global Supervised Refinement G – Gabor Functions Remarks: • Random filter and no filter learning achieve decent performance • Both Rectification and Supervised fine tuning improved performance • Two-stage systems are better than single-stage models • Unsupervised training does not significantly improve performance if both rectification and normalization are used • Performance of Gabor Filters were worse than random filters 45 Experiment: NORB Dataset Remarks: • Rectification and Normalization makes a significant improvement when samples are low • As the number of samples increases, improvement with Rectification and Normalization becomes insignificant • Random filters performs much worse on large number of labeled samples 46 Experiment: MNIST Dataset • Two-stage feature extraction architecture was used • The parameters are first trained using PSD • Classifier is initialized randomly and the whole system is fine tuned in supervised mode • A test error rate of 0.53% was observed – best known error rate on MNIST without distortions or preprocessing 47 Coming back to the same Questions! • How do the non-linearities following filter banks influence recognition accuracy? - Yes – Rectification improves performance possibly due to i) non-polar features improves recognition or ii) it prohibits cancellations of neighbors during pooling layer – Normalization also enables performance improvement and makes supervised learning faster by contrast enhancement • Does unsupervised or supervised learning of filter banks improve performance over hard-wired or random filters? - Yes – Random filter shows good performance in the limit of small training set sizes where the optimal stimuli for random filters are similar to that trained filters – Global supervised learning of filters yield good results if proper nonlinearities are used • Is there any benefit of using a 2-stage feature extractor as opposed to single stage feature extractor? - Yes – The experiments show that 2-stage feature extractor performs much better compared to single stage feature extractor models. 48 Questions 49 Extra Slides 50 Hybrid Monte Carlo (HMC) Sampler • Model samples are obtained by simulating a physical system • Particles are subjected to potential and kinetic energies • Velocities are sampled from a univariate Gaussian to obtain state1 • State of the particles follow conservation of Hamiltonian H(s,φ) Hamiltonian Dynamics Leap-Frog Discretization: • n-steps of Leap-Frog Algorithm applied to state1 (s,φ) to obtain state2 • Acceptance is performed based on Pacc(state1, state2) Neal RM, Proabilistic inference using Markov Chain Monte Carlo Methods, Technical Report, U Toronto, 1993 51