Learning Deep Energy Models

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

similar documents