Hopfield networks and energy equations for neural networks

 As we discussed in Chapter 3, Building Blocks of Deep Neural Networks, Hebbian Learning states, "Neurons that fire together, wire together", "and many models, including the multi-layer perceptron, made use of this idea in order to develop learning rules. One of these models was the Hopfield network, developed in the 1970-80s by several researchers. In this network, each "neuron" is connected to every other by a symmetric weight, but no self-connections (there are only connections between neurons, no self-loops).

Unlike the multi-layer perceptrons and other architectures we studied in Chapter 3, Building Blocks of Deep Neural Networks, the Hopfield network is an undirected graph, since the edges go "both ways."

The neurons in the Hopfield network take on binary values, either (-1, 1) or (0, 1), as a thresholded version of the tanh or sigmoidal activation function:

The threshold values (sigma) never change during training; to update the weights, a "Hebbian" approach is to use a set of n binary patterns (configurations of all the neurons) and update as:

where n is the number of patterns, and e is the binary activations of neurons i and j in a particular configuration. Looking at this equation, you can see that if the neurons share a configuration, the connection between them is strengthened, while if they are opposite signs (one neuron has a sign of +1, the other -1), it is weakened. Following this rule to iteratively strengthen or weaken a connection leads the network to converge to a stable configuration that resembles a "memory" for a particular activation of the network, given some input. This represents a model for associative memory in biological organisms- the kind of memory that links unrelated ideas, just as the neurons in the Hopifield network are linked together.

Besides representing biological memory, Hopfield networks also have an interesting parallel to electromagnetism. If we consider each neuron as a particle or "charge," we can describe the model in terms of a "free energy" equation that represents how the particles in this system mutually repulse/attract each other and where on the distribution of potential configurations the system lies relative to equilibrium:

where we is the weights between neurons i and j, s is the "states" of those neurons (either 1, "on," or -1, "off"), and sigma is the threshold of each neuron (for example, the value that its total inputs must exceed to set it to "on"). When the Hopfield network is in its final configuration, it also minimizes the value of the energy function computed for the network, which is lowered by units with an identical state(s) being connected strongly(w). The probability associated with a particular configuration is given by the Gibbs measuer:

Here, Z(B) is a normalizing constant that represents all possible configurations of the newtwork, in the same respect as the normalizing constant in the Bayesian probability function you saw in Chapter 1, An Introduction to Generative AI: "Drawing" Data from Model.

Also notice in the energy function definition that the state of neuron is only affected by local connections (rather than the state of every other neuron in the network, regardless of if it is connected); this is also known as the Markov property, since the state is "memoryless," depending only on its immediate "past" (neighbors). In fact, the Hammersly-Clifford therem states that any distribution having this same memoryless property can be represented using the Gibbs measure.

Restricted Bolzmann Machines: generating pixels with statistical mechanics

 The neural network model that we will apply to the MNIST data has its origins in earlier research on how neurons in the mammalian brain might work together to transmit signals and encode patterns as memories. By using analogies to statistical mechanics in physics, this section will show you how simple networks can "learn" the distribution of image data and be used as building blocks for larger networks.

Retrieving and loading the MNIST dataset in TensorFlow

 The first step in training our own DBN is to construct our dataset. This section will show you how to transform the MNIST data into a convenient format that allows you to train a neural network, using some of TensorFlow 2's built-in functions for simplicity.

Let's start by loading the MNIST dataset in TensorFlow. As the MNIST data has been used for many deep learning benchmarks, TensorFlow2 already has convenient utilities for loading and formatting this data. To doo so, we need to first install the tensorflow-dataset library;

pip install tensorflow-datasets

After installing the package, we need to import it along with the required dependencies:

from __future__ import absolute_import

from __future__ import division

from __future__ import print_function

import matplotlib.phlot as plt

import numpy as np

import tesorflow.compat.v2 as tf

import tensorflow_datasets as tfds

Now we can download the MNIST data locally from Google Cloud Storage(GCS) using the builder functionality:

mnist_builder = tfds.builder("mnist")

mnist_builder.download_and_prepare()

The dataset will now be available on disk on our machine. As noted earlier, this data is divided into a training and test dataset, which you can verify by taking a look at the info command:

info = mnist_builder.info

print(info)

This gives the following output:

tfds.core.DatasetInfo(

    name='mnist',

    version=3.0.1

    description='The MNIST database of handwitten digits.',

    homepage='http://yann.lecun.com/exdb/mnist/',

    features=FeaturesDict({

        'image': Image(shape=(28, 28, 1), dtype=tf.unit8),

        'label': ClassLabel(shape=(), dtype=tf.int64, num_classes=10),

    }),

    total_num_examples=70000,

    splits={

        'test':10000,

        'train':60000,

    },

    supervised_keys=('image', 'label'),

    citation="""@article{lecun2010mnist,

        title={MNIST handwritten digit database},

        author={LeCun, Yann and Cortes, Corinna and Burges, CJ},

        journal={ATT Labs [Online], Available: http://yann.lecun.com/exdb/mnist},

        volume={2},

        year={2010}

        }""",

    redistribution_info=,

)

As you can see, the test dataset has 10,000 examples, the training dataset has 60,000 examples, and the images are 28*28 pixels with a label from one of 10 classes (0 to 9).

Let's start by taking at the training dataset:

mnist_train = mnist_builder.as_dataset(split="train")

We can visually plot some examples using the show_examples function:

fig = tds.show_examples(info, mnist_train)

This gives the following figure:

You can also see more clearly here the grayscale edges on the numbers where the anti-aliasing was applied to the original dataset to make the edges seem less jagged (the colors have also been flipped from the original example in Figure 4.1).

We can also plot an individual image by taking one element from the dataset, reshaping it to a 28*28 array, casting it as a 32-bit float, and plotting it in grayscale:

flatten_image = partial(flatten_image, label=True)

for image, label in mnist_train.map(flatten_image).take(1):

    plt.imshow(image.numpy().reshape(28,28).astype(np.float32),cmap=plt.get_cmap("gray"))

    print("Label: %d" % label.numpy())

This gives the following figure:

This in nice for visual inspection, but for our experiments in this chapter, we will actually need to flatten these images into a vector. To do so, we can use the map() function, and verify that the dataset is now flattened; note that we also need to cast to a float for use in the RBM later. The RBM also assumes binary (0 or1) inputs, so we need to rescale the pixels, which range from 0 to 256 to the range 0 to 1:

def flatten_image(x, label=True):

    if label:

        return (tf.divide(tf.dtypes.cast(tf.reshape(x["image"],(1,28*28)), tf.float32), 256.0), x["label"])

    else:

        return (tf.divide(tf.dtypes.cast(tf.reshape(x["image"],(1,28*28)), tf.float32), 256.0))

for image, label in mnist_train.map(flatten_image).take(1):

    plt.imshow(image.numpy().astype(np.float32), cmap=plt.get_cmap("gray"))

    print("Label: %d" % label.numpy())

This gives a 784*1 vector, which is the "flattened" version of the pixels of the digit "4":

Now that we have the MNIST data as a series of vectors, we are ready to start implementing an RBM to process this data and ultimately create a model capable of generating new images.

The MNIST database

 In developing the DBN model, we will use a dataset that we have discussed before - the MNIST database, which contains digital images of hand-drawn digits from 0 to 9. This database is combination of two sets of earlier images from the National Institute of Standards and Technology(NIST): Special Database 1(digits written by US high school students) and Special Database 3(written by US Census Bureau employees), the sum of which is split into 60,000 training images and 10,000 test images.

The original images in the dataset were all block and white, while the modified dataset normalized them to fit into a 20*20-pixel bounding box and removed jagged edges using anti-aliasing, leading to intermediary grayscale values in cleaned images; they are padded for a final resolution of 28*28 pixels.

In the original NIST dataset, all the training images came from bureau employees, while the test dataset came from high school students, and the modified version mixes the two groups in the training and test sets to provide a less biased population for training machine learning algorithms.

An early application of Support Vector Machines(SMVs) to this dataset yielded an error rate of 0.8%, while the latest deep learning models have shown error rates as low as 0.23%. You should note that these figures were obtained due to not only the discrimination algorithms used but also "data augmentation" tricks such as creating additional translated images where the digit has been shifted by several pixels, thus increasing the number of data examples for the algorithm to learn from. Because of its wide availability, this dataset has become a benchmark for many machine learning models, including Deep Neural Networks.

The dataset was also the benchmark for a breakthrough in training multi-layer neural networks in 2006, in which an error rate of 1.25% was achieved(without image translation, as in the preceding examples). In this chapter, we will example in detail how this breakthrough was achieved using a generative model, and explore how to build our own DBN that can generate MNIST digits.

4. Teaching Networks to Generate Digits

 In the previous chapter, we covered the building blocks of neural network models. In this chapter, our first project will recreate one of the most groundbreaking models in the history of deep learning, Deep Belief Network(DBN). DBN was one of the first multi-layer networks for which a feasible learning algorithm was developed. Besides being of historical interest, this model is connected to the topic of this book because the learning algorithm makes use of a generative model in order to pre-train the neural network weights into a reasonable configuration prior to backprogagation.

In this chapter, we will cover:

- How to load the Modified National Institute of Standards and Technology(MNIST) dataset and transform it using TensorFlow 2's Dataset API.

- How a Restricted Boltzmann Machine(RBM) - a simple neural network - is trained by minimizing an "energe" equation that resembles formulas from physics to generate images.

- How to stack several RBMs to make a DBN and apply forward and backward passes to pre-train this network to generate image data.

- How to implement an end-to-end classifier by combining this pre-training with backpropagation "fine-tuning" using the TensorFlow 2 API.

Summary

 In this chapter, we've covered the basic vocabulary of deep learning - how initial research into perceptrons and LMPs led to simple learning rules being abandoned for backpropagation. We also looked at specialized neural network architectures such as CNNs, based on the visual cortex, and recurrent networks, specialized for sqequence modeling. Finally, we examined variants of the gradient descent algorithm proposed originally for backpropagation, which have advantages such as momentum, and described weight initialization schemes that place the parameters of the network in a range that is easier to navigate to local minimum.

With this context in place, we are all set to dive into projects in generative modeling, beginning with the generation of MNIST digit using Deep Belief Networks in Chapter 4, Teaching Networks to Generate Digits.

Xavier initialization

 As noted previously, in earlier research it was common to initialize weights in a neural network with some range of random values. Breakthroughs in the training of Deep Belief Networks in 2006, as you will see in Chapter 4, Teaching Networks to Generate Digits, used pre-training (through a generative modeling approach) to initialize weights before performing standard backpropagation.

If you've ever used a layer in the TensorFlow Keras module, you will notice that the default initialization for layer weights draws from either a truncated normal or uniform distribution. Where does this choice come from? As I described previously, one of the challenges with deep networks using sigmoidal or hyperbolic activation functions is that they tend to become saturated, since the values for these functions are capped with very large or negative input. We might interpret the challenge of initializing networks then as keeping weights in such a range that they don't saturate the neuron's output. Another way to understand this is to assume that the input and output values of the neuron have similar variance; the signal is not massively amplifying or diminishing while passing through the neuron.

In practice, for a linear neuron, y = ws + h, we could compute the variance of the input and output as:

var(y) = var(ws + b)

The b is constant, so we are left with:

var(y) = var(w)var(x) +  var(x)E(x)2 + var(x)E(w)2 = var(w)var(x)

Since there are N elements in the weight matrix, and we want var(y) to equal var(x), this gives:

1 = Nvar(x),var(w) = 1/N

Therefore, for a weight matrix w, we can use a truncated normal or uniform distribution with variance 1/N (the average number of input and output units, so the number of weights). Variations have also been applied to ReLU units. these methods are referred to by their original authors' names as Xavier or He initialization.

In summary, we've reviewed several common optimizers used under the hood in TensorFlow2, and discussed how they improve upon the basic form of SGD. We've also discussed how clever weight initialization schemes work together with these optimizers to allow us to train ever more complex models.