Early CNNs

 This idea of columns inspired early research into CNN architectures. Instead of learning individual weights between units as in a feedforward network, this architecture uses shared weights within a group of neurons specialized to detect a specific edge in an image. The initial layer of the network (denoted H1) consists of 12 groups of 64 neurons each. Each of these groups is derived by passing a 5 * 5 grid over the 16 * 16-pixel input image; each of the 64 5*5 grids in this group share the same weights, but are tired to different spatial regions of the input. You can see that there must be 64 neurons in each group to cover the input image if their receptive fields overlap by two pixels.

When combined, these 12 groups of neurons in layer H1 form 12 8*8 grids representing the presence or absence of a particular edge within a part of the image - the 8 * 8 grid is effectively a down-sampled version of the image(Figure 3.9). This weight sharing makes intutive sense in that the kernel represented by the weight is specified to detect a distinct color and/or shape, regardless of where it appears in the image. An effect of this down-sampling is a degree of positional invariance; we only know the edge occurred somewhere within a resion of the image, but not the exact location due to the reduced resolution from downsampling. Because they are computed by multiplying a 5*5 matrix(kernel) with a part of the image, an operation used in image blurring and other transformations, these 5*5 input features are known as convlutional kernels, and give the network its name.

Once we have these 128*8 downsampled versions of the image, the next layer(H2) also has 12 groups of neurons; here, the kernels are 5*5*8 - they traverse the surface of an 8*8 map from H1, across 8 of the 12 troups. We need 16 neurons of these 5*5*8 groups since a 5*5 grid can be moved over four times up and down on an 8 * 8 grid to cover all the pixels in the 8 * 8 grid.

Just like deeper cells in the visual cortex, the deeper layers in the network integrate across multiple columns to combine information from different edge detectors together.

Finally, the third hidden layer of this network (H3) contains all-to-all connections between 30 hidden units and the 12 * 16  units in the H2, just as in a traditional feedforward network; a final output of 10 unit classifies the input image as one of 10 hand-drawn digits.

Through weight sharing, the overall number of free parameters in this network is reduced, though it is still large in absolute terms. While backpropagation was used successfully for this task, it required a carefully designed network for a rather limited set of images with a restricted set of outcomes - for real-world applications, such as detection objects from hundreds or thousands of possible categories, other approaches would be necessary.

Networks for seeing: Convolutional architectures

 As noted at the beginning of this chapter, one of the inspirations for deep neural network models is the biological nervous system. As researchers attempted to design computer vision systems that would mimic the functioning of the vissual system, they turned to the architecture of the retina, as revealed by physiological studies by neurobiologists David Huber and Torsten Weisel in the 1960s. As previously described, the physiologist Santiago Ramon Y Cajal provided visual evidence that neural structures such as the retina are arranged in vertical networks:

Huber and Weisel studied the retinal system in cats, showing how their perception of shapes is composed of the activity of individual cells arranged in a column. Each column of cells is designed to detect a specific orientation of an edge in an input image; images of complex shapes are stitched together from these simpler images.

Varieties of networks: Convolution and recursive

 Up until now we've primarily discussed the basics of neural networks by referencing feedforward networks, where every input is connected to every output in each layer. 

While these feedforward networks are useful for illustrating how deep networks are trained, they are only on class of a broader set of architectures used in modern applications, including generative models, Thus, before covering some of the techniques that make training large networks practical, let's review these alternative deep models.

The shortfalls of backpropagation

 While the backpropagation procedure provides a way to update interior weights within the network in a principled way, it has several shortcomings that make deep networks difficult to use in practice. One is the problem of vanishing gradients. In out derivation of the backpropagation formulas, you saw that gradients for weights depper in the network are product of successive partial derivatives from higher layers. In our example, we used the sigmoid function; if we plot out the value of the sigmoid and its first derivative, we can see a potential problem:

As the value of the sigmoid function increase or decrease towards the extremes (0 or 1, representing being either "off" or "on"), the values of the gradient vanish to near zero. This means that the updates to w and b, which are products of these gradients from hidden activation functions y, shrink towards zero, making the weights change little between iterations and making the parameters of the hidden layer neurons change very slowly during backpropagation. Clearly one problem here is that the sigmoid function saturates; thus, choosing another nonliearity might circumvent this problem (this is indeed one of the solutions that was proposed as the ReLU, as we'll cover later).

Another problem is more subtle, and has to do with how the network utilizes its available free parameters. As you saw in Chapter 1, An Introduction to Generative AI: "Drawing" Data from Models, a posterior probability of a variable can be computed as a product of a likelihood and a prior distribution. We can see deep neural networks as a graphical representation of this kind of probability: the ouput of the neuraon, depending updon its parameters, is a product of all the input values and the distributions on those inputs (the priors). A problem occurs when those values become tightly coupled. As an illustration, consider the competing hypotheses for a headache:

If a patient has cancer, the evidence is so overwhelming that whether they have a could or not profides no additional value; in essence, the vlaue of the two prior hypotheses becomes coupled because of the influence of one. This makes it intractable to compute the relative contribution of different parameters, particularly in a deep network; we will cover this problem in our discussion of Restricted Boltman Machine and Deep Belief Networks in Chapter 4, Teaching Networks to Generate Digits. As we will describe in more detail in that chapter, a 2006 study showed how to counteract this effect, and was one of the first demonstrations of tractable inference in deep neural networks, a breakthrough that relied upon a generative model that produced images of hand-drawn digits.

Beyond these concerns, other challenges in the more widespread adaption of neural networks in the 1990s and early 200-s were the availability of methods such as Support Vector Machines, Gradient and Stochastic Gradient Bootstring Models, Random Forests, and even penalized regression methods such as LASSO and Elastic Net, for classification and regression tasks.

While, in theory, deep neural networks had potentially greater representational power than these models since they built hierarchical representations of the input data through successive layers in contrast to the "shallow" representation given by a single transformation such as a regression weight or decision tree, in practice the Challenges of training deep networks made these "shallo" methods more attrative for practical applications. This was coupled with the fact that larger networks required tuning thousands or even millions of parameters, requiring larg-scale matrix calculations that were infeasible before the explosion of cheap compute resources - including GPUs and TPUs especially suited to rapid matrix calculations - available from cloud vendors made these experiments practical.

Now that we've covered the basics of training simple network architectures, let's turn to more complex models that will form the building blocks of many of the generative models in the rest of the book: CNNs and sequence models (RNNs, LSTMs, and other).

Backpropagation in practice

 While it is useful to go through this derivation in order to understand how the update rules for a deep neural network are derived, this would clearly quickly become unwieldy for large networks and complex architectures. It's fortunate, therefore, that TensorFlow 2 handles the computation of these gradients automatically. During the initialization of the model, each gradient is computed as an intermediate node between tensors and operations in the graph: as an example, see Figure 3.4:

The left side of the preceding figure shows a cost function C computed from the output of a Rectified Linear Unit (ReLU) - a type of neuron function we'll cover later in this chapter), which in turn is computed from multiplying a weight vector by an input x and adding a bias term b. On the right, you can see that this graph has been augmented by TensorFlow to compute all the intermediate gradients required for backpropagation as part of the overall control flow.

After storing these intermediate values, the task of combining them, as shown in the calculation in Figure 3.4, into a complete gradient through recursive operation falls to the GradientTape API. Under the hood, TensorFlow uses a method called reversemode automatic differentiation to compute gradients; it holds the dependent variable (the output y) fixed, and recursively computes backwards to the beginning of the network the required gradients.

For example, let's consider a neural network of the following form:

If we want to compute the derivative of the output y with respect to an input x we need to repeatedly substitute the outermost expression.

Thus, to compute the desired gradient we need to just traverse the graph from top to bottom, storing each intermediate gradient as we calculate it. These values are stored on a record, referred to as a tape in reference to early computers in which information was stored on a magnetic taps, which is then used to replay the values for calcuation. The alternative would be to use forward-mode automatic differentiation, computing, from bottom to top. This requires two instead of one pass(for each branch feeding into the final value), but is conceptually simpler to implement and doesn't require the storage memory of reverse mode. More importantly, though, reverse-mode minics the derivation of backpropagation that I described earlier.

The taps (aslo known as the Wengert Tape, after one of its devcelopers) is actually a data structure that you can access in the TensorFlow Core API. As an example, import the core library:

from __future__ import absolute_import, division, print_function, unicode_literals

import tensorflow as tf

The tape is then available using the tf.gradientTape() method, with which you can evaluate gradients with respect to intermediate values within the graph:

x = tf.one((2,2))

    with tf.GradientTape() as t:

        t.watch(x)

        y = tf.reduce_sum(x)

        z = tf.mutiply(y,y)

    # use the tap to compute the derivative of z with respect to the

    # intermediate value y.

    dz_dy = t.gredient(z,y)

    # note that the resulting derivative, 2*y, = sum(x)  *2 = 8

    assert bz_dy.numpy() == 8.0

By defualt, the memory resources used by GradientTape() are released once gradient() is called; however, you can also use the persistent argument to store these results:

x = tf.constant(3.0)

with tf.GradientTape(persistent = true) as t:

    t.watch(x)

    y = x * x

    z = y * y

dz_dx = t.gradient(z, x) # 108.0 (4*x ^3 at x =3)

dy_dx = t.gradient(y, x) # 6.0

Now that you've seen how TensorFlow computes gradients in practice to evaluate backpropagation, let's return to the details of how the backpropagation technique evolved over time in response to challenges in practical implementation.

Multi-layer perceptrons and backpropagation

 While large research funding for neural networks declined until the 1980s after the publication of Perceptrons, researchers still recongnized that these models had value, particularly when assembled into multi-layer networks, each composed of several perceptron units. Indeed, when the mathematical form of the output function (that is, the output of the model) was relaxed to take on manu forms (such as a linear function or a sigmoid), these networks could solve both regression and classification problems, with theoretical results showing that 3-layer networks could effectively approximate any output. However, none of this work address the practical limitations of computing the solutions to these models, with rules such as the perceptron learning algorithm described earlier proving a great limitation to the applied use of them.

Renewed interest in neural networks came with the popularization of the backpropagation algorithm, which, while discovered in the 1960s, was not widely applied to neural networks until the 1980s, following serveral studies highlighting it usefulness for learning the weights in these models. As you saw with the perceptron model, a learning urle to update weights is relatively easy to derive as long as there are no "hidden" layers. The input is transformed once by the perceptron to compute an output value, meaning the weights can be directly tuned to yield the desired output. When there are hidden layers between the input and output, the problem becomes more complex: when do we change the internal weights to compute the activations that feed into the final output? How do we modify them in relation to the input weights?

Then insight of the backpropagation technique is that we can use the chain rule from calculus to efficiently compute the derivatives of each parameter of a network with respect to a loss function and, combined with a learning rule, this provides a scalable way to train multilayer networks.

Let's illustrate backpropagation with an example: consider a network like the one shown in Figure 3.3. Assume that the output in the final layer is computed using a sigmoidal function, which yields a value between 0 and 1:

Furthermore, the value y, the sum of the inputs to the final neuron, is a weighted sum of the sigmoidal inputs of the hidden units:

We also need a notion of when the network is performing well or badly at its task. A straightforward error function to use here is squared loss:

where yhat is the estimated value (from the output of the model) and y is the real value, summed over all the input examples J and the output of the network K (where K=1, since there is only a single output value). Backpropagation begins with a "forwar pass" where we compute the values of all the outputs in the inner and outer layers, to obtain the estimated values of yhat. We then proceed with a backward step to compute gradients to update the weights.

Our overall objective is to compute partial derivatives for the weights w and bias terms b in each neuron:&E/&w and &E/&b, which will allow us to compute the updates for b and w. Towards this goal, let's start by computing the update rule for  the inputs in the final neuron; we want to date the partial derivative of the error E with respect to each of these inputs(in this example there are five, corresponding to the five hidden layer neurons), using the chain rule:

which for an individual example is just the difference between the input and output value. we need to take the partial derivative of the sigmoid function:

Putting it all together, we have:

If we want to compute the gradient for a particular parameter of x, such as a weight w or bias term b, we need on more step:

We already know the first term and x depends on w only through the inputs from the lower layers y since it is a linear function, so we obtain:

If  we want to compute this derivative for one of the neurons in the hidden layer, we likewise take the partial derivative with respect to this input y, which is simply:

So, in total we can sum over all units that feed into this hidden layer:

We can repeat this process recursively for any units in deeper layers to obtain the desired update rule, since we now know how to calculate the gradients for y or w at any layer. This makes the process of updating weights efficient since once we h ave computed the gradients through the backward pass we can combine consecutive gradients through the layers to get the required gradient at any depth of the network.

Now that we have the gradients for each w (or other parameter of the neuron we might want to calculate), how can we make a "learning rule" to update the weight? In their paper, Hinton et al. noted tat we could apply an update to the model parameters after computing gradients on each sample batch but suggested instead applying an update cakculated after averaging over all samples. The gradient represents the direction in which the error function is changing with the greatest magnitude with respect to the parameters; thus, to update, we want to push the weight in the opposite direction, with (w) the update, and e a small value (a step size):

Then at each time t during training we update the weight using this calculated gradient:

where alpha is a decay parameter to weight the contribution of prior updates ranging from 0 to 1. Following this procedure, we would initialize the weights in the network with some small random values, choose a step size e and iterate with forward and backward passes, along with updates to the parameters, until the loss function reaches some desired value.

Now that we have described the formal mathematics behind backpropagation, let us look at how it is implemented in practice in software packages such as TensorFlow 2.

From TLUs to tuning perceptrons

 Besides these limitations for representing the XOR and XNOR operations, there are additional simplifications that cap the representational power of the TLU model; the weights are fixed, and the output can only be binary (0 or 1). Clearly, for a system such as a neuron to "learn," it needs to respond to the environment and determine the relevance of different inputs based on feedback from prior experiences. This idea was captured in the 1949 book Organization of Behavior by Canadian Psychologist Donald Hebb, who proposed that the activity of nearby neuraonal cells would tend to synchronize over time, sometimes paraphrased at Hebb's Law: Neurons that fire together wire together. Building on Hubb's proposal that weights changed over time, researcher Frank Rosenblatt of the Cornell Aeronautical Laboratory proposed the perceptron model in the 1950s. He replaced the fixed weights in the TLU model with adaptive weights and added a bias term, giving a new function:

We note that the inputs I have been denoted X to underscore the fact that they could be any value, not just binary 0 or 1. Combining Hebb's observations with the TLU model, the weights of the perceptron would be updated according to a simple learning rule:

1. Start with a set of J samples x(1).....x(j). These samples all have a label y which is 0 or 1, giving labeled data(y, x)(1)...(y,x)(j). These samples could have either a single value, in which case the perceptron has a single input , or be a vector with length N and indices i for multi-value input.

2. Initialize all weights w to a small random value or 0.

3. Compute the estimated value, yhat, for all the examples x using the perceptron function.

4. Update the weights using a learning rate r to more closely match the input to the desired output for each step t in training:

wi(t+1) = wi(t) + r(yi - yhati)xji, for all J samples and Nfeatures. 

Conceptually, note that if y is 0 and the target is 1, we want to increase the value of the weight by some increment r; likewike, if the target is 0 and the estimate is 1, we want to decrease the weight so the inputs do not exceed the threshold.

5. Repeat step 3-4 until the difference between the prediced and actual ouputs, y and yhat, falls below some desired threshold. In the case of a non zero bias term, b, an update can be computed as well using a similar formula.

While simple, you can appreciate that many patterns could be learned from such a clasifier, though still not the XOR function, However, by combining serveral perceptrons into multiple layers, these units could represent any simple Boolean function, and indeed McCulloch and Pitts had previously speculated on combining such simple units into a universal computeatation engine, or Turing Machine, that could represent any operation in a standard programming language. However, the preceding learning algorithm operates on each unit independently, meaning it could be extended to networks composed of many layers of perceptrons.

however, the 1969 book Percetrons, by MIT computer scientists Marvin Minksy and Seymour Papert, demonstrated that a three-layer feed-forward network required complete (non-zero weight) connections between at least one of these units (in the first layer) and all inputs to compute all possible logical outputs. This meant that instead of having a very sparese structure, like biological neurons, which are only oconnected to a few of their neighbors, these computational modles required very dense connections.

While connective sparsity has been incorporated in later architectures, such as CNNs, such dense connections remain a feature of many models too, particularly in the fully connected layers that oftern form the secound to last hidden layers in models. In addition to these models being computationally unwieldy on the hardware of the day, the observation that spare models could not compute all logical operations was interpreted more broadly by the research community as Perceptrons cannot compute XOR. While erroneous, this message led to a drought in funding for AI in subsequent years, a period sometimes refferred to as the AI Winter.

The next revolution in neural network research would require a more efficient way to compute the required parameters updated in complex models, a technique that would become known as backpropagation.