Deep Networks: Overview
In the previous sections, you constructed a 3-layer neural network comprising an input, hidden and output layer. While fairly effective for MNIST, this 3-layer model is a fairly shallow network; by this, we mean that the features (hidden layer activations a(2)) are computed using only "one layer" of computation (the hidden layer).
In this section, we begin to discuss deep neural networks, meaning ones in which we have multiple hidden layers; this will allow us to compute much more complex features of the input. Because each hidden layer computes a non-linear transformation of the previous layer, a deep network can have significantly greater representational power (i.e., can learn significantly more complex functions) than a shallow one.
Note that when training a deep network, it is important to use a non-linear activation function in each hidden layer. This is because multiple layers of linear functions would itself compute only a linear function of the input (i.e., composing multiple linear functions together results in just another linear function), and thus be no more expressive than using just a single layer of hidden units.
Advantages of deep networks
Why do we want to use a deep network? The primary advantage is that it can compactly represent a significantly larger set of fuctions than shallow networks. Formally, one can show that there are functions which a k-layer network can represent compactly (with a number of hidden units that is polynomial in the number of inputs), that a (k − 1)-layer network cannot represent unless it has an exponentially large number of hidden units.
To take a simple example, consider building a boolean circuit/network to compute the parity (or XOR) of n input bits. Suppose each node in the network can compute either the logical OR of its inputs (or the OR of the negation of the inputs), or compute the logical AND. If we have a network with only one input, one hidden, and one output layer, the parity function would require a number of nodes that is exponential in the input size n. If however we are allowed a deeper network, then the network/circuit size can be only polynomial in n.
By using a deep network, in the case of images, one can also start to learn part-whole decompositions. For example, the first layer might learn to group together pixels in an image in order to detect edges (as seen in the earlier exercises). The second layer might then group together edges to detect longer contours, or perhaps detect simple "parts of objects." An even deeper layer might then group together these contours or detect even more complex features.
Finally, cortical computations (in the brain) also have multiple layers of processing. For example, visual images are processed in multiple stages by the brain, by cortical area "V1", followed by cortical area "V2" (a different part of the brain), and so on.
Difficulty of training deep architectures
While the theoretical benefits of deep networks in terms of their compactness and expressive power have been appreciated for many decades, until recently researchers had little success training deep architectures.
The main learning algorithm that researchers were using was to randomly initialize the weights of a deep network, and then train it using a labeled training set using a supervised learning objective, for example by applying gradient descent to try to drive down the training error. However, this usually did not work well. There were several reasons for this.
Availability of data
With the method described above, one relies only on labeled data for training. However, labeled data is often scarce, and thus for many problems it is difficult to get enough examples to fit the parameters of a complex model. For example, given the high degree of expressive power of deep networks, training on insufficient data would also result in overfitting.
Training a shallow network (with 1 hidden layer) using supervised learning usually resulted in the parameters converging to reasonable values; but when we are training a deep network, this works much less well. In particular, training a neural network using supervised learning involves solving a highly non-convex optimization problem (say, minimizing the training error as a function of the network parameters ). In a deep network, this problem turns out to be rife with bad local optima, and training with gradient descent (or methods like conjugate gradient and L-BFGS) no longer work well.
Diffusion of gradients
There is an additional technical reason, pertaining to the gradients becoming very small, that explains why gradient descent (and related algorithms like L-BFGS) do not work well on a deep networks with randomly initialized weights. Specifically, when using backpropagation to compute the derivatives, the gradients that are propagated backwards (from the output layer to the earlier layers of the network) rapidly diminish in magnitude as the depth of the network increases. As a result, the derivative of the overall cost with respect to the weights in the earlier layers is very small. Thus, when using gradient descent, the weights of the earlier layers change slowly, and the earlier layers fail to learn much. This problem is often called the "diffusion of gradients."
A closely related problem to the diffusion of gradients is that if the last few layers in a neural network have a large enough number of neurons, it may be possible for them to model the labeled data alone without the help of the earlier layers. Hence, training the entire network at once with all the layers randomly initialized ends up giving similar performance to training a shallow network (the last few layers) on corrupted input (the result of the processing done by the earlier layers).
Greedy layer-wise training
How can we train a deep network? One method that has seen some success is the greedy layer-wise training method. We describe this method in detail in later sections, but briefly, the main idea is to train the layers of the network one at a time, so that we first train a network with 1 hidden layer, and only after that is done, train a network with 2 hidden layers, and so on. At each step, we take the old network with k − 1 hidden layers, and add an additional k-th hidden layer (that takes as input the previous hidden layer k − 1 that we had just trained). Training can either be supervised (say, with classification error as the objective function on each step), but more frequently it is unsupervised (as in an autoencoder; details to provided later). The weights from training the layers individually are then used to initialize the weights in the final/overall deep network, and only then is the entire architecture "fine-tuned" (i.e., trained together to optimize the labeled training set error).
The success of greedy layer-wise training has been attributed to a number of factors:
Availability of data
While labeled data can be expensive to obtain, unlabeled data is cheap and plentiful. The promise of self-taught learning is that by exploiting the massive amount of unlabeled data, we can learn much better models. By using unlabeled data to learn a good initial value for the weights in all the layers (except for the final classification layer that maps to the outputs/predictions), our algorithm is able to learn and discover patterns from massively more amounts of data than purely supervised approaches. This often results in much better classifiers being learned.
Better local optima
After having trained the network on the unlabeled data, the weights are now starting at a better location in parameter space than if they had been randomly initialized. We can then further fine-tune the weights starting from this location. Empirically, it turns out that gradient descent from this location is much more likely to lead to a good local minimum, because the unlabeled data has already provided a significant amount of "prior" information about what patterns there are in the input data.
In the next section, we will describe the specific details of how to go about implementing greedy layer-wise training.