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The only thing hidden is that computing the cost involves a call to the cost method for the output layer; that code is elsewhere in network3.

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But that code is short and simple, anyway. By averaging over these functions, we will be able to compute accuracies on the entire validation and test data sets. The remainder of the SGD method is self-explanatory - we simply iterate over the epochs, repeatedly training the network on mini-batches of training data, and computing the validation and test accuracies. Okay, we've now understood the most important pieces of code in network3.

Let's take a brief look at the entire program. You don't need to read through this in detail, but you may enjoy glancing over it, and perhaps diving down into any pieces that strike your fancy. The best way to really understand it is, of course, by modifying it, adding extra features, or refactoring anything you think could be done more elegantly. After the code, there are some problems which contain a few starter suggestions for things to do.

In particular, it's easy to make the mistake of pulling data off the GPU, which can slow things down a lot. I've tried to avoid this. With that said, this code can certainly be sped up quite a bit further with careful optimization of Theano's configuration. See the Theano documentation for more details. Supports several layer types fully connected, convolutional, max pooling, softmax , and activation functions sigmoid, tanh, and rectified linear units, with more easily added. When run on a CPU, this program is much faster than network. However, unlike network. Because the code is based on Theano, the code is different in many ways from network.

However, where possible I have tried to maintain consistency with the earlier programs. In particular, the API is similar to network2. Note that I have focused on making the code simple, easily readable, and easily modifiable. It is not optimized, and omits many desirable features. Written for Theano 0. This allows Theano to copy the data to the GPU, if one is available. A more sophisticated implementation would separate the two, but for our purposes we'll always use them together, and it simplifies the code, so it makes sense to combine them. RandomStreams np. RandomState 0.

Mathematical Modelling: Theory and Applications

At present, the SGD method requires the user to manually choose the number of epochs to train for. Earlier in the book we discussed an automated way of selecting the number of epochs to train for, known as early stopping. Modify network3. Hint: After working on this problem for a while, you may find it useful to see the discussion at this link. Earlier in the chapter I described a technique for expanding the training data by applying small rotations, skewing, and translation.

Plausible Neural Networks for Biological Modelling

Note: Unless you have a tremendous amount of memory, it is not practical to explicitly generate the entire expanded data set. So you should consider alternate approaches. A shortcoming of the current code is that it provides few diagnostic tools. Can you think of any diagnostics to add that would make it easier to understand to what extent a network is overfitting? Add them.

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  5. We've used the same initialization procedure for rectified linear units as for sigmoid and tanh neurons. Our argument for that initialization was specific to the sigmoid function. Consider a network made entirely of rectified linear units including outputs. How does this change if the final layer is a softmax? What do you think of using the sigmoid initialization procedure for the rectified linear units? Can you think of a better initialization procedure?

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    Note: This is a very open-ended problem, not something with a simple self-contained answer. Still, considering the problem will help you better understand networks containing rectified linear units. Our analysis of the unstable gradient problem was for sigmoid neurons. How does the analysis change for networks made up of rectified linear units? Can you think of a good way of modifying such a network so it doesn't suffer from the unstable gradient problem?

    Note: The word good in the second part of this makes the problem a research problem. It's actually easy to think of ways of making such modifications. But I haven't investigated in enough depth to know of a really good technique. Recent progress in image recognition.

    The Math Behind Neural Networks (01)

    In , the year MNIST was introduced, it took weeks to train a state-of-the-art workstation to achieve accuracies substantially worse than those we can achieve using a GPU and less than an hour of training. Thus, MNIST is no longer a problem that pushes the limits of available technique; rather, the speed of training means that it is a problem good for teaching and learning purposes.

    Meanwhile, the focus of research has moved on, and modern work involves much more challenging image recognition problems. In this section, I briefly describe some recent work on image recognition using neural networks. The section is different to most of the book. Through the book I've focused on ideas likely to be of lasting interest - ideas such as backpropagation, regularization, and convolutional networks. I've tried to avoid results which are fashionable as I write, but whose long-term value is unknown.

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    6. In science, such results are more often than not ephemera which fade and have little lasting impact. Given this, a skeptic might say: "well, surely the recent progress in image recognition is an example of such ephemera? In another two or three years, things will have moved on. So surely these results are only of interest to a few specialists who want to compete at the absolute frontier? Why bother discussing it? Such a skeptic is right that some of the finer details of recent papers will gradually diminish in perceived importance.

      With that said, the past few years have seen extraordinary improvements using deep nets to attack extremely difficult image recognition tasks. Imagine a historian of science writing about computer vision in the year They will identify the years to and probably a few years beyond as a time of huge breakthroughs, driven by deep convolutional nets. That doesn't mean deep convolutional nets will still be used in , much less detailed ideas such as dropout, rectified linear units, and so on.

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      But it does mean that an important transition is taking place, right now, in the history of ideas. It's a bit like watching the discovery of the atom, or the invention of antibiotics: invention and discovery on a historic scale. And so while we won't dig down deep into details, it's worth getting some idea of the exciting discoveries currently being made.