AI News, Document worth reading: &#8220;Does putting your emotions into words make you feel better? Measuring the minute-scale dynamics of emotions from onlinedata&#8221;

Document worth reading: &#8220;Does putting your emotions into words make you feel better? Measuring the minute-scale dynamics of emotions from onlinedata&#8221;

&#8216;Quantum Equilibrium-Disequilibrium&#8217;: Asset Price Dynamics, Symmetry Breaking, and Defaults as Dissipative Instantons• Effective Caching for the Secure Content Distribution in Information-Centric Networking• Self-Adaptive Systems in Organic Computing: Strategies for Self-Improvement• A Hybrid Recommender System for Patient-Doctor Matchmaking in Primary Care• Evolutionary optimisation of neural network models for fish collective behaviours in mixed groups of robots and zebrafish• Random forest prediction of Alzheimer&#8217;s disease using pairwise selection from time series data• VerIDeep: Verifying Integrity of Deep Neural Networks through Sensitive-Sample Fingerprinting• Temporal starvation in multi-channel CSMA networks: an analytical framework• Who Falls for Online Political Manipulation?• The frog model on trees with drift• Code-Mixed Sentiment Analysis Using Machine Learning and Neural Network Approaches• $α$-Approximation Density-based Clustering of Multi-valued 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to Attrition Bias in Big Data: Evidence from a Randomized Experiment• A note on partial rejection sampling for the hard disks model in the plane• Blue Phase: Optimal Network Traffic Control for Legacy and Autonomous Vehicles• A survey of data transfer and storage techniques in prevalent cryptocurrencies and suggested improvements• Code-division multiplexed resistive pulse sensor networks for spatio-temporal detection of particles in microfluidic devices• Classes of graphs with e-positive chromatic symmetric function• Distinctiveness, complexity, and repeatability of online signature templates• End-to-end Active Object Tracking and Its Real-world Deployment via Reinforcement Learning• A note on the critical barrier for the survival of $α-$stable branching random walk with absorption• On the Convergence of AdaGrad with Momentum for Training Deep Neural Networks• Linear time algorithm to check the singularity of block graphs• Efficient Measurement on Programmable SwitchesUsing Probabilistic Recirculation• DeepWrinkles: Accurate and Realistic Clothing Modeling• (De)localization of Fermions in Correlated-Spin Background• Learning and Inference on Generative Adversarial Quantum Circuits• WonDerM: Skin Lesion Classification with Fine-tuned Neural Networks• Power Minimization Based Joint Task Scheduling and Resource Allocation in Downlink C-RAN• Stochastic $R_0$ Tensors to Stochastic Tensor Complementarity Problems• Hybrid approach for transliteration of Algerian arabizi: a primary study• Spin systems on Bethe lattices• On the optimal designs for the prediction of complex Ornstein-Uhlenbeck processes• The Moment-SOS hierarchy• Stability for Intersecting Families of Perfect Matchings• Weakly-Supervised Attention and Relation Learning for Facial Action Unit Detection• The Evolution of Sex Chromosomes through the Baldwin Effect• Cross-location wind speed forecasting for wind energy applications using machine learning based models• Deep Learning Based Speed 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objects using recursive image fusion• Dynamic all scores matrices for LCS score• Ektelo: A Framework for Defining Differentially-Private Computations• Existence of symmetric maximal noncrossing collections of $k$-element sets• Finding a Small Number of Colourful Components• Choosing the optimal multi-point iterative method for the Colebrook flow friction equation &#8212;

Deep learning

Deep learning (also known as deep structured learning or hierarchical learning) is part of a broader family of machine learning methods based on learning data representations, as opposed to task-specific algorithms.

Deep learning architectures such as deep neural networks, deep belief networks and recurrent neural networks have been applied to fields including computer vision, speech recognition, natural language processing, audio recognition, social network filtering, machine translation, bioinformatics, drug design and board game programs, where they have produced results comparable to and in some cases superior to human experts.&#91;4&#93;&#91;5&#93;&#91;6&#93;

Deep learning models are vaguely inspired by information processing and communication patterns in biological nervous systems yet have various differences from the structural and functional properties of biological brains, which make them incompatible with neuroscience evidences.&#91;7&#93;&#91;8&#93;&#91;9&#93;

Most modern deep learning models are based on an artificial neural network, although they can also include propositional formulas or latent variables organized layer-wise in deep generative models such as the nodes in Deep Belief Networks and Deep Boltzmann Machines.&#91;11&#93;

No universally agreed upon threshold of depth divides shallow learning from deep learning, but most researchers agree that deep learning involves CAP depth &gt;

For supervised learning tasks, deep learning methods obviate feature engineering, by translating the data into compact intermediate representations akin to principal components, and derive layered structures that remove redundancy in representation.

The universal approximation theorem concerns the capacity of feedforward neural networks with a single hidden layer of finite size to approximate continuous functions.&#91;15&#93;&#91;16&#93;&#91;17&#93;&#91;18&#93;&#91;19&#93;

By 1991 such systems were used for recognizing isolated 2-D hand-written digits, while recognizing 3-D objects was done by matching 2-D images with a handcrafted 3-D object model.

But while Neocognitron required a human programmer to hand-merge features, Cresceptron learned an open number of features in each layer without supervision, where each feature is represented by a convolution kernel.

In 1994, André de Carvalho, together with Mike Fairhurst and David Bisset, published experimental results of a multi-layer boolean neural network, also known as a weightless neural network, composed of a 3-layers self-organising feature extraction neural network module (SOFT) followed by a multi-layer classification neural network module (GSN), which were independently trained.

In 1995, Brendan Frey demonstrated that it was possible to train (over two days) a network containing six fully connected layers and several hundred hidden units using the wake-sleep algorithm, co-developed with Peter Dayan and Hinton.&#91;39&#93;

Simpler models that use task-specific handcrafted features such as Gabor filters and support vector machines (SVMs) were a popular choice in the 1990s and 2000s, because of ANNs' computational cost and a lack of understanding of how the brain wires its biological networks.

These methods never outperformed non-uniform internal-handcrafting Gaussian mixture model/Hidden Markov model (GMM-HMM) technology based on generative models of speech trained discriminatively.&#91;45&#93;

The principle of elevating 'raw' features over hand-crafted optimization was first explored successfully in the architecture of deep autoencoder on the 'raw' spectrogram or linear filter-bank features in the late 1990s,&#91;48&#93;

Many aspects of speech recognition were taken over by a deep learning method called long short-term memory (LSTM), a recurrent neural network published by Hochreiter and Schmidhuber in 1997.&#91;50&#93;

showed how a many-layered feedforward neural network could be effectively pre-trained one layer at a time, treating each layer in turn as an unsupervised restricted Boltzmann machine, then fine-tuning it using supervised backpropagation.&#91;58&#93;

The impact of deep learning in industry began in the early 2000s, when CNNs already processed an estimated 10% to 20% of all the checks written in the US, according to Yann LeCun.&#91;67&#93;

was motivated by the limitations of deep generative models of speech, and the possibility that given more capable hardware and large-scale data sets that deep neural nets (DNN) might become practical.

However, it was discovered that replacing pre-training with large amounts of training data for straightforward backpropagation when using DNNs with large, context-dependent output layers produced error rates dramatically lower than then-state-of-the-art Gaussian mixture model (GMM)/Hidden Markov Model (HMM) and also than more-advanced generative model-based systems.&#91;59&#93;&#91;70&#93;

offering technical insights into how to integrate deep learning into the existing highly efficient, run-time speech decoding system deployed by all major speech recognition systems.&#91;10&#93;&#91;72&#93;&#91;73&#93;

In 2010, researchers extended deep learning from TIMIT to large vocabulary speech recognition, by adopting large output layers of the DNN based on context-dependent HMM states constructed by decision trees.&#91;75&#93;&#91;76&#93;&#91;77&#93;&#91;72&#93;

In 2009, Nvidia was involved in what was called the “big bang” of deep learning, “as deep-learning neural networks were trained with Nvidia graphics processing units (GPUs).”&#91;78&#93;

In 2014, Hochreiter's group used deep learning to detect off-target and toxic effects of environmental chemicals in nutrients, household products and drugs and won the 'Tox21 Data Challenge' of NIH, FDA and NCATS.&#91;87&#93;&#91;88&#93;&#91;89&#93;

Although CNNs trained by backpropagation had been around for decades, and GPU implementations of NNs for years, including CNNs, fast implementations of CNNs with max-pooling on GPUs in the style of Ciresan and colleagues were needed to progress on computer vision.&#91;80&#93;&#91;81&#93;&#91;34&#93;&#91;90&#93;&#91;2&#93;

In November 2012, Ciresan et al.'s system also won the ICPR contest on analysis of large medical images for cancer detection, and in the following year also the MICCAI Grand Challenge on the same topic.&#91;92&#93;

In 2013 and 2014, the error rate on the ImageNet task using deep learning was further reduced, following a similar trend in large-scale speech recognition.

For example, in image recognition, they might learn to identify images that contain cats by analyzing example images that have been manually labeled as 'cat' or 'no cat' and using the analytic results to identify cats in other images.

Over time, attention focused on matching specific mental abilities, leading to deviations from biology such as backpropagation, or passing information in the reverse direction and adjusting the network to reflect that information.

Neural networks have been used on a variety of tasks, including computer vision, speech recognition, machine translation, social network filtering, playing board and video games and medical diagnosis.

Despite this number being several order of magnitude less than the number of neurons on a human brain, these networks can perform many tasks at a level beyond that of humans (e.g., recognizing faces, playing 'Go'&#91;99&#93;

The extra layers enable composition of features from lower layers, potentially modeling complex data with fewer units than a similarly performing shallow network.&#91;11&#93;

The training process can be guaranteed to converge in one step with a new batch of data, and the computational complexity of the training algorithm is linear with respect to the number of neurons involved.&#91;115&#93;&#91;116&#93;

that involve multi-second intervals containing speech events separated by thousands of discrete time steps, where one time step corresponds to about 10 ms.

All major commercial speech recognition systems (e.g., Microsoft Cortana, Xbox, Skype Translator, Amazon Alexa, Google Now, Apple Siri, Baidu and iFlyTek voice search, and a range of Nuance speech products, etc.) are based on deep learning.&#91;10&#93;&#91;122&#93;&#91;123&#93;&#91;124&#93;

DNNs have proven themselves capable, for example, of a) identifying the style period of a given painting, b) 'capturing' the style of a given painting and applying it in a visually pleasing manner to an arbitrary photograph, and c) generating striking imagery based on random visual input fields.&#91;128&#93;&#91;129&#93;

Word embedding, such as word2vec, can be thought of as a representational layer in a deep learning architecture that transforms an atomic word into a positional representation of the word relative to other words in the dataset;

Finding the appropriate mobile audience for mobile advertising is always challenging, since many data points must be considered and assimilated before a target segment can be created and used in ad serving by any ad server.&#91;161&#93;&#91;162&#93;

'Deep anti-money laundering detection system can spot and recognize relationships and similarities between data and, further down the road, learn to detect anomalies or classify and predict specific events'.

Deep learning is closely related to a class of theories of brain development (specifically, neocortical development) proposed by cognitive neuroscientists in the early 1990s.&#91;165&#93;&#91;166&#93;&#91;167&#93;&#91;168&#93;

These developmental models share the property that various proposed learning dynamics in the brain (e.g., a wave of nerve growth factor) support the self-organization somewhat analogous to the neural networks utilized in deep learning models.

Like the neocortex, neural networks employ a hierarchy of layered filters in which each layer considers information from a prior layer (or the operating environment), and then passes its output (and possibly the original input), to other layers.

Other researchers have argued that unsupervised forms of deep learning, such as those based on hierarchical generative models and deep belief networks, may be closer to biological reality.&#91;172&#93;&#91;173&#93;

Such techniques lack ways of representing causal relationships (...) have no obvious ways of performing logical inferences, and they are also still a long way from integrating abstract knowledge, such as information about what objects are, what they are for, and how they are typically used.

systems, like Watson (...) use techniques like deep learning as just one element in a very complicated ensemble of techniques, ranging from the statistical technique of Bayesian inference to deductive reasoning.'&#91;187&#93;

As an alternative to this emphasis on the limits of deep learning, one author speculated that it might be possible to train a machine vision stack to perform the sophisticated task of discriminating between 'old master' and amateur figure drawings, and hypothesized that such a sensitivity might represent the rudiments of a non-trivial machine empathy.&#91;188&#93;

In further reference to the idea that artistic sensitivity might inhere within relatively low levels of the cognitive hierarchy, a published series of graphic representations of the internal states of deep (20-30 layers) neural networks attempting to discern within essentially random data the images on which they were trained&#91;190&#93;

Learning a grammar (visual or linguistic) from training data would be equivalent to restricting the system to commonsense reasoning that operates on concepts in terms of grammatical production rules and is a basic goal of both human language acquisition&#91;196&#93;

Such a manipulation is termed an “adversarial attack.” In 2016 researchers used one ANN to doctor images in trial and error fashion, identify another's focal points and thereby generate images that deceived it.

Another group showed that certain psychedelic spectacles could fool a facial recognition system into thinking ordinary people were celebrities, potentially allowing one person to impersonate another.

ANNs can however be further trained to detect attempts at deception, potentially leading attackers and defenders into an arms race similar to the kind that already defines the malware defense industry.

ANNs have been trained to defeat ANN-based anti-malware software by repeatedly attacking a defense with malware that was continually altered by a genetic algorithm until it tricked the anti-malware while retaining its ability to damage the target.&#91;198&#93;

Deep learning

In the last chapter we learned that deep neural networks are often much harder to train than shallow neural networks.

We'll also look at the broader picture, briefly reviewing recent progress on using deep nets for image recognition, speech recognition, and other applications.

We'll work through a detailed example - code and all - of using convolutional nets to solve the problem of classifying handwritten digits from the MNIST data set:

As we go we'll explore many powerful techniques: convolutions, pooling, the use of GPUs to do far more training than we did with our shallow networks, the algorithmic expansion of our training data (to reduce overfitting), the use of the dropout technique (also to reduce overfitting), the use of ensembles of networks, and others.

We conclude our discussion of image recognition with a survey of some of the spectacular recent progress using networks (particularly convolutional nets) to do image recognition.

We'll briefly survey other models of neural networks, such as recurrent neural nets and long short-term memory units, and how such models can be applied to problems in speech recognition, natural language processing, and other areas.

And we'll speculate about the future of neural networks and deep learning, ranging from ideas like intention-driven user interfaces, to the role of deep learning in artificial intelligence.

For the $28 \times 28$ pixel images we've been using, this means our network has $784$ ($= 28 \times 28$) input neurons.

Our earlier networks work pretty well: we've obtained a classification accuracy better than 98 percent, using training and test data from the MNIST handwritten digit data set.

But the seminal paper establishing the modern subject of convolutional networks was a 1998 paper, 'Gradient-based learning applied to document recognition', by Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner.

LeCun has since made an interesting remark on the terminology for convolutional nets: 'The [biological] neural inspiration in models like convolutional nets is very tenuous.

That's why I call them 'convolutional nets' not 'convolutional neural nets', and why we call the nodes 'units' and not 'neurons' '.

Despite this remark, convolutional nets use many of the same ideas as the neural networks we've studied up to now: ideas such as backpropagation, gradient descent, regularization, non-linear activation functions, and so on.

In a convolutional net, it'll help to think instead of the inputs as a $28 \times 28$ square of neurons, whose values correspond to the $28 \times 28$ pixel intensities we're using as inputs:

To be more precise, each neuron in the first hidden layer will be connected to a small region of the input neurons, say, for example, a $5 \times 5$ region, corresponding to $25$ input pixels.

So, for a particular hidden neuron, we might have connections that look like this: That region in the input image is called the local receptive field for the hidden neuron.

To illustrate this concretely, let's start with a local receptive field in the top-left corner: Then we slide the local receptive field over by one pixel to the right (i.e., by one neuron), to connect to a second hidden neuron:

Note that if we have a $28 \times 28$ input image, and $5 \times 5$ local receptive fields, then there will be $24 \times 24$ neurons in the hidden layer.

This is because we can only move the local receptive field $23$ neurons across (or $23$ neurons down), before colliding with the right-hand side (or bottom) of the input image.

In this chapter we'll mostly stick with stride length $1$, but it's worth knowing that people sometimes experiment with different stride lengths* *As was done in earlier chapters, if we're interested in trying different stride lengths then we can use validation data to pick out the stride length which gives the best performance.

The same approach may also be used to choose the size of the local receptive field - there is, of course, nothing special about using a $5 \times 5$ local receptive field.

In general, larger local receptive fields tend to be helpful when the input images are significantly larger than the $28 \times 28$ pixel MNIST images..

In other words, for the $j, k$th hidden neuron, the output is: \begin{eqnarray} \sigma\left(b + \sum_{l=0}^4 \sum_{m=0}^4 w_{l,m} a_{j+l, k+m} \right).

Informally, think of the feature detected by a hidden neuron as the kind of input pattern that will cause the neuron to activate: it might be an edge in the image, for instance, or maybe some other type of shape.

To see why this makes sense, suppose the weights and bias are such that the hidden neuron can pick out, say, a vertical edge in a particular local receptive field.

To put it in slightly more abstract terms, convolutional networks are well adapted to the translation invariance of images: move a picture of a cat (say) a little ways, and it's still an image of a cat* *In fact, for the MNIST digit classification problem we've been studying, the images are centered and size-normalized.

One of the early convolutional networks, LeNet-5, used $6$ feature maps, each associated to a $5 \times 5$ local receptive field, to recognize MNIST digits.

Let's take a quick peek at some of the features which are learned* *The feature maps illustrated come from the final convolutional network we train, see here.:

Each map is represented as a $5 \times 5$ block image, corresponding to the $5 \times 5$ weights in the local receptive field.

By comparison, suppose we had a fully connected first layer, with $784 = 28 \times 28$ input neurons, and a relatively modest $30$ hidden neurons, as we used in many of the examples earlier in the book.

That, in turn, will result in faster training for the convolutional model, and, ultimately, will help us build deep networks using convolutional layers.

Incidentally, the name convolutional comes from the fact that the operation in Equation (125)\begin{eqnarray} \sigma\left(b + \sum_{l=0}^4 \sum_{m=0}^4 w_{l,m} a_{j+l, k+m} \right) \nonumber\end{eqnarray}$('#margin_223550267310_reveal').click(function() {$('#margin_223550267310').toggle('slow', function() {});});

A little more precisely, people sometimes write that equation as $a^1 = \sigma(b + w * a^0)$, where $a^1$ denotes the set of output activations from one feature map, $a^0$ is the set of input activations, and $*$ is called a convolution operation.

In particular, I'm using 'feature map' to mean not the function computed by the convolutional layer, but rather the activation of the hidden neurons output from the layer.

In max-pooling, a pooling unit simply outputs the maximum activation in the $2 \times 2$ input region, as illustrated in the following diagram:

Note that since we have $24 \times 24$ neurons output from the convolutional layer, after pooling we have $12 \times 12$ neurons.

So if there were three feature maps, the combined convolutional and max-pooling layers would look like:

Here, instead of taking the maximum activation of a $2 \times 2$ region of neurons, we take the square root of the sum of the squares of the activations in the $2 \times 2$ region.

It's similar to the architecture we were just looking at, but has the addition of a layer of $10$ output neurons, corresponding to the $10$ possible values for MNIST digits ('0', '1', '2', etc):

Problem Backpropagation in a convolutional network The core equations of backpropagation in a network with fully-connected layers are (BP1)\begin{eqnarray} \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j) \nonumber\end{eqnarray}$('#margin_511945174620_reveal').click(function() {$('#margin_511945174620').toggle('slow', function() {});});-(BP4)\begin{eqnarray} \frac{\partial C}{\partial w^l_{jk}} = a^{l-1}_k \delta^l_j \nonumber\end{eqnarray}$('#margin_896578903066_reveal').click(function() {$('#margin_896578903066').toggle('slow', function() {});});

Suppose we have a network containing a convolutional layer, a max-pooling layer, and a fully-connected output layer, as in the network discussed above.

The program we'll use to do this is called network3.py, and it's an improved version of the programs network.py and network2.py developed in earlier chapters* *Note also that network3.py incorporates ideas from the Theano library's documentation on convolutional neural nets (notably the implementation of LeNet-5), from Misha Denil's implementation of dropout, and from Chris Olah..

But now that we understand those details, for network3.py we're going to use a machine learning library known as Theano* *See Theano: A CPU and GPU Math Expression Compiler in Python, by James Bergstra, Olivier Breuleux, Frederic Bastien, Pascal Lamblin, Ravzan Pascanu, Guillaume Desjardins, Joseph Turian, David Warde-Farley, and Yoshua Bengio (2010).

The examples which follow were run using Theano 0.6* *As I release this chapter, the current version of Theano has changed to version 0.7.

Note that the code in the script simply duplicates and parallels the discussion in this section.Note also that throughout the section I've explicitly specified the number of training epochs.

In practice, it's worth using early stopping, that is, tracking accuracy on the validation set, and stopping training when we are confident the validation accuracy has stopped improving.: &gt;&gt;&gt;

Using the validation data to decide when to evaluate the test accuracy helps avoid overfitting to the test data (see this earlier discussion of the use of validation data).

Your results may vary slightly, since the network's weights and biases are randomly initialized* *In fact, in this experiment I actually did three separate runs training a network with this architecture.

This $97.80$ percent accuracy is close to the $98.04$ percent accuracy obtained back in Chapter 3, using a similar network architecture and learning hyper-parameters.

Second, while the final layer in the earlier network used sigmoid activations and the cross-entropy cost function, the current network uses a softmax final layer, and the log-likelihood cost function.

I haven't made this switch for any particularly deep reason - mostly, I've done it because softmax plus log-likelihood cost is more common in modern image classification networks.

In this architecture, we can think of the convolutional and pooling layers as learning about local spatial structure in the input training image, while the later, fully-connected layer learns at a more abstract level, integrating global information from across the entire image.

filter_shape=(20, 1, 5, 5),

poolsize=(2, 2)),

validation_data, test_data)

Can we improve on the $98.78$ percent classification accuracy?

filter_shape=(20, 1, 5, 5),

poolsize=(2, 2)),

filter_shape=(40, 20, 5, 5),

poolsize=(2, 2)),

validation_data, test_data)

In fact, you can think of the second convolutional-pooling layer as having as input $12 \times 12$ 'images', whose 'pixels' represent the presence (or absence) of particular localized features in the original input image.

The output from the previous layer involves $20$ separate feature maps, and so there are $20 \times 12 \times 12$ inputs to the second convolutional-pooling layer.

In fact, we'll allow each neuron in this layer to learn from all $20 \times 5 \times 5$ input neurons in its local receptive field.

More informally: the feature detectors in the second convolutional-pooling layer have access to all the features from the previous layer, but only within their particular local receptive field* *This issue would have arisen in the first layer if the input images were in color.

In that case we'd have 3 input features for each pixel, corresponding to red, green and blue channels in the input image.

So we'd allow the feature detectors to have access to all color information, but only within a given local receptive field..

Problem Using the tanh activation function Several times earlier in the book I've mentioned arguments that the tanh function may be a better activation function than the sigmoid function.

Try training the network with tanh activations in the convolutional and fully-connected layers* *Note that you can pass activation_fn=tanh as a parameter to the ConvPoolLayer and FullyConnectedLayer classes..

Try plotting the per-epoch validation accuracies for both tanh- and sigmoid-based networks, all the way out to $60$ epochs.

If your results are similar to mine, you'll find the tanh networks train a little faster, but the final accuracies are very similar.

Can you get a similar training speed with the sigmoid, perhaps by changing the learning rate, or doing some rescaling* *You may perhaps find inspiration in recalling that $\sigma(z) = (1+\tanh(z/2))/2$.?

Try a half-dozen iterations on the learning hyper-parameters or network architecture, searching for ways that tanh may be superior to the sigmoid.

Personally, I did not find much advantage in switching to tanh, although I haven't experimented exhaustively, and perhaps you may find a way.

In any case, in a moment we will find an advantage in switching to the rectified linear activation function, and so we won't go any deeper into the use of tanh.

Using rectified linear units: The network we've developed at this point is actually a variant of one of the networks used in the seminal 1998 paper* *'Gradient-based learning applied to document recognition', by Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner (1998).

filter_shape=(20, 1, 5, 5),

poolsize=(2, 2),

activation_fn=ReLU),

filter_shape=(40, 20, 5, 5),

poolsize=(2, 2),

activation_fn=ReLU),

However, across all my experiments I found that networks based on rectified linear units consistently outperformed networks based on sigmoid activation functions.

The reason for that recent adoption is empirical: a few people tried rectified linear units, often on the basis of hunches or heuristic arguments* *A common justification is that $\max(0, z)$ doesn't saturate in the limit of large $z$, unlike sigmoid neurons, and this helps rectified linear units continue learning.

A simple way of expanding the training data is to displace each training image by a single pixel, either up one pixel, down one pixel, left one pixel, or right one pixel.

filter_shape=(20, 1, 5, 5),

poolsize=(2, 2),

activation_fn=ReLU),

filter_shape=(40, 20, 5, 5),

poolsize=(2, 2),

activation_fn=ReLU),

Just to remind you of the flavour of some of the results in that earlier discussion: in 2003 Simard, Steinkraus and Platt* *Best Practices for Convolutional Neural Networks Applied to Visual Document Analysis, by Patrice Simard, Dave Steinkraus, and John Platt (2003).

improved their MNIST performance to $99.6$ percent using a neural network otherwise very similar to ours, using two convolutional-pooling layers, followed by a hidden fully-connected layer with $100$ neurons.

There were a few differences of detail in their architecture - they didn't have the advantage of using rectified linear units, for instance - but the key to their improved performance was expanding the training data.

filter_shape=(20, 1, 5, 5),

poolsize=(2, 2),

activation_fn=ReLU),

filter_shape=(40, 20, 5, 5),

poolsize=(2, 2),

activation_fn=ReLU),

filter_shape=(20, 1, 5, 5),

poolsize=(2, 2),

activation_fn=ReLU),

filter_shape=(40, 20, 5, 5),

poolsize=(2, 2),

activation_fn=ReLU),

Using this, we obtain an accuracy of $99.60$ percent, which is a substantial improvement over our earlier results, especially our main benchmark, the network with $100$ hidden neurons, where we achieved $99.37$ percent.

In fact, I tried experiments with both $300$ and $1,000$ hidden neurons, and obtained (very slightly) better validation performance with $1,000$ hidden neurons.

Why we only applied dropout to the fully-connected layers: If you look carefully at the code above, you'll notice that we applied dropout only to the fully-connected section of the network, not to the convolutional layers.

But apart from that, they used few other tricks, including no convolutional layers: it was a plain, vanilla network, of the kind that, with enough patience, could have been trained in the 1980s (if the MNIST data set had existed), given enough computing power.

In particular, we saw that the gradient tends to be quite unstable: as we move from the output layer to earlier layers the gradient tends to either vanish (the vanishing gradient problem) or explode (the exploding gradient problem).

In particular, in our final experiments we trained for $40$ epochs using a data set $5$ times larger than the raw MNIST training data.

I've occasionally heard people adopt a deeper-than-thou attitude, holding that if you're not keeping-up-with-the-Joneses in terms of number of hidden layers, then you're not really doing deep learning.

To speed that process up you may find it helpful to revisit Chapter 3's discussion of how to choose a neural network's hyper-parameters, and perhaps also to look at some of the further reading suggested in that section.

Here's the code (discussion below)* *Note added November 2016: several readers have noted that in the line initializing self.w, I set scale=np.sqrt(1.0/n_out), when the arguments of Chapter 3 suggest a better initialization may be scale=np.sqrt(1.0/n_in).

np.random.normal(

loc=0.0, scale=np.sqrt(1.0/n_out), size=(n_in, n_out)),

dtype=theano.config.floatX),

dtype=theano.config.floatX),

I use the name inpt rather than input because input is a built-in function in Python, and messing with built-ins tends to cause unpredictable behavior and difficult-to-diagnose bugs.

So self.inpt_dropout and self.output_dropout are used during training, while self.inpt and self.output are used for all other purposes, e.g., evaluating accuracy on the validation and test data.

prev_layer, layer = self.layers[j-1], self.layers[j]

prev_layer.output, prev_layer.output_dropout, self.mini_batch_size)

Now, this isn't a Theano tutorial, and so we won't get too deeply into what it means that these are symbolic variables* *The Theano documentation provides a good introduction to Theano.

0.5*lmbda*l2_norm_squared/num_training_batches

self.x:

training_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size],

self.y:

training_y[i*self.mini_batch_size: (i+1)*self.mini_batch_size]

self.x:

validation_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size],

self.y:

validation_y[i*self.mini_batch_size: (i+1)*self.mini_batch_size]

self.x:

test_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size],

self.y:

test_y[i*self.mini_batch_size: (i+1)*self.mini_batch_size]

self.x:

test_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size]

iteration = num_training_batches*epoch+minibatch_index

if iteration

print(&quot;Training mini-batch number {0}&quot;.format(iteration))

cost_ij = train_mb(minibatch_index)

if (iteration+1)

validation_accuracy = np.mean(

[validate_mb_accuracy(j) for j in xrange(num_validation_batches)])

print(&quot;Epoch {0}: validation accuracy {1:.2

epoch, validation_accuracy))

if validation_accuracy &gt;= best_validation_accuracy:

print(&quot;This is the best validation accuracy to date.&quot;)

best_validation_accuracy = validation_accuracy

best_iteration = iteration

if test_data:

test_accuracy = np.mean(

[test_mb_accuracy(j) for j in xrange(num_test_batches)])

print(&#39;The corresponding test accuracy is {0:.2

test_accuracy))

0.5*lmbda*l2_norm_squared/num_training_batches

In these lines we symbolically set up the regularized log-likelihood cost function, compute the corresponding derivatives in the gradient function, as well as the corresponding parameter updates.

With all these things defined, the stage is set to define the train_mb function, a Theano symbolic function which uses the updates to update the Network parameters, given a mini-batch index.

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.

prev_layer, layer = self.layers[j-1], self.layers[j]

prev_layer.output, prev_layer.output_dropout, self.mini_batch_size)

0.5*lmbda*l2_norm_squared/num_training_batches

self.x:

training_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size],

self.y:

training_y[i*self.mini_batch_size: (i+1)*self.mini_batch_size]

self.x:

validation_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size],

self.y:

validation_y[i*self.mini_batch_size: (i+1)*self.mini_batch_size]

self.x:

test_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size],

self.y:

test_y[i*self.mini_batch_size: (i+1)*self.mini_batch_size]

self.x:

test_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size]

iteration = num_training_batches*epoch+minibatch_index

if iteration % 1000 == 0:

print(&quot;Training mini-batch number {0}&quot;.format(iteration))

cost_ij = train_mb(minibatch_index)

if (iteration+1) % num_training_batches == 0:

validation_accuracy = np.mean(

[validate_mb_accuracy(j) for j in xrange(num_validation_batches)])

print(&quot;Epoch {0}: validation accuracy {1:.2%}&quot;.format(

epoch, validation_accuracy))

if validation_accuracy &gt;= best_validation_accuracy:

print(&quot;This is the best validation accuracy to date.&quot;)

best_validation_accuracy = validation_accuracy

best_iteration = iteration

if test_data:

test_accuracy = np.mean(

[test_mb_accuracy(j) for j in xrange(num_test_batches)])

print(&#39;The corresponding test accuracy is {0:.2%}&#39;.format(

test_accuracy))

activation_fn=sigmoid):

of filters, the number of input feature maps, the filter height, and the

poolsize is a tuple of length 2, whose entries are the y and

np.random.normal(loc=0, scale=np.sqrt(1.0/n_out), size=filter_shape),

dtype=theano.config.floatX),

np.random.normal(loc=0, scale=1.0, size=(filter_shape[0],)),

dtype=theano.config.floatX),

pooled_out + self.b.dimshuffle(&#39;x&#39;, 0, &#39;x&#39;, &#39;x&#39;))

np.random.normal(

loc=0.0, scale=np.sqrt(1.0/n_out), size=(n_in, n_out)),

dtype=theano.config.floatX),

dtype=theano.config.floatX),

Earlier in the book we discussed an automated way of selecting the number of epochs to train for, known as early stopping.

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.

Note: Unless you have a tremendous amount of memory, it is not practical to explicitly generate the entire expanded data set.

Show that rescaling all the weights in the network by a constant factor $c > 0$ simply rescales the outputs by a factor $c^{L-1}$, where $L$ is the number of layers.

Still, considering the problem will help you better understand networks containing rectified linear units.

Note: The word good in the second part of this makes the problem a research problem.

In 1998, 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.

With that said, the past few years have seen extraordinary improvements using deep nets to attack extremely difficult image recognition tasks.

They will identify the years 2011 to 2015 (and probably a few years beyond) as a time of huge breakthroughs, driven by deep convolutional nets.

The 2012 LRMD paper: Let me start with a 2012 paper* *Building high-level features using large scale unsupervised learning, by Quoc Le, Marc'Aurelio Ranzato, Rajat Monga, Matthieu Devin, Kai Chen, Greg Corrado, Jeff Dean, and Andrew Ng (2012).

Note that the detailed architecture of the network used in the paper differed in many details from the deep convolutional networks we've been studying.

Details about ImageNet are available in the original ImageNet paper, ImageNet: a large-scale hierarchical image database, by Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei (2009).:

If you're looking for a challenge, I encourage you to visit ImageNet's list of hand tools, which distinguishes between beading planes, block planes, chamfer planes, and about a dozen other types of plane, amongst other categories.

The 2012 KSH paper: The work of LRMD was followed by a 2012 paper of Krizhevsky, Sutskever and Hinton (KSH)* *ImageNet classification with deep convolutional neural networks, by Alex Krizhevsky, Ilya Sutskever, and Geoffrey E.

By this top-$5$ criterion, KSH's deep convolutional network achieved an accuracy of $84.7$ percent, vastly better than the next-best contest entry, which achieved an accuracy of $73.8$ percent.

The input layer contains $3 \times 224 \times 224$ neurons, representing the RGB values for a $224 \times 224$ image.

The feature maps are split into two groups of $48$ each, with the first $48$ feature maps residing on one GPU, and the second $48$ feature maps residing on the other GPU.

Their respectives parameters are: (3) $384$ feature maps, with $3 \times 3$ local receptive fields, and $256$ input channels;

A Theano-based implementation has also been developed* *Theano-based large-scale visual recognition with multiple GPUs, by Weiguang Ding, Ruoyan Wang, Fei Mao, and Graham Taylor (2014)., with the code available here.

As in 2012, it involved a training set of $1.2$ million images, in $1,000$ categories, and the figure of merit was whether the top $5$ predictions included the correct category.

The winning team, based primarily at Google* *Going deeper with convolutions, by Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich (2014)., used a deep convolutional network with $22$ layers of neurons.

GoogLeNet achieved a top-5 accuracy of $93.33$ percent, a giant improvement over the 2013 winner (Clarifai, with $88.3$ percent), and the 2012 winner (KSH, with $84.7$ percent).

In 2014 a team of researchers wrote a survey paper about the ILSVRC competition* *ImageNet large scale visual recognition challenge, by Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C.

...the task of labeling images with 5 out of 1000 categories quickly turned out to be extremely challenging, even for some friends in the lab who have been working on ILSVRC and its classes for a while.

In the end I realized that to get anywhere competitively close to GoogLeNet, it was most efficient if I sat down and went through the painfully long training process and the subsequent careful annotation process myself...

Some images are easily recognized, while some images (such as those of fine-grained breeds of dogs, birds, or monkeys) can require multiple minutes of concentrated effort.

In other words, an expert human, working painstakingly, was with great effort able to narrowly beat the deep neural network.

In fact, Karpathy reports that a second human expert, trained on a smaller sample of images, was only able to attain a $12.0$ percent top-5 error rate, significantly below GoogLeNet's performance.

One encouraging practical set of results comes from a team at Google, who applied deep convolutional networks to the problem of recognizing street numbers in Google's Street View imagery* *Multi-digit Number Recognition from Street View Imagery using Deep Convolutional Neural Networks, by Ian J.

And they go on to make the broader claim: 'We believe with this model we have solved [optical character recognition] for short sequences [of characters] for many applications.'

For instance, a 2013 paper* *Intriguing properties of neural networks, by Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus (2013) showed that deep networks may suffer from what are effectively blind spots.

The existence of the adversarial negatives appears to be in contradiction with the network’s ability to achieve high generalization performance.

The explanation is that the set of adversarial negatives is of extremely low probability, and thus is never (or rarely) observed in the test set, yet it is dense (much like the rational numbers), and so it is found near virtually every test case.

For example, one recent paper* *Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images, by Anh Nguyen, Jason Yosinski, and Jeff Clune (2014).

shows that given a trained network it's possible to generate images which look to a human like white noise, but which the network classifies as being in a known category with a very high degree of confidence.

If you read the neural networks literature, you'll run into many ideas we haven't discussed: recurrent neural networks, Boltzmann machines, generative models, transfer learning, reinforcement learning, and so on, on and on $\ldots$ and on!

One way RNNs are currently being used is to connect neural networks more closely to traditional ways of thinking about algorithms, ways of thinking based on concepts such as Turing machines and (conventional) programming languages.

A 2014 paper developed an RNN which could take as input a character-by-character description of a (very, very simple!) Python program, and use that description to predict the output.

For example, an approach based on deep nets has achieved outstanding results on large vocabulary continuous speech recognition.

And another system based on deep nets has been deployed in Google's Android operating system (for related technical work, see Vincent Vanhoucke's 2012-2015 papers).

Many other ideas used in feedforward nets, ranging from regularization techniques to convolutions to the activation and cost functions used, are also useful in recurrent nets.

Deep belief nets, generative models, and Boltzmann machines: Modern interest in deep learning began in 2006, with papers explaining how to train a type of neural network known as a deep belief network (DBN)* *See A fast learning algorithm for deep belief nets, by Geoffrey Hinton, Simon Osindero, and Yee-Whye Teh (2006), as well as the related work in Reducing the dimensionality of data with neural networks, by Geoffrey Hinton and Ruslan Salakhutdinov (2006)..

A generative model like a DBN can be used in a similar way, but it's also possible to specify the values of some of the feature neurons and then 'run the network backward', generating values for the input activations.

And the ability to do unsupervised learning is extremely interesting both for fundamental scientific reasons, and - if it can be made to work well enough - for practical applications.

Active areas of research include using neural networks to do natural language processing (see also this informative review paper), machine translation, as well as perhaps more surprising applications such as music informatics.

In many cases, having read this book you should be able to begin following recent work, although (of course) you'll need to fill in gaps in presumed background knowledge.

It combines deep convolutional networks with a technique known as reinforcement learning in order to learn to play video games well (see also this followup).

The idea is to use the convolutional network to simplify the pixel data from the game screen, turning it into a simpler set of features, which can be used to decide which action to take: 'go left', 'go down', 'fire', and so on.

What is particularly interesting is that a single network learned to play seven different classic video games pretty well, outperforming human experts on three of the games.

But looking past the surface gloss, consider that this system is taking raw pixel data - it doesn't even know the game rules!

Google CEO Larry Page once described the perfect search engine as understanding exactly what [your queries] mean and giving you back exactly what you want.

In this vision, instead of responding to users' literal queries, search will use machine learning to take vague user input, discern precisely what was meant, and take action on the basis of those insights.

Over the next few decades, thousands of companies will build products which use machine learning to make user interfaces that can tolerate imprecision, while discerning and acting on the user's true intent.

Inspired user interface design is hard, and I expect many companies will take powerful machine learning technology and use it to build insipid user interfaces.

Machine learning, data science, and the virtuous circle of innovation: Of course, machine learning isn't just being used to build intention-driven interfaces.

But I do want to mention one consequence of this fashion that is not so often remarked: over the long run it's possible the biggest breakthrough in machine learning won't be any single conceptual breakthrough.

If a company can invest 1 dollar in machine learning research and get 1 dollar and 10 cents back reasonably rapidly, then a lot of money will end up in machine learning research.

So, for example, Conway's law suggests that the design of a Boeing 747 aircraft will mirror the extended organizational structure of Boeing and its contractors at the time the 747 was designed.

If the application's dashboard is supposed to be integrated with some machine learning algorithm, the person building the dashboard better be talking to the company's machine learning expert.

I won't define 'deep ideas' precisely, but loosely I mean the kind of idea which is the basis for a rich field of enquiry.

The backpropagation algorithm and the germ theory of disease are both good examples.: think of things like the germ theory of disease, for instance, or the understanding of how antibodies work, or the understanding that the heart, lungs, veins and arteries form a complete cardiovascular system.

Instead of a monolith, we have fields within fields within fields, a complex, recursive, self-referential social structure, whose organization mirrors the connections between our deepest insights.

Deep learning is the latest super-special weapon I've heard used in such arguments* *Interestingly, often not by leading experts in deep learning, who have been quite restrained.

And there is paper after paper leveraging the same basic set of ideas: using stochastic gradient descent (or a close variation) to optimize a cost function.

Build your First Deep Learning Neural Network Model using Keras in Python

I have chosen my today’s topic as Neural Network because it is most the fascinating learning model in the world of data science and starters in Data Science think that Neural Network is difficult and its understanding requires knowledge of neurons, perceptron and blaahhhh…There is nothing like that, I have been working on Neural Network for quite a few month now and realized that it is so easy.

The fact of the matter is Keras is built on top of Tensorflow and Theano so this 2 insane library will be running in back-end whenever you run the program in Keras.

Deep understanding of NN(you can skip this if you don’t want to learn in depth) Now you can see that Country names are replaced by 0,1 and 2 while male and female are replaced by 0 and 1.

Dummy variable is difficult concept if you read in depth but don’t take tension, I have found this simple resource which will help you in understanding.

In Machine Learning, we always divide our data into training and testing part meaning that we train our model on training data and then we check the accuracy of a model on testing data.

Here we are using rectifier(relu) function in our hidden layer and Sigmoid function in our output layer as we want binary result from output layer but if the number of categories in output layer is more than 2 then use SoftMax function.

First argument is Optimizer, this is nothing but the algorithm you wanna use to find optimal set of weights(Note that in step 9 we just initialized weights now we are applying some sort of algorithm which will optimize weights in turn making out neural network more powerful.

Since out dependent variable is binary, we will have to use logarithmic loss function called ‘binary_crossentropy’, if our dependent variable has more than 2 categories in output then use ‘categorical_crossentropy’.

Using neural nets to recognize handwritten digits

Simple intuitions about how we recognize shapes - 'a 9 has a loop at the top, and a vertical stroke in the bottom right' - turn out to be not so simple to express algorithmically.

As a prototype it hits a sweet spot: it's challenging - it's no small feat to recognize handwritten digits - but it's not so difficult as to require an extremely complicated solution, or tremendous computational power.

But along the way we'll develop many key ideas about neural networks, including two important types of artificial neuron (the perceptron and the sigmoid neuron), and the standard learning algorithm for neural networks, known as stochastic gradient descent.

Today, it's more common to use other models of artificial neurons - in this book, and in much modern work on neural networks, the main neuron model used is one called the sigmoid neuron.

A perceptron takes several binary inputs, $x_1, x_2, \ldots$, and produces a single binary output: In the example shown the perceptron has three inputs, $x_1, x_2, x_3$.

The neuron's output, $0$ or $1$, is determined by whether the weighted sum $\sum_j w_j x_j$ is less than or greater than some threshold value.

To put it in more precise algebraic terms: \begin{eqnarray} \mbox{output} & = & \left\{ \begin{array}{ll} 0 & \mbox{if } \sum_j w_j x_j \leq \mbox{ threshold} \\ 1 & \mbox{if } \sum_j w_j x_j > \mbox{ threshold} \end{array} \right.

And it should seem plausible that a complex network of perceptrons could make quite subtle decisions: In this network, the first column of perceptrons - what we'll call the first layer of perceptrons - is making three very simple decisions, by weighing the input evidence.

The first change is to write $\sum_j w_j x_j$ as a dot product, $w \cdot x \equiv \sum_j w_j x_j$, where $w$ and $x$ are vectors whose components are the weights and inputs, respectively.

Using the bias instead of the threshold, the perceptron rule can be rewritten: \begin{eqnarray} \mbox{output} = \left\{ \begin{array}{ll} 0 & \mbox{if } w\cdot x + b \leq 0 \\ 1 & \mbox{if } w\cdot x + b > 0 \end{array} \right.

This requires computing the bitwise sum, $x_1 \oplus x_2$, as well as a carry bit which is set to $1$ when both $x_1$ and $x_2$ are $1$, i.e., the carry bit is just the bitwise product $x_1 x_2$: To get an equivalent network of perceptrons we replace all the NAND gates by perceptrons with two inputs, each with weight $-2$, and an overall bias of $3$.

Note that I've moved the perceptron corresponding to the bottom right NAND gate a little, just to make it easier to draw the arrows on the diagram: One notable aspect of this network of perceptrons is that the output from the leftmost perceptron is used twice as input to the bottommost perceptron.

(If you don't find this obvious, you should stop and prove to yourself that this is equivalent.) With that change, the network looks as follows, with all unmarked weights equal to -2, all biases equal to 3, and a single weight of -4, as marked: Up to now I've been drawing inputs like $x_1$ and $x_2$ as variables floating to the left of the network of perceptrons.

In fact, it's conventional to draw an extra layer of perceptrons - the input layer - to encode the inputs: This notation for input perceptrons, in which we have an output, but no inputs, is a shorthand.

Then the weighted sum $\sum_j w_j x_j$ would always be zero, and so the perceptron would output $1$ if $b > 0$, and $0$ if $b \leq 0$.

Instead of explicitly laying out a circuit of NAND and other gates, our neural networks can simply learn to solve problems, sometimes problems where it would be extremely difficult to directly design a conventional circuit.

If it were true that a small change in a weight (or bias) causes only a small change in output, then we could use this fact to modify the weights and biases to get our network to behave more in the manner we want.

In fact, a small change in the weights or bias of any single perceptron in the network can sometimes cause the output of that perceptron to completely flip, say from $0$ to $1$.

We'll depict sigmoid neurons in the same way we depicted perceptrons: Just like a perceptron, the sigmoid neuron has inputs, $x_1, x_2, \ldots$.

Instead, it's $\sigma(w \cdot x+b)$, where $\sigma$ is called the sigmoid function* *Incidentally, $\sigma$ is sometimes called the logistic function, and this new class of neurons called logistic neurons.

\tag{3}\end{eqnarray} To put it all a little more explicitly, the output of a sigmoid neuron with inputs $x_1,x_2,\ldots$, weights $w_1,w_2,\ldots$, and bias $b$ is \begin{eqnarray} \frac{1}{1+\exp(-\sum_j w_j x_j-b)}.

In fact, there are many similarities between perceptrons and sigmoid neurons, and the algebraic form of the sigmoid function turns out to be more of a technical detail than a true barrier to understanding.

var data = d3.range(sample).map(function(d){ return { x: x1(d), y: s(x1(d))};

var y = d3.scale.linear() .domain([0, 1]) .range([height, 0]);

}) var graph = d3.select('#sigmoid_graph') .append('svg') .attr('width', width + m[1] + m[3]) .attr('height', height + m[0] + m[2]) .append('g') .attr('transform', 'translate(' + m[3] + ',' + m[0] + ')');

var xAxis = d3.svg.axis() .scale(x) .tickValues(d3.range(-4, 5, 1)) .orient('bottom') graph.append('g') .attr('class', 'x axis') .attr('transform', 'translate(0, ' + height + ')') .call(xAxis);

var yAxis = d3.svg.axis() .scale(y) .tickValues(d3.range(0, 1.01, 0.2)) .orient('left') .ticks(5) graph.append('g') .attr('class', 'y axis') .call(yAxis);

graph.append('text') .attr('class', 'x label') .attr('text-anchor', 'end') .attr('x', width/2) .attr('y', height+35) .text('z');

graph.append('text') .attr('x', (width / 2)) .attr('y', -10) .attr('text-anchor', 'middle') .style('font-size', '16px') .text('sigmoid function');

var data = d3.range(sample).map(function(d){ return { x: x1(d), y: s(x1(d))};

var y = d3.scale.linear() .domain([0,1]) .range([height, 0]);

}) var graph = d3.select('#step_graph') .append('svg') .attr('width', width + m[1] + m[3]) .attr('height', height + m[0] + m[2]) .append('g') .attr('transform', 'translate(' + m[3] + ',' + m[0] + ')');

var xAxis = d3.svg.axis() .scale(x) .tickValues(d3.range(-4, 5, 1)) .orient('bottom') graph.append('g') .attr('class', 'x axis') .attr('transform', 'translate(0, ' + height + ')') .call(xAxis);

var yAxis = d3.svg.axis() .scale(y) .tickValues(d3.range(0, 1.01, 0.2)) .orient('left') .ticks(5) graph.append('g') .attr('class', 'y axis') .call(yAxis);

graph.append('text') .attr('class', 'x label') .attr('text-anchor', 'end') .attr('x', width/2) .attr('y', height+35) .text('z');

graph.append('text') .attr('x', (width / 2)) .attr('y', -10) .attr('text-anchor', 'middle') .style('font-size', '16px') .text('step function');

If $\sigma$ had in fact been a step function, then the sigmoid neuron would be a perceptron, since the output would be $1$ or $0$ depending on whether $w\cdot x+b$ was positive or negative* *Actually, when $w \cdot x +b = 0$ the perceptron outputs $0$, while the step function outputs $1$.

The smoothness of $\sigma$ means that small changes $\Delta w_j$ in the weights and $\Delta b$ in the bias will produce a small change $\Delta \mbox{output}$ in the output from the neuron.

In fact, calculus tells us that $\Delta \mbox{output}$ is well approximated by \begin{eqnarray} \Delta \mbox{output} \approx \sum_j \frac{\partial \, \mbox{output}}{\partial w_j} \Delta w_j + \frac{\partial \, \mbox{output}}{\partial b} \Delta b, \tag{5}\end{eqnarray} where the sum is over all the weights, $w_j$, and $\partial \, \mbox{output} / \partial w_j$ and $\partial \, \mbox{output} /\partial b$ denote partial derivatives of the $\mbox{output}$ with respect to $w_j$ and $b$, respectively.

While the expression above looks complicated, with all the partial derivatives, it's actually saying something very simple (and which is very good news): $\Delta \mbox{output}$ is a linear function of the changes $\Delta w_j$ and $\Delta b$ in the weights and bias.

If it's the shape of $\sigma$ which really matters, and not its exact form, then why use the particular form used for $\sigma$ in Equation (3)\begin{eqnarray} \sigma(z) \equiv \frac{1}{1+e^{-z}} \nonumber\end{eqnarray}$('#margin_778862672352_reveal').click(function() {$('#margin_778862672352').toggle('slow', function() {});});?

In fact, later in the book we will occasionally consider neurons where the output is $f(w \cdot x + b)$ for some other activation function $f(\cdot)$.

The main thing that changes when we use a different activation function is that the particular values for the partial derivatives in Equation (5)\begin{eqnarray} \Delta \mbox{output} \approx \sum_j \frac{\partial \, \mbox{output}}{\partial w_j} \Delta w_j + \frac{\partial \, \mbox{output}}{\partial b} \Delta b \nonumber\end{eqnarray}$('#margin_726336021933_reveal').click(function() {$('#margin_726336021933').toggle('slow', function() {});});

It turns out that when we compute those partial derivatives later, using $\sigma$ will simplify the algebra, simply because exponentials have lovely properties when differentiated.

But in practice we can set up a convention to deal with this, for example, by deciding to interpret any output of at least $0.5$ as indicating a '9', and any output less than $0.5$ as indicating 'not a 9'.

Exercises Sigmoid neurons simulating perceptrons, part I $\mbox{}$ Suppose we take all the weights and biases in a network of perceptrons, and multiply them by a positive constant, $c > 0$.

Show that the behaviour of the network doesn't change.Sigmoid neurons simulating perceptrons, part II $\mbox{}$ Suppose we have the same setup as the last problem - a network of perceptrons.

Suppose the weights and biases are such that $w \cdot x + b \neq 0$ for the input $x$ to any particular perceptron in the network.

Now replace all the perceptrons in the network by sigmoid neurons, and multiply the weights and biases by a positive constant $c > 0$.

Suppose we have the network: As mentioned earlier, the leftmost layer in this network is called the input layer, and the neurons within the layer are called input neurons.

The term 'hidden' perhaps sounds a little mysterious - the first time I heard the term I thought it must have some deep philosophical or mathematical significance - but it really means nothing more than 'not an input or an output'.

For example, the following four-layer network has two hidden layers: Somewhat confusingly, and for historical reasons, such multiple layer networks are sometimes called multilayer perceptrons or MLPs, despite being made up of sigmoid neurons, not perceptrons.

If the image is a $64$ by $64$ greyscale image, then we'd have $4,096 = 64 \times 64$ input neurons, with the intensities scaled appropriately between $0$ and $1$.

The output layer will contain just a single neuron, with output values of less than $0.5$ indicating 'input image is not a 9', and values greater than $0.5$ indicating 'input image is a 9 '.

A trial segmentation gets a high score if the individual digit classifier is confident of its classification in all segments, and a low score if the classifier is having a lot of trouble in one or more segments.

So instead of worrying about segmentation we'll concentrate on developing a neural network which can solve the more interesting and difficult problem, namely, recognizing individual handwritten digits.

As discussed in the next section, our training data for the network will consist of many $28$ by $28$ pixel images of scanned handwritten digits, and so the input layer contains $784 = 28 \times 28$ neurons.

The input pixels are greyscale, with a value of $0.0$ representing white, a value of $1.0$ representing black, and in between values representing gradually darkening shades of grey.

A seemingly natural way of doing that is to use just $4$ output neurons, treating each neuron as taking on a binary value, depending on whether the neuron's output is closer to $0$ or to $1$.

The ultimate justification is empirical: we can try out both network designs, and it turns out that, for this particular problem, the network with $10$ output neurons learns to recognize digits better than the network with $4$ output neurons.

In a similar way, let's suppose for the sake of argument that the second, third, and fourth neurons in the hidden layer detect whether or not the following images are present:

Of course, that's not the only sort of evidence we can use to conclude that the image was a $0$ - we could legitimately get a $0$ in many other ways (say, through translations of the above images, or slight distortions).

Assume that the first $3$ layers of neurons are such that the correct output in the third layer (i.e., the old output layer) has activation at least $0.99$, and incorrect outputs have activation less than $0.01$.

We'll use the MNIST data set, which contains tens of thousands of scanned images of handwritten digits, together with their correct classifications.

To make this a good test of performance, the test data was taken from a different set of 250 people than the original training data (albeit still a group split between Census Bureau employees and high school students).

For example, if a particular training image, $x$, depicts a $6$, then $y(x) = (0, 0, 0, 0, 0, 0, 1, 0, 0, 0)^T$ is the desired output from the network.

We use the term cost function throughout this book, but you should note the other terminology, since it's often used in research papers and other discussions of neural networks.

\tag{6}\end{eqnarray} Here, $w$ denotes the collection of all weights in the network, $b$ all the biases, $n$ is the total number of training inputs, $a$ is the vector of outputs from the network when $x$ is input, and the sum is over all training inputs, $x$.

If we instead use a smooth cost function like the quadratic cost it turns out to be easy to figure out how to make small changes in the weights and biases so as to get an improvement in the cost.

Even given that we want to use a smooth cost function, you may still wonder why we choose the quadratic function used in Equation (6)\begin{eqnarray} C(w,b) \equiv \frac{1}{2n} \sum_x \|

This is a well-posed problem, but it's got a lot of distracting structure as currently posed - the interpretation of $w$ and $b$ as weights and biases, the $\sigma$ function lurking in the background, the choice of network architecture, MNIST, and so on.

And for neural networks we'll often want far more variables - the biggest neural networks have cost functions which depend on billions of weights and biases in an extremely complicated way.

We could do this simulation simply by computing derivatives (and perhaps some second derivatives) of $C$ - those derivatives would tell us everything we need to know about the local 'shape' of the valley, and therefore how our ball should roll.

So rather than get into all the messy details of physics, let's simply ask ourselves: if we were declared God for a day, and could make up our own laws of physics, dictating to the ball how it should roll, what law or laws of motion could we pick that would make it so the ball always rolled to the bottom of the valley?

To make this question more precise, let's think about what happens when we move the ball a small amount $\Delta v_1$ in the $v_1$ direction, and a small amount $\Delta v_2$ in the $v_2$ direction.

Calculus tells us that $C$ changes as follows: \begin{eqnarray} \Delta C \approx \frac{\partial C}{\partial v_1} \Delta v_1 + \frac{\partial C}{\partial v_2} \Delta v_2.

To figure out how to make such a choice it helps to define $\Delta v$ to be the vector of changes in $v$, $\Delta v \equiv (\Delta v_1, \Delta v_2)^T$, where $T$ is again the transpose operation, turning row vectors into column vectors.

We denote the gradient vector by $\nabla C$, i.e.: \begin{eqnarray} \nabla C \equiv \left( \frac{\partial C}{\partial v_1}, \frac{\partial C}{\partial v_2} \right)^T.

In fact, it's perfectly fine to think of $\nabla C$ as a single mathematical object - the vector defined above - which happens to be written using two symbols.

With these definitions, the expression (7)\begin{eqnarray} \Delta C \approx \frac{\partial C}{\partial v_1} \Delta v_1 + \frac{\partial C}{\partial v_2} \Delta v_2 \nonumber\end{eqnarray}$('#margin_832985330775_reveal').click(function() {$('#margin_832985330775').toggle('slow', function() {});});

\tag{9}\end{eqnarray} This equation helps explain why $\nabla C$ is called the gradient vector: $\nabla C$ relates changes in $v$ to changes in $C$, just as we'd expect something called a gradient to do.

In particular, suppose we choose \begin{eqnarray} \Delta v = -\eta \nabla C, \tag{10}\end{eqnarray} where $\eta$ is a small, positive parameter (known as the learning rate).

\nabla C \|^2 \geq 0$, this guarantees that$\Delta C \leq 0$, i.e.,$C$will always decrease, never increase, if we change$v$according to the prescription in (10)\begin{eqnarray} \Delta v = -\eta \nabla C \nonumber\end{eqnarray}$('#margin_39079991636_reveal').click(function() {$('#margin_39079991636').toggle('slow', function() {});});. to compute a value for$\Delta v$, then move the ball's position$v$by that amount: \begin{eqnarray} v \rightarrow v' = v -\eta \nabla C. To make gradient descent work correctly, we need to choose the learning rate$\eta$to be small enough that Equation (9)\begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v \nonumber\end{eqnarray}$('#margin_663076476028_reveal').click(function() {$('#margin_663076476028').toggle('slow', function() {});}); In practical implementations,$\eta$is often varied so that Equation (9)\begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v \nonumber\end{eqnarray}$('#margin_362932327599_reveal').click(function() {$('#margin_362932327599').toggle('slow', function() {});}); Then the change$\Delta C$in$C$produced by a small change$\Delta v = (\Delta v_1, \ldots, \Delta v_m)^T$is \begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v, \tag{12}\end{eqnarray} where the gradient$\nabla C$is the vector \begin{eqnarray} \nabla C \equiv \left(\frac{\partial C}{\partial v_1}, \ldots, \frac{\partial C}{\partial v_m}\right)^T. \tag{13}\end{eqnarray} Just as for the two variable case, we can choose \begin{eqnarray} \Delta v = -\eta \nabla C, \tag{14}\end{eqnarray} and we're guaranteed that our (approximate) expression (12)\begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v \nonumber\end{eqnarray}$('#margin_398945612724_reveal').click(function() {$('#margin_398945612724').toggle('slow', function() {});}); This gives us a way of following the gradient to a minimum, even when$C$is a function of many variables, by repeatedly applying the update rule \begin{eqnarray} v \rightarrow v' = v-\eta \nabla C. The rule doesn't always work - several things can go wrong and prevent gradient descent from finding the global minimum of$C$, a point we'll return to explore in later chapters. But, in practice gradient descent often works extremely well, and in neural networks we'll find that it's a powerful way of minimizing the cost function, and so helping the net learn. It can be proved that the choice of$\Delta v$which minimizes$\nabla C \cdot \Delta v$is$\Delta v = - \eta \nabla C$, where$\eta = \epsilon / \|\nabla C\|$is determined by the size constraint$\|\Delta v\|

Hint: If you're not already familiar with the Cauchy-Schwarz inequality, you may find it helpful to familiarize yourself with it.

If there are a million such $v_j$ variables then we'd need to compute something like a trillion (i.e., a million squared) second partial derivatives* *Actually, more like half a trillion, since $\partial^2 C/ \partial v_j \partial v_k = \partial^2 C/ \partial v_k \partial v_j$.

The idea is to use gradient descent to find the weights $w_k$ and biases $b_l$ which minimize the cost in Equation (6)\begin{eqnarray} C(w,b) \equiv \frac{1}{2n} \sum_x \|

In other words, our 'position' now has components $w_k$ and $b_l$, and the gradient vector $\nabla C$ has corresponding components $\partial C / \partial w_k$ and $\partial C / \partial b_l$.

Writing out the gradient descent update rule in terms of components, we have \begin{eqnarray} w_k & \rightarrow & w_k' = w_k-\eta \frac{\partial C}{\partial w_k} \tag{16}\\ b_l & \rightarrow & b_l' = b_l-\eta \frac{\partial C}{\partial b_l}.

In practice, to compute the gradient $\nabla C$ we need to compute the gradients $\nabla C_x$ separately for each training input, $x$, and then average them, $\nabla C = \frac{1}{n} \sum_x \nabla C_x$.

To make these ideas more precise, stochastic gradient descent works by randomly picking out a small number $m$ of randomly chosen training inputs.

Provided the sample size $m$ is large enough we expect that the average value of the $\nabla C_{X_j}$ will be roughly equal to the average over all $\nabla C_x$, that is, \begin{eqnarray} \frac{\sum_{j=1}^m \nabla C_{X_{j}}}{m} \approx \frac{\sum_x \nabla C_x}{n} = \nabla C, \tag{18}\end{eqnarray} where the second sum is over the entire set of training data.

Swapping sides we get \begin{eqnarray} \nabla C \approx \frac{1}{m} \sum_{j=1}^m \nabla C_{X_{j}}, \tag{19}\end{eqnarray} confirming that we can estimate the overall gradient by computing gradients just for the randomly chosen mini-batch.

Then stochastic gradient descent works by picking out a randomly chosen mini-batch of training inputs, and training with those, \begin{eqnarray} w_k & \rightarrow & w_k' = w_k-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial w_k} \tag{20}\\ b_l & \rightarrow & b_l' = b_l-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial b_l}, \tag{21}\end{eqnarray} where the sums are over all the training examples $X_j$ in the current mini-batch.

And, in a similar way, the mini-batch update rules (20)\begin{eqnarray} w_k & \rightarrow & w_k' = w_k-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial w_k} \nonumber\end{eqnarray}$('#margin_255037324417_reveal').click(function() {$('#margin_255037324417').toggle('slow', function() {});});

and (21)\begin{eqnarray} b_l & \rightarrow & b_l' = b_l-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial b_l} \nonumber\end{eqnarray}$('#margin_141169455106_reveal').click(function() {$('#margin_141169455106').toggle('slow', function() {});});

We can think of stochastic gradient descent as being like political polling: it's much easier to sample a small mini-batch than it is to apply gradient descent to the full batch, just as carrying out a poll is easier than running a full election.

For example, if we have a training set of size $n = 60,000$, as in MNIST, and choose a mini-batch size of (say) $m = 10$, this means we'll get a factor of $6,000$ speedup in estimating the gradient!

Of course, the estimate won't be perfect - there will be statistical fluctuations - but it doesn't need to be perfect: all we really care about is moving in a general direction that will help decrease $C$, and that means we don't need an exact computation of the gradient.

In practice, stochastic gradient descent is a commonly used and powerful technique for learning in neural networks, and it's the basis for most of the learning techniques we'll develop in this book.

That is, given a training input, $x$, we update our weights and biases according to the rules $w_k \rightarrow w_k' = w_k - \eta \partial C_x / \partial w_k$ and $b_l \rightarrow b_l' = b_l - \eta \partial C_x / \partial b_l$.

Name one advantage and one disadvantage of online learning, compared to stochastic gradient descent with a mini-batch size of, say, $20$.

In neural networks the cost $C$ is, of course, a function of many variables - all the weights and biases - and so in some sense defines a surface in a very high-dimensional space.

I won't go into more detail here, but if you're interested then you may enjoy reading this discussion of some of the techniques professional mathematicians use to think in high dimensions.

We'll leave the test images as is, but split the 60,000-image MNIST training set into two parts: a set of 50,000 images, which we'll use to train our neural network, and a separate 10,000 image validation set.

We won't use the validation data in this chapter, but later in the book we'll find it useful in figuring out how to set certain hyper-parameters of the neural network - things like the learning rate, and so on, which aren't directly selected by our learning algorithm.

When I refer to the 'MNIST training data' from now on, I'll be referring to our 50,000 image data set, not the original 60,000 image data set* *As noted earlier, the MNIST data set is based on two data sets collected by NIST, the United States' National Institute of Standards and Technology.

for x, y in zip(sizes[:-1], sizes[1:])]

So, for example, if we want to create a Network object with 2 neurons in the first layer, 3 neurons in the second layer, and 1 neuron in the final layer, we'd do this with the code: net = Network([2, 3, 1])

The biases and weights in the Network object are all initialized randomly, using the Numpy np.random.randn function to generate Gaussian distributions with mean $0$ and standard deviation $1$.

Note that the Network initialization code assumes that the first layer of neurons is an input layer, and omits to set any biases for those neurons, since biases are only ever used in computing the outputs from later layers.

The big advantage of using this ordering is that it means that the vector of activations of the third layer of neurons is: \begin{eqnarray} a' = \sigma(w a + b).

(This is called vectorizing the function $\sigma$.) It's easy to verify that Equation (22)\begin{eqnarray} a' = \sigma(w a + b) \nonumber\end{eqnarray}$('#margin_552886241220_reveal').click(function() {$('#margin_552886241220').toggle('slow', function() {});});

gives the same result as our earlier rule, Equation (4)\begin{eqnarray} \frac{1}{1+\exp(-\sum_j w_j x_j-b)} \nonumber\end{eqnarray}$('#margin_7421600236_reveal').click(function() {$('#margin_7421600236').toggle('slow', function() {});});, for computing the output of a sigmoid neuron.

in component form, and verify that it gives the same result as the rule (4)\begin{eqnarray} \frac{1}{1+\exp(-\sum_j w_j x_j-b)} \nonumber\end{eqnarray}$('#margin_347257101140_reveal').click(function() {$('#margin_347257101140').toggle('slow', function() {});});

We then add a feedforward method to the Network class, which, given an input a for the network, returns the corresponding output* *It is assumed that the input a is an (n, 1) Numpy ndarray, not a (n,) vector.

Although using an (n,) vector appears the more natural choice, using an (n, 1) ndarray makes it particularly easy to modify the code to feedforward multiple inputs at once, and that is sometimes convenient.

All the method does is applies Equation (22)\begin{eqnarray} a' = \sigma(w a + b) \nonumber\end{eqnarray}$('#margin_335258165235_reveal').click(function() {$('#margin_335258165235').toggle('slow', function() {});});

training_data[k:k+mini_batch_size]

for k in xrange(0, n, mini_batch_size)]

self.update_mini_batch(mini_batch, eta)

print &quot;Epoch {0}: {1} / {2}&quot;.format(

j, self.evaluate(test_data), n_test)

print &quot;Epoch {0} complete&quot;.format(j)

This is done by the code self.update_mini_batch(mini_batch, eta), which updates the network weights and biases according to a single iteration of gradient descent, using just the training data in mini_batch.

delta_nabla_b, delta_nabla_w = self.backprop(x, y)

nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]

nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]

for w, nw in zip(self.weights, nabla_w)]

for b, nb in zip(self.biases, nabla_b)]

Most of the work is done by the line delta_nabla_b, delta_nabla_w = self.backprop(x, y)

The self.backprop method makes use of a few extra functions to help in computing the gradient, namely sigmoid_prime, which computes the derivative of the $\sigma$ function, and self.cost_derivative, which I won't describe here.

for x, y in zip(sizes[:-1], sizes[1:])]

training_data[k:k+mini_batch_size]

for k in xrange(0, n, mini_batch_size)]

self.update_mini_batch(mini_batch, eta)

print &quot;Epoch {0}: {1} / {2}&quot;.format(

j, self.evaluate(test_data), n_test)

print &quot;Epoch {0} complete&quot;.format(j)

delta_nabla_b, delta_nabla_w = self.backprop(x, y)

nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]

nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]

for w, nw in zip(self.weights, nabla_w)]

for b, nb in zip(self.biases, nabla_b)]

activations = [x] # list to store all the activations, layer by layer

zs = [] # list to store all the z vectors, layer by layer

# l = 1 means the last layer of neurons, l = 2 is the

delta = np.dot(self.weights[-l+1].transpose(), delta) * sp

for (x, y) in test_data]

Finally, we'll use stochastic gradient descent to learn from the MNIST training_data over 30 epochs, with a mini-batch size of 10, and a learning rate of $\eta = 3.0$, &gt;&gt;&gt;

As was the case earlier, if you're running the code as you read along, you should be warned that it takes quite a while to execute (on my machine this experiment takes tens of seconds for each training epoch), so it's wise to continue reading in parallel while the code executes.

At least in this case, using more hidden neurons helps us get better results* *Reader feedback indicates quite some variation in results for this experiment, and some training runs give results quite a bit worse.

Using the techniques introduced in chapter 3 will greatly reduce the variation in performance across different training runs for our networks..

(If making a change improves things, try doing more!) If we do that several times over, we'll end up with a learning rate of something like $\eta = 1.0$ (and perhaps fine tune to $3.0$), which is close to our earlier experiments.

Exercise Try creating a network with just two layers - an input and an output layer, no hidden layer - with 784 and 10 neurons, respectively.

The data structures used to store the MNIST data are described in the documentation strings - it's straightforward stuff, tuples and lists of Numpy ndarray objects (think of them as vectors if you're not familiar with ndarrays): &quot;&quot;&quot;mnist_loader~~~~~~~~~~~~A library to load the MNIST image data.

In some sense, the moral of both our results and those in more sophisticated papers, is that for some problems: sophisticated algorithm $\leq$ simple learning algorithm + good training data.

We could attack this problem the same way we attacked handwriting recognition - by using the pixels in the image as input to a neural network, with the output from the network a single neuron indicating either 'Yes, it's a face' or 'No, it's not a face'.

The end result is a network which breaks down a very complicated question - does this image show a face or not - into very simple questions answerable at the level of single pixels.

It does this through a series of many layers, with early layers answering very simple and specific questions about the input image, and later layers building up a hierarchy of ever more complex and abstract concepts.

Comparing a deep network to a shallow network is a bit like comparing a programming language with the ability to make function calls to a stripped down language with no ability to make such calls.

Neural Network Toolbox

Neural Network Toolbox™ provides algorithms, pretrained models, and apps to create, train, visualize, and simulate both shallow and deep neural networks.

Deep learning networks include convolutional neural networks (ConvNets, CNNs), directed acyclic graph (DAG) network topologies, and autoencoders for image classification, regression, and feature learning.

For small training sets, you can quickly apply deep learning by performing transfer learning with pretrained deep network models (including Inception-v3, ResNet-50, ResNet-101, GoogLeNet, AlexNet, VGG-16, and VGG-19) and models imported from TensorFlow™ Keras or Caffe.

But what *is* a Neural Network? | Deep learning, chapter 1

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