AI News, BOOK REVIEW: Simple multi layer neural network implementation [closed]

Simple multi layer neural network implementation [closed]

If your inputs are stored in a 2d numpy matrix, where each row corresponds to one sample and the columns correspond to the attributes of your samples, you can propagate through the network like this: (assuming logistic sigmoid function as activation function) The last element in activations will be the output of your network, but be careful, you need to omit the additional column for the biases, so your output will be activations[-1][:,:-1].


Here is our new feed forward code which accepts matrices instead of scalar inputs.

Using dot product, we multiply the input matrix by the weights connecting them to the neurons in the next layer.

The result is a new matrix, Zh which has a column for every neuron in the hidden layer and a row for every observation in our dataset.

In the hidden layer activation step, we apply the ReLU activation function np.maximum(0,Z) to every cell in the new matrix.

In the section on linear classification we computed scores for different visual categories given the image using the formula \( s = W x \), where \(W\) was a matrix and \(x\) was an input column vector containing all pixel data of the image.

In the case of CIFAR-10, \(x\) is a [3072x1] column vector, and \(W\) is a [10x3072] matrix, so that the output scores is a vector of 10 class scores.

There are several choices we could make for the non-linearity (which we’ll study below), but this one is a common choice and simply thresholds all activations that are below zero to zero.

Notice that the non-linearity is critical computationally - if we left it out, the two matrices could be collapsed to a single matrix, and therefore the predicted class scores would again be a linear function of the input.

three-layer neural network could analogously look like \( s = W_3 \max(0, W_2 \max(0, W_1 x)) \), where all of \(W_3, W_2, W_1\) are parameters to be learned.

The area of Neural Networks has originally been primarily inspired by the goal of modeling biological neural systems, but has since diverged and become a matter of engineering and achieving good results in Machine Learning tasks.

Approximately 86 billion neurons can be found in the human nervous system and they are connected with approximately 10^14 - 10^15 synapses.

The idea is that the synaptic strengths (the weights \(w\)) are learnable and control the strength of influence (and its direction: excitory (positive weight) or inhibitory (negative weight)) of one neuron on another.

Based on this rate code interpretation, we model the firing rate of the neuron with an activation function \(f\), which represents the frequency of the spikes along the axon.

Historically, a common choice of activation function is the sigmoid function \(\sigma\), since it takes a real-valued input (the signal strength after the sum) and squashes it to range between 0 and 1.

An example code for forward-propagating a single neuron might look as follows: In other words, each neuron performs a dot product with the input and its weights, adds the bias and applies the non-linearity (or activation function), in this case the sigmoid \(\sigma(x) = 1/(1+e^{-x})\).

As we saw with linear classifiers, a neuron has the capacity to “like” (activation near one) or “dislike” (activation near zero) certain linear regions of its input space.

With this interpretation, we can formulate the cross-entropy loss as we have seen in the Linear Classification section, and optimizing it would lead to a binary Softmax classifier (also known as logistic regression).

The regularization loss in both SVM/Softmax cases could in this biological view be interpreted as gradual forgetting, since it would have the effect of driving all synaptic weights \(w\) towards zero after every parameter update.

The sigmoid non-linearity has the mathematical form \(\sigma(x) = 1 / (1 + e^{-x})\) and is shown in the image above on the left.

The sigmoid function has seen frequent use historically since it has a nice interpretation as the firing rate of a neuron: from not firing at all (0) to fully-saturated firing at an assumed maximum frequency (1).

Also note that the tanh neuron is simply a scaled sigmoid neuron, in particular the following holds: \( \tanh(x) = 2 \sigma(2x) -1 \).

Other types of units have been proposed that do not have the functional form \(f(w^Tx + b)\) where a non-linearity is applied on the dot product between the weights and the data.

TLDR: “What neuron type should I use?” Use the ReLU non-linearity, be careful with your learning rates and possibly monitor the fraction of “dead” units in a network.

For regular neural networks, the most common layer type is the fully-connected layer in which neurons between two adjacent layers are fully pairwise connected, but neurons within a single layer share no connections.

Working with the two example networks in the above picture: To give you some context, modern Convolutional Networks contain on orders of 100 million parameters and are usually made up of approximately 10-20 layers (hence deep learning).

The full forward pass of this 3-layer neural network is then simply three matrix multiplications, interwoven with the application of the activation function: In the above code, W1,W2,W3,b1,b2,b3 are the learnable parameters of the network.

Notice also that instead of having a single input column vector, the variable x could hold an entire batch of training data (where each input example would be a column of x) and then all examples would be efficiently evaluated in parallel.

Neural Networks work well in practice because they compactly express nice, smooth functions that fit well with the statistical properties of data we encounter in practice, and are also easy to learn using our optimization algorithms (e.g.

Similarly, the fact that deeper networks (with multiple hidden layers) can work better than a single-hidden-layer networks is an empirical observation, despite the fact that their representational power is equal.

As an aside, in practice it is often the case that 3-layer neural networks will outperform 2-layer nets, but going even deeper (4,5,6-layer) rarely helps much more.

We could train three separate neural networks, each with one hidden layer of some size and obtain the following classifiers: In the diagram above, we can see that Neural Networks with more neurons can express more complicated functions.

For example, the model with 20 hidden neurons fits all the training data but at the cost of segmenting the space into many disjoint red and green decision regions.

The subtle reason behind this is that smaller networks are harder to train with local methods such as Gradient Descent: It’s clear that their loss functions have relatively few local minima, but it turns out that many of these minima are easier to converge to, and that they are bad (i.e.

Conversely, bigger neural networks contain significantly more local minima, but these minima turn out to be much better in terms of their actual loss.

In practice, what you find is that if you train a small network the final loss can display a good amount of variance - in some cases you get lucky and converge to a good place but in some cases you get trapped in one of the bad minima.

How to Implement the Backpropagation Algorithm From Scratch In Python

The backpropagation algorithm is the classical feed-forward artificial neural network.

The principle of the backpropagation approach is to model a given function by modifying internal weightings of input signals to produce an expected output signal.

The system is trained using a supervised learning method, where the error between the system’s output and a known expected output is presented to the system and used to modify its internal state.

The scale for each numeric input value vary, so some data normalization may be required for use with algorithms that weight inputs like the backpropagation algorithm.

Update, download the dataset in CSV format directly: This tutorial is broken down into 6 parts: These steps will provide the foundation that you need to implement the backpropagation algorithm from scratch and apply it to your own predictive modeling problems.

We will need to store additional properties for a neuron during training, therefore we will use a dictionary to represent each neuron and store properties by names such as ‘weights‘

You can see that for the hidden layer we create n_hidden neurons and each neuron in the hidden layer has n_inputs + 1 weights, one for each input column in a dataset and an additional one for the bias.

We can calculate an output from a neural network by propagating an input signal through each layer until the output layer outputs its values.

It is the technique we will need to generate predictions during training that will need to be corrected, and it is the method we will need after the network is trained to make predictions on new data.

Where weight is a network weight, input is an input, i is the index of a weight or an input and bias is a special weight that has no input to multiply with (or you can think of the input as always being 1.0).

We can transfer an activation function using the sigmoid function as follows: Where e is the base of the natural logarithms (Euler’s number).

You can also see that we collect the outputs for a layer in an array named new_inputs that becomes the array inputs and is used as inputs for the following layer.

We define our network inline with one hidden neuron that expects 2 input values and an output layer with two neurons.

These errors are then propagated backward through the network from the output layer to the hidden layer, assigning blame for the error and updating weights as they go.

The math for backpropagating error is rooted in calculus, but we will remain high level in this section and focus on what is calculated and how rather than why the calculations take this particular form.

We are using the sigmoid transfer function, the derivative of which can be calculated as follows: Below is a function named transfer_derivative() that implements this equation.

The first step is to calculate the error for each output neuron, this will give us our error signal (input) to propagate backwards through the network.

The error for a given neuron can be calculated as follows: Where expected is the expected output value for the neuron, output is the output value for the neuron and transfer_derivative() calculates the slope of the neuron’s output value, as shown above.

The back-propagated error signal is accumulated and then used to determine the error for the neuron in the hidden layer, as follows: Where error_j is the error signal from the jth neuron in the output layer, weight_k is the weight that connects the kth neuron to the current neuron and output is the output for the current neuron.

You can see that the error signal for neurons in the hidden layer is accumulated from neurons in the output layer where the hidden neuron number j is also the index of the neuron’s weight in the output layer neuron[‘weights’][j].

This involves multiple iterations of exposing a training dataset to the network and for each row of data forward propagating the inputs, backpropagating the error and updating the network weights.

This part is broken down into two sections: Once errors are calculated for each neuron in the network via the back propagation method above, they can be used to update weights.

Network weights are updated as follows: Where weight is a given weight, learning_rate is a parameter that you must specify, error is the error calculated by the backpropagation procedure for the neuron and input is the input value that caused the error.

This increases the likelihood of the network finding a good set of weights across all layers rather than the fastest set of weights that minimize error (called premature convergence).

Below is a function named update_weights() that updates the weights for a network given an input row of data, a learning rate and assume that a forward and backward propagation have already been performed.

Below is a function that implements the training of an already initialized neural network with a given training dataset, learning rate, fixed number of epochs and an expected number of output values.

We can put together an example that includes everything we’ve seen so far including network initialization and train a network on a small dataset.

We can put this together with our code above for forward propagating input and with our small contrived dataset to test making predictions with an already-trained network.

For this we will use the helper function load_csv() to load the file, str_column_to_float() to convert string numbers to floats and str_column_to_int() to convert the class column to integer values.

It is generally good practice to normalize input values to the range of the chosen transfer function, in this case, the sigmoid function that outputs values between 0 and 1.

new function named back_propagation() was developed to manage the application of the Backpropagation algorithm, first initializing a network, training it on the training dataset and then using the trained network to make predictions on a test dataset.

You can see that backpropagation and the chosen configuration achieved a mean classification accuracy of about 93% which is dramatically better than the Zero Rule algorithm that did slightly better than 28% accuracy.

It involves subtracting the mean across every individual feature in the data, and has the geometric interpretation of centering the cloud of data around the origin along every dimension.

It only makes sense to apply this preprocessing if you have a reason to believe that different input features have different scales (or units), but they should be of approximately equal importance to the learning algorithm. In

case of images, the relative scales of pixels are already approximately equal (and in range from 0 to 255), so it is not strictly necessary to perform this additional preprocessing step.

Then, we can compute the covariance matrix that tells us about the correlation structure in the data: The (i,j) element of the data covariance matrix contains the covariance between i-th and j-th dimension of the data.

To decorrelate the data, we project the original (but zero-centered) data into the eigenbasis: Notice that the columns of U are a set of orthonormal vectors (norm of 1, and orthogonal to each other), so they can be regarded as basis vectors.

This is also sometimes refereed to as Principal Component Analysis (PCA) dimensionality reduction: After this operation, we would have reduced the original dataset of size [N x D] to one of size [N x 100], keeping the 100 dimensions of the data that contain the most variance.

The geometric interpretation of this transformation is that if the input data is a multivariable gaussian, then the whitened data will be a gaussian with zero mean and identity covariance matrix.

One weakness of this transformation is that it can greatly exaggerate the noise in the data, since it stretches all dimensions (including the irrelevant dimensions of tiny variance that are mostly noise) to be of equal size in the input.

Note that we do not know what the final value of every weight should be in the trained network, but with proper data normalization it is reasonable to assume that approximately half of the weights will be positive and half of them will be negative.

The idea is that the neurons are all random and unique in the beginning, so they will compute distinct updates and integrate themselves as diverse parts of the full network.

The implementation for one weight matrix might look like W = 0.01* np.random.randn(D,H), where randn samples from a zero mean, unit standard deviation gaussian.

With this formulation, every neuron’s weight vector is initialized as a random vector sampled from a multi-dimensional gaussian, so the neurons point in random direction in the input space.

That is, the recommended heuristic is to initialize each neuron’s weight vector as: w = np.random.randn(n) / sqrt(n), where n is the number of its inputs.

The sketch of the derivation is as follows: Consider the inner product \(s = \sum_i^n w_i x_i\) between the weights \(w\) and input \(x\), which gives the raw activation of a neuron before the non-linearity.

And since \(\text{Var}(aX) = a^2\text{Var}(X)\) for a random variable \(X\) and a scalar \(a\), this implies that we should draw from unit gaussian and then scale it by \(a = \sqrt{1/n}\), to make its variance \(1/n\).

In this paper, the authors end up recommending an initialization of the form \( \text{Var}(w) = 2/(n_{in} + n_{out}) \) where \(n_{in}, n_{out}\) are the number of units in the previous layer and the next layer.

A more recent paper on this topic, Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification by He et al., derives an initialization specifically for ReLU neurons, reaching the conclusion that the variance of neurons in the network should be \(2.0/n\).

This gives the initialization w = np.random.randn(n) * sqrt(2.0/n), and is the current recommendation for use in practice in the specific case of neural networks with ReLU neurons.

Another way to address the uncalibrated variances problem is to set all weight matrices to zero, but to break symmetry every neuron is randomly connected (with weights sampled from a small gaussian as above) to a fixed number of neurons below it.

For ReLU non-linearities, some people like to use small constant value such as 0.01 for all biases because this ensures that all ReLU units fire in the beginning and therefore obtain and propagate some gradient.

However, it is not clear if this provides a consistent improvement (in fact some results seem to indicate that this performs worse) and it is more common to simply use 0 bias initialization.

A recently developed technique by Ioffe and Szegedy called Batch Normalization alleviates a lot of headaches with properly initializing neural networks by explicitly forcing the activations throughout a network to take on a unit gaussian distribution at the beginning of the training.

In the implementation, applying this technique usually amounts to insert the BatchNorm layer immediately after fully connected layers (or convolutional layers, as we’ll soon see), and before non-linearities.

It is common to see the factor of \(\frac{1}{2}\) in front because then the gradient of this term with respect to the parameter \(w\) is simply \(\lambda w\) instead of \(2 \lambda w\).

Lastly, notice that during gradient descent parameter update, using the L2 regularization ultimately means that every weight is decayed linearly: W += -lambda * W towards zero.

L1 regularization is another relatively common form of regularization, where for each weight \(w\) we add the term \(\lambda \mid w \mid\) to the objective.

Another form of regularization is to enforce an absolute upper bound on the magnitude of the weight vector for every neuron and use projected gradient descent to enforce the constraint.

In practice, this corresponds to performing the parameter update as normal, and then enforcing the constraint by clamping the weight vector \(\vec{w}\) of every neuron to satisfy \(\Vert \vec{w} \Vert_2 <

Vanilla dropout in an example 3-layer Neural Network would be implemented as follows: In the code above, inside the train_step function we have performed dropout twice: on the first hidden layer and on the second hidden layer.

It can also be shown that performing this attenuation at test time can be related to the process of iterating over all the possible binary masks (and therefore all the exponentially many sub-networks) and computing their ensemble prediction.

Since test-time performance is so critical, it is always preferable to use inverted dropout, which performs the scaling at train time, leaving the forward pass at test time untouched.

Inverted dropout looks as follows: There has a been a large amount of research after the first introduction of dropout that tries to understand the source of its power in practice, and its relation to the other regularization techniques.

As we already mentioned in the Linear Classification section, it is not common to regularize the bias parameters because they do not interact with the data through multiplicative interactions, and therefore do not have the interpretation of controlling the influence of a data dimension on the final objective.

For example, a binary classifier for each category independently would take the form: where the sum is over all categories \(j\), and \(y_{ij}\) is either +1 or -1 depending on whether the i-th example is labeled with the j-th attribute, and the score vector \(f_j\) will be positive when the class is predicted to be present and negative otherwise.

A binary logistic regression classifier has only two classes (0,1), and calculates the probability of class 1 as: Since the probabilities of class 1 and 0 sum to one, the probability for class 0 is \(P(y = 0 \mid x;

The expression above can look scary but the gradient on \(f\) is in fact extremely simple and intuitive: \(\partial{L_i} / \partial{f_j} = y_{ij} - \sigma(f_j)\) (as you can double check yourself by taking the derivatives).

The L2 norm squared would compute the loss for a single example of the form: The reason the L2 norm is squared in the objective is that the gradient becomes much simpler, without changing the optimal parameters since squaring is a monotonic operation.

For example, if you are predicting star rating for a product, it might work much better to use 5 independent classifiers for ratings of 1-5 stars instead of a regression loss.

If you’re certain that classification is not appropriate, use the L2 but be careful: For example, the L2 is more fragile and applying dropout in the network (especially in the layer right before the L2 loss) is not a great idea.

How the backpropagation algorithm works

The backpropagation algorithm was originally introduced in the 1970s, but its importance wasn't fully appreciated until a famous 1986 paper by David Rumelhart, Geoffrey Hinton, and Ronald Williams.

That paper describes several neural networks where backpropagation works far faster than earlier approaches to learning, making it possible to use neural nets to solve problems which had previously been insoluble.

At the heart of backpropagation is an expression for the partial derivative $\partial C / \partial w$ of the cost function $C$ with respect to any weight $w$ (or bias $b$) in the network.

Warm up: a fast matrix-based approach to computing the output from a neural network Before discussing backpropagation, let's warm up with a fast matrix-based algorithm to compute the output from a neural network.

So, for example, the diagram below shows the weight on a connection from the fourth neuron in the second layer to the second neuron in the third layer of a network: This notation is cumbersome at first, and it does take some work to master.

The following diagram shows examples of these notations in use: With these notations, the activation $a^{l}_j$ of the $j^{\rm th}$ neuron in the $l^{\rm th}$ layer is related to the activations in the $(l-1)^{\rm th}$ layer by the equation (compare Equation (4)\begin{eqnarray} \frac{1}{1+\exp(-\sum_j w_j x_j-b)} \nonumber\end{eqnarray}$('#margin_266827233159_reveal').click(function() {$('#margin_266827233159').toggle('slow', function() {});});

and surrounding discussion in the last chapter) \begin{eqnarray} a^{l}_j = \sigma\left( \sum_k w^{l}_{jk} a^{l-1}_k + b^l_j \right), \tag{23}\end{eqnarray} where the sum is over all neurons $k$ in the $(l-1)^{\rm th}$ layer.

The entries of the weight matrix $w^l$ are just the weights connecting to the $l^{\rm th}$ layer of neurons, that is, the entry in the $j^{\rm th}$ row and $k^{\rm th}$ column is $w^l_{jk}$.

The last ingredient we need to rewrite (23)\begin{eqnarray} a^{l}_j = \sigma\left( \sum_k w^{l}_{jk} a^{l-1}_k + b^l_j \right) \nonumber\end{eqnarray}$('#margin_810655220635_reveal').click(function() {$('#margin_810655220635').toggle('slow', function() {});});

As an example, if we have the function $f(x) = x^2$ then the vectorized form of $f$ has the effect \begin{eqnarray} f\left(\left[ \begin{array}{c} 2 \\ 3 \end{array} \right] \right) = \left[ \begin{array}{c} f(2) \\ f(3) \end{array} \right] = \left[ \begin{array}{c} 4 \\ 9 \end{array} \right], \tag{24}\end{eqnarray} that is, the vectorized $f$ just squares every element of the vector.

With these notations in mind, Equation (23)\begin{eqnarray} a^{l}_j = \sigma\left( \sum_k w^{l}_{jk} a^{l-1}_k + b^l_j \right) \nonumber\end{eqnarray}$('#margin_195068543076_reveal').click(function() {$('#margin_195068543076').toggle('slow', function() {});});

\tag{25}\end{eqnarray} This expression gives us a much more global way of thinking about how the activations in one layer relate to activations in the previous layer: we just apply the weight matrix to the activations, then add the bias vector, and finally apply the $\sigma$ function* *By the way, it's this expression that motivates the quirk in the $w^l_{jk}$ notation mentioned earlier.

If we used $j$ to index the input neuron, and $k$ to index the output neuron, then we'd need to replace the weight matrix in Equation (25)\begin{eqnarray} a^{l} = \sigma(w^l a^{l-1}+b^l) \nonumber\end{eqnarray}$('#margin_423204946796_reveal').click(function() {$('#margin_423204946796').toggle('slow', function() {});});

It's also worth noting that $z^l$ has components $z^l_j = \sum_k w^l_{jk} a^{l-1}_k+b^l_j$, that is, $z^l_j$ is just the weighted input to the activation function for neuron $j$ in layer $l$.

The two assumptions we need about the cost function The goal of backpropagation is to compute the partial derivatives $\partial C / \partial w$ and $\partial C / \partial b$ of the cost function $C$ with respect to any weight $w$ or bias $b$ in the network.

The reason we need this assumption is because what backpropagation actually lets us do is compute the partial derivatives $\partial C_x / \partial w$ and $\partial C_x / \partial b$ for a single training example.

The second assumption we make about the cost is that it can be written as a function of the outputs from the neural network: For example, the quadratic cost function satisfies this requirement, since the quadratic cost for a single training example $x$ may be written as \begin{eqnarray} C = \frac{1}{2} \|y-a^L\|^2 = \frac{1}{2} \sum_j (y_j-a^L_j)^2, \tag{27}\end{eqnarray} and thus is a function of the output activations.

As an example, \begin{eqnarray} \left[\begin{array}{c} 1 \\ 2 \end{array}\right] \odot \left[\begin{array}{c} 3 \\ 4\end{array} \right] = \left[ \begin{array}{c} 1 * 3 \\ 2 * 4 \end{array} \right] = \left[ \begin{array}{c} 3 \\ 8 \end{array} \right].

Backpropagation will give us a way of computing $\delta^l$ for every layer, and then relating those errors to the quantities of real interest, $\partial C / \partial w^l_{jk}$ and $\partial C / \partial b^l_j$.

So we'll stick with $\delta^l_j = \frac{\partial C}{\partial z^l_j}$ as our measure of error* *In classification problems like MNIST the term 'error' is sometimes used to mean the classification failure rate.

As an example, in the case of the quadratic cost we have $\nabla_a C = (a^L-y)$, and so the fully matrix-based form of (BP1)\begin{eqnarray} \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j) \nonumber\end{eqnarray}$('#margin_817957397658_reveal').click(function() {$('#margin_817957397658').toggle('slow', function() {});});

An equation for the error $\delta^l$ in terms of the error in the next layer, $\delta^{l+1}$: In particular \begin{eqnarray} \delta^l = ((w^{l+1})^T \delta^{l+1}) \odot \sigma'(z^l), \tag{BP2}\end{eqnarray} where $(w^{l+1})^T$ is the transpose of the weight matrix $w^{l+1}$ for the $(l+1)^{\rm th}$ layer.

When we apply the transpose weight matrix, $(w^{l+1})^T$, we can think intuitively of this as moving the error backward through the network, giving us some sort of measure of the error at the output of the $l^{\rm th}$ layer.

An equation for the rate of change of the cost with respect to any bias in the network: In particular: \begin{eqnarray} \frac{\partial C}{\partial b^l_j} = \delta^l_j.

An equation for the rate of change of the cost with respect to any weight in the network: In particular: \begin{eqnarray} \frac{\partial C}{\partial w^l_{jk}} = a^{l-1}_k \delta^l_j.

The equation can be rewritten in a less index-heavy notation as \begin{eqnarray} \frac{\partial C}{\partial w} = a_{\rm in} \delta_{\rm out}, \tag{32}\end{eqnarray} where it's understood that $a_{\rm in}$ is the activation of the neuron input to the weight $w$, and $\delta_{\rm out}$ is the error of the neuron output from the weight $w$.

Zooming in to look at just the weight $w$, and the two neurons connected by that weight, we can depict this as: A nice consequence of Equation (32)\begin{eqnarray} \frac{\partial C}{\partial w} = a_{\rm in} \delta_{\rm out} \nonumber\end{eqnarray}$('#margin_328556804909_reveal').click(function() {$('#margin_328556804909').toggle('slow', function() {});});

There are other insights along these lines which can be obtained from (BP1)\begin{eqnarray} \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j) \nonumber\end{eqnarray}$('#margin_988408147203_reveal').click(function() {$('#margin_988408147203').toggle('slow', function() {});});-(BP4)\begin{eqnarray} \frac{\partial C}{\partial w^l_{jk}} = a^{l-1}_k \delta^l_j \nonumber\end{eqnarray}$('#margin_472943649145_reveal').click(function() {$('#margin_472943649145').toggle('slow', function() {});});.

And so the lesson is that a weight in the final layer will learn slowly if the output neuron is either low activation ($\approx 0$) or high activation ($\approx 1$).

And this, in turn, means that any weights input to a saturated neuron will learn slowly* *This reasoning won't hold if ${w^{l+1}}^T \delta^{l+1}$ has large enough entries to compensate for the smallness of $\sigma'(z^l_j)$.

Keeping the four equations (BP1)\begin{eqnarray} \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j) \nonumber\end{eqnarray}$('#margin_357408049891_reveal').click(function() {$('#margin_357408049891').toggle('slow', function() {});});-(BP4)\begin{eqnarray} \frac{\partial C}{\partial w^l_{jk}} = a^{l-1}_k \delta^l_j \nonumber\end{eqnarray}$('#margin_938065140080_reveal').click(function() {$('#margin_938065140080').toggle('slow', function() {});});

Problem Alternate presentation of the equations of backpropagation: I've stated the equations of backpropagation (notably (BP1)\begin{eqnarray} \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j) \nonumber\end{eqnarray}$('#margin_77625141718_reveal').click(function() {$('#margin_77625141718').toggle('slow', function() {});});

may be rewritten as \begin{eqnarray} \delta^L = \Sigma'(z^L) \nabla_a C, \tag{33}\end{eqnarray} where $\Sigma'(z^L)$ is a square matrix whose diagonal entries are the values $\sigma'(z^L_j)$, and whose off-diagonal entries are zero.

\tag{34}\end{eqnarray} (3) By combining observations (1) and (2) show that \begin{eqnarray} \delta^l = \Sigma'(z^l) (w^{l+1})^T \ldots \Sigma'(z^{L-1}) (w^L)^T \Sigma'(z^L) \nabla_a C \tag{35}\end{eqnarray} For readers comfortable with matrix multiplication this equation may be easier to understand than (BP1)\begin{eqnarray} \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j) \nonumber\end{eqnarray}$('#margin_649073791210_reveal').click(function() {$('#margin_649073791210').toggle('slow', function() {});});

Proof of the four fundamental equations (optional) We'll now prove the four fundamental equations (BP1)\begin{eqnarray} \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j) \nonumber\end{eqnarray}$('#margin_901201540049_reveal').click(function() {$('#margin_901201540049').toggle('slow', function() {});});-(BP4)\begin{eqnarray} \frac{\partial C}{\partial w^l_{jk}} = a^{l-1}_k \delta^l_j \nonumber\end{eqnarray}$('#margin_953233747393_reveal').click(function() {$('#margin_953233747393').toggle('slow', function() {});});.

\tag{36}\end{eqnarray} Applying the chain rule, we can re-express the partial derivative above in terms of partial derivatives with respect to the output activations, \begin{eqnarray} \delta^L_j = \sum_k \frac{\partial C}{\partial a^L_k} \frac{\partial a^L_k}{\partial z^L_j}, \tag{37}\end{eqnarray} where the sum is over all neurons $k$ in the output layer.

\tag{38}\end{eqnarray} Recalling that $a^L_j = \sigma(z^L_j)$ the second term on the right can be written as $\sigma'(z^L_j)$, and the equation becomes \begin{eqnarray} \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j), \tag{39}\end{eqnarray} which is just (BP1)\begin{eqnarray} \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j) \nonumber\end{eqnarray}$('#margin_337471468169_reveal').click(function() {$('#margin_337471468169').toggle('slow', function() {});});, in component form.

Next, we'll prove (BP2)\begin{eqnarray} \delta^l = ((w^{l+1})^T \delta^{l+1}) \odot \sigma'(z^l) \nonumber\end{eqnarray}$('#margin_762022419444_reveal').click(function() {$('#margin_762022419444').toggle('slow', function() {});});, which gives an equation for the error $\delta^l$ in terms of the error in the next layer, $\delta^{l+1}$.

We can do this using the chain rule, \begin{eqnarray} \delta^l_j & = & \frac{\partial C}{\partial z^l_j} \tag{40}\\ & = & \sum_k \frac{\partial C}{\partial z^{l+1}_k} \frac{\partial z^{l+1}_k}{\partial z^l_j} \tag{41}\\ & = & \sum_k \frac{\partial z^{l+1}_k}{\partial z^l_j} \delta^{l+1}_k, \tag{42}\end{eqnarray} where in the last line we have interchanged the two terms on the right-hand side, and substituted the definition of $\delta^{l+1}_k$.

\tag{44}\end{eqnarray} Substituting back into (42)\begin{eqnarray} & = & \sum_k \frac{\partial z^{l+1}_k}{\partial z^l_j} \delta^{l+1}_k \nonumber\end{eqnarray}$('#margin_83640522976_reveal').click(function() {$('#margin_83640522976').toggle('slow', function() {});});

Exercises Backpropagation with a single modified neuron Suppose we modify a single neuron in a feedforward network so that the output from the neuron is given by $f(\sum_j w_j x_j + b)$, where $f$ is some function other than the sigmoid.

How should we modify the backpropagation algorithm in this case?Backpropagation with linear neurons Suppose we replace the usual non-linear $\sigma$ function with $\sigma(z) = z$ throughout the network.

In particular, given a mini-batch of $m$ training examples, the following algorithm applies a gradient descent learning step based on that mini-batch: Input a set of training examples For each training example $x$: Set the corresponding input activation $a^{x,1}$, and perform the following steps: Feedforward: For each $l = 2, 3, \ldots, L$ compute $z^{x,l} = w^l a^{x,l-1}+b^l$ and $a^{x,l} = \sigma(z^{x,l})$.

Gradient descent: For each $l = L, L-1, \ldots, 2$ update the weights according to the rule $w^l \rightarrow w^l-\frac{\eta}{m} \sum_x \delta^{x,l} (a^{x,l-1})^T$, and the biases according to the rule $b^l \rightarrow b^l-\frac{\eta}{m} \sum_x \delta^{x,l}$.

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) which uses the backprop method to figure out the partial derivatives $\partial C_x / \partial b^l_j$ and $\partial C_x / \partial w^l_{jk}$.

This change is made to take advantage of a feature of Python, namely the use of negative list indices to count backward from the end of a list, so, e.g., l[-3] is the third last entry in a list l.

The idea is that instead of beginning with a single input vector, $x$, we can begin with a matrix $X = [x_1 x_2 \ldots x_m]$ whose columns are the vectors in the mini-batch.

(On my laptop, for example, the speedup is about a factor of two when run on MNIST classification problems like those we considered in the last chapter.) In practice, all serious libraries for backpropagation use this fully matrix-based approach or some variant.

An obvious way of doing that is to use the approximation \begin{eqnarray} \frac{\partial C}{\partial w_{j}} \approx \frac{C(w+\epsilon e_j)-C(w)}{\epsilon}, \tag{46}\end{eqnarray} where $\epsilon > 0$ is a small positive number, and $e_j$ is the unit vector in the $j^{\rm th}$ direction.

In other words, we can estimate $\partial C / \partial w_j$ by computing the cost $C$ for two slightly different values of $w_j$, and then applying Equation (46)\begin{eqnarray} \frac{\partial C}{\partial w_{j}} \approx \frac{C(w+\epsilon e_j)-C(w)}{\epsilon} \nonumber\end{eqnarray}$('#margin_453863887131_reveal').click(function() {$('#margin_453863887131').toggle('slow', function() {});});.

What's clever about backpropagation is that it enables us to simultaneously compute all the partial derivatives $\partial C / \partial w_j$ using just one forward pass through the network, followed by one backward pass through the network.

It's plausible because the dominant computational cost in the forward pass is multiplying by the weight matrices, while in the backward pass it's multiplying by the transposes of the weight matrices.

Compare that to the million and one forward passes we needed for the approach based on (46)\begin{eqnarray} \frac{\partial C}{\partial w_{j}} \approx \frac{C(w+\epsilon e_j)-C(w)}{\epsilon} \nonumber\end{eqnarray}$('#margin_825729805689_reveal').click(function() {$('#margin_825729805689').toggle('slow', function() {});});!

And so even though backpropagation appears superficially more complex than the approach based on (46)\begin{eqnarray} \frac{\partial C}{\partial w_{j}} \approx \frac{C(w+\epsilon e_j)-C(w)}{\epsilon} \nonumber\end{eqnarray}$('#margin_213340340123_reveal').click(function() {$('#margin_213340340123').toggle('slow', function() {});});, it's actually much, much faster.

To improve our intuition about what the algorithm is doing, let's imagine that we've made a small change $\Delta w^l_{jk}$ to some weight in the network, $w^l_{jk}$: That change in weight will cause a change in the output activation from the corresponding neuron: That, in turn, will cause a change in all the activations in the next layer: Those changes will in turn cause changes in the next layer, and then the next, and so on all the way through to causing a change in the final layer, and then in the cost function: The change $\Delta C$ in the cost is related to the change $\Delta w^l_{jk}$ in the weight by the equation \begin{eqnarray} \Delta C \approx \frac{\partial C}{\partial w^l_{jk}} \Delta w^l_{jk}.

We'll concentrate on the way just a single one of those activations is affected, say $a^{l+1}_q$, In fact, it'll cause the following change: \begin{eqnarray} \Delta a^{l+1}_q \approx \frac{\partial a^{l+1}_q}{\partial a^l_j} \Delta a^l_j.

\tag{49}\end{eqnarray} Substituting in the expression from Equation (48)\begin{eqnarray} \Delta a^l_j \approx \frac{\partial a^l_j}{\partial w^l_{jk}} \Delta w^l_{jk} \nonumber\end{eqnarray}$('#margin_881292020317_reveal').click(function() {$('#margin_881292020317').toggle('slow', function() {});});, we get: \begin{eqnarray} \Delta a^{l+1}_q \approx \frac{\partial a^{l+1}_q}{\partial a^l_j} \frac{\partial a^l_j}{\partial w^l_{jk}} \Delta w^l_{jk}.

If the path goes through activations $a^l_j, a^{l+1}_q, \ldots, a^{L-1}_n, a^L_m$ then the resulting expression is \begin{eqnarray} \Delta C \approx \frac{\partial C}{\partial a^L_m} \frac{\partial a^L_m}{\partial a^{L-1}_n} \frac{\partial a^{L-1}_n}{\partial a^{L-2}_p} \ldots \frac{\partial a^{l+1}_q}{\partial a^l_j} \frac{\partial a^l_j}{\partial w^l_{jk}} \Delta w^l_{jk}, \tag{51}\end{eqnarray} that is, we've picked up a $\partial a / \partial a$ type term for each additional neuron we've passed through, as well as the $\partial C/\partial a^L_m$ term at the end.

To compute the total change in $C$ it is plausible that we should sum over all the possible paths between the weight and the final cost, i.e., \begin{eqnarray} \Delta C \approx \sum_{mnp\ldots q} \frac{\partial C}{\partial a^L_m} \frac{\partial a^L_m}{\partial a^{L-1}_n} \frac{\partial a^{L-1}_n}{\partial a^{L-2}_p} \ldots \frac{\partial a^{l+1}_q}{\partial a^l_j} \frac{\partial a^l_j}{\partial w^l_{jk}} \Delta w^l_{jk}, \tag{52}\end{eqnarray} where we've summed over all possible choices for the intermediate neurons along the path.

we see that \begin{eqnarray} \frac{\partial C}{\partial w^l_{jk}} = \sum_{mnp\ldots q} \frac{\partial C}{\partial a^L_m} \frac{\partial a^L_m}{\partial a^{L-1}_n} \frac{\partial a^{L-1}_n}{\partial a^{L-2}_p} \ldots \frac{\partial a^{l+1}_q}{\partial a^l_j} \frac{\partial a^l_j}{\partial w^l_{jk}}.

\tag{53}\end{eqnarray} Now, Equation (53)\begin{eqnarray} \frac{\partial C}{\partial w^l_{jk}} = \sum_{mnp\ldots q} \frac{\partial C}{\partial a^L_m} \frac{\partial a^L_m}{\partial a^{L-1}_n} \frac{\partial a^{L-1}_n}{\partial a^{L-2}_p} \ldots \frac{\partial a^{l+1}_q}{\partial a^l_j} \frac{\partial a^l_j}{\partial w^l_{jk}} \nonumber\end{eqnarray}$('#margin_779763129343_reveal').click(function() {$('#margin_779763129343').toggle('slow', function() {});});

First, you could derive explicit expressions for all the individual partial derivatives in Equation (53)\begin{eqnarray} \frac{\partial C}{\partial w^l_{jk}} = \sum_{mnp\ldots q} \frac{\partial C}{\partial a^L_m} \frac{\partial a^L_m}{\partial a^{L-1}_n} \frac{\partial a^{L-1}_n}{\partial a^{L-2}_p} \ldots \frac{\partial a^{l+1}_q}{\partial a^l_j} \frac{\partial a^l_j}{\partial w^l_{jk}} \nonumber\end{eqnarray}$('#margin_131353667164_reveal').click(function() {$('#margin_131353667164').toggle('slow', function() {});});.

Or, to put it slightly differently, the backpropagation algorithm is a clever way of keeping track of small perturbations to the weights (and biases) as they propagate through the network, reach the output, and then affect the cost.

In Equation (53)\begin{eqnarray} \frac{\partial C}{\partial w^l_{jk}} = \sum_{mnp\ldots q} \frac{\partial C}{\partial a^L_m} \frac{\partial a^L_m}{\partial a^{L-1}_n} \frac{\partial a^{L-1}_n}{\partial a^{L-2}_p} \ldots \frac{\partial a^{l+1}_q}{\partial a^l_j} \frac{\partial a^l_j}{\partial w^l_{jk}} \nonumber\end{eqnarray}$('#margin_311957842940_reveal').click(function() {$('#margin_311957842940').toggle('slow', function() {});});

If you don't have this idea, and instead continue using the activations $a^{l+1}_q$, the proof you obtain turns out to be slightly more complex than the proof given earlier in the chapter.

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