# Training a Neural Network

In this example, we’ll be training a neural network using particle swarm optimization. For this we’ll be using the standard global-best PSO `pyswarms.single.GBestPSO` for optimizing the network’s weights and biases. This aims to demonstrate how the API is capable of handling custom-defined functions.

For this example, we’ll try to classify the three iris species in the Iris Dataset.

    [1]:

    # Import modules
    import numpy as np
    import matplotlib.pyplot as plt
    from sklearn.datasets import load_iris

    # Import PySwarms
    import pyswarms as ps

    # Some more magic so that the notebook will reload external python modules;
    # see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
    %load_ext autoreload
    %autoreload 2

First, we’ll load the dataset from `scikit-learn`. The Iris Dataset contains 3 classes for each of the iris species (iris setosa, iris virginica, and iris versicolor). It has 50 samples per class with 150 samples in total, making it a very balanced dataset. Each sample is characterized by four features (or dimensions): sepal length, sepal width, petal length, petal width.

## Load the iris dataset

    [9]:

    data = load_iris()

    [10]:

    # Store the features as X and the labels as y
    X = data.data
    y = data.target

## Constructing a custom objective function

Recall that neural networks can simply be seen as a mapping function from one space to another. For now, we’ll build a simple neural network with the following characteristics: * Input layer size: 4 * Hidden layer size: 20 (activation: \(\tanh(x)\)) * Output layer size: 3 (activation: \(softmax(x)\))

Things we’ll do: 1. Create a `forward_prop` method that will do forward propagation for one particle. 2. Create an overhead objective function `f()` that will compute `forward_prop()` for the whole swarm.

What we’ll be doing then is to create a swarm with a number of dimensions equal to the weights and biases. We will unroll these parameters into an n-dimensional array, and have each particle take on different values. Thus, each particle represents a candidate neural network with its own weights and bias. When feeding back to the network, we will reconstruct the learned weights and biases.

When rolling-back the parameters into weights and biases, it is useful to recall the shape and bias matrices: * Shape of input-to-hidden weight matrix: (4, 20) * Shape of input-to-hidden bias array: (20, ) * Shape of hidden-to-output weight matrix: (20, 3) * Shape of hidden-to-output bias array: (3, )

By unrolling them together, we have \((4 * 20) + (20 * 3) + 20 + 3 = 163\) parameters, or 163 dimensions for each particle in the swarm.

The negative log-likelihood will be used to compute for the error between the ground-truth values and the predictions. Also, because PSO doesn’t rely on the gradients, we’ll not be performing backpropagation (this may be a good thing or bad thing under some circumstances).

Now, let’s write the forward propagation procedure as our objective function. Let \(X\) be the input, \(z_l\) the pre-activation at layer \(l\), and \(a_l\) the activation for layer \(l\):

## Neural network architecture

    [11]:

    n_inputs = 4
    n_hidden = 20
    n_classes = 3

    num_samples = 150

    [16]:

    def logits_function(p):
        """ Calculate roll-back the weights and biases

        Inputs
        ------
        p: np.ndarray
            The dimensions should include an unrolled version of the
            weights and biases.

        Returns
        -------
        numpy.ndarray of logits for layer 2

        """
        # Roll-back the weights and biases
        W1 = p[0:80].reshape((n_inputs,n_hidden))
        b1 = p[80:100].reshape((n_hidden,))
        W2 = p[100:160].reshape((n_hidden,n_classes))
        b2 = p[160:163].reshape((n_classes,))

        # Perform forward propagation
        z1 = X.dot(W1) + b1  # Pre-activation in Layer 1
        a1 = np.tanh(z1)     # Activation in Layer 1
        logits = a1.dot(W2) + b2 # Pre-activation in Layer 2
        return logits          # Logits for Layer 2

    [22]:

    # Forward propagation
    def forward_prop(params):
        """Forward propagation as objective function

        This computes for the forward propagation of the neural network, as
        well as the loss.

        Inputs
        ------
        params: np.ndarray
            The dimensions should include an unrolled version of the
            weights and biases.

        Returns
        -------
        float
            The computed negative log-likelihood loss given the parameters
        """

        logits = logits_function(params)

        # Compute for the softmax of the logits
        exp_scores = np.exp(logits)
        probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)

        # Compute for the negative log likelihood

        corect_logprobs = -np.log(probs[range(num_samples), y])
        loss = np.sum(corect_logprobs) / num_samples

        return loss

Now that we have a method to do forward propagation for one particle (or for one set of dimensions), we can then create a higher-level method to compute `forward_prop()` to the whole swarm:

    [23]:

    def f(x):
        """Higher-level method to do forward_prop in the
        whole swarm.

        Inputs
        ------
        x: numpy.ndarray of shape (n_particles, dimensions)
            The swarm that will perform the search

        Returns
        -------
        numpy.ndarray of shape (n_particles, )
            The computed loss for each particle
        """
        n_particles = x.shape[0]
        j = [forward_prop(x[i]) for i in range(n_particles)]
        return np.array(j)

## Performing PSO on the custom-function

Now that everything has been set-up, we just call our global-best PSO and run the optimizer as usual. For now, we’ll just set the PSO parameters arbitrarily.

    [ ]:

    %%time
    # Initialize swarm
    options = {'c1': 0.5, 'c2': 0.3, 'w':0.9}

    # Call instance of PSO
    dimensions = (n_inputs * n_hidden) + (n_hidden * n_classes) + n_hidden + n_classes
    optimizer = ps.single.GlobalBestPSO(n_particles=100, dimensions=dimensions, options=options)

    # Perform optimization
    cost, pos = optimizer.optimize(f, iters=1000)

## Checking the accuracy

We can then check the accuracy by performing forward propagation once again to create a set of predictions. Then it’s only a simple matter of matching which one’s correct or not. For the `logits`, we take the `argmax`. Recall that the softmax function returns probabilities where the whole vector sums to 1. We just take the one with the highest probability then treat it as the network’s prediction.

Moreover, we let the best position vector found by the swarm be the weight and bias parameters of the network.

    [20]:

    def predict(pos):
        """
        Use the trained weights to perform class predictions.

        Inputs
        ------
        pos: numpy.ndarray
            Position matrix found by the swarm. Will be rolled
            into weights and biases.
        """
        logits = logits_function(pos)
        y_pred = np.argmax(logits, axis=1)
        return y_pred

And from this we can just compute for the accuracy. We perform predictions, compare an equivalence to the ground-truth value `y`, and get the mean.

    [21]:

    (predict(pos) == y).mean()

    [21]:

    0.9866666666666667