Compute Cost

We will use Binary Cross Entropy loss function (aka log loss). Here, \(y\) is the true label and \(\hat{y}\) is the predicted output.

\[ cost = – 1/N\sum_{i=1}^{N} y_{i}\log(\hat{y}_{i}) + (1 – y_{i})(\log(1 – \hat{y}_{i})) \]

The computeCost() function takes as arguments the input matrix \(X\), the true labels \(y\) and a cache. cache is the output of the forward pass that we calculated above. To calculate the error, we will only use the final output A2 from the cache.

computeCost <- function(X, y, cache) {
    m <- dim(X)[2]
    A2 <- cache$A2
    logprobs <- (log(A2) * y) + (log(1-A2) * (1-y))
    cost <- -sum(logprobs/m)
    return (cost)
}
cost <- computeCost(X_train, y_train, fwd_prop)
cost
## [1] 0.693

Backpropagation

Now comes the best part of this all: backpropagation!

We’ll write a function that will calculate the gradient of the loss function with respect to the parameters. Generally, in a deep network, we have something like the following.

The above figure has two hidden layers. During backpropagation (red boxes), we use the output cached during forward propagation (purple boxes). Our neural net has only one hidden layer. More specifically, we have the following:

To compute backpropagation, we write a function that takes as arguments an input matrix X, the train labels y, the output activations from the forward pass as cache, and a list of layer_sizes. The three outputs \((dW^{[l]}, db^{[l]}, dA^{[l-1]})\) are computed using the input \(dZ^{[l]}\) where \(l\) is the layer number.

We first differentiate the loss function with respect to the weight \(W\) of the current layer.

\[ dW^{[l]} = \frac{\partial \mathcal{L} }{\partial W^{[l]}} = \frac{1}{m} dZ^{[l]} A^{[l-1] T} \tag{8}\]

Then we differentiate the loss function with respect to the bias \(b\) of the current layer. \[ db^{[l]} = \frac{\partial \mathcal{L} }{\partial b^{[l]}} = \frac{1}{m} \sum_{i = 1}^{m} dZ^{[l](i)}\tag{9}\]

Once we have these, we calculate the derivative of the previous layer with respect to \(A\), the output + activation from the previous layer.

\[ dA^{[l-1]} = \frac{\partial \mathcal{L} }{\partial A^{[l-1]}} = W^{[l] T} dZ^{[l]} \tag{10}\]

Because we only have a single hidden layer, we first calculate the gradients for the final (output) layer and then the middle (hidden) layer. In other words, the gradients for the weights that lie between the output and hidden layer are calculated first. Using this (and chain rule), gradients for the weights that lie between the hidden and input layer are calculated next.

Finally, we return a list of gradient matrices. These gradients tell us the the small value by which we should increase/decrease our weights such that the loss decreases. Here are the equations for the gradients. I’ve calculated them for you so you don’t differentiate anything. We’ll directly use these values -

• \(dZ^{[2]} = A^{[2]} - Y\)
• \(dW^{[2]} = \frac{1}{m} dZ^{[2]}A^{[1]^T}\)
• \(db^{[2]} = \frac{1}{m}\sum dZ^{[2]}\)
• \(dZ^{[1]} = W^{[2]^T} * g^{[1]'} Z^{[1]}\) where \(g\) is the activation function.
• \(dW^{[1]} = \frac{1}{m}dZ^{[1]}X^{T}\)
• \(db^{[1]} = \frac{1}{m}\sum dZ^{[1]}\)

If you would like to know more about the math involved in constructing these equations, please see the references below.

backwardPropagation <- function(X, y, cache, params, list_layer_size){
    
    m <- dim(X)[2]
    
    n_x <- list_layer_size$n_x
    n_h <- list_layer_size$n_h
    n_y <- list_layer_size$n_y

    A2 <- cache$A2
    A1 <- cache$A1
    W2 <- params$W2

    dZ2 <- A2 - y
    dW2 <- 1/m * (dZ2 %*% t(A1)) 
    db2 <- matrix(1/m * sum(dZ2), nrow = n_y)
    db2_new <- matrix(rep(db2, m), nrow = n_y)
    
    dZ1 <- (t(W2) %*% dZ2) * (1 - A1^2)
    dW1 <- 1/m * (dZ1 %*% t(X))
    db1 <- matrix(1/m * sum(dZ1), nrow = n_h)
    db1_new <- matrix(rep(db1, m), nrow = n_h)
    
    grads <- list("dW1" = dW1, 
                  "db1" = db1,
                  "dW2" = dW2,
                  "db2" = db2)
    
    return(grads)
}

As you can see below, the shapes of the gradients are the same as their corresponding weights i.e. W1 has the same shape as dW1 and so on. This is important because we are going to use these gradients to update our actual weights.

back_prop <- backwardPropagation(X_train, y_train, fwd_prop, init_params, layer_size)
lapply(back_prop, function(x) dim(x))
## $dW1
## [1] 4 2
## 
## $db1
## [1] 4 1
## 
## $dW2
## [1] 1 4
## 
## $db2
## [1] 1 1

Update Parameters

From the gradients calculated by the backwardPropagation(), we update our weights using the updateParameters() function. The updateParameters() function takes as arguments the gradients, network parameters, and a learning rate.

Why a learning rate? Because sometimes the weight updates (gradients) are too large and because of that we miss the minima completely. Learning rate is a hyper-parameter that is set by us, the user, to control the impact of weight updates. The value of learning rate lies between \(0\) and \(1\). This learning rate is multiplied with the gradients before being subtracted from the weights.The weights are updated as follows where the learning rate is defined by \(\alpha\).

• \(W^{[2]} = W^{[2]} - \alpha * dW^{[2]}\)
• \(b^{[2]} = b^{[2]} - \alpha * db^{[2]}\)
• \(W^{[1]} = W^{[1]} - \alpha * dW^{[1]}\)
• \(b^{[1]} = b^{[1]} - \alpha * db^{[1]}\)

Updated parameters are returned by updateParameters() function. We take the gradients, weight parameters, and a learning rate as the input. grads and params are calculated above while we choose the learning_rate.

updateParameters <- function(grads, params, learning_rate){

    W1 <- params$W1
    b1 <- params$b1
    W2 <- params$W2
    b2 <- params$b2
    
    dW1 <- grads$dW1
    db1 <- grads$db1
    dW2 <- grads$dW2
    db2 <- grads$db2
    
    
    W1 <- W1 - learning_rate * dW1
    b1 <- b1 - learning_rate * db1
    W2 <- W2 - learning_rate * dW2
    b2 <- b2 - learning_rate * db2
    
    updated_params <- list("W1" = W1,
                           "b1" = b1,
                           "W2" = W2,
                           "b2" = b2)
    
    return (updated_params)
}

As we can see, the weights still maintain their original shape. This means we’ve done things correctly till this point.

update_params <- updateParameters(back_prop, init_params, learning_rate = 0.01)
lapply(update_params, function(x) dim(x))
## $W1
## [1] 4 2
## 
## $b1
## [1] 4 1
## 
## $W2
## [1] 1 4
## 
## $b2
## [1] 1 1

Train the Model

Now that we have all our components, let’s go ahead write a function that will train our model.

We will use all the functions we have written above in the following order.

1. Run forward propagation
2. Calculate loss
3. Calculate gradients
4. Update parameters
5. Repeat

This trainModel() function takes as arguments the input matrix X, the true labels y, and the number of epochs.

1. Get the sizes for layers and initialize random parameters.
2. Initialize a vector called cost_history which we’ll use to store the calculated loss value per epoch.
3. Run a for-loop:
   • Run forward prop.
   • Calculate loss.
   • Update parameters.
   • Replace the current parameters with updated parameters.

This function returns the updated parameters which we’ll use to run our model inference. It also returns the cost_history vector.

trainModel <- function(X, y, num_iteration, hidden_neurons, lr){
    
    layer_size <- getLayerSize(X, y, hidden_neurons)
    init_params <- initializeParameters(X, layer_size)
    cost_history <- c()
    for (i in 1:num_iteration) {
        fwd_prop <- forwardPropagation(X, init_params, layer_size)
        cost <- computeCost(X, y, fwd_prop)
        back_prop <- backwardPropagation(X, y, fwd_prop, init_params, layer_size)
        update_params <- updateParameters(back_prop, init_params, learning_rate = lr)
        init_params <- update_params
        cost_history <- c(cost_history, cost)
        
        if (i %% 10000 == 0) cat("Iteration", i, " | Cost: ", cost, "\n")
    }
    
    model_out <- list("updated_params" = update_params,
                      "cost_hist" = cost_history)
    return (model_out)
}

Now that we’ve defined our function to train, let’s run it! We’re going to train our model, with 40 hidden neurons, for 60000 epochs with a learning rate of 0.9. We will print out the loss after every 10000 epochs.

EPOCHS = 60000
HIDDEN_NEURONS = 40
LEARNING_RATE = 0.9

train_model <- trainModel(X_train, y_train, hidden_neurons = HIDDEN_NEURONS, num_iteration = EPOCHS, lr = LEARNING_RATE)
## Iteration 10000  | Cost:  0.3724 
## Iteration 20000  | Cost:  0.4081 
## Iteration 30000  | Cost:  0.3273 
## Iteration 40000  | Cost:  0.4671 
## Iteration 50000  | Cost:  0.4479 
## Iteration 60000  | Cost:  0.3074

Logistic Regression

Before we go ahead and test our neural net, let’s quickly train a simple logistic regression model so that we can compare its performance with our neural net. Since, a logistic regression model can learn only linear boundaries, it will not fit the data well. A neural-network on the other hand will.

We’ll use the glm() function in R to build this model.

lr_model <- glm(y ~ x1 + x2, data = train)
lr_model
## 
## Call:  glm(formula = y ~ x1 + x2, data = train)
## 
## Coefficients:
## (Intercept)           x1           x2  
##     0.51697      0.00889     -0.05207  
## 
## Degrees of Freedom: 319 Total (i.e. Null);  317 Residual
## Null Deviance:       80 
## Residual Deviance: 76.4  AIC: 458

Let’s now make generate predictions of the logistic regression model on the test set.

lr_pred <- round(as.vector(predict(lr_model, test[, 1:2])))
lr_pred
##  [1] 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 1 1 1 1 1
## [39] 1 1 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 0
## [77] 1 1 1 0

Test the Model

Finally, it’s time to make predictions. To do that -

1. First get the layer sizes.
2. Run forward propagation.
3. Return the prediction.

During inference time, we do not need to perform backpropagation as you can see below. We only perform forward propagation and return the final output from our neural network. (Note that instead of randomly initializing parameters, we’re using the trained parameters here. )

makePrediction <- function(X, y, hidden_neurons){
    layer_size <- getLayerSize(X, y, hidden_neurons)
    params <- train_model$updated_params
    fwd_prop <- forwardPropagation(X, params, layer_size)
    pred <- fwd_prop$A2
    
    return (pred)
}

After obtaining our output probabilities (Sigmoid), we round-off those to obtain output labels.

y_pred <- makePrediction(X_test, y_test, HIDDEN_NEURONS)
y_pred <- round(y_pred)

Here are the true labels and the predicted labels.

## Neural Net: 
##  1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 1 1 1 1 1 0 1 1 0 0 1 1 0 1
## Ground Truth: 
##  0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 1 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 1 1 0 1
## Logistic Reg: 
##  1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 0 1 1 1 0

tb_nn <- table(y_test, y_pred)
tb_lr <- table(y_test, lr_pred)

cat("NN Confusion Matrix: \n")
## NN Confusion Matrix:
tb_nn
##       y_pred
## y_test  0  1
##      0 34 10
##      1  7 29
cat("\nLR Confusion Matrix: \n")
## 
## LR Confusion Matrix:
tb_lr
##       lr_pred
## y_test  0  1
##      0 14 30
##      1 18 18

calculate_stats <- function(tb, model_name) {
  acc <- (tb[1] + tb[4])/(tb[1] + tb[2] + tb[3] + tb[4])
  recall <- tb[4]/(tb[4] + tb[3])
  precision <- tb[4]/(tb[4] + tb[2])
  f1 <- 2 * ((precision * recall) / (precision + recall))
  
  cat(model_name, ": \n")
  cat("\tAccuracy = ", acc*100, "%.")
  cat("\n\tPrecision = ", precision*100, "%.")
  cat("\n\tRecall = ", recall*100, "%.")
  cat("\n\tF1 Score = ", f1*100, "%.\n\n")
}

## Neural Network : 
##  Accuracy =  78.75 %.
##  Precision =  80.56 %.
##  Recall =  74.36 %.
##  F1 Score =  77.33 %.
## Logistic Regression : 
##  Accuracy =  40 %.
##  Precision =  50 %.
##  Recall =  37.5 %.
##  F1 Score =  42.86 %.

Conclusion

In this two-part series, we’ve built a neural net from scratch with a vectorized implementation of backpropagation. We went through the entire life cycle of training a model; right from data pre-processing to model evaluation. Along the way, we learned about the mathematics that makes a neural-network. We went over basic concepts of linear algebra and calculus and implemented them as functions. We saw how to initialize weights, perform forward propagation, gradient descent, and back-propagation.

We learned about the ability of a neural net to fit to non-linear data and understood the importance of the role activation functions play in it. We trained a neural net and compared it’s performance to a logistic-regression model. We visualized the decision boundaries of both these models and saw how a neural-net was able to fit better than logistic regression. We learned about metrics like Precision, Recall, F1-Score, and Accuracy by evaluating our models against them.

You should now have a pretty solid understanding of how neural-networks are built.

I hope you had as much fun reading as I had while writing this! If I’ve made a mistake somewhere, I’d love to hear about it so I can correct it. Suggestions and constructive criticism are welcome. 🙂

References

Here is a short list of two intermediate level and two beginner level references for the mathematics underlying neural networks.

Intermediate

• The Matrix Calculus You Need for Deep Learning - Parr and Howard (2018)
• Deep Learning: An Introduction for Applied Mathematicians - Higham and Higham (2018)

Beginner

• Deep Learning course by Andrew NG on Coursera. It can be audited for free.
  
• Grant Sanderson’s YouTube channel. Here are the 4 relevant playlists. diff eq, linear algebra, calculus, neural nets.