Backpropagation in Neural Networks

Introduction

Welcome back to "Training Neural Networks: The Backpropagation Algorithm"! You've made excellent progress so far, having learned about loss functions in our first lesson and gradient descent in our second. Today, we're diving into the heart of neural network training: backpropagation . In our previous lesson, we explored how gradient descent updates weights by moving in the direction opposite to the gradient of the loss function. But we left an important question unanswered: How do we actually calculate these gradients in a neural network with multiple layers and thousands or even millions of parameters? That's where backpropagation comes in. Backpropagation (short for "backward propagation of errors") is an efficient algorithm for computing these gradients. Today, we'll focus specifically on implementing the backward pass for a single dense layer, which will form the building block for training complete neural networks. By the end of this lesson, you'll understand how to: Calculate derivatives for different activation functions . Store necessary values during the forward pass . Implement the backward pass to calculate gradients. Connect these gradients to the gradient descent algorithm we learned previously. Let's embark on this crucial step in our neural network journey!

Understanding the Chain Rule for Backpropagation

f(g(x))

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The DenseLayer Class Structure

Now, let's look at the structure of our DenseLayer class using R6. This class encapsulates both the forward pass (which we've seen in previous lessons) and the backward pass (which we're focusing on today).

library(R6)

DenseLayer <- R6Class("DenseLayer",
  public = list(
    n_inputs = NULL,
    n_neurons = NULL,
    activation_fn_name = NULL,
    weight_init_strategy = NULL,
    weight_init_scale = NULL,
    weights = NULL,
    biases = NULL,
    activation_fn = NULL,
    activation_derivative_fn = NULL,
    inputs = NULL,
    z = NULL,
    output = NULL,
    d_weights = NULL,
    d_biases = NULL,
    
    initialize = function(n_inputs, n_neurons, activation_fn_name = 'sigmoid',
                         weight_init_strategy = 'random_scaled', weight_init_scale = 0.01) {
      self$n_inputs <- n_inputs
      self$n_neurons <- n_neurons
      self$activation_fn_name <- activation_fn_name
      self$weight_init_strategy <- weight_init_strategy
      self$weight_init_scale <- weight_init_scale
      
      # Initialize weights and biases
      if (weight_init_strategy == 'random_scaled') {
        self$weights <- matrix(rnorm(n_inputs * n_neurons), nrow = n_inputs, ncol = n_neurons) * weight_init_scale
      }
      self$biases <- matrix(0, nrow = 1, ncol = n_neurons)
      
      # Set activation function and its derivative
      if (activation_fn_name == 'sigmoid') {
        self$activation_fn <- sigmoid
        self$activation_derivative_fn <- sigmoid_derivative
      } else if (activation_fn_name == 'relu') {
        self$activation_fn <- relu
        self$activation_derivative_fn <- relu_derivative
      } else if (activation_fn_name == 'linear') {
        self$activation_fn <- linear
        self$activation_derivative_fn <- linear_derivative
      } else {
        stop(paste("Unsupported activation:", activation_fn_name))
      }
      
      # Variables to store values needed for backpropagation
      self$inputs <- NULL
      self$z <- NULL
      self$output <- NULL
      self$d_weights <- NULL
      self$d_biases <- NULL
    }
  )
)

The key added components to notice:

Activation Derivatives: We store not only the chosen activation_fn but also its activation_derivative_fn.
Caching Variables:
- self$inputs will store the inputs to the layer.
- self$z will store the pre-activation outputs.
- self$output will store the post-activation outputs.
- self$d_weights and self$d_biases will store the gradients of weights and biases.

This caching of intermediate values is crucial for backpropagation. We need to know these values during the backward pass to correctly compute the gradients.

The Backward Pass: Calculating Gradients

Now we come to the heart of backpropagation: the backward method. This method calculates the gradients that will be used to update the weights during gradient descent.

# Add this method to the DenseLayer class
DenseLayer$set("public", "backward", function(d_loss_wrt_layer_output) {
  # Gradient of loss w.r.t. pre-activation output (dL/dz)
  # dL/dz = dL/dy * dy/dz, where y = activation(z)
  d_activation_output <- d_loss_wrt_layer_output * self$activation_derivative_fn(self$output)
  
  # Gradient of loss w.r.t. weights (dL/dW)
  # dL/dW = dL/dz * dz/dW = d_activation_output * self$inputs.T
  self$d_weights <- t(self$inputs) %*% d_activation_output
  
  # Gradient of loss w.r.t. biases (dL/dB)
  # dL/dB = dL/dz * dz/dB = d_activation_output * 1 (summed over samples)
  self$d_biases <- matrix(colSums(d_activation_output), nrow = 1)
  
  # Gradient of loss w.r.t. inputs of this layer (dL/dX_current_layer)
  # This will be d_loss_wrt_layer_output for the *previous* layer
  # dL/dX = dL/dz * dz/dX = d_activation_output . self$weights.T
  d_loss_wrt_prev_layer_output <- d_activation_output %*% t(self$weights)
  
  return(d_loss_wrt_prev_layer_output)
})

Let's break down what's happening:

Input: d_loss_wrt_layer_output is the gradient of the loss with respect to this layer's output. For the output layer, this comes directly from the loss function. For hidden layers, it's passed backward from the next layer.
Gradient w.r.t. pre-activation output: We first calculate $\frac{dL}{dz}$ by applying the chain rule: $\frac{dL}{dz} = \frac{dL}{dy} \cdot \frac{dy}{dz}$ , where y is the activation output and z is the pre-activation output.
Gradient w.r.t. weights: We calculate $\frac{dL}{dW}$ using the chain rule again: $\frac{dL}{dW} = \frac{dL}{dz} \cdot \frac{dz}{dW}$ . Since $z = X \cdot W + b$ , we have $\frac{dz}{dW} = X^T$ , leading to $\frac{dL}{dW} = X^T \cdot \frac{dL}{dz}$ .
Gradient w.r.t. biases: Similarly, $\frac{dL}{db} = \frac{dL}{dz} \cdot \frac{dz}{db}$ . Since $\frac{dz}{db} = 1$ , we sum $\frac{dL}{dz}$ over all samples.
Gradient w.r.t. inputs: Finally, we calculate $\frac{dL}{dX}$ to pass backward to the previous layer: $\frac{dL}{dX} = \frac{dL}{dz} \cdot \frac{dz}{dX}$ . Since $\frac{dz}{dX} = W^T$ , we get $\frac{dL}{dX} = \frac{dL}{dz} \cdot W^T$ .

Each of these calculations is a direct application of the chain rule, and together they form the core of the backpropagation algorithm.

Backpropagation in Action: A Practical Example

Let's now see how our backpropagation implementation works in practice with a simple example:

# Create a single layer: 2 inputs, 3 neurons, sigmoid activation
layer <- DenseLayer$new(n_inputs = 2, n_neurons = 3, activation_fn_name = 'sigmoid')

X_sample <- matrix(c(0.5, -0.2), nrow = 1, ncol = 2)  # 1 sample, 2 features
cat("Input X:\n")
print(X_sample)

# Forward pass
layer_output <- layer$forward(X_sample)
cat("Layer output (after sigmoid):\n")
print(layer_output)

# Assume a dummy gradient from a hypothetical next layer or loss function
# This is d(Loss)/d(layer_output)
# Shape must match layer_output: (n_samples, n_neurons_in_layer)
dummy_d_loss_wrt_layer_output <- matrix(c(0.1, -0.2, 0.05), nrow = 1, ncol = 3)
cat("Dummy dL/d(layer_output):\n")
print(dummy_d_loss_wrt_layer_output)

# Backward pass
d_loss_wrt_input <- layer$backward(dummy_d_loss_wrt_layer_output)

cat("\nCalculated Gradients:\n")
cat("  dL/d_weights (shape", dim(layer$d_weights), "):\n")
print(layer$d_weights)
cat("  dL/d_biases (shape", dim(layer$d_biases), "):\n")
print(layer$d_biases)
cat("  dL/d_inputs (to pass to prev layer) (shape", dim(d_loss_wrt_input), "):\n")
print(d_loss_wrt_input)

# Create a single layer: 2 inputs, 3 neurons, sigmoid activation
layer <- DenseLayer$new(n_inputs = 2, n_neurons = 3, activation_fn_name = 'sigmoid')

X_sample <- matrix(c(0.5, -0.2), nrow = 1, ncol = 2)  # 1 sample, 2 features
cat("Input X:\n")
print(X_sample)

# Forward pass
layer_output <- layer$forward(X_sample)
cat("Layer output (after sigmoid):\n")
print(layer_output)

# Assume a dummy gradient from a hypothetical next layer or loss function
# This is d(Loss)/d(layer_output)
# Shape must match layer_output: (n_samples, n_neurons_in_layer)
dummy_d_loss_wrt_layer_output <- matrix(c(0.1, -0.2, 0.05), nrow = 1, ncol = 3)
cat("Dummy dL/d(layer_output):\n")
print(dummy_d_loss_wrt_layer_output)

# Backward pass
d_loss_wrt_input <- layer$backward(dummy_d_loss_wrt_layer_output)

cat("\nCalculated Gradients:\n")
cat("  dL/d_weights (shape", dim(layer$d_weights), "):\n")
print(layer$d_weights)
cat("  dL/d_biases (shape", dim(layer$d_biases), "):\n")
print(layer$d_biases)
cat("  dL/d_inputs (to pass to prev layer) (shape", dim(d_loss_wrt_input), "):\n")
print(d_loss_wrt_input)

This example:

Creates a single DenseLayer with 2 inputs and 3 neurons.
Performs a forward pass with a sample input.
Simulates receiving gradients from the next layer using a dummy gradient.
Performs a backward pass using this gradient.
Prints the computed gradients for weights, biases, and inputs.

Output Discussion

When we run this code, we get the following output:

Input X:
     [,1] [,2]
[1,]  0.5 -0.2

Layer output (after sigmoid):
          [,1]      [,2]      [,3]
[1,] 0.5000116 0.4984150 0.5013022

Dummy dL/d(layer_output):
     [,1] [,2] [,3]
[1,]  0.1 -0.2 0.05

Calculated Gradients:
  dL/d_weights (shape 2 3 ):
           [,1]        [,2]        [,3]
[1,] 0.01250000 -0.02499975  0.00624996
[2,] -0.00500000  0.00999990 -0.00249998

  dL/d_biases (shape 1 3 ):
           [,1]        [,2]        [,3]
[1,] 0.02500000 -0.04999950  0.01249992

  dL/d_inputs (to pass to prev layer) (shape 1 2 ):
            [,1]        [,2]
[1,] 0.0006935802 -0.0001824598

Input X:
     [,1] [,2]
[1,]  0.5 -0.2

Layer output (after sigmoid):
          [,1]      [,2]      [,3]
[1,] 0.5000116 0.4984150 0.5013022

Dummy dL/d(layer_output):
     [,1] [,2] [,3]
[1,]  0.1 -0.2 0.05

Calculated Gradients:
  dL/d_weights (shape 2 3 ):
           [,1]        [,2]        [,3]
[1,] 0.01250000 -0.02499975  0.00624996
[2,] -0.00500000  0.00999990 -0.00249998

  dL/d_biases (shape 1 3 ):
           [,1]        [,2]        [,3]
[1,] 0.02500000 -0.04999950  0.01249992

  dL/d_inputs (to pass to prev layer) (shape 1 2 ):
            [,1]        [,2]
[1,] 0.0006935802 -0.0001824598

Looking at this output:

Our input is a single sample with two features: [0.5, -0.2].
The forward pass produces outputs around 0.5 (since our weights are initialized close to zero, the sigmoid of values near zero is about 0.5).
We provide a dummy gradient [0.1, -0.2, 0.05] representing how the loss would change if each output neuron's value changed slightly.
The backward pass calculates:
- Gradients for each weight (d_weights).
- Gradients for each bias (d_biases).
- Gradients to pass to the previous layer (d_loss_wrt_input).

This example demonstrates the full cycle of forward and backward passes for a single layer. In a complete neural network, we would perform this process for each layer, starting from the output and working backward (hence the name backpropagation).

# --- Activation Functions & Derivatives --- sigmoid <- function(x) { res <- 1 / (1 + exp(-x)) if (is.matrix(x)) { dim(res) <- dim(x) } return(res) } sigmoid_derivative <- function(output) { output * (1 - output) # output = sigmoid(z) } relu <- function(x) { res <- pmax(0, x) if (is.matrix(x)) { dim(res) <- dim(x) } return(res) } relu_derivative <- function(output) { ifelse(output > 0, 1, 0) # output = relu(z) } linear <- function(x) { x } linear_derivative <- function(output) { array(1, dim = dim(output)) # output = linear(z) }

# Add this method to the DenseLayer class DenseLayer$set("public", "backward", function(d_loss_wrt_layer_output) { # Gradient of loss w.r.t. pre-activation output (dL/dz) # dL/dz = dL/dy * dy/dz, where y = activation(z) d_activation_output <- d_loss_wrt_layer_output * self$activation_derivative_fn(self$output) # Gradient of loss w.r.t. weights (dL/dW) # dL/dW = dL/dz * dz/dW = d_activation_output * self$inputs.T self$d_weights <- t(self$inputs) %*% d_activation_output # Gradient of loss w.r.t. biases (dL/dB) # dL/dB = dL/dz * dz/dB = d_activation_output * 1 (summed over samples) self$d_biases <- matrix(colSums(d_activation_output), nrow = 1) # Gradient of loss w.r.t. inputs of this layer (dL/dX_current_layer) # This will be d_loss_wrt_layer_output for the *previous* layer # dL/dX = dL/dz * dz/dX = d_activation_output . self$weights.T d_loss_wrt_prev_layer_output <- d_activation_output %*% t(self$weights) return(d_loss_wrt_prev_layer_output) })