What is Backpropagation?

Backpropagation is a widely used algorithm in machine learning for training artificial neural networks. It is a gradient computation technique that allows the network to learn from its mistakes by adjusting the weights of its connections between neurons. Backpropagation works by propagating errors backward through the network and updating weights based on those errors. The optimization step, such as using stochastic gradient descent (SGD), follows the computation of gradients.

In this article, we will explore what backpropagation is and how it works in detail. We will also discuss some common issues that arise when using backpropagation and ways to mitigate them.

Backpropagation Introduction

Backpropagation refers to the process of computing gradients of a loss function $\mathcal{L}$ with respect to each weight $w$ in a neural network. The gradients $\frac{\partial \mathcal{L}}{\partial w}$ are then used to update these weights so that they better predict outputs given inputs during the training phase.

The goal of backpropagation is to minimize the difference between predicted outputs and actual targets (labels) as measured by some loss function such as mean squared error $\text{MSE}$ or cross-entropy loss.

The process involves two phases:

1. Forward Propagation:

Forward propagation is the initial phase in the backpropagation algorithm, where input data $X$ is passed through the neural network, layer by layer, to generate the final output. The input data is transformed as it travels through each layer of the network, undergoing linear and non-linear operations defined by the weights $w$ and activation functions $f$ associated with each layer.

2. Backward Propagation

In which the error signal $\delta$ computed from the output layer is propagated backwards through all layers using chain rule derivatives until reaching the input layer. The error signal is computed as the gradient of the loss function with respect to activations (pre-activation values). Gradient updates are applied based on the computed values for each weight parameter $w$ along the way.

The calculated gradients are then used with an optimization algorithm such as stochastic gradient descent (SGD), Adam optimizer etc., to update model parameters like bias $b$ or weight $w$ values so that they better fit data provided during the training phase.

How Does BackPropagation Work?

The backpropagation algorithm works by calculating partial derivatives of the cost function $J$ with respect to every parameter (weights/biases) $\frac{\partial J}{\partial w}$ , $\frac{\partial J}{\partial b}$ within our neural network. These partial derivatives are then used to update the parameters, so that the cost function is minimized.

The process of backpropagation can be broken down into several steps:

Forward Pass

In this step, input data is fed through each layer of the neural network and transformed using activation functions $f$ and weights $w$ . The output $a$ from each layer serves as input to the next layer until we reach the final output.

It’s important to distinguish between pre-activation values (the weighted sums) and activation values (the result of applying activation functions). This distinction is crucial in understanding how the backward pass works.

Compute Loss

Once we have produced an output from our model, we compare it with actual labels (targets) using a loss function $\mathcal{L}$ such as mean squared error $\text{MSE}$ or cross-entropy loss $H(y, \hat{y})$ . This gives us a measure of how well our model performed on this particular example.

Backward Pass

In this step, we calculate gradients for each parameter (weights/biases) $\frac{\partial \mathcal{L}}{\partial w}$ , $\frac{\partial \mathcal{L}}{\partial b}$ involved in our model by computing partial derivatives with respect to the loss function calculated earlier via chain rule derivatives.

The computed gradients are then used in an optimization algorithm like stochastic gradient descent (SGD), Adam optimizer etc., to update weight values $w$ so that they better fit data provided during the training phase.

The process repeats itself over many iterations until convergence is achieved, i.e., when no further improvements can be made based upon current set parameter values within the tolerance level specified beforehand by the user or chosen automatically based upon some criteria such as validation accuracy etc.

Common Issues When Using Backpropagation:

Vanishing/Exploding Gradients

During backpropagation, gradients $\nabla$ may become very small or very large, which makes learning difficult since updates become too small/large, leading to oscillations around the optimal solution instead of finding it directly.

Vanishing gradients are common when using activation functions like sigmoid or tanh, especially in deep networks, as gradients become very small in early layers. Using ReLU activation functions and Xavier/He initialization helps mitigate this issue. Exploding gradients can occur in deeper networks, where gradients grow excessively large. Gradient clipping is often used to address this.

Overfitting

If our model has too many parameters or if we train for too long, it may start overfitting. Overfitting occurs when the neural network becomes too complex and starts memorizing the training data instead of generalizing to new data. This can be addressed through techniques like early stopping, regularization methods etc.

Saddle Points (instead of Local Minima)

In deep learning, neural networks are less likely to get stuck in local minima and more likely to hover around saddle points, where gradients are flat in many directions. This can slow down convergence. Optimization techniques like momentum or Adam can help speed up convergence in these cases.

Gradient Descent Variants

There are many variants of gradient descent such as stochastic gradient descent (SGD), mini-batch gradient descent, and batch gradient descent. Each variant has its own advantages and disadvantages depending on the size of the dataset, complexity of the model, etc., so researchers should choose wisely based on their specific needs.

Conclusion:

Backpropagation is a widely used algorithm for training artificial neural networks. It involves computing gradients with respect to each weight in a neural network and updating these weights to minimize the difference between predicted outputs and actual targets, as measured by some loss function like mean squared error or cross-entropy loss.

The process involves two phases: forward propagation, where input data passes through different layers while being transformed at each layer until the final output is generated at the end; and backward propagation, where the error signal computed from the output layer is propagated backwards through all layers using chain rule derivatives until reaching the input layer, where gradient updates are applied based upon computed values for each weight parameter involved along the way.

Backpropagation is not without its challenges, such as vanishing/exploding gradients, overfitting, and saddle points, which require careful consideration when designing models that incorporate this algorithm into their architecture.