Deep Learning Optimization: How Gradient Descent Works

Updated September 13, 2024

Is your deep learning model stuck? Learn how gradient descent, the workhorse of neural network training, helps you find the optimal solution!

Deep learning boils down to optimization, tweaking millions of parameters to minimize a loss function. Gradient descent is a powerful algorithm that navigates this complex landscape.

Loss Function: Gauging Imperfection

Imagine a landscape where the height represents the 'loss' (error) of your neural network. Our goal is to find the lowest point in this landscape, which corresponds to the set of network parameters that produce more accurate outcomes. The loss function acts as a gauge for how well our network performs on a dataset. Minimizing this function is key to training effective models.

Let's simplify and assume our network only has two adjustable parameters or weights.

Visualizing the Loss Function Landscape

The x and y axes represent the two weights, and the z axis shows the loss value for each combination. We want to find the weight values corresponding to the “minima,” the lowest point on this landscape.

Initially, our network performs poorly due to random weight initialization, corresponding to a high point on the loss surface. Gradient descent helps us to 'descend' towards the minima, iteratively improving our network.

Gradient Descent: Finding the Steepest Path Down

Gradient descent is like finding the quickest way down a hill. It calculates the gradient, which points in the direction of the steepest ascent. By moving in the opposite direction, we descend towards the minimum loss.

Here's how it works:

• Calculate the gradient: Determine the direction of the steepest increase.
• Adjust the weights: Update weights by moving opposite to the gradient direction.
• Repeat: Repeat the process until the loss is minimized.

Here is a critical point: The trajectory is confined to the x-y plane.

Learning Rate: Pacing Our Descent

The learning rate determines the step size during each iteration. It's crucial to choose it wisely as a small learning rate can lead to slow convergence, while with a learning rate that is too big, the model may overshoot.

Choosing the Right Learning Rate:

• Large Learning Rate: Overshoots the minimum, leading to oscillations.
• Small Learning Rate: Slow convergence, potentially getting stuck.

Understanding the Gradient Descent Equation

The core update rule in gradient descent is:

w = w - alpha * gradient(loss, w)

Where:

• w is the weights vector.
• alpha is the learning rate.
• gradient(loss, w) is the gradient of the loss function with respect to the weights.

This equation is applied simultaneously to every individual weight.

Challenges: Local Minima and Saddle Points

Gradient descent isn't always smooth sailing. Two major challenges can hinder its effectiveness:

• Local Minima: The algorithm gets stuck in a local minimum, which isn't the optimal solution.
• Saddle Points: The algorithm plateaus at a saddle point, mistaking it for a minimum.

Neural network loss functions are non-convex functions, meaning they have a lot of imperfections that we should keep in mind.

Escaping Traps with Randomness

To overcome those previous challenges we can infuse randomness into the process using Stochastic Gradient Descent (SGD).

From Batch to Stochastic Gradient Descent

Stochastic Gradient Descent (SGD) introduces randomness by updating the network's weights based on the gradient calculated from a single, randomly chosen training example.

Here is the update rule:

w = w - alpha * gradient(loss_i, w)

Where: loss_i is the loss calculated for a single data point.

This intentional "noise" helps to avoid local minima.