Master Deep Learning: A Beginner's Guide to Optimization with Gradient Descent

Deep learning hinges on solving complex optimization challenges. At its core, a neural network functions as an intricate mathematical equation, fine-tuned through millions of parameters to provide solutions. Think of image classification: a network like AlexNet transforms image data into class scores. Training these networks involves minimizing a "loss function" – a measure of how far the network's performance deviates from perfection.

Prerequisites: Start Optimizing Now!

This beginner-friendly introduction to deep learning optimization requires no prior experience. Get ready to dive in!

Understanding the Loss Function in Neural Networks

Imagine a simplified neural network with just two adjustable parameters. In reality, networks could have billions! Consider a 2D loss function:

Loss function contour

This simplified contour visualizes the loss function. The x and y axes represent the two weights, while the z axis displays the corresponding loss value. Our objective? Find the weight values that minimize the loss, reaching the elusive minima.

Initially, the randomly initialized weights cause poor performance – like a network misclassifying cats. This corresponds to a high point on the loss function contour. The challenge is to navigate down to the "valley" representing the minima.

Demystifying Gradient Descent for Deep Learning

Upon weight initialization, we start at a specific point on the loss landscape. Gradient descent identifies the direction of the steepest decline in the loss function. By moving in the opposite direction of the gradient we iteratively approach the minima. The gradient points toward the steepest ascent.

The higher dimensional cousin of derivative offers the direction of steepest ascent. Visualizing this requires imagining a plane tangent to the current point. The gradient indicates the steepest ascent on this plane, and its opposite reveals the steepest descent. This 'descent' down the gradient is the core principle behind the algorithm.

Learning Rate: The Key to Effective Optimization

The learning rate determines the step size during each iteration. Crucially, this parameter needs careful tuning.

• Too high: Overshooting the minima, leading to endless bouncing.
• Too Low: Extremely slow training and vulnerability to being trapped in suboptimal minima.

With the gradient and learning rate defined, we proceed iteratively, recomputing the gradient at each new position. The gradient's magnitude indicates the steepness of the slope. As we approach the minima, the gradient diminishes, ideally becoming zero at the minimum point.

In reality, convergence typically means oscillating within the minima's vicinity. Once the loss plateaus, the training is considered to have "converged."

The Correct Way to Visualize Gradient Descent

Many tutorials depict gradient descent misleadingly, showing movement in the z-direction (loss). The trajectory is confined to the x-y plane, containing the weights, not in the z direction (loss). Gradients descent moves only with the weights.

Example of a correctly displayed gradient descent trajectory

Gradient Descent: The Essential Equations

The fundamental update equation for each iteration is:

Where:

• w is the weight vector.
• α represents the learning rate.
• The gradient (∇) indicates the direction of steepest ascent.

This update is simultaneously done for all the weights.

A variable learning rate, implemented through Simulated Annealing, can initially use large step sizes before gradually slowing down. This is implemented through decaying the learning rate as training continues.

Challenge #1: Avoiding Local Minima in Deep Neural Networks

Loss functions in neural networks are rarely convex. Instead, they have multiple local minima, which may not be optimal for the networks performance.

If the weights are initialized at point A, the algorithm converges to the local minima, without any way to jump to global minima.

Challenge #2: Navigating Saddle Points with Gradient Descent in Deep Learning

Saddle points, which are minima in one direction and maxima in another, can stall gradient descent. The algorithm may oscillate, creating a false sense of convergence.

Saddle point visualized

Gradient Descent Optimization: Harnessing the Power of Randomness

Randomness is injected by utilising Stochastic Gradient Descent (SGD). Rather than computing the gradients across all training examples at once (Batch Gradient Descent), SGD calculates the gradient of the loss for only one randomly selected example.

The update rule for SGD is:

At each step, this "one-example-loss" gradient points in a slightly different direction than the "all-example-loss" gradient. The introduction of randomness helps steer clear of local minima and saddle points.