Discovering Machine Learning: A Comprehensive Introduction to Optimization Algorithms

Disclosure: Please note that the content in this blog was written with the assistance of OpenAI's ChatGPT language model.

What is Optimizer in Neural Nets?

An optimizer is a function or algorithm that adjusts the parameters of your model to minimize the error (or 'loss') the model generates on your data. In other words, optimizers shape and mold your model into its most accurate possible form by futzing with the weights. They do this by feeding the loss back through the network in the opposite direction, adjusting the weights against the gradient (known as backpropagation).

The objective of an optimizer is to find the minimum of the loss function. This can be understood as finding the best parameters of the model that reduce the error made for the seen data, as well as provide a model that generalizes well to unseen data.

Let's take the example of the most basic optimizer - Gradient Descent. The mathematical notation for Gradient Descent can be expressed as follows:

theta = theta - learning_rate * d(loss)/d(theta)

Here,

• 'theta' represents the parameters (or weights) of the model
• 'learning_rate' is a tuning parameter in the optimization process (often set between 0 and 1)
• 'd(loss)/d(theta)' is the derivative (or gradient) of the loss function with respect to the parameters

Let's consider a simple quadratic function f(x) = (x-3)^2 as our loss function. We will try to minimize this function using Gradient Descent. Our 'theta' in this case is 'x', and our learning_rate is 0.1.

Step 1: Initialize 'x' randomly. Let's start with x=0.

Step 2: Compute the derivative of f(x) with respect to 'x'. The derivative is 2*(x-3).

Step 3: Update 'x' using the formula above: x = x - learning_rate * derivative. With our learning_rate of 0.1, and our derivative at x=0, our update rule becomes: x = 0 - 0.1 * -6 = 0.6

Step 4: Repeat Step 2 and Step 3 until convergence. The value of 'x' that we converge upon will be the minimum of the function.

This is a very simplified example of how an optimizer works. In a deep learning context, the loss function is not a simple quadratic and can have many local minima, making optimization much more challenging. But the core idea remains the same: to iteratively adjust parameters in the direction that reduces the loss. Different optimizers like Stochastic Gradient Descent (SGD), RMSprop, Adam, etc., employ different strategies to navigate the loss function in the most efficient manner.

There are several optimization algorithms used in the field of machine learning and deep learning. Here are a few of the most popular ones:

• Gradient Descent (GD): This is the most basic form of optimization algorithm, which updates the weights by moving in the direction of the negative gradient of the function. It uses the whole dataset to compute the gradient at every step, which makes it computationally expensive for large datasets.
• Stochastic Gradient Descent (SGD): Unlike GD, which uses all data to make a single update, SGD uses only a single data point (chosen randomly) to make an update. This makes it faster but also noisier, as the updates are not based on an accurate estimate of the gradient.
• Mini-batch Gradient Descent: This is a compromise between GD and SGD. It uses a subset (or 'mini-batch') of the dataset to compute the gradient at every step. This makes it faster than GD, but less noisy than SGD.
• Momentum: This is a variant of SGD that takes into account the previous gradients to smooth out the updates. It adds a fraction of the direction of the previous step to a current step. This helps to accelerate SGD in the relevant direction and dampens the oscillation.
• Nesterov Accelerated Gradient (NAG): This is a variant of SGD with Momentum where the gradient is calculated for the expected future parameters rather than the current parameters.
• Adagrad: This is an algorithm for gradient-based optimization that adapts the learning rate to the parameters, performing larger updates for infrequent and smaller updates for frequent parameters.
• RMSprop: This is an optimizer that utilizes the magnitude of the recent gradient to normalize the gradient. It tries to resolve Adagrad's radically diminishing learning rates by using a moving average of the squared gradient.
• Adam (Adaptive Moment Estimation): Adam is another method that computes adaptive learning rates for each parameter. In addition to storing an exponentially decaying average of past squared gradients like Adagrad and RMSprop, Adam also keeps an exponentially decaying average of past gradients, similar to momentum.
• Adamax: It is a variant of Adam based on the infinity norm.
• Nadam: It is a variant of Adam that incorporates Nesterov Momentum.

These are just a few examples, and there are many other optimization algorithms. Each one has its strengths and weaknesses, and the choice of an optimizer often depends on the specific needs of the task at hand.