Understanding the Adam Optimization Algorithm: A Deep Dive into the Formulas

The Adam optimization algorithm has become popular for training machine learning and deep learning models due to its efficiency and adaptability. Developed by Diederik Kingma and Jimmy Ba, Adam combines the advantages of the Momentum and RMSprop optimization algorithms. In this post, we will focus on understanding the formulas behind the Adam optimization algorithm, breaking down its components step by step to provide a comprehensive understanding of its inner workings.

Background

Gradient-based optimization algorithms use the gradients of the loss function concerning the model parameters to update these parameters iteratively, minimizing the loss function. While gradient descent is the most basic optimization algorithm, it has limitations such as sensitivity to learning rate choice, slow convergence, and difficulty navigating noisy or sparse gradients.

Several optimization algorithms have been proposed to address these limitations, including Momentum, Nesterov Accelerated Gradient (NAG), AdaGrad, and RMSprop. The Adam optimization algorithm was introduced to combine the best features of Momentum and RMSprop while tackling their shortcomings.

The Adam Algorithm Formulas

The Adam algorithm computes adaptive learning rates for each parameter using the first and second moments of the gradients. Let’s break down the formulas involved in the Adam algorithm:

• Initialize the model parameters (θ), learning rate (α), and hyper-parameters (β1, β2, and ε).
• Compute the gradients (g) of the loss function (L) with respect to the model parameters.

• Update the first moment estimates (m):

• Update the second moment estimates (v):

O is a Hadamard product

• Correct the bias in the first (m_hat) and second (v_hat) moment estimates for the current iteration (t):

• Compute the adaptive learning rates (α_t):

• Update the model parameters using the adaptive learning rates:

This is a MATLAB implementation of the Adam optimization algorithm as described above. This implementation can be easily adapted for other loss functions and machine learning models.

function [W,b,M,V] = Adam(W, b, dW, db, alpha, M, V, iT)
    beta1   = 0.9;
    beta2   = 0.999;
    epsilon = 1e-8;

    params = [W;b];
    grads  = [dW;db];

    M  = beta1*M + (1-beta1)*grads;
    V  = beta2*V + (1-beta2)*grads.^2;

    M2 = M / (1-beta1^iT);
    V2 = V / (1-beta2^iT);

    alpha   = alpha*sqrt(1-beta2^iT)/(1-beta1^iT);

    params  = params - alpha*M2 ./ (sqrt(V2)+epsilon);

    W = params(1:end-1,:);
    b = params(end,:);
end

Implementing Adam in MATLAB

Here, we demonstrated a basic MATLAB implementation of the Adam optimization algorithm for minimizing the loss function in Iris dataset classification using a simple neural network model. This implementation can be easily adapted for other loss functions and machine learning models.

Understanding the Adam Optimization Algorithm

Advantages of Adam

The Adam optimization algorithm has several advantages over other gradient-based optimization algorithms:

• Adaptive learning rates: Adam computes individual learning rates for each parameter, speeding up convergence and improving the quality of the final solution.
• Suitable for noisy gradients: Adam performs well in cases with noisy gradients, such as training deep learning models with mini-batches.
• Low memory requirements: Adam requires only two additional variables for each parameter, making it memory-efficient.
• Robust to the choice of hyperparameters: Adam is relatively insensitive to the choice of hyperparameters, making it easy to use in practice.

Conclusion

The Adam algorithm is a powerful and versatile optimization technique for training machine learning and deep learning models. By combining the best features of Momentum and RMSprop, it offers adaptive learning rates, faster convergence, and robustness to the choice of hyperparameters. If you are looking for an optimization algorithm that is easy to use, efficient, and effective, the Adam optimization method is an excellent choice.

This article has provided a gentle introduction to the Adam optimization algorithm, explained its key concepts, and demonstrated how to implement it in MATLAB from scratch. Whether you are new to optimization algorithms or an experienced practitioner, we hope this introduction to the Adam optimization algorithm and its MATLAB implementation has been helpful and informative.

By following the guidance provided in this article, you can better understand the Adam algorithm and implement it in MATLAB to optimize your machine learning models. The provided MATLAB code can serve as a starting point for further exploration and adaptation to other loss functions and model types. Happy optimizing!