Hello! Today, we will explore the ADAM (Adaptive Moment Estimation) algorithm. This advanced optimization algorithm is a favorite among machine learning practitioners as it combines the advantages of two other extensions of Stochastic Gradient Descent (SGD): Root Mean Square Propagation (RMSprop) and Adaptive Gradient Algorithm (AdaGrad). Our primary focus today is understanding ADAM, and we will also build it from scratch in C++ to optimize multivariable functions.

Before we dive into ADAM, let us recall that classic gradient descent methods like SGD and even sophisticated versions like Momentum and RMSProp have some limitations. These limitations relate to sensitivity to learning rates, the issue of vanishing gradients, and the absence of individual adaptive learning rates for different parameters.

ADAM, a promising choice for an optimization algorithm, combines the merits of RMSProp and AdaGrad. It maintains a per-parameter learning rate adapted based on the average of recent magnitudes of the gradients for the weights (similar to RMSProp) and the average of recent gradients (like Momentum). This mechanism enables the algorithm to traverse quickly over the low gradient regions and slow down near the optimal points.

For ADAM, we modify the update rule of SGD, introducing two additional hyperparameters, beta1 and beta2. The hyperparameter beta1 controls the exponential decay rate for the first-moment estimates (similar to Momentum), while beta2 controls the exponential decay rate for the second-moment estimates (similar to RMSProp). The standard ADAM algorithm always includes bias correction for these moment estimates, which is crucial for proper convergence, especially in the early stages of training.

The mathematical expression, including bias correction, is as follows:

Let's now consolidate the ADAM concept into C++ code, using the bias-corrected version. We will define an ADAM function, which takes the gradients, the decay rates beta1 and beta2, a numerical constant epsilon, the learning rate, previous estimates of m and v (initialized to 0), and the current epoch as input, and returns the updated parameters, along with the updated m and v.

Note: The bias correction is essential and is always included in the standard ADAM algorithm. This ensures that the moment estimates are unbiased, especially during the initial steps of optimization.

Now, let's test ADAM by finding the minimum of a multivariable function f(x, y) = x^2 + y^2. The corresponding gradients are df/dx = 2*x and df/dy = 2*y. With an initial starting point at (x, y) = (3, 4), selected reasonable values for beta1=0.9, beta2=0.999, epsilon=1e-8, learning_rate=0.001 and an epoch size of 150, we can start minimizing our function.

ADAM (Adaptive Moment Estimation) optimizer is generally more efficient than many other optimization algorithms such as SGD (Stochastic Gradient Descent) or RMSprop.

Overall, while the actual efficiency of ADAM compared to other optimizing algorithms can depend on the specific task or dataset, it often performs well in terms of both speed and accuracy across a variety of tasks.

Congratulations! You've now understood ADAM and how to code it in C++. With its sound mathematical foundations and impressive empirical results, ADAM constitutes an excellent stepping-stone into the fascinating world of machine learning optimization.

Remember, practice solidifies comprehension and consolidates understanding. Remember to attempt the upcoming hands-on exercises to reinforce these new burgeoning concepts. Until next time, happy coding!