Adaptive Gradient Algorithm (AdaGrad)

AdaGrad is a renowned optimization algorithm utilized extensively in Machine Learning (ML) and Deep Learning (DL). Brought to light in 2011 by Duchi, Hazan, and Singer, its primary function is the alteration of the learning rate throughout the training period. The unique characteristic of AdaGrad is that it tweaks the learning rate for each model parameter, depending on that parameter's past gradients.

The process involves the computation of the learning rate as an aggregate of the gradients' squares over a period, each one corresponding to a parameter. The outcome is a reduction in the learning rate for parameters with prominent gradients and an elevation in the learning rate for parameters with mild gradients. This distinct methodology aids in the adjustment of the learning rate to the loss function's geometry, which, in return, enables quicker convergence in steep gradient directions while maintaining conservation in flatter gradient directions. Resultantly, enhancing convergence speed and improving generalization.

That said, the AdaGrad method is not without its drawbacks. One key issue is that the accumulated gradient magnitudes can drastically inflate over time, leading to an incredibly low effective learning rate, potentially stifling the learning process. Modern optimization algorithms like Adam and RMSProp tackle this issue by pairing their adaptive learning rate approach with other techniques to keep the gradient magnitudes' growth in check over time.

Highlighting Gradient Descent, it's a key optimization methodology in ML and DL used to find optimum model parameters' values. It is an iterative method aimed to minimize a loss function that encapsulates the gap between predicted and factual outputs of the model. Subgradient Descent is an iteration of gradient descent, utilized when the loss function cannot be differentiated at some points.

Under these circumstances, the function's gradient is unknown, but a subgradient can be calculated. With each iteration, the subgradient descent method chooses a subgradient g of the loss function and updates the current optimal solution estimate in the negative subgradient's direction. Despite being a tad slower than ordinary gradient descent, by carefully determining the step size, it can still converge to the ideal solution.

Mainly, there are three variants of Gradient Descent:

1. Batch Gradient Descent: In this type, at each step, the gradient is computed using the entire dataset, adjusting parameters by moving towards the loss function's negative gradient.
2. Stochastic Gradient Descent (SGD): Under this variant, the gradient is estimated at each step using a single randomly chosen sample from the dataset, making the process faster but at the expense of accuracy.
3. Mini-batch Gradient Descent: This combines both batch gradient descent and SGD. It uses a small batch of randomly selected samples from the dataset to calculate the gradient, finding equilibrium between SGD's speed and batch gradient descent's computation cost.
   

Usage of AdaGrad comes with several advantages:

1. Simplicity: AdaGrad is a simple optimization technique that can be implemented on various models with ease.
2. Auto-adaption: It eliminates the need for manual fine-tuning of hyperparameters as it adjusts the learning rate per parameter automatically.
3. Adaptive learning rate: It allows the learning rate to be tweaked depending on past gradients of each parameter, reducing the risk of overshooting the ideal solution.
4. Efficient with noisy data: AdaGrad smoothens the noisy data impact by reducing learning rates for parameters with large gradients.
5. Effective with sparse data: It's especially efficient in dealing with sparse data, common in NLP and recommendation systems, by assigning them higher learning rates.
   

Conclusively, considering AdaGrad's potential adaptability to sparse, noisy data or data with a large number of parameters, it can serve as a potent optimization tool in ML and DL.