Challenges with Standard Gradient Descent

While standard Gradient Descent, often referred to as Batch Gradient Descent (BGD), provides a solid theoretical foundation for optimization, it runs into significant practical hurdles when applied to the large datasets and complex models common in deep learning. Calculating the gradient requires summing contributions from every single data point in the training set before making a single parameter update.

Computational Cost and Memory Requirements

The most immediate challenge is the sheer computational expense. Deep learning models often train on datasets containing millions or even billions of examples. Consider a dataset with $N$ examples. To perform one update step using BGD, you need to:

Perform a forward pass for all $N$ examples to compute predictions.
Compute the loss for all $N$ examples.
Perform a backward pass (backpropagation) for all $N$ examples to find the gradient of the loss with respect to each parameter.
Average these gradients across all $N$ examples.
Update the model parameters using this average gradient.

This entire process must be repeated for each update step (epoch). The cost of calculating the gradient scales linearly with the size of the dataset, $N$ . When $N$ is very large, each step becomes incredibly slow, making training times prohibitively long.

Furthermore, depending on the implementation and model complexity, calculating gradients for the entire dataset might require loading a significant amount of data into memory, which can exceed the capacity of available hardware (like GPUs).

Inefficiency Due to Data Redundancy

Large datasets often contain redundant information. Many data points might be similar and produce gradients pointing in roughly the same direction. BGD processes all these similar examples in every single step, even though a smaller sample might provide a good enough estimate of the true gradient direction. This exhaustive computation over redundant data makes each update step less efficient than it could be. Imagine learning to recognize cats by looking at 1 million nearly identical pictures of the same cat before adjusting your understanding; you could likely learn faster by looking at a smaller, more diverse set of cat pictures.

Navigation of the Loss: Local Minima and Saddle Points

The loss landscapes of deep neural networks are highly complex and non-convex, featuring numerous local minima (points that are optimal within a neighborhood but not globally) and saddle points (points where the gradient is zero, but which are not minima).

Batch Gradient Descent calculates the exact gradient based on the entire dataset. While this provides a 'true' direction of steepest descent for the current parameters over the whole dataset, this smooth, averaged gradient might lack the 'noise' needed to escape certain undesirable regions of the loss surface.

Local Minima: If BGD converges to a local minimum with a significantly higher loss than the global minimum, the model's performance will be suboptimal. The deterministic nature of the BGD update might not provide the necessary perturbation to jump out of such a valley.
Saddle Points: Saddle points are even more common in high-dimensional spaces like those defined by neural network parameters. At a saddle point, the gradient is zero, causing BGD to potentially stop updating or slow down dramatically, even though it's not at a minimum. Some dimensions might curve upwards (like a minimum), while others curve downwards (like a maximum). BGD can struggle to roll off the saddle point effectively.

A loss surface showing a global minimum (low point), a local minimum (higher valley), and a saddle point (center, flat locally but curving down in one direction and up in another). BGD might get stuck in the local minimum or slow down at the saddle point.

While optimization algorithms can't guarantee finding the global minimum, the deterministic nature of BGD makes it potentially more susceptible to getting permanently trapped compared to methods introducing more randomness.

Inability to Perform Online Learning

Batch Gradient Descent requires the entire dataset to be available before any learning can occur. This makes it unsuitable for scenarios where data arrives sequentially, known as online learning. In such cases, the model needs to adapt as new data comes in, without storing and reprocessing all historical data for every update.

These limitations, particularly the computational cost and the potential for getting stuck, highlight the need for more practical optimization algorithms in deep learning. This motivates the move towards methods that use smaller subsets of data for each update, leading us to Stochastic Gradient Descent and Mini-batch Gradient Descent.

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References

Deep Learning, Ian Goodfellow, Yoshua Bengio, and Aaron Courville, 2016 (MIT Press) - A fundamental textbook that covers optimization algorithms, including the theoretical basis and practical difficulties of Batch Gradient Descent in deep learning.
Neural Networks Part 3: Learning and Evaluation, Andrej Karpathy, Justin Johnson, and Fei-Fei Li, 2023 (Stanford University) - Stanford CS231n course notes offer practical observations on the challenges of training deep neural networks, including the computational demands and memory limits of Batch Gradient Descent, and the complexity of the loss surface.
The Loss Landscape of Neural Networks, Anna Choromanska, Mikael Henaff, Michael Mathieu, Gerard Ben Arous, Yann LeCun, 2015 Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, Vol. 38 (PMLR) - Presents theoretical and empirical examination of the loss landscape of neural networks, suggesting that in high dimensions, local minima often resemble the global minimum, but saddle points are a more significant problem.