How do optimizers like Adam and RMSprop work?

Optimizers like Adam and RMSprop are algorithms that adjust the parameters of a neural network during training to minimize the loss function. They improve upon basic stochastic gradient descent (SGD) by adapting the learning rate for each parameter dynamically, which speeds up convergence and handles challenges like sparse or noisy gradients. While they share similarities, they differ in how they compute and apply these adaptive updates.

RMSprop (Root Mean Square Propagation) addresses SGD’s limitation of using a single learning rate for all parameters. It maintains a moving average of the squared gradients for each parameter, which helps normalize the learning rate. Specifically, it computes an exponentially decaying average of past squared gradients and divides the current gradient by the square root of this average. This scaling reduces the learning rate for parameters with large historical gradients and increases it for those with smaller gradients. For example, in a network processing images, RMSprop might slow down updates for frequently activated convolutional filters (which have large gradients) while speeding up updates for less active ones. The decay rate (often set to 0.9) controls how quickly past gradients are forgotten, and a small epsilon (e.g., 1e-8) prevents division by zero. RMSprop is particularly effective for non-stationary problems, such as training recurrent neural networks (RNNs), where gradient magnitudes vary widely over time.

Adam (Adaptive Moment Estimation) combines ideas from RMSprop and momentum-based optimization. It maintains two moving averages: one for the gradients (first moment) and another for the squared gradients (second moment). These averages are bias-corrected to account for their initialization at zero. The first moment acts like momentum, smoothing noisy gradients, while the second moment scales the learning rate adaptively, similar to RMSprop. For instance, in training a transformer model, Adam’s momentum helps navigate flat regions of the loss landscape, while the adaptive scaling handles layers with varying gradient scales. Adam’s hyperparameters (e.g., beta1=0.9, beta2=0.999) are often left at default values, making it a popular “out-of-the-box” choice. However, it can sometimes converge to suboptimal solutions in problems requiring precise learning rate tuning, such as fine-tuning pre-trained models.

In practice, RMSprop is preferred for tasks with non-stationary objectives (e.g., reinforcement learning), while Adam is widely used for general deep learning due to its robustness. Developers should experiment with both: for example, using Adam for convolutional networks and RMSprop for RNNs. Both optimizers require setting a base learning rate, but Adam’s adaptive nature often reduces the need for extensive tuning. Understanding their mechanics helps debug training issues, such as divergence (which might require lowering the learning rate) or slow convergence (which could benefit from momentum adjustments).