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Momentum-based Gradient Optimizer - ML

Last Updated : 11 Jul, 2025

Momentum-based gradient optimizers are used to optimize the training of machine learning models. They are more advanced than the classic gradient descent method and helps to accelerate the training process especially for large-scale datasets and deep neural networks.

By incorporating a "momentum" term these optimizers can navigate the loss surface more efficiently leading to faster convergence, reduced oscillations and better overall performance.

Before understanding momentum-based optimizers it’s important to understand the traditional gradient descent method. In gradient descent the model's weights are updated by taking small steps in the direction of the negative gradient of the loss function. Mathematically its updates weight using:

w_{t+1} = w_t - \eta \nabla L(w_t)

Where:

• w_t ​ is the weight at time step t
• \eta is the learning rate
• \nabla L(w_t) is the gradient of the loss function with respect to the weights.

This method works well but it suffers from issues such as slow convergence and the tendency to get stuck in local minima especially in high-dimensional spaces. This is where momentum-based optimization can be useful.

Momentum is a concept from physics where an object’s motion depends not only on the current force but also on its previous velocity. In the context of gradient optimization it refers to a method that smoothens the optimization trajectory by adding a term that helps the optimizer remember the past gradients.

In mathematical terms the momentum-based gradient descent updates can be described as:

v_{t+1} = \beta v_t + (1 - \beta) \nabla L(w_t)

w_{t+1} = w_t - \eta v_{t+1}

Where:

• v_t is the velocity i.e a running average of gradients
• \beta is the momentum factor, typically a value between 0 and 1 (often around 0.9)
• \nabla L(w_t) is the current gradient of the loss function
• \eta is the learning rate

• Learning Rate ( \eta ): The learning rate determines the size of the step taken during each update. It plays a crucial role in both standard gradient descent and momentum-based optimizers.
• Momentum Factor ( \beta ): This controls how much of the past gradients are remembered in the current update. A value close to 1 means the optimizer will have more inertia while a value closer to 0 means less reliance on past gradients.

1. Velocity Update: The velocity v_t ​ is updated by considering both the previous velocity which represents the momentum and the current gradient. The momentum factor \beta controls the contribution of the previous velocity to the current update.
2. Weight Update: The weights are updated using the velocity v_{t+1} ​ which is a weighted average of the past gradients and the current gradient.

There are several variations of momentum-based optimizers each with slight modifications to the basic momentum algorithm:

Nesterov momentum is an advanced form of momentum-based optimization. It modifies the update rule by calculating the gradient at the upcoming position rather than the current position of the weights.

The update rule becomes:

v_{t+1} = \beta v_t + \nabla L(w_t - \eta \beta v_t)

w_{t+1} = w_t - \eta v_{t+1}

NAG is considered more efficient than classical momentum because it has a better understanding of the future trajectory, leading to even faster convergence and better performance in some cases.

AdaMomentum combines the concept of adaptive learning rates with momentum. It adjusts the momentum term based on the recent gradients making the optimizer more sensitive to the landscape of the loss function. This can help in fine-tuning the convergence process.

Although not strictly a momentum-based optimizer in the traditional sense RMSProp incorporates a form of momentum by adapting the learning rate for each parameter. It’s particularly effective when dealing with non-stationary objectives, such as in training recurrent neural networks (RNNs).

1. Faster Convergence: It helps to accelerate the convergence by considering past gradients, which helps the model navigate through flat regions more efficiently.
2. Reduces Oscillation: Traditional gradient descent can oscillate when there are steep gradients in some directions and flat gradients in others. Momentum reduces this oscillation by maintaining the direction of previous updates.
3. Improved Generalization: By smoothing the optimization process, momentum-based methods can lead to better generalization on unseen data, preventing overfitting.
4. Helps Avoid Local Minima: The momentum term can help the optimizer escape from local minima by maintaining a strong enough "velocity" to continue moving past these suboptimal points.

1. Choosing Hyperparameters: Selecting the appropriate values for the learning rate and momentum factor can be challenging. Typically a momentum factor of 0.9 is common but it may vary based on the specific problem or dataset.
2. Potential for Over-Accumulation: If the momentum term becomes too large it can lead to the optimizer overshooting the minimum, especially in the presence of noisy gradients.
3. Initial Momentum: When momentum is initialized it can have a significant impact on the convergence rate. Poor initialization can lead to slow or erratic optimization behavior.

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