Nag as optimiser in deep learning - day 36

Nag as optimiser in deep learning – day 36

September 2, 2024
Posted by ingoampt

Nesterov Accelerated Gradient (NAG): A Comprehensive Overview

Introduction to Nesterov Accelerated Gradient

Nesterov Accelerated Gradient (NAG), also known as Nesterov Momentum, is an advanced optimization technique introduced by Yurii Nesterov in the early 1980s. It is an enhancement of the traditional momentum-based optimization used in gradient descent, designed to accelerate the convergence rate of the optimization process, particularly in the context of deep learning and complex optimization problems.

How NAG Works

The core idea behind NAG is the introduction of a “look-ahead” step before calculating the gradient, which allows for a more accurate and responsive update of parameters. In traditional momentum methods, the gradient is computed at the current position of the parameters, which might lead to less efficient convergence if the trajectory is not perfectly aligned with the optimal path. NAG, however, calculates the gradient at a position slightly ahead, based on the accumulated momentum, thus allowing the algorithm to “correct” its course more effectively if it is heading towards a suboptimal direction.

The NAG update rule can be summarized as follows:

Look-ahead Step: Compute a preliminary update based on the momentum.
Gradient Calculation: Evaluate the gradient at this look-ahead position.
Momentum Update: Adjust the momentum based on the newly computed gradient.
Parameter Update: Finally, update the parameters using the adjusted momentum.

This approach reduces the chances of overshooting the minimum and enhances the stability of the convergence process.

Comparing NAG with Traditional Momentum

Momentum and NAG are both designed to accelerate gradient descent by smoothing the optimization path and helping the algorithm overcome obstacles such as local minima or flat regions. However, they differ in their approach and efficiency:

Momentum: In the traditional momentum method, the gradient is computed at the current parameter position. The update rule for momentum is:

$v_t = \gamma v_{t-1} + \eta \nabla J(\theta_t)$

$\theta_{t+1} = \theta_t - v_t$

Here, $v_t$ is the velocity (momentum) term, $\gamma$ is the momentum coefficient, $\eta$ is the learning rate, and $\nabla J(\theta_t)$ is the gradient at the current parameter position $\theta_t$ . The traditional momentum method can overshoot the minimum, especially if the learning rate or momentum term is too high, leading to instability.
Nesterov Accelerated Gradient: NAG modifies the momentum update by first predicting where the current momentum will take the parameters, and then calculating the gradient at this predicted location. The update rules for NAG are:

$\theta' = \theta_t - \gamma v_{t-1}$

$v_t = \gamma v_{t-1} + \eta \nabla J(\theta')$

$\theta_{t+1} = \theta_t - v_t$

This look-ahead mechanism allows NAG to correct its path more effectively and reduces the risk of overshooting, leading to faster and more stable convergence.

In essence, while both methods aim to accelerate convergence, NAG generally provides better stability and responsiveness due to its anticipatory step, making it more suitable for deep learning tasks where the loss surface can be highly non-convex with many flat regions and local minima.

Mathematical Foundation of NAG

The mathematical foundation of NAG lies in its ability to leverage the accumulated momentum more effectively by predicting where the momentum will take the parameters before applying the gradient update. This mechanism can be broken down into the following steps:

Velocity Update:

$v_t = \gamma v_{t-1} + \eta \nabla J(\theta_t - \gamma v_{t-1})$

Here, the velocity $v_t$ is updated based on the momentum from the previous step $v_{t-1}$ , the learning rate $\eta$ , and the gradient at the look-ahead position $\theta_t - \gamma v_{t-1}$ .
Parameter Update:

$\theta_{t+1} = \theta_t - v_t$

The parameters $\theta$ are then updated by subtracting the velocity term $v_t$ , which has been adjusted using the look-ahead gradient.

Proof of NAG’s Effectiveness: A Simple Example

Let’s consider a simple quadratic objective function:

$J(\theta) = \frac{1}{2} \theta^2$

The gradient of this function is:

$\nabla J(\theta) = \theta$

Suppose the learning rate $\eta = 0.1$ and momentum $\gamma = 0.9$ . We initialize $\theta_0 = 1.0$ and $v_0 = 0$ .

Iteration 1

Traditional Momentum:
- Compute the gradient: $\nabla J(\theta_0) = \theta_0 = 1.0$
- Update the velocity: $v_1 = \gamma v_0 + \eta \nabla J(\theta_0) = 0.9 \times 0 + 0.1 \times 1.0 = 0.1$
- Update the parameters: $\theta_1 = \theta_0 - v_1 = 1.0 - 0.1 = 0.9$
Nesterov Accelerated Gradient:
- Look-ahead step: $\theta' = \theta_0 - \gamma v_0 = 1.0$ (since $v_0 = 0$ )
- Compute the gradient at look-ahead position: $\nabla J(\theta') = \theta' = 1.0$
- Update the velocity: $v_1 = \gamma v_0 + \eta \nabla J(\theta') = 0.9 \times 0 + 0.1 \times 1.0 = 0.1$
- Update the parameters: $\theta_1 = \theta_0 - v_1 = 1.0 - 0.1 = 0.9$

Iteration 2

Traditional Momentum:
- Compute the gradient: $\nabla J(\theta_1) = \theta_1 = 0.9$
- Update the velocity: $v_2 = \gamma v_1 + \eta \nabla J(\theta_1) = 0.9 \times 0.1 + 0.1 \times 0.9 = 0.18$
- Update the parameters: $\theta_2 = \theta_1 - v_2 = 0.9 - 0.18 = 0.72$
- Nesterov Accelerated Gradient:
  - Look-ahead step: $\theta' = \theta_1 - \gamma v_1 = 0.9 - 0.9 \times 0.1 = 0.81$
  - Compute the gradient at the look-ahead position: $\nabla J(\theta') = \theta' = 0.81$
  - Update the velocity: $v_2 = \gamma v_1 + \eta \nabla J(\theta') = 0.9 \times 0.1 + 0.1 \times 0.81 = 0.171$
  - Update the parameters: $\theta_2 = \theta_1 - v_2 = 0.9 - 0.171 = 0.729$
Iteration 3
- Traditional Momentum:
  - Compute the gradient: $\nabla J(\theta_2) = \theta_2 = 0.72$
  - Update the velocity: $v_3 = \gamma v_2 + \eta \nabla J(\theta_2) = 0.9 \times 0.18 + 0.1 \times 0.72 = 0.234$
  - Update the parameters: $\theta_3 = \theta_2 - v_3 = 0.72 - 0.234 = 0.486$
- Nesterov Accelerated Gradient:
  - Look-ahead step: $\theta' = \theta_2 - \gamma v_2 = 0.729 - 0.9 \times 0.171 = 0.576$
  - Compute the gradient at the look-ahead position: $\nabla J(\theta') = \theta' = 0.576$
  - Update the velocity: $v_3 = \gamma v_2 + \eta \nabla J(\theta') = 0.9 \times 0.171 + 0.1 \times 0.576 = 0.2091$
  - Update the parameters: $\theta_3 = \theta_2 - v_3 = 0.729 - 0.2091 = 0.5199$
As we can see from these iterations, Nesterov Accelerated Gradient (NAG) adjusts its updates by considering the future position of the parameters, leading to potentially faster convergence and a more accurate path towards the optimum. NAG takes into account where the momentum will take the parameters and adjusts the gradient accordingly, making it more robust, particularly in scenarios with complex loss landscapes.

Practical Implementation Example in Code

Here is an example of how NAG can be implemented in Python using TensorFlow:
```
import tensorflow as tf

# Define the model
model = tf.keras.models.Sequential([
    tf.keras.layers.Dense(64, activation='relu', input_shape=(input_dim,)),
    tf.keras.layers.Dense(1)
])

# Compile the model using NAG
optimizer = tf.keras.optimizers.SGD(learning_rate=0.001, momentum=0.9, nesterov=True)
model.compile(optimizer=optimizer, loss='mean_squared_error')

# Train the model
model.fit(X_train, y_train, epochs=100, batch_size=32, validation_data=(X_val, y_val))

# Evaluate the model
loss = model.evaluate(X_test, y_test)
print(f'Test loss: {loss}')
```
In this example, we use the TensorFlow library to build a simple neural network model. We configure the SGD optimizer with Nesterov Accelerated Gradient by setting the nesterov=True flag. This allows us to leverage the benefits of NAG, including faster convergence and improved stability during training.

Conclusion

Nesterov Accelerated Gradient (NAG) offers significant improvements over traditional momentum by incorporating a look-ahead mechanism, leading to faster and more stable convergence. Its mathematical foundation and practical implementations make it a powerful tool in the optimization toolbox, particularly useful for deep learning tasks where the optimization landscape is complex. Understanding the differences between NAG and traditional momentum, as well as the mathematics behind it, provides valuable insights into why NAG is often preferred in practice.

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