What is Nesterov Accelerated Gradient?

The Nesterov Accelerated Gradient (NAG) is an optimization algorithm used in machine learning, mainly in training neural networks. The Nesterov Accelerated Gradient (NAG) approach is a potent tool for accelerated convergence and enhanced performance within machine learning and deep learning optimization techniques. This algorithm has become quite popular since it can solve some shortcomings of conventional gradient descent techniques.

This Answer will delve deep into the complexities of NAG to explore its functionality and benefits.

Introduction to gradient descent

Understanding gradient descent is crucial before diving into NAG, as it is the foundation for many machine learning optimization techniques. Gradient descent is an iterative optimization process that modifies a model’s parameters to minimize a loss function. The algorithm updates the parameters in the opposite direction as the gradient at each iteration to minimize the loss. It does this by computing the gradient of the loss function concerning the parameters.

Although gradient descent works well, it has several drawbacks, especially for complex, high-dimensional optimization issues. The slow pace of convergence is one of the main problems, particularly in areas with shallow gradients. Furthermore, gradient descent may oscillate or overshoot the ideal solution, especially if the learning rate is very high.

Understanding Nesterov Accelerated Gradient

NAG introduces the notion of momentum, strengthening the basis of conventional gradient descent. In directions where the gradient is constant, momentum enables the algorithm to accelerate; momentum dampens oscillations in directions with large curvature.

Where:

$θ_{t}$ represents the parameters at iteration $t$ .
$v_{t}$ is the velocity or momentum at iteration $t$ .
$α$ is the learning rate.
$μ$ is the momentum term.
$∇f(θ_{t−1}+μv_{t−1})$ is the gradient of the loss function evaluated at the lookahead position.

Working

Let’s break down the working of NAG based on the mathematical formulation provided earlier:

Initialization

The algorithm first sets the velocity ( $v$ ) and model parameters ( $θ$ ) to zero or tiny random values.

Lookahead position calculation

The algorithm determines the lookahead location $(θ_{t−1}+μv_{t−1})$ at each iteration $t$ .

Gradient calculation

At this lookahead location $(θ_{t−1}+μv_{t−1}),$ the gradient of the loss function is calculated. This gradient represents the direction the parameters will probably go in the following iteration.

Velocity update

The velocity ( $v_{t}$ ) at iteration $t$ is updated using the current velocity ( $v_{t−1}$ ), the gradient at the lookahead position, and the learning rate ( $α$ ). Using this equation:

Key components

Momentum term ( $μ$ ): Controls the influence of the previous velocity on the current update. It determines how much of the previous velocity is retained and how much is updated based on the new gradient.
Learning rate ( $α$ ): Determines the step size taken in parameter space during each iteration. It influences convergence speed but must be carefully chosen to avoid overshooting or instability.

The fundamental concept of NAG involves calculating the loss function’s gradient at a place ahead of the present position. By doing this, the algorithm predicts the future travel path and modifies the velocity accordingly. This lookahead update’s ability to reduce oscillations and overshooting tendencies results in faster convergence and more stable optimization.

Example code

NAG is implemented with random values below:

import numpy as np
import matplotlib.pyplot as plt
def nesterov_gradient_descent(x_init, f_grad, lr=0.01, momentum=0.9, num_iterations=100):
    x = np.copy(x_init)     # Copy the initial position
    v = np.zeros_like(x)    # Copy the initial velocity
    trajectory = [np.copy(x)]   
    for _ in range(num_iterations):
        v_prev = np.copy(v)
        v = momentum * v - lr * f_grad(x + momentum * v)
        x += -momentum * v_prev + (1 + momentum) * v
        trajectory.append(np.copy(x))
    return x, trajectory
def f(x):
    return x**4 + 3*x**3 + 2*x**2 - 5*x + 6
def f_grad(x):
    return 4*x**3 + 9*x**2 + 4*x - 5
# Define the initial point
x_init = np.array([3.0])
lr = 0.01
momentum = 0.9 
num_iterations = 100
# Run Nesterov Accelerated Gradient descent
x_opt, trajectory = nesterov_gradient_descent(x_init, f_grad, lr, momentum, num_iterations)
# Visualize the function and the trajectory
x_vals = np.linspace(-4, 2, 400)
y_vals = f(x_vals)
plt.figure(figsize=(10, 6))
plt.plot(x_vals, y_vals, label='Function')
plt.scatter([point[0] for point in trajectory], [f(point) for point in trajectory], c='r', label='Trajectory')
plt.title('Nesterov Accelerated Gradient Descent')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.legend()
plt.grid(True)
plt.show()

Advantages of Nesterov Accelerated Gradient

Faster convergence: Unlike conventional gradient descent techniques, NAG speeds convergence by combining momentum and lookahead updates.
Improved robustness: Particularly in high-dimensional areas, the momentum term strengthens optimization by mitigating oscillations and overshooting.
Parameter tuning: Compared to other optimization techniques, the NAG requires tuning fewer hyperparameters, streamlining the optimization procedure.
Better generalization: NAG often produces models with superior generalization performance because of its capacity to converge more quickly and prevent local minima.

NAG is a potent optimization technique that overcomes some of the drawbacks of conventional gradient descent techniques. It provides better generalization, faster convergence, and enhanced robustness by utilizing momentum and lookahead updates. It has thus gained popularity as a method for optimizing deep neural networks and other complex machine-learning models. To optimize machine learning processes more quickly and effectively, practitioners must comprehend the fundamentals of Nesterov Accelerated Gradient.

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