The Nadam optimizer is a powerful and efficient algorithm for training deep neural networks. It combines the advantages of the Nesterov Accelerated Gradient (NAG) and the Adaptive Moment Estimation (Adam) optimizers, resulting in improved convergence speed and parameter estimation accuracy. This article provides a detailed mathematical foundation of the Nadam optimizer, explaining its key concepts and mathematical underpinnings.

### Nesterov Accelerated Gradient (NAG)

NAG is a momentum-based optimizer that aims to accelerate convergence by taking a lookahead into the future gradient direction. It modifies the standard gradient descent update rule by adding a momentum term to the gradient calculation:

“`

v_t = βv_{t-1} + (1 – β)g_t

θ_t = θ_{t-1} – α(v_t + β(v_t – v_{t-1}))

“`

where:

* v_t is the momentum term

* β is the momentum coefficient (typically 0.9)

* g_t is the gradient at time step t

* θ_t is the model parameter vector

* α is the learning rate

### Adaptive Moment Estimation (Adam)

Adam is an optimizer that estimates the first and second moments of the gradients to adjust the learning rate for each parameter. It calculates a running average of the gradients (m_t) and squared gradients (v_t):

“`

m_t = β_1m_{t-1} + (1 – β_1)g_t

v_t = β_2v_{t-1} + (1 – β_2)g_t^2

“`

where:

* β_1 and β_2 are the exponential decay rates for the moments (typically 0.9 and 0.999, respectively)

The adaptive learning rate is then computed using the estimated moments:

“`

θ_t = θ_{t-1} – α * m_t / (√(v_t) + ε)

“`

where ε is a small constant to prevent division by zero.

### Nadam Optimizer

The Nadam optimizer combines NAG and Adam to leverage the advantages of both algorithms. It uses the momentum term of NAG to accelerate convergence and the adaptive learning rate of Adam to improve parameter estimation. The update rule for the Nadam optimizer is as follows:

“`

v_t = γ * v_{t-1} + (1 – γ)g_t

m_t = β_1m_{t-1} + (1 – β_1)g_t

v_t = β_2v_{t-1} + (1 – β_2)g_t^2

θ_t = θ_{t-1} – α * m_t / (√(v_t) + ε)

“`

where:

* γ = β_1 * β_2

### Implementation

The Nadam optimizer can be implemented in popular deep learning frameworks such as TensorFlow and PyTorch. The following code snippet shows an example implementation in Python using TensorFlow:

“`

import tensorflow as tf

class NadamOptimizer(tf.optimizers.Optimizer):

def __init__(self, learning_rate=0.001, beta_1=0.9, beta_2=0.999, epsilon=1e-8):

super().__init__(name=Nadam)

self._learning_rate = learning_rate

self._beta_1 = beta_1

self._beta_2 = beta_2

self._epsilon = epsilon

def _create_slots(self, var_list):

for var in var_list:

self.add_slot(var, v, dtype=tf.float32, initializer=tf.zeros_initializer())

self.add_slot(var, m, dtype=tf.float32, initializer=tf.zeros_initializer())

self.add_slot(var, v_hat, dtype=tf.float32, initializer=tf.zeros_initializer())

def _apply_dense(self, grad, var):

v = self.get_slot(var, v)

m = self.get_slot(var, m)

v_hat = self.get_slot(var, v_hat)

beta_1 = tf.cast(self._beta_1, grad.dtype)

beta_2 = tf.cast(self._beta_2, grad.dtype)

gamma = beta_1 * beta_2

lr = self._learning_rate

# Update moments

v = beta_1 * v + (1 – beta_1) * grad

m = beta_1 * m + (1 – beta_1) * grad

v_hat = beta_2 * v_hat + (1 – beta_2) * tf.square(grad)

# Update parameter

v_hat_corrected = v_hat / (1 – tf.pow(gamma,

var.assign_sub(lr * m / (tf.sqrt(v_hat_corrected) + self._epsilon)))

return tf.no_op()

def get_config(self):

config = super().get_config()

config[learning_rate]= self._learning_rate

config[beta_1] = self._beta_1

config[beta_2] = self._beta_2

config[epsilon]= self._epsilon

return config

“`

### Conclusion

The Nadam optimizer is an efficient and robust optimization algorithm that combines the strengths of NAG and Adam. It provides faster convergence and improved parameter estimation accuracy, making it a popular choice for training deep neural networks. Understanding the mathematical foundation of the Nadam optimizer enables practitioners to optimize its performance and effectively solve complex optimization problems.

Kind regards,

J.O. Schneppat