Writing Popular Machine Learning Optimizers From Scratch on Python

1. Writing popular Machine Learning Optimizers from scratch on Python

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This blog will include some mathematical and theoritical representation along with Python codes from scratch. Most of the codes and formulas are taken from different resources and I have given links to them also.

This post is related to below posts(these posts depends on this post):

Before begining,

If you want to check the files first, then please follow this link ML From Basics.

2. Contains

(Optimizers Code were referenced from here.)

  • Gradient Descent
  • Momentum
  • RMS Prop
  • ADAM
  • ADAGrad
  • ADADelta

2.1 Initialize our class

  class Optimizer:
    def __init__(self, layers, name=None, learning_rate = 0.01, mr=0.001):
        self.name = name
        self.learning_rate = learning_rate
        self.mr = mr
        keys = ["sgd", "iterative", "momentum", "rmsprop", "adagrad", "adam", "adamax", "adadelta"]
        values = [self.sgd, self.iterative, self.momentum, self.rmsprop, self.adagrad, self.adam, self.adamax, self.adadelta]
        self.opt_dict = {keys[i]:values[i] for i in range(len(keys))}
        if name != None and name in keys:
            self.opt_dict[name](layers=layers, training=False)

We are using the reference of every optimizer method. The object of this class will be made while compiling the model and at the same time, the reference of the optimizer is taken from the opt_dict. The boolean training sets all the terms(to 0) that are required for weight optimization.

Let me take some notation form the book Tensorflow for Dummies by Matthew Scarpino. Most of the formulas and concepts are taken from this book.

  • The set of trainable variables is denoted θ. The values of the variables at Step t is denoted θt.
  • The loss, which is a mathematical relationship containing the model’s variables, is denoted J(θ). The gradient of the loss is ∇J(θ).
  • The learning rate, denoted η, is a value that affects how much θj changes from step to step.

Before diving into algorithm and comparing it with code, let us understand that, I have done addition with all delta terms because I have already taken minus of delta terms.

Gradient Descent

Weight update term for all units is:

\begin{equation} \triangle w_{ji} = \alpha \delta_j x_{ji} \end{equation}

\begin{equation} \ when\ momentum\ term\ is\ applied, \end{equation}

\begin{equation} \triangle w_{ji}(n) = \beta \delta_j x_{ji} + \triangle w_{ji}(n-1) \end{equation}

\begin{equation} \ \beta\ is\ momentum\ rate \end{equation}

\begin{equation} \delta_j\ formula\ varies\ with\ the\ unit\ being\ output\ or\ internal. \end{equation}

\begin{equation} w_{ji} = w_{ji} - \triangle w_{ji} \ \end{equation}

OR more simple representation:

$$ \begin{equation} \theta_t = \theta_t - \alpha \triangle J(\theta) \end{equation} $$

This shows how the model’s variables change with each training operation. As training continues, ∇J(θ) should approach zero, which means that each new set of variables should be approximately equal to the previous set. At this point, optimization has completed because the optimizer has converged to a minimum. But Gradient Descent always suffers from convergence because the loss might never reach global minimum and oscillate between values.

2.3 Momentum Optimizer

This optimizer tries to eliminate the previous problem of Oscilation between values by introducing momentum term.

\begin{equation} v_t = \beta v_{t-1} - \alpha \triangle J(\theta) \end{equation} \begin{equation} \theta = \theta + v_t \end{equation}

    def momentum(self, layers, learning_rate=0.1, beta1=0.9, weight_decay=0.0005, training=True):
        learning_rate = self.learning_rate
        for l in layers:
            if l.parameters !=0:
                if training:
                    l.weights_momentum = beta1 * l.weights_momentum + learning_rate * 
                                         l.delta_weights-weight_decay * learning_rate*l.weights
                    l.biases_momentum = beta1 * l.biases_momentum + learning_rate * 
                                        l.delta_biases-weight_decay *learning_rate*l.biases
                    l.weights_momentum = 0
                    l.biases_momentum = 0

The term weight_decay and beta1 is not present on original Momentum Algorithm but it helps to slowly converge the loss towards global minima.

2.4 Adagrad

  • The learning rate changes from variable to variable and from step to step. The learning rate at the tth step for the ith variable is denoted .
  • Adagrad methods compute subgradients instead of gradients. A subgradient is a generalization of a gradient that applies to non-differentiable functions. This means AdaGrad methods can optimize both differentiable and non-differentiable functions.

The learning rate will be:

\begin{equation} \alpha_{t,i} = \frac{\alpha}{\sqrt G_{t,ii}} \end{equation}

In this equation, Gt, ii is the ith element of the diagonal of the matrix formed by taking the outer product of the sub-gradient of the loss with itself. After computing the learning rates, the optimizer updates the variables:

\begin{equation} \theta_{t,i} = \theta_{t-1,i} - \alpha g_t \end{equation}

Shortcoming of this optimizer is that, the learning rate eventually becomes 0 and training stops.

    def adagrad(self, layers, learning_rate=0.01, beta1=0.9, epsilon=1e-8, training=True):
        for l in layers:
            if l.parameters != 0:
                if training:
                    l.weights_adagrad += l.delta_weights ** 2
                    l.weights += learning_rate * (l.delta_weights/np.sqrt(l.weights_adagrad+epsilon))
                    l.biases_adagrad += l.delta_biases ** 2
                    l.biases += learning_rate * (l.delta_biases/np.sqrt(l.biases_adagrad+epsilon))
                    l.weights_adagrad = 0
                    l.biases_adagrad = 0

2.5 RMS Prop

  • This algorithm is devised by Geoffrey Hinton.
  • This algorithm uses different learning rate for different parameters by using moving average of squared gradient. It utilizes the magnitude of recent gradient to normalize gradient.
  • Divides learning rate by the average of exponential decay of squared gradients.

\begin{equation} \theta_{t+1} = \theta_t - \frac{\alpha}{\sqrt{(1-\beta) g^2_{t-1} + \beta g_t + \epsilon}} * g_t \end{equation}

Where g_t is a delta term for parameter 𝜃.

    def rmsprop(self, layers, learning_rate=0.001, beta1=0.9, epsilon=1e-8, training=True):
        for l in layers:
            if l.parameters !=0:
                if training:
                    l.weights_rms = beta1*l.weights_rms + (1-beta1)*(l.delta_weights ** 2)
                    l.weights += learning_rate * (l.delta_weights/np.sqrt(l.weights_rms + epsilon))
                    l.biases_rms = beta1*l.biases_rms + (1-beta1)*(l.delta_biases ** 2)
                    l.biases += learning_rate * (l.delta_biases/np.sqrt(l.biases_rms + epsilon))
                    l.weights_rms = 0
                    l.biases_rms = 0

2.6 Adam Optimizer

The Adam (Adaptive Moment Estimation) algorithm closely resembles the Adagrad algorithm in many respects. It also resembles the Momentum algorithm because it takes two factors into account:

  • The first moment vector: Scales the gradient by 1-beta1
  • The second moment vector: Scales the square of the gradient by 1-beta2

These moment vectors are denoted mt and vt, respectively. The following equations show how their values change from step to step:

$$ m_t = \beta_1 m_{t-1} + (1-\beta_1) \triangle J(\theta) \

$$ v_t = \beta_2 v_{t-1} + (1-\beta_2)[\triangle J(\theta)]^2 \

$$ m^{\prime}_t = \frac{m_t}{1-\beta^t_1} \

$$ v^{\prime}_t = \frac{v_t}{1-\beta^t_2} \

$$ \theta_{t+1} = \theta_t - \frac{\alpha}{\sqrt{v^{\prime}_t} + \epsilon} * m^{\prime}_t $$

Where 𝜖 is a small value which is used to prevent from divide by 0.

    def adam(self, layers, learning_rate=0.001, beta1=0.9, beta2=0.999, epsilon=1e-8, decay=0, training=True):
        for l in layers:
            if l.parameters != 0:
                if training:
                    l.t += 1
                    if l.t == 1:
                        l.pdelta_biases = 0
                        l.pdelta_weights = 0
                    l.weights_adam1 = beta1 * l.weights_adam1 + (1-beta1)*l.delta_weights
                    l.weights_adam2 = beta2 * l.weights_adam2 + (1-beta2)*(l.delta_weights**2)
                    mcap = l.weights_adam1/(1-beta1**l.t)
                    vcap = l.weights_adam2/(1-beta2**l.t)
                    l.delta_weights = mcap/(np.sqrt(vcap) + epsilon)
                    l.weights += l.pdelta_weights * self.mr + learning_rate * l.delta_weights
                    l.pdelta_weights = l.delta_weights * 0
                    l.biases_adam1 = beta1 * l.biases_adam1 + (1-beta1)*l.delta_biases
                    l.biases_adam2 = beta2 * l.biases_adam2 + (1-beta2)*(l.delta_biases**2)
                    mcap = l.biases_adam1/(1-beta1**l.t)
                    vcap = l.biases_adam2/(1-beta2**l.t)
                    l.delta_biases = mcap/(np.sqrt(vcap) +epsilon)
                    l.biases += l.pdelta_biases * self.mr + learning_rate * l.delta_biases
                    l.pdelta_biases = l.delta_biases * 0                    
                    l.t = 0
                    l.weights_adam1 = 0
                    l.weights_adam2 = 0
                    l.biases_adam1 = 0
                    l.biases_adam2 = 0

2.7 Adamax

Please refer to the Sebastian Ruder's site for more explanation.

This is slight variation of Adam optimizer.

$$ u_t = \beta_2^{\infty}+ (1-\beta_2^\infty) * abs(g_t)^\infty $$ $$ \ = max(\beta_2 * v_{t-1}, abs(g_t)) \
$$ $$ \text now, \\ $$ $$ \theta_{t+1} = \theta_t - \frac{\alpha}{u_t} * m^{\prime}_t $$

    def adamax(self, layers, learning_rate=0.002, beta1=0.9, beta2=0.999, epsilon=1e-8, training=True):
        for l in layers:
            if l.parameters != 0:
                if training:
                    l.weights_m = beta1*l.weights_m + (1-beta1)*l.delta_weights
                    l.weights_v = np.maximum(beta2*l.weights_v, abs(l.delta_weights))
                    l.weights += (self.learning_rate/(1-beta1))*(l.weights_m/(l.weights_v+epsilon))                    
                    l.biases_m = beta1*l.biases_m + (1-beta1)*l.delta_biases
                    l.biases_v = np.maximum(beta2*l.biases_v, abs(l.delta_biases))
                    l.biases += (self.learning_rate/(1-beta1))*(l.biases_m/(l.biases_v+epsilon))                    
                    l.weights_m = 0
                    l.biases_m = 0
                    l.weights_v = 0
                    l.biases_v = 0

2.8 Adadelta

  • We don't use learning rate here. But the ratio of the running average of the previous time steps to the current gradient is used.
  • This algorithm tries to reduce learning rate monotonically. This is extended version of Adagrad.

$$ \begin{equation} \theta_{t+1} = \theta_t + \triangle \theta_t \
\triangle \theta = - \frac{RMS[\triangle \theta_t-1]}{RMS[g_t]}.g_t \end{equation} $$

Where gt is the gradient term.

    def adadelta(self, layers, learning_rate=0.01, beta1=0.9, epsilon=1e-8, training=True):
        for l in layers:
            if l.parameters != 0:
                if training:
                    l.weights_v = beta1*l.weights_v + (1-beta1)*(l.delta_weights ** 2)
                    l.delta_weights = np.sqrt((l.weights_m+epsilon)/(l.weights_v+epsilon))*l.delta_weights
                    l.weights_m = beta1*l.weights_m + (1-beta1)*(l.delta_weights)
                    l.weights += l.delta_weights                    
                    l.biases_v = beta1*l.biases_v + (1-beta1)*(l.delta_biases ** 2)
                    l.delta_biases = np.sqrt((l.biases_m+epsilon)/(l.biases_v+epsilon))*l.delta_biases
                    l.biases_m = beta1*l.biases_m+ (1-beta1)*(l.delta_biases)
                    l.biases += l.delta_biases                    
                    l.weights_m = 0
                    l.biases_m = 0
                    l.weights_v = 0
                    l.biases_v = 0


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