Week 40: Gradient descent methods (continued) and start Neural networks

Plans for week 40

Lecture Monday September 30, 2024

Suggested readings and videos

Lab sessions Tuesday and Wednesday

Summary from last week, using gradient descent methods, limitations

• Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
• GD is sensitive to initial conditions. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.
• Gradients are computationally expensive to calculate for large datasets. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, \( E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2 \); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over all \( n \) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called "mini batches". This has the added benefit of introducing stochasticity into our algorithm.
• GD is very sensitive to choices of learning rates. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would adaptively choose the learning rates to match the landscape.
• GD treats all directions in parameter space uniformly. Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.
• GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.

Simple implementation of GD for OLS, Ridge and Lasso

Last week we studied both several gradient methods. With and without an update of the learning. We summarize some of these here for the methods we hvae studied in project one, without the inclusion of momentum.

from random import random, seed
import numpy as np

# the number of datapoints with a 2nd-order polynomial
n = 100
x = 2*np.random.rand(n,1)
y = 4+3*x+5*x*x
# Design matrix including the intercept
# No scaling of data of and all data used for training 
X = np.c_[np.ones((n,1)), x, x*x]
# Learning rate and number of iterations
eta = 0.05
Niterations = 100

# OLS part
beta_OLS = np.random.randn(3,1)
gradient = np.zeros(3)
for iter in range(Niterations):
    gradient = (2.0/n)*X.T @ (X @ beta_OLS-y)
    beta_OLS -= eta*gradient
print('Parameters for OLS using gradient descent')    
print(beta_OLS)

#Ridge and Lasso parameter Lambda
Lambda  = 0.01
Id = n*Lambda* np.eye((X.T @ X).shape[0])
# Gradient descent with  Ridge
beta_Ridge = np.random.randn(3,1)
gradient = np.zeros(3)
for iter in range(Niterations):
    gradients = 2.0/n*X.T @ (X @ beta_Ridge-y)+2*Lambda*beta_Ridge
    beta_Ridge -= eta*gradients
print('Parameters for Ridge using gradient descent')    
print(beta_Ridge)

# Gradient descent with Lasso
beta_Lasso = np.random.randn(3,1)
gradient = np.zeros(3)
for iter in range(Niterations):
    gradients = 2.0/n*X.T @ (X @ beta_Lasso-y)+Lambda*np.sign(beta_Lasso)
    beta_Lasso -= eta*gradients
print('Parameters for Lasso using gradient descent')    
print(beta_Lasso)

But none of these can compete with Newton's method

Note that we here have introduced automatic differentiation

# Using Newton's method
from random import random, seed
import numpy as np
import autograd.numpy as np
from autograd import grad

def CostOLS(beta):
    return (1.0/n)*np.sum((y-X @ beta)**2)

n = 100
x = 2*np.random.rand(n,1)
y = 4+3*x+5*x*x

X = np.c_[np.ones((n,1)), x, x*x]
XT_X = X.T @ X
beta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
print("Own inversion")
print(beta_linreg)
# Hessian matrix
H = (2.0/n)* XT_X
# Note that here the Hessian does not depend on the parameters beta
invH = np.linalg.pinv(H)
beta = np.random.randn(3,1)
Niterations = 5
# define the gradient
training_gradient = grad(CostOLS)

for iter in range(Niterations):
    gradients = training_gradient(beta)
    beta -= invH @ gradients
    print(iter,gradients[0],gradients[1])
print("beta from own Newton code")
print(beta)

Gradient descent and Logistic regression

Finally, we complete these examples by adding a simple code for Logistic regression. Note the more general approach with a class for the method. Here we use a so-called AND gate for our data set.

import numpy as np
class LogisticRegression:
    def __init__(self, learning_rate=0.01, num_iterations=1000):
        self.learning_rate = learning_rate
        self.num_iterations = num_iterations
        self.beta_logreg = None
    def sigmoid(self, z):
        return 1 / (1 + np.exp(-z))
    def GDfit(self, X, y):
        n_data, num_features = X.shape
        self.beta_logreg = np.zeros(num_features)
        for _ in range(self.num_iterations):
            linear_model = X @ self.beta_logreg
            y_predicted = self.sigmoid(linear_model)
            # Gradient calculation
            gradient = (X.T @ (y_predicted - y))/n_data
            # Update beta_logreg
            self.beta_logreg -= self.learning_rate*gradient
    def predict(self, X):
        linear_model = X @ self.beta_logreg
        y_predicted = self.sigmoid(linear_model)
        return [1 if i >= 0.5 else 0 for i in y_predicted]
# Example usage
if __name__ == "__main__":
    # Sample data
    X = np.array([[0, 0], [1, 0], [0, 1], [1, 1]])
    y = np.array([0, 0, 0, 1])  # This is an AND gate
    model = LogisticRegression(learning_rate=0.01, num_iterations=1000)
    model.GDfit(X, y)
    predictions = model.predict(X)
    print("Predictions:", predictions)

Overview video on Stochastic Gradient Descent

There are several reasons for using stochastic gradient descent. Some of these are:

1. Efficiency: Updates weights more frequently using a single or a small batch of samples, which speeds up convergence.
2. Hopefully avoid Local Minima
3. Memory Usage: Requires less memory compared to computing gradients for the entire dataset.

Batches and mini-batches

In gradient descent we compute the cost function and its gradient for all data points we have.

In large-scale applications such as the ILSVRC challenge, the training data can have on order of millions of examples. Hence, it seems wasteful to compute the full cost function over the entire training set in order to perform only a single parameter update. A very common approach to addressing this challenge is to compute the gradient over batches of the training data. For example, a typical batch could contain some thousand examples from an entire training set of several millions. This batch is then used to perform a parameter update.

Stochastic Gradient Descent (SGD)

In stochastic gradient descent, the extreme case is the case where we have only one batch, that is we include the whole data set.

This process is called Stochastic Gradient Descent (SGD) (or also sometimes on-line gradient descent). This is relatively less common to see because in practice due to vectorized code optimizations it can be computationally much more efficient to evaluate the gradient for 100 examples, than the gradient for one example 100 times. Even though SGD technically refers to using a single example at a time to evaluate the gradient, you will hear people use the term SGD even when referring to mini-batch gradient descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD for “Batch gradient descent” are rare to see), where it is usually assumed that mini-batches are used. The size of the mini-batch is a hyperparameter but it is not very common to cross-validate or bootstrap it. It is usually based on memory constraints (if any), or set to some value, e.g. 32, 64 or 128. We use powers of 2 in practice because many vectorized operation implementations work faster when their inputs are sized in powers of 2.

In our notes with SGD we mean stochastic gradient descent with mini-batches.

Stochastic Gradient Descent

Stochastic gradient descent (SGD) and variants thereof address some of the shortcomings of the Gradient descent method discussed above.

The underlying idea of SGD comes from the observation that the cost function, which we want to minimize, can almost always be written as a sum over \( n \) data points \( \{\mathbf{x}_i\}_{i=1}^n \),

 
$$ C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, \mathbf{\beta}). $$

Computation of gradients

This in turn means that the gradient can be computed as a sum over \( i \)-gradients

 
$$ \nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, \mathbf{\beta}). $$

Stochasticity/randomness is introduced by only taking the gradient on a subset of the data called minibatches. If there are \( n \) data points and the size of each minibatch is \( M \), there will be \( n/M \) minibatches. We denote these minibatches by \( B_k \) where \( k=1,\cdots,n/M \).

SGD example

As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) and we choose to have \( M=5 \) minibathces, then each minibatch contains two data points. In particular we have \( B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 = (\mathbf{x}_9,\mathbf{x}_{10}) \). Note that if you choose \( M=1 \) you have only a single batch with all data points and on the other extreme, you may choose \( M=n \) resulting in a minibatch for each datapoint, i.e \( B_k = \mathbf{x}_k \).

The idea is now to approximate the gradient by replacing the sum over all data points with a sum over the data points in one the minibatches picked at random in each gradient descent step

 
$$ \nabla_{\beta} C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, \mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, \mathbf{\beta}). $$

The gradient step

Thus a gradient descent step now looks like

 
$$ \beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, \mathbf{\beta}) $$

where \( k \) is picked at random with equal probability from \( [1,n/M] \). An iteration over the number of minibathces (n/M) is commonly referred to as an epoch. Thus it is typical to choose a number of epochs and for each epoch iterate over the number of minibatches, as exemplified in the code below.

Simple example code

import numpy as np 

n = 100 #100 datapoints 
M = 5   #size of each minibatch
m = int(n/M) #number of minibatches
n_epochs = 10 #number of epochs

j = 0
for epoch in range(1,n_epochs+1):
    for i in range(m):
        k = np.random.randint(m) #Pick the k-th minibatch at random
        #Compute the gradient using the data in minibatch Bk
        #Compute new suggestion for 
        j += 1

Taking the gradient only on a subset of the data has two important benefits. First, it introduces randomness which decreases the chance that our opmization scheme gets stuck in a local minima. Second, if the size of the minibatches are small relative to the number of datapoints (\( M < n \)), the computation of the gradient is much cheaper since we sum over the datapoints in the \( k-th \) minibatch and not all \( n \) datapoints.

When do we stop?

A natural question is when do we stop the search for a new minimum? One possibility is to compute the full gradient after a given number of epochs and check if the norm of the gradient is smaller than some threshold and stop if true. However, the condition that the gradient is zero is valid also for local minima, so this would only tell us that we are close to a local/global minimum. However, we could also evaluate the cost function at this point, store the result and continue the search. If the test kicks in at a later stage we can compare the values of the cost function and keep the \( \beta \) that gave the lowest value.

Slightly different approach

Another approach is to let the step length \( \gamma_j \) depend on the number of epochs in such a way that it becomes very small after a reasonable time such that we do not move at all. Such approaches are also called scaling. There are many such ways to scale the learning rate and discussions here. See also https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1 for a discussion of different scaling functions for the learning rate.

Time decay rate

As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function

 
$$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$

 
goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in time \( t \).

In this way we can fix the number of epochs, compute \( \beta \) and evaluate the cost function at the end. Repeating the computation will give a different result since the scheme is random by design. Then we pick the final \( \beta \) that gives the lowest value of the cost function.

import numpy as np 

def step_length(t,t0,t1):
    return t0/(t+t1)

n = 100 #100 datapoints 
M = 5   #size of each minibatch
m = int(n/M) #number of minibatches
n_epochs = 500 #number of epochs
t0 = 1.0
t1 = 10

gamma_j = t0/t1
j = 0
for epoch in range(1,n_epochs+1):
    for i in range(m):
        k = np.random.randint(m) #Pick the k-th minibatch at random
        #Compute the gradient using the data in minibatch Bk
        #Compute new suggestion for beta
        t = epoch*m+i
        gamma_j = step_length(t,t0,t1)
        j += 1

print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))

Code with a Number of Minibatches which varies

In the code here we vary the number of mini-batches.

# Importing various packages
from math import exp, sqrt
from random import random, seed
import numpy as np
import matplotlib.pyplot as plt

n = 100
x = 2*np.random.rand(n,1)
y = 4+3*x+np.random.randn(n,1)

X = np.c_[np.ones((n,1)), x]
XT_X = X.T @ X
theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)
# Hessian matrix
H = (2.0/n)* XT_X
EigValues, EigVectors = np.linalg.eig(H)
print(f"Eigenvalues of Hessian Matrix:{EigValues}")

theta = np.random.randn(2,1)
eta = 1.0/np.max(EigValues)
Niterations = 1000


for iter in range(Niterations):
    gradients = 2.0/n*X.T @ ((X @ theta)-y)
    theta -= eta*gradients
print("theta from own gd")
print(theta)

xnew = np.array([[0],[2]])
Xnew = np.c_[np.ones((2,1)), xnew]
ypredict = Xnew.dot(theta)
ypredict2 = Xnew.dot(theta_linreg)

n_epochs = 50
M = 5   #size of each minibatch
m = int(n/M) #number of minibatches
t0, t1 = 5, 50

def learning_schedule(t):
    return t0/(t+t1)

theta = np.random.randn(2,1)

for epoch in range(n_epochs):
# Can you figure out a better way of setting up the contributions to each batch?
    for i in range(m):
        random_index = M*np.random.randint(m)
        xi = X[random_index:random_index+M]
        yi = y[random_index:random_index+M]
        gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)
        eta = learning_schedule(epoch*m+i)
        theta = theta - eta*gradients
print("theta from own sdg")
print(theta)

plt.plot(xnew, ypredict, "r-")
plt.plot(xnew, ypredict2, "b-")
plt.plot(x, y ,'ro')
plt.axis([0,2.0,0, 15.0])
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.title(r'Random numbers ')
plt.show()

Replace or not

In the above code, we have use replacement in setting up the mini-batches. The discussion here may be useful.

Momentum based GD

The stochastic gradient descent (SGD) is almost always used with a momentum or inertia term that serves as a memory of the direction we are moving in parameter space. This is typically implemented as follows

 
$$ \begin{align} \mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\ \boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}, \tag{1} \end{align} $$

where we have introduced a momentum parameter \( \gamma \), with \( 0\le\gamma\le 1 \), and for brevity we dropped the explicit notation to indicate the gradient is to be taken over a different mini-batch at each step. We call this algorithm gradient descent with momentum (GDM). From these equations, it is clear that \( \mathbf{v}_t \) is a running average of recently encountered gradients and \( (1-\gamma)^{-1} \) sets the characteristic time scale for the memory used in the averaging procedure. Consistent with this, when \( \gamma=0 \), this just reduces down to ordinary SGD as discussed earlier. An equivalent way of writing the updates is

 
$$ \Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t), $$

where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).

More on momentum based approaches

Let us try to get more intuition from these equations. It is helpful to consider a simple physical analogy with a particle of mass \( m \) moving in a viscous medium with drag coefficient \( \mu \) and potential \( E(\mathbf{w}) \). If we denote the particle's position by \( \mathbf{w} \), then its motion is described by

 
$$ m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}). $$

We can discretize this equation in the usual way to get

 
$$ m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}). $$

Rearranging this equation, we can rewrite this as

 
$$ \Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t. $$

Momentum parameter

Notice that this equation is identical to previous one if we identify the position of the particle, \( \mathbf{w} \), with the parameters \( \boldsymbol{\theta} \). This allows us to identify the momentum parameter and learning rate with the mass of the particle and the viscous drag as:

 
$$ \gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}. $$

Thus, as the name suggests, the momentum parameter is proportional to the mass of the particle and effectively provides inertia. Furthermore, in the large viscosity/small learning rate limit, our memory time scales as \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).

Why is momentum useful? SGD momentum helps the gradient descent algorithm gain speed in directions with persistent but small gradients even in the presence of stochasticity, while suppressing oscillations in high-curvature directions. This becomes especially important in situations where the landscape is shallow and flat in some directions and narrow and steep in others. It has been argued that first-order methods (with appropriate initial conditions) can perform comparable to more expensive second order methods, especially in the context of complex deep learning models.

These beneficial properties of momentum can sometimes become even more pronounced by using a slight modification of the classical momentum algorithm called Nesterov Accelerated Gradient (NAG).

In the NAG algorithm, rather than calculating the gradient at the current parameters, \( \nabla_\theta E(\boldsymbol{\theta}_t) \), one calculates the gradient at the expected value of the parameters given our current momentum, \( \nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \). This yields the NAG update rule

 
$$ \begin{align} \mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\ \boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}. \tag{2} \end{align} $$

One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of \( \gamma \).

Second moment of the gradient

In stochastic gradient descent, with and without momentum, we still have to specify a schedule for tuning the learning rates \( \eta_t \) as a function of time. As discussed in the context of Newton's method, this presents a number of dilemmas. The learning rate is limited by the steepest direction which can change depending on the current position in the landscape. To circumvent this problem, ideally our algorithm would keep track of curvature and take large steps in shallow, flat directions and small steps in steep, narrow directions. Second-order methods accomplish this by calculating or approximating the Hessian and normalizing the learning rate by the curvature. However, this is very computationally expensive for extremely large models. Ideally, we would like to be able to adaptively change the step size to match the landscape without paying the steep computational price of calculating or approximating Hessians.

During the last decade a number of methods have been introduced that accomplish this by tracking not only the gradient, but also the second moment of the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and ADAM.

RMS prop

In RMS prop, in addition to keeping a running average of the first moment of the gradient, we also keep track of the second moment denoted by \( \mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2] \). The update rule for RMS prop is given by

 
$$ \begin{align} \mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) \tag{3}\\ \mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\ \boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber \end{align} $$

where \( \beta \) controls the averaging time of the second moment and is typically taken to be about \( \beta=0.9 \), \( \eta_t \) is a learning rate typically chosen to be \( 10^{-3} \), and \( \epsilon\sim 10^{-8} \) is a small regularization constant to prevent divergences. Multiplication and division by vectors is understood as an element-wise operation. It is clear from this formula that the learning rate is reduced in directions where the norm of the gradient is consistently large. This greatly speeds up the convergence by allowing us to use a larger learning rate for flat directions.

ADAM optimizer

A related algorithm is the ADAM optimizer. In ADAM, we keep a running average of both the first and second moment of the gradient and use this information to adaptively change the learning rate for different parameters. The method isefficient when working with large problems involving lots data and/or parameters. It is a combination of the gradient descent with momentum algorithm and the RMSprop algorithm discussed above.

In addition to keeping a running average of the first and second moments of the gradient (i.e. \( \mathbf{m}_t=\mathbb{E}[\mathbf{g}_t] \) and \( \mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t] \), respectively), ADAM performs an additional bias correction to account for the fact that we are estimating the first two moments of the gradient using a running average (denoted by the hats in the update rule below). The update rule for ADAM is given by (where multiplication and division are once again understood to be element-wise operations below)

 
$$ \begin{align} \mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) \tag{4}\\ \mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\ \mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\ \boldsymbol{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\ \boldsymbol{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\ \boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \boldsymbol{\mathbf{m}}_t \over \sqrt{\boldsymbol{\mathbf{s}}_t} +\epsilon}, \nonumber \\ \tag{5} \end{align} $$

where \( \beta_1 \) and \( \beta_2 \) set the memory lifetime of the first and second moment and are typically taken to be \( 0.9 \) and \( 0.99 \) respectively, and \( \eta \) and \( \epsilon \) are identical to RMSprop.

Like in RMSprop, the effective step size of a parameter depends on the magnitude of its gradient squared. To understand this better, let us rewrite this expression in terms of the variance \( \boldsymbol{\sigma}_t^2 = \boldsymbol{\mathbf{s}}_t - (\boldsymbol{\mathbf{m}}_t)^2 \). Consider a single parameter \( \theta_t \). The update rule for this parameter is given by

 
$$ \Delta \theta_{t+1}= -\eta_t { \boldsymbol{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}. $$

Algorithms and codes for Adagrad, RMSprop and Adam

The algorithms we have implemented are well described in the text by Goodfellow, Bengio and Courville, chapter 8.

The codes which implement these algorithms are discussed after our presentation of automatic differentiation.

AdaGrad algorithm, taken from Goodfellow et al

RMSProp algorithm, taken from Goodfellow et al

ADAM algorithm, taken from Goodfellow et al

Practical tips

• Randomize the data when making mini-batches. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.
• Transform your inputs. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the cost function is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case.
• Monitor the out-of-sample performance. Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This early stopping significantly improves performance in many settings.
• Adaptive optimization methods don't always have good generalization. Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.

Geron's text, see chapter 11, has several interesting discussions.

Automatic differentiation

Automatic differentiation (AD), also called algorithmic differentiation or computational differentiation,is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.

Automatic differentiation is neither:

• Symbolic differentiation, nor
• Numerical differentiation (the method of finite differences).

Symbolic differentiation can lead to inefficient code and faces the difficulty of converting a computer program into a single expression, while numerical differentiation can introduce round-off errors in the discretization process and cancellation

Python has tools for so-called automatic differentiation. Consider the following example

 
$$ f(x) = \sin\left(2\pi x + x^2\right) $$

which has the following derivative

 
$$ f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) $$

Using autograd we have

import autograd.numpy as np

# To do elementwise differentiation:
from autograd import elementwise_grad as egrad 

# To plot:
import matplotlib.pyplot as plt 


def f(x):
    return np.sin(2*np.pi*x + x**2)

def f_grad_analytic(x):
    return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)

# Do the comparison:
x = np.linspace(0,1,1000)

f_grad = egrad(f)

computed = f_grad(x)
analytic = f_grad_analytic(x)

plt.title('Derivative computed from Autograd compared with the analytical derivative')
plt.plot(x,computed,label='autograd')
plt.plot(x,analytic,label='analytic')

plt.xlabel('x')
plt.ylabel('y')
plt.legend()

plt.show()

print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic))))

Using autograd

Here we experiment with what kind of functions Autograd is capable of finding the gradient of. The following Python functions are just meant to illustrate what Autograd can do, but please feel free to experiment with other, possibly more complicated, functions as well.

import autograd.numpy as np
from autograd import grad

def f1(x):
    return x**3 + 1

f1_grad = grad(f1)

# Remember to send in float as argument to the computed gradient from Autograd!
a = 1.0

# See the evaluated gradient at a using autograd:
print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))

# Compare with the analytical derivative, that is f1'(x) = 3*x**2 
grad_analytical = 3*a**2
print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))

Autograd with more complicated functions

To differentiate with respect to two (or more) arguments of a Python function, Autograd need to know at which variable the function if being differentiated with respect to.

import autograd.numpy as np
from autograd import grad
def f2(x1,x2):
    return 3*x1**3 + x2*(x1 - 5) + 1

# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
f2_grad_x1 = grad(f2,0)

# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
f2_grad_x2 = grad(f2,1)

x1 = 1.0
x2 = 3.0 

print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
print("-"*30)

# Compare with the analytical derivatives:

# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
f2_grad_x1_analytical = 9*x1**2 + x2

# Derivative of f2 w.r.t x2 is: x1 - 5:
f2_grad_x2_analytical = x1 - 5

# See the evaluated derivations:
print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))

print()

print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))

Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.

More complicated functions using the elements of their arguments directly

import autograd.numpy as np
from autograd import grad
def f3(x): # Assumes x is an array of length 5 or higher
    return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2

f3_grad = grad(f3)

x = np.linspace(0,4,5)

# Print the computed gradient:
print("The computed gradient of f3 is: ", f3_grad(x))

# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])

# Print the analytical gradient:
print("The analytical gradient of f3 is: ", f3_grad_analytical)

Note that in this case, when sending an array as input argument, the output from Autograd is another array. This is the true gradient of the function, as opposed to the function in the previous example. By using arrays to represent the variables, the output from Autograd might be easier to work with, as the output is closer to what one could expect form a gradient-evaluting function.

Functions using mathematical functions from Numpy

import autograd.numpy as np
from autograd import grad
def f4(x):
    return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)

f4_grad = grad(f4)

x = 2.7

# Print the computed derivative:
print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))

# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi

# Print the analytical gradient:
print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))

More autograd

import autograd.numpy as np
from autograd import grad
def f5(x):
    if x >= 0:
        return x**2
    else:
        return -3*x + 1

f5_grad = grad(f5)

x = 2.7

# Print the computed derivative:
print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))

And with loops

import autograd.numpy as np
from autograd import grad
def f6_for(x):
    val = 0
    for i in range(10):
        val = val + x**i
    return val

def f6_while(x):
    val = 0
    i = 0
    while i < 10:
        val = val + x**i
        i = i + 1
    return val

f6_for_grad = grad(f6_for)
f6_while_grad = grad(f6_while)

x = 0.5

# Print the computed derivaties of f6_for and f6_while
print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))

import autograd.numpy as np
from autograd import grad
# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9
# The analytical derivative is: sum(i*x**(i-1)) 
f6_grad_analytical = 0
for i in range(10):
    f6_grad_analytical += i*x**(i-1)

print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))

Using recursion

import autograd.numpy as np
from autograd import grad

def f7(n): # Assume that n is an integer
    if n == 1 or n == 0:
        return 1
    else:
        return n*f7(n-1)

f7_grad = grad(f7)

n = 2.0

print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))

# The function f7 is an implementation of the factorial of n.
# By using the product rule, one can find that the derivative is:

f7_grad_analytical = 0
for i in range(int(n)-1):
    tmp = 1
    for k in range(int(n)-1):
        if k != i:
            tmp *= (n - k)
    f7_grad_analytical += tmp

print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))

Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.

Using Autograd with OLS

We conclude the part on optmization by showing how we can make codes for linear regression and logistic regression using autograd. The first example shows results with ordinary leats squares.

# Using Autograd to calculate gradients for OLS
from random import random, seed
import numpy as np
import autograd.numpy as np
import matplotlib.pyplot as plt
from autograd import grad

def CostOLS(beta):
    return (1.0/n)*np.sum((y-X @ beta)**2)

n = 100
x = 2*np.random.rand(n,1)
y = 4+3*x+np.random.randn(n,1)

X = np.c_[np.ones((n,1)), x]
XT_X = X.T @ X
theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)
# Hessian matrix
H = (2.0/n)* XT_X
EigValues, EigVectors = np.linalg.eig(H)
print(f"Eigenvalues of Hessian Matrix:{EigValues}")

theta = np.random.randn(2,1)
eta = 1.0/np.max(EigValues)
Niterations = 1000
# define the gradient
training_gradient = grad(CostOLS)

for iter in range(Niterations):
    gradients = training_gradient(theta)
    theta -= eta*gradients
print("theta from own gd")
print(theta)

xnew = np.array([[0],[2]])
Xnew = np.c_[np.ones((2,1)), xnew]
ypredict = Xnew.dot(theta)
ypredict2 = Xnew.dot(theta_linreg)

plt.plot(xnew, ypredict, "r-")
plt.plot(xnew, ypredict2, "b-")
plt.plot(x, y ,'ro')
plt.axis([0,2.0,0, 15.0])
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.title(r'Random numbers ')
plt.show()

Same code but now with momentum gradient descent

# Using Autograd to calculate gradients for OLS
from random import random, seed
import numpy as np
import autograd.numpy as np
import matplotlib.pyplot as plt
from autograd import grad

def CostOLS(beta):
    return (1.0/n)*np.sum((y-X @ beta)**2)

n = 100
x = 2*np.random.rand(n,1)
y = 4+3*x#+np.random.randn(n,1)

X = np.c_[np.ones((n,1)), x]
XT_X = X.T @ X
theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)
# Hessian matrix
H = (2.0/n)* XT_X
EigValues, EigVectors = np.linalg.eig(H)
print(f"Eigenvalues of Hessian Matrix:{EigValues}")

theta = np.random.randn(2,1)
eta = 1.0/np.max(EigValues)
Niterations = 30

# define the gradient
training_gradient = grad(CostOLS)

for iter in range(Niterations):
    gradients = training_gradient(theta)
    theta -= eta*gradients
    print(iter,gradients[0],gradients[1])
print("theta from own gd")
print(theta)

# Now improve with momentum gradient descent
change = 0.0
delta_momentum = 0.3
for iter in range(Niterations):
    # calculate gradient
    gradients = training_gradient(theta)
    # calculate update
    new_change = eta*gradients+delta_momentum*change
    # take a step
    theta -= new_change
    # save the change
    change = new_change
    print(iter,gradients[0],gradients[1])
print("theta from own gd wth momentum")
print(theta)

Including Stochastic Gradient Descent with Autograd

In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using autograd.

# Using Autograd to calculate gradients using SGD
# OLS example
from random import random, seed
import numpy as np
import autograd.numpy as np
import matplotlib.pyplot as plt
from autograd import grad

# Note change from previous example
def CostOLS(y,X,theta):
    return np.sum((y-X @ theta)**2)

n = 100
x = 2*np.random.rand(n,1)
y = 4+3*x+np.random.randn(n,1)

X = np.c_[np.ones((n,1)), x]
XT_X = X.T @ X
theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)
# Hessian matrix
H = (2.0/n)* XT_X
EigValues, EigVectors = np.linalg.eig(H)
print(f"Eigenvalues of Hessian Matrix:{EigValues}")

theta = np.random.randn(2,1)
eta = 1.0/np.max(EigValues)
Niterations = 1000

# Note that we request the derivative wrt third argument (theta, 2 here)
training_gradient = grad(CostOLS,2)

for iter in range(Niterations):
    gradients = (1.0/n)*training_gradient(y, X, theta)
    theta -= eta*gradients
print("theta from own gd")
print(theta)

xnew = np.array([[0],[2]])
Xnew = np.c_[np.ones((2,1)), xnew]
ypredict = Xnew.dot(theta)
ypredict2 = Xnew.dot(theta_linreg)

plt.plot(xnew, ypredict, "r-")
plt.plot(xnew, ypredict2, "b-")
plt.plot(x, y ,'ro')
plt.axis([0,2.0,0, 15.0])
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.title(r'Random numbers ')
plt.show()

n_epochs = 50
M = 5   #size of each minibatch
m = int(n/M) #number of minibatches
t0, t1 = 5, 50
def learning_schedule(t):
    return t0/(t+t1)

theta = np.random.randn(2,1)

for epoch in range(n_epochs):
# Can you figure out a better way of setting up the contributions to each batch?
    for i in range(m):
        random_index = M*np.random.randint(m)
        xi = X[random_index:random_index+M]
        yi = y[random_index:random_index+M]
        gradients = (1.0/M)*training_gradient(yi, xi, theta)
        eta = learning_schedule(epoch*m+i)
        theta = theta - eta*gradients
print("theta from own sdg")
print(theta)

Same code but now with momentum gradient descent

# Using Autograd to calculate gradients using SGD
# OLS example
from random import random, seed
import numpy as np
import autograd.numpy as np
import matplotlib.pyplot as plt
from autograd import grad

# Note change from previous example
def CostOLS(y,X,theta):
    return np.sum((y-X @ theta)**2)

n = 100
x = 2*np.random.rand(n,1)
y = 4+3*x+np.random.randn(n,1)

X = np.c_[np.ones((n,1)), x]
XT_X = X.T @ X
theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)
# Hessian matrix
H = (2.0/n)* XT_X
EigValues, EigVectors = np.linalg.eig(H)
print(f"Eigenvalues of Hessian Matrix:{EigValues}")

theta = np.random.randn(2,1)
eta = 1.0/np.max(EigValues)
Niterations = 100

# Note that we request the derivative wrt third argument (theta, 2 here)
training_gradient = grad(CostOLS,2)

for iter in range(Niterations):
    gradients = (1.0/n)*training_gradient(y, X, theta)
    theta -= eta*gradients
print("theta from own gd")
print(theta)


n_epochs = 50
M = 5   #size of each minibatch
m = int(n/M) #number of minibatches
t0, t1 = 5, 50
def learning_schedule(t):
    return t0/(t+t1)

theta = np.random.randn(2,1)

change = 0.0
delta_momentum = 0.3

for epoch in range(n_epochs):
    for i in range(m):
        random_index = M*np.random.randint(m)
        xi = X[random_index:random_index+M]
        yi = y[random_index:random_index+M]
        gradients = (1.0/M)*training_gradient(yi, xi, theta)
        eta = learning_schedule(epoch*m+i)
        # calculate update
        new_change = eta*gradients+delta_momentum*change
        # take a step
        theta -= new_change
        # save the change
        change = new_change
print("theta from own sdg with momentum")
print(theta)

Similar (second order function now) problem but now with AdaGrad

# Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent
# OLS example
from random import random, seed
import numpy as np
import autograd.numpy as np
import matplotlib.pyplot as plt
from autograd import grad

# Note change from previous example
def CostOLS(y,X,theta):
    return np.sum((y-X @ theta)**2)

n = 1000
x = np.random.rand(n,1)
y = 2.0+3*x +4*x*x

X = np.c_[np.ones((n,1)), x, x*x]
XT_X = X.T @ X
theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)


# Note that we request the derivative wrt third argument (theta, 2 here)
training_gradient = grad(CostOLS,2)
# Define parameters for Stochastic Gradient Descent
n_epochs = 50
M = 5   #size of each minibatch
m = int(n/M) #number of minibatches
# Guess for unknown parameters theta
theta = np.random.randn(3,1)

# Value for learning rate
eta = 0.01
# Including AdaGrad parameter to avoid possible division by zero
delta  = 1e-8
for epoch in range(n_epochs):
    Giter = 0.0
    for i in range(m):
        random_index = M*np.random.randint(m)
        xi = X[random_index:random_index+M]
        yi = y[random_index:random_index+M]
        gradients = (1.0/M)*training_gradient(yi, xi, theta)
        Giter += gradients*gradients
        update = gradients*eta/(delta+np.sqrt(Giter))
        theta -= update
print("theta from own AdaGrad")
print(theta)

Running this code we note an almost perfect agreement with the results from matrix inversion.

RMSprop for adaptive learning rate with Stochastic Gradient Descent

# Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent
# OLS example
from random import random, seed
import numpy as np
import autograd.numpy as np
import matplotlib.pyplot as plt
from autograd import grad

# Note change from previous example
def CostOLS(y,X,theta):
    return np.sum((y-X @ theta)**2)

n = 1000
x = np.random.rand(n,1)
y = 2.0+3*x +4*x*x# +np.random.randn(n,1)

X = np.c_[np.ones((n,1)), x, x*x]
XT_X = X.T @ X
theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)


# Note that we request the derivative wrt third argument (theta, 2 here)
training_gradient = grad(CostOLS,2)
# Define parameters for Stochastic Gradient Descent
n_epochs = 50
M = 5   #size of each minibatch
m = int(n/M) #number of minibatches
# Guess for unknown parameters theta
theta = np.random.randn(3,1)

# Value for learning rate
eta = 0.01
# Value for parameter rho
rho = 0.99
# Including AdaGrad parameter to avoid possible division by zero
delta  = 1e-8
for epoch in range(n_epochs):
    Giter = 0.0
    for i in range(m):
        random_index = M*np.random.randint(m)
        xi = X[random_index:random_index+M]
        yi = y[random_index:random_index+M]
        gradients = (1.0/M)*training_gradient(yi, xi, theta)
	# Accumulated gradient
	# Scaling with rho the new and the previous results
        Giter = (rho*Giter+(1-rho)*gradients*gradients)
	# Taking the diagonal only and inverting
        update = gradients*eta/(delta+np.sqrt(Giter))
	# Hadamard product
        theta -= update
print("theta from own RMSprop")
print(theta)

And finally ADAM

# Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent
# OLS example
from random import random, seed
import numpy as np
import autograd.numpy as np
import matplotlib.pyplot as plt
from autograd import grad

# Note change from previous example
def CostOLS(y,X,theta):
    return np.sum((y-X @ theta)**2)

n = 1000
x = np.random.rand(n,1)
y = 2.0+3*x +4*x*x# +np.random.randn(n,1)

X = np.c_[np.ones((n,1)), x, x*x]
XT_X = X.T @ X
theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)


# Note that we request the derivative wrt third argument (theta, 2 here)
training_gradient = grad(CostOLS,2)
# Define parameters for Stochastic Gradient Descent
n_epochs = 50
M = 5   #size of each minibatch
m = int(n/M) #number of minibatches
# Guess for unknown parameters theta
theta = np.random.randn(3,1)

# Value for learning rate
eta = 0.01
# Value for parameters beta1 and beta2, see https://arxiv.org/abs/1412.6980
beta1 = 0.9
beta2 = 0.999
# Including AdaGrad parameter to avoid possible division by zero
delta  = 1e-7
iter = 0
for epoch in range(n_epochs):
    first_moment = 0.0
    second_moment = 0.0
    iter += 1
    for i in range(m):
        random_index = M*np.random.randint(m)
        xi = X[random_index:random_index+M]
        yi = y[random_index:random_index+M]
        gradients = (1.0/M)*training_gradient(yi, xi, theta)
        # Computing moments first
        first_moment = beta1*first_moment + (1-beta1)*gradients
        second_moment = beta2*second_moment+(1-beta2)*gradients*gradients
        first_term = first_moment/(1.0-beta1**iter)
        second_term = second_moment/(1.0-beta2**iter)
	# Scaling with rho the new and the previous results
        update = eta*first_term/(np.sqrt(second_term)+delta)
        theta -= update
print("theta from own ADAM")
print(theta)

And Logistic Regression

import autograd.numpy as np
from autograd import grad

def sigmoid(x):
    return 0.5 * (np.tanh(x / 2.) + 1)

def logistic_predictions(weights, inputs):
    # Outputs probability of a label being true according to logistic model.
    return sigmoid(np.dot(inputs, weights))

def training_loss(weights):
    # Training loss is the negative log-likelihood of the training labels.
    preds = logistic_predictions(weights, inputs)
    label_probabilities = preds * targets + (1 - preds) * (1 - targets)
    return -np.sum(np.log(label_probabilities))

# Build a toy dataset.
inputs = np.array([[0.52, 1.12,  0.77],
                   [0.88, -1.08, 0.15],
                   [0.52, 0.06, -1.30],
                   [0.74, -2.49, 1.39]])
targets = np.array([True, True, False, True])

# Define a function that returns gradients of training loss using Autograd.
training_gradient_fun = grad(training_loss)

# Optimize weights using gradient descent.
weights = np.array([0.0, 0.0, 0.0])
print("Initial loss:", training_loss(weights))
for i in range(100):
    weights -= training_gradient_fun(weights) * 0.01

print("Trained loss:", training_loss(weights))

Introducing JAX

Presently, instead of using autograd, we recommend using JAX

JAX is Autograd and XLA (Accelerated Linear Algebra)), brought together for high-performance numerical computing and machine learning research. It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more.

Getting started with Jax, note the way we import numpy

import jax
import jax.numpy as jnp
import numpy as np
import matplotlib.pyplot as plt

from jax import grad as jax_grad

A warm-up example

def function(x):
    return x**2

def analytical_gradient(x):
    return 2*x

def gradient_descent(starting_point, learning_rate, num_iterations, solver="analytical"):
    x = starting_point
    trajectory_x = [x]
    trajectory_y = [function(x)]

    if solver == "analytical":
        grad = analytical_gradient    
    elif solver == "jax":
        grad = jax_grad(function)
        x = jnp.float64(x)
        learning_rate = jnp.float64(learning_rate)

    for _ in range(num_iterations):
        
        x = x - learning_rate * grad(x)
        trajectory_x.append(x)
        trajectory_y.append(function(x))

    return trajectory_x, trajectory_y

x = np.linspace(-5, 5, 100)
plt.plot(x, function(x), label="f(x)")

descent_x, descent_y = gradient_descent(5, 0.1, 10, solver="analytical")
jax_descend_x, jax_descend_y = gradient_descent(5, 0.1, 10, solver="jax")

plt.plot(descent_x, descent_y, label="Gradient descent", marker="o")
plt.plot(jax_descend_x, jax_descend_y, label="JAX", marker="x")

A more advanced example

backend = np

def function(x):
    return x*backend.sin(x**2 + 1)

def analytical_gradient(x):
    return backend.sin(x**2 + 1) + 2*x**2*backend.cos(x**2 + 1)


x = np.linspace(-5, 5, 100)
plt.plot(x, function(x), label="f(x)")

descent_x, descent_y = gradient_descent(1, 0.01, 300, solver="analytical")

# Change the backend to JAX
backend = jnp
jax_descend_x, jax_descend_y = gradient_descent(1, 0.01, 300, solver="jax")

plt.scatter(descent_x, descent_y, label="Gradient descent", marker="v", s=10, color="red") 
plt.scatter(jax_descend_x, jax_descend_y, label="JAX", marker="x", s=5, color="black")

Introduction to Neural networks

Artificial neural networks are computational systems that can learn to perform tasks by considering examples, generally without being programmed with any task-specific rules. It is supposed to mimic a biological system, wherein neurons interact by sending signals in the form of mathematical functions between layers. All layers can contain an arbitrary number of neurons, and each connection is represented by a weight variable.

Artificial neurons

The field of artificial neural networks has a long history of development, and is closely connected with the advancement of computer science and computers in general. A model of artificial neurons was first developed by McCulloch and Pitts in 1943 to study signal processing in the brain and has later been refined by others. The general idea is to mimic neural networks in the human brain, which is composed of billions of neurons that communicate with each other by sending electrical signals. Each neuron accumulates its incoming signals, which must exceed an activation threshold to yield an output. If the threshold is not overcome, the neuron remains inactive, i.e. has zero output.

This behaviour has inspired a simple mathematical model for an artificial neuron.

 
$$ \begin{equation} y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u) \tag{6} \end{equation} $$

Here, the output \( y \) of the neuron is the value of its activation function, which have as input a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neurons.

Conceptually, it is helpful to divide neural networks into four categories:

1. general purpose neural networks for supervised learning,
2. neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),
3. neural networks for sequential data such as Recurrent Neural Networks (RNNs), and
4. neural networks for unsupervised learning such as Deep Boltzmann Machines.

In natural science, DNNs and CNNs have already found numerous applications. In statistical physics, they have been applied to detect phase transitions in 2D Ising and Potts models, lattice gauge theories, and different phases of polymers, or solving the Navier-Stokes equation in weather forecasting. Deep learning has also found interesting applications in quantum physics. Various quantum phase transitions can be detected and studied using DNNs and CNNs, topological phases, and even non-equilibrium many-body localization. Representing quantum states as DNNs quantum state tomography are among some of the impressive achievements to reveal the potential of DNNs to facilitate the study of quantum systems.

In quantum information theory, it has been shown that one can perform gate decompositions with the help of neural.

The applications are not limited to the natural sciences. There is a plethora of applications in essentially all disciplines, from the humanities to life science and medicine.

Neural network types

An artificial neural network (ANN), is a computational model that consists of layers of connected neurons, or nodes or units. We will refer to these interchangeably as units or nodes, and sometimes as neurons.

It is supposed to mimic a biological nervous system by letting each neuron interact with other neurons by sending signals in the form of mathematical functions between layers. A wide variety of different ANNs have been developed, but most of them consist of an input layer, an output layer and eventual layers in-between, called hidden layers. All layers can contain an arbitrary number of nodes, and each connection between two nodes is associated with a weight variable.

Neural networks (also called neural nets) are neural-inspired nonlinear models for supervised learning. As we will see, neural nets can be viewed as natural, more powerful extensions of supervised learning methods such as linear and logistic regression and soft-max methods we discussed earlier.

Feed-forward neural networks

The feed-forward neural network (FFNN) was the first and simplest type of ANNs that were devised. In this network, the information moves in only one direction: forward through the layers.

Nodes are represented by circles, while the arrows display the connections between the nodes, including the direction of information flow. Additionally, each arrow corresponds to a weight variable (figure to come). We observe that each node in a layer is connected to all nodes in the subsequent layer, making this a so-called fully-connected FFNN.

Convolutional Neural Network

A different variant of FFNNs are convolutional neural networks (CNNs), which have a connectivity pattern inspired by the animal visual cortex. Individual neurons in the visual cortex only respond to stimuli from small sub-regions of the visual field, called a receptive field. This makes the neurons well-suited to exploit the strong spatially local correlation present in natural images. The response of each neuron can be approximated mathematically as a convolution operation. (figure to come)

Convolutional neural networks emulate the behaviour of neurons in the visual cortex by enforcing a local connectivity pattern between nodes of adjacent layers: Each node in a convolutional layer is connected only to a subset of the nodes in the previous layer, in contrast to the fully-connected FFNN. Often, CNNs consist of several convolutional layers that learn local features of the input, with a fully-connected layer at the end, which gathers all the local data and produces the outputs. They have wide applications in image and video recognition.

Recurrent neural networks

So far we have only mentioned ANNs where information flows in one direction: forward. Recurrent neural networks on the other hand, have connections between nodes that form directed cycles. This creates a form of internal memory which are able to capture information on what has been calculated before; the output is dependent on the previous computations. Recurrent NNs make use of sequential information by performing the same task for every element in a sequence, where each element depends on previous elements. An example of such information is sentences, making recurrent NNs especially well-suited for handwriting and speech recognition.

Other types of networks

There are many other kinds of ANNs that have been developed. One type that is specifically designed for interpolation in multidimensional space is the radial basis function (RBF) network. RBFs are typically made up of three layers: an input layer, a hidden layer with non-linear radial symmetric activation functions and a linear output layer (''linear'' here means that each node in the output layer has a linear activation function). The layers are normally fully-connected and there are no cycles, thus RBFs can be viewed as a type of fully-connected FFNN. They are however usually treated as a separate type of NN due the unusual activation functions.

Multilayer perceptrons

One uses often so-called fully-connected feed-forward neural networks with three or more layers (an input layer, one or more hidden layers and an output layer) consisting of neurons that have non-linear activation functions.

Such networks are often called multilayer perceptrons (MLPs).

Why multilayer perceptrons?

According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function.

Note that the requirements on the activation function only applies to the hidden layer, the output nodes are always assumed to be linear, so as to not restrict the range of output values.

Illustration of a single perceptron model and a multi-perceptron model

Examples of XOR, OR and AND gates

Let us first try to fit various gates using standard linear regression. The gates we are thinking of are the classical XOR, OR and AND gates, well-known elements in computer science. The tables here show how we can set up the inputs \( x_1 \) and \( x_2 \) in order to yield a specific target \( y_i \).

"""
Simple code that tests XOR, OR and AND gates with linear regression
"""

import numpy as np
# Design matrix
X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
print(f"The X.TX  matrix:{X.T @ X}")
Xinv = np.linalg.pinv(X.T @ X)
print(f"The invers of X.TX  matrix:{Xinv}")

# The XOR gate 
yXOR = np.array( [ 0, 1 ,1, 0])
ThetaXOR  = Xinv @ X.T @ yXOR
print(f"The values of theta for the XOR gate:{ThetaXOR}")
print(f"The linear regression prediction  for the XOR gate:{X @ ThetaXOR}")


# The OR gate 
yOR = np.array( [ 0, 1 ,1, 1])
ThetaOR  = Xinv @ X.T @ yOR
print(f"The values of theta for the OR gate:{ThetaOR}")
print(f"The linear regression prediction  for the OR gate:{X @ ThetaOR}")


# The OR gate 
yAND = np.array( [ 0, 0 ,0, 1])
ThetaAND  = Xinv @ X.T @ yAND
print(f"The values of theta for the AND gate:{ThetaAND}")
print(f"The linear regression prediction  for the AND gate:{X @ ThetaAND}")

What is happening here?

Does Logistic Regression do a better Job?

"""
Simple code that tests XOR and OR gates with linear regression
and logistic regression
"""

import matplotlib.pyplot as plt
from sklearn.linear_model import LogisticRegression
import numpy as np

# Design matrix
X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
print(f"The X.TX  matrix:{X.T @ X}")
Xinv = np.linalg.pinv(X.T @ X)
print(f"The invers of X.TX  matrix:{Xinv}")

# The XOR gate 
yXOR = np.array( [ 0, 1 ,1, 0])
ThetaXOR  = Xinv @ X.T @ yXOR
print(f"The values of theta for the XOR gate:{ThetaXOR}")
print(f"The linear regression prediction  for the XOR gate:{X @ ThetaXOR}")


# The OR gate 
yOR = np.array( [ 0, 1 ,1, 1])
ThetaOR  = Xinv @ X.T @ yOR
print(f"The values of theta for the OR gate:{ThetaOR}")
print(f"The linear regression prediction  for the OR gate:{X @ ThetaOR}")


# The OR gate 
yAND = np.array( [ 0, 0 ,0, 1])
ThetaAND  = Xinv @ X.T @ yAND
print(f"The values of theta for the AND gate:{ThetaAND}")
print(f"The linear regression prediction  for the AND gate:{X @ ThetaAND}")

# Now we change to logistic regression


# Logistic Regression
logreg = LogisticRegression()
logreg.fit(X, yOR)
print("Test set accuracy with Logistic Regression for OR gate: {:.2f}".format(logreg.score(X,yOR)))

logreg.fit(X, yXOR)
print("Test set accuracy with Logistic Regression for XOR gate: {:.2f}".format(logreg.score(X,yXOR)))


logreg.fit(X, yAND)
print("Test set accuracy with Logistic Regression for AND gate: {:.2f}".format(logreg.score(X,yAND)))

Not exactly impressive, but somewhat better.

Adding Neural Networks

# and now neural networks with Scikit-Learn and the XOR

from sklearn.neural_network import MLPClassifier
from sklearn.datasets import make_classification
X, yXOR = make_classification(n_samples=100, random_state=1)
FFNN = MLPClassifier(random_state=1, max_iter=300).fit(X, yXOR)
FFNN.predict_proba(X)
print(f"Test set accuracy with Feed Forward Neural Network  for XOR gate:{FFNN.score(X, yXOR)}")

Mathematical model

The output \( y \) is produced via the activation function \( f \)

 
$$ y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z), $$

This function receives \( x_i \) as inputs. Here the activation \( z=(\sum_{i=1}^n w_ix_i+b_i) \). In an FFNN of such neurons, the inputs \( x_i \) are the outputs of the neurons in the preceding layer. Furthermore, an MLP is fully-connected, which means that each neuron receives a weighted sum of the outputs of all neurons in the previous layer.

Mathematical model

First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),

 
$$ \begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1 \tag{7} \end{equation} $$

Here \( b_i \) is the so-called bias which is normally needed in case of zero activation weights or inputs. How to fix the biases and the weights will be discussed below. The value of \( z_i^1 \) is the argument to the activation function \( f_i \) of each node \( i \), The variable \( M \) stands for all possible inputs to a given node \( i \) in the first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as

 
$$ \begin{equation} y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right) \tag{8} \end{equation} $$

where we assume that all nodes in the same layer have identical activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions. In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,

 
$$ \begin{equation} y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right) \tag{9} \end{equation} $$

where \( N_l \) is the number of nodes in layer \( l \). When the output of all the nodes in the first hidden layer are computed, the values of the subsequent layer can be calculated and so forth until the output is obtained.

Mathematical model

The output of neuron \( i \) in layer 2 is thus,

 
$$ \begin{align} y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) \tag{10}\\ &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right] \tag{11} \end{align} $$

where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads

 
$$ \begin{align} y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) \tag{12}\\ &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right) + b_1^3\right] \tag{13} \end{align} $$

Mathematical model

We can generalize this expression to an MLP with \( l \) hidden layers. The complete functional form is,

 
$$ \begin{align} &y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] && \tag{14} \end{align} $$

which illustrates a basic property of MLPs: The only independent variables are the input values \( x_n \).

Mathematical model

This confirms that an MLP, despite its quite convoluted mathematical form, is nothing more than an analytic function, specifically a mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow \hat{y} \in \mathbb{R}^m \).

Furthermore, the flexibility and universality of an MLP can be illustrated by realizing that the expression is essentially a nested sum of scaled activation functions of the form

 
$$ \begin{equation} f(x) = c_1 f(c_2 x + c_3) + c_4 \tag{15} \end{equation} $$

where the parameters \( c_i \) are weights and biases. By adjusting these parameters, the activation functions can be shifted up and down or left and right, change slope or be rescaled which is the key to the flexibility of a neural network.

Matrix-vector notation

We can introduce a more convenient notation for the activations in an A NN.

Additionally, we can represent the biases and activations as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.

We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors. With this notation, the sum becomes a matrix-vector multiplication, and we can write the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as

 
$$ \begin{equation} \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) = f_2\left(\left[\begin{array}{ccc} w^2_{11} &w^2_{12} &w^2_{13} \\ w^2_{21} &w^2_{22} &w^2_{23} \\ w^2_{31} &w^2_{32} &w^2_{33} \\ \end{array} \right] \cdot \left[\begin{array}{c} y^1_1 \\ y^1_2 \\ y^1_3 \\ \end{array}\right] + \left[\begin{array}{c} b^2_1 \\ b^2_2 \\ b^2_3 \\ \end{array}\right]\right). \tag{16} \end{equation} $$

Matrix-vector notation and activation

The activation of node \( i \) in layer 2 is

 
$$ \begin{equation} y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right). \tag{17} \end{equation} $$

This is not just a convenient and compact notation, but also a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications and vector additions that are used as input to the activation functions. For each operation \( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.

Activation functions

A property that characterizes a neural network, other than its connectivity, is the choice of activation function(s). As described in, the following restrictions are imposed on an activation function for a FFNN to fulfill the universal approximation theorem

• Non-constant
• Bounded
• Monotonically-increasing
• Continuous

Activation functions, Logistic and Hyperbolic ones

The second requirement excludes all linear functions. Furthermore, in a MLP with only linear activation functions, each layer simply performs a linear transformation of its inputs.

Regardless of the number of layers, the output of the NN will be nothing but a linear function of the inputs. Thus we need to introduce some kind of non-linearity to the NN to be able to fit non-linear functions Typical examples are the logistic Sigmoid

 
$$ f(x) = \frac{1}{1 + e^{-x}}, $$

and the hyperbolic tangent function

 
$$ f(x) = \tanh(x) $$

Relevance

The sigmoid function are more biologically plausible because the output of inactive neurons are zero. Such activation function are called one-sided. However, it has been shown that the hyperbolic tangent performs better than the sigmoid for training MLPs. has become the most popular for deep neural networks

"""The sigmoid function (or the logistic curve) is a 
function that takes any real number, z, and outputs a number (0,1).
It is useful in neural networks for assigning weights on a relative scale.
The value z is the weighted sum of parameters involved in the learning algorithm."""

import numpy
import matplotlib.pyplot as plt
import math as mt

z = numpy.arange(-5, 5, .1)
sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
sigma = sigma_fn(z)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, sigma)
ax.set_ylim([-0.1, 1.1])
ax.set_xlim([-5,5])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('sigmoid function')

plt.show()

"""Step Function"""
z = numpy.arange(-5, 5, .02)
step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
step = step_fn(z)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, step)
ax.set_ylim([-0.5, 1.5])
ax.set_xlim([-5,5])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('step function')

plt.show()

"""Sine Function"""
z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
t = numpy.sin(z)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, t)
ax.set_ylim([-1.0, 1.0])
ax.set_xlim([-2*mt.pi,2*mt.pi])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('sine function')

plt.show()

"""Plots a graph of the squashing function used by a rectified linear
unit"""
z = numpy.arange(-2, 2, .1)
zero = numpy.zeros(len(z))
y = numpy.max([zero, z], axis=0)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, y)
ax.set_ylim([-2.0, 2.0])
ax.set_xlim([-2.0, 2.0])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('Rectified linear unit')

plt.show()