Week 2 Lab

Andrew Beam, PhD

Department of Epidemiology

Harvard T.H. Chan School of Public Health

twitter: @AndrewLBeam

In class assignment:

PERCEPTRON BY HAND

PERCEPTRONS

Let's say we'd like to have a single neural learn a simple function

y

X_1
X_2
X1 X2 y
0 0 0
0 1 1
1 0 1
1 1 1
w_2
w_1
b

Observations

PERCEPTRONS

How do we make a prediction for each observations?

y

X_1
X_2
X1 X2 y
0 0 0
0 1 1
1 0 1
1 1 1
w_2
w_1
b

Assume we have the following values

w1 w2 b
1 -1 -0.5

Observations

Predictions

For the first observation:

Assume we have the following values

w1 w2 b
1 -1 -0.5
X_1 = 0, X_2 = 0, y =0

Predictions

For the first observation:

Assume we have the following values

w1 w2 b
1 -1 -0.5
X_1 = 0, X_2 = 0, y =0

First compute the weighted sum:

h = w_1*X_1 + w_2*X_2 + b
h = 1*0 + -1*0 + (-0.5)
h = -0.5

Predictions

For the first observation:

Assume we have the following values

w1 w2 b
1 -1 -0.5
X_1 = 0, X_2 = 0, y =0

First compute the weighted sum:

h = w_1*X_1 + w_2*X_2 + b
h = 1*0 + -1*0 + -0.5
h = -0.5

Transform to probability:

p = \frac{1}{1+\exp(-h)}
p = \frac{1}{1+\exp(-0.5)}
p = 0.38

Predictions

For the first observation:

Assume we have the following values

w1 w2 b
1 -1 -0.5
X_1 = 0, X_2 = 0, y =0

First compute the weighted sum:

h = w_1*X_1 + w_2*X_2 + b
h = 1*0 + -1*0 + -0.5
h = -0.5

Transform to probability:

p = \frac{1}{1+\exp(-h)}
p = \frac{1}{1+\exp(-0.5)}
p = 0.38

Round to get prediction:

\hat{y} = round(p)
\hat{y} = 0

Predictions

Putting it all together:

h = w_1*X_1 + w_2*X_2 + b
p = \frac{1}{1+\exp(-h)}
\hat{y} = round(p)

Assume we have the following values

w1 w2 b
1 -1 -0.5
X1 X2 y h p
0 0 0 -0.5 0.38 0
0 1 1 -1.5 0.18 0
1 0 1 0.5 .62 1
1 1 1 -0.5 0.38 0
\hat{y}

Fill out this table

Room for Improvement

Let's define how we want to measure the network's performance.

There are many ways, but let's use squared-error:

(y - p)^2

Room for Improvement

Let's define how we want to measure the network's performance.

There are many ways, but let's use squared-error:

Now we need to find values for                  that make this error as small as possible

(y - p)^2
w_1, w_2, b

The Backpropagation Algorithm

h = w_1*X_1 + w_2*X_2 + b
p = \frac{1}{1+\exp(-h)}

Our perceptron performs the following computations

\ell = (y - p)^2

And we want to minimize this quantity

The Backpropagation Algorithm

h = w_1*X_1 + w_2*X_2 + b
p = \frac{1}{1+\exp(-h)}

Our perception performs the following computations

\ell = (y - p)^2

And we want to minimize this quantity

We'll compute the gradients for each parameter by "back-propagating" errors through each component of the network

The Backpropagation Algorithm

For      we need to compute

h = w_1*X_1 + w_2*X_2 + b

Computations

p = \frac{1}{1+\exp(-h)}

Loss

w_1
\frac{\partial \ell}{\partial w_1}
\ell = (y - p)^2

To get there, we will use the chain rule

\frac{\partial \ell}{\partial w_1} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial h}*\frac{\partial h}{\partial w_1}

This is "backprop"

The Backpropagation Algorithm

Let's break it into pieces

h = w_1*X_1 + w_2*X_2 + b

Computations

p = \frac{1}{1+\exp(-h)}

Loss

\ell = (y - p)^2
\frac{\partial \ell}{\partial w_1} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial h}*\frac{\partial h}{\partial w_1}
\frac{\partial \ell}{\partial p} = ?

The Backpropagation Algorithm

Let's break it into pieces

h = w_1*X_1 + w_2*X_2 + b

Computations

p = \frac{1}{1+\exp(-h)}

Loss

\ell = (y - p)^2
\frac{\partial \ell}{\partial w_1} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial h}*\frac{\partial h}{\partial w_1}
\frac{\partial \ell}{\partial p} = 2*(p-y)

The Backpropagation Algorithm

Let's break it into pieces

h = w_1*X_1 + w_2*X_2 + b

Computations

p = \frac{1}{1+\exp(-h)}

Loss

\ell = (y - p)^2
\frac{\partial \ell}{\partial w_1} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial h}*\frac{\partial h}{\partial w_1}
\frac{\partial \ell}{\partial p} = 2*(p-y)
\frac{\partial p}{\partial h} = ?

The Backpropagation Algorithm

Let's break it into pieces

h = w_1*X_1 + w_2*X_2 + b

Computations

p = \frac{1}{1+\exp(-h)}

Loss

\ell = (y - p)^2
\frac{\partial \ell}{\partial w_1} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial h}*\frac{\partial h}{\partial w_1}
\frac{\partial \ell}{\partial p} = 2*(p-y)
\frac{\partial p}{\partial h} = p*(1-p)

The Backpropagation Algorithm

Let's break it into pieces

h = w_1*X_1 + w_2*X_2 + b

Computations

p = \frac{1}{1+\exp(-h)}

Loss

\ell = (y - p)^2
\frac{\partial \ell}{\partial w_1} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial h}*\frac{\partial h}{\partial w_1}
\frac{\partial \ell}{\partial p} = 2*(p-y)
\frac{\partial p}{\partial h} = p*(1-p)
\frac{\partial h}{\partial w} = ?

The Backpropagation Algorithm

Let's break it into pieces

h = w_1*X_1 + w_2*X_2 + b

Computations

p = \frac{1}{1+\exp(-h)}

Loss

\ell = (y - p)^2
\frac{\partial \ell}{\partial w_1} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial h}*\frac{\partial h}{\partial w_1}
\frac{\partial \ell}{\partial p} = 2*(p-y)
\frac{\partial p}{\partial h} = p*(1-p)
\frac{\partial h}{\partial w_1} = X_1

The Backpropagation Algorithm

Let's break it into pieces

h = w_1*X_1 + w_2*X_2 + b

Computations

p = \frac{1}{1+\exp(-h)}

Loss

\ell = (y - p)^2
\frac{\partial \ell}{\partial w_1} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial h}*\frac{\partial h}{\partial w_1}
\frac{\partial \ell}{\partial p} = 2*(p-y)
\frac{\partial p}{\partial h} = p*(1-p)
\frac{\partial h}{\partial w_1} = X_1
\frac{\partial \ell}{\partial w_1} = 2*(p-y)*p*(1-p)*X_1

Putting it all together

Gradient Descent with Backprop

gw_1 = \eta*(p - y)*(p*(1-p)*X_1)

1) Compute the gradient for

w^{new}_1 = w^{old}_1 - \frac{1}{N}\sum gw_1

2) Update

w_1
w_1
\eta

is the learning rate

For some number of iterations we will:

3) Repeat until "convergence"

Learning Rules for each Parameter

gw_1 = (p - y)*(p*(1-p)*X_1)
gw_2 = (p - y)*(p*(1-p)*X_2)
g_b = (p - y)*(p*(1-p))

Gradient for

Gradient for

Gradient for

w^{new}_1 = w^{old}_1 - \eta*\frac{1}{N}\sum gw_1

Update for

Update for

Update for

w^{new}_2 = w^{old}_2 - \eta*\frac{1}{N}\sum gw_2
b^{new} = b^{old} - \eta*\frac{1}{N}\sum g_b
w_1
w_1
w_2
w_2
b
b
\eta

is the learning rate

GPU COMPUTING

Training neural nets = large matrix multiplications

GPUs = Massively parallel linear algebra devices

Special computer chips known as graphics processing units (GPUs) make training huge models on large data tractable

GPUs vs. CPUs

CPUs

1000s of number crunchers

GPUs

General

Purpose

Computation

REGULARIZATION

One of the biggest problems with neural networks is overfitting.

Regularization schemes combat overfitting in a variety of different ways

REGULARIZATION

Learning means solving the following optimization problem:

\text{argmin}_{W} \ \ell(y, f(X))

where f(X) = neural net output

REGULARIZATION

One way to regularize is introduce penalties and change

\text{argmin}_{W} \ \ell(y, f(X))

REGULARIZATION

\text{argmin}_{W} \ \ell(y, f(X))

to

\text{argmin}_{W} \ \ell(y, f(X)) + \lambda R(W)

One way to regularize is introduce penalties and change

REGULARIZATION

A familiar why to regularize is introduce penalties and change

\text{argmin}_{W} \ \ell(y, f(X))

to

\text{argmin}_{W} \ \ell(y, f(X)) + \lambda R(W)

where R(W) is often the L1 or L2 norm of W. These are the well known ridge and LASSO penalties, referred to as weight decay by neural net community

L2 REGULARIZATION

We can limit the size of the L2 norm of the weight vector:

\text{argmin}_{W} \ \ell(y, f(X)) + \lambda ||W||_2

where

||W||_2 = \sum^p_{j=1} w_j^2

L1/L2 REGULARIZATION

We can limit the size of the L2 norm of the weight vector:

\text{argmin}_{W} \ \ell(y, f(X)) + \lambda ||W||_2

where

||W||_2 = \sum^p_{j=1} w_j^2

We can do the same for the L1 norm. What do these penalties do?

SHRINKAGE

L1 and L2 penalties shrink the weights towards 0

L2 Penalty

L1 Penalty

Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. The elements of statistical learning. Vol. 1. New York: Springer series in statistics, 2001.

SHRINKAGE

L1 and L2 penalties shrink the weights towards 0

Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. The elements of statistical learning. Vol. 1. New York: Springer series in statistics, 2001.

SHRINKAGE

L1 and L2 penalties shrink the weights towards 0

Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. The elements of statistical learning. Vol. 1. New York: Springer series in statistics, 2001.

Why is this a "good" idea?

STOCHASTIC REGULARIZATION

Often, we will inject noise into the neural network during training. By far the most popular way to do this is dropout

STOCHASTIC REGULARIZATION

Often, we will inject noise into the neural network during training. By far the most popular way to do this is dropout

Given a hidden layer, we are going to set each element of the hidden layer to 0 with probability p each SGD update.

STOCHASTIC REGULARIZATION

One way to think of this is the network is trained by bagged versions of the network. Bagging reduces variance.

STOCHASTIC REGULARIZATION

One way to think of this is the network is trained by bagged versions of the network. Bagging reduces variance.​

Others have argued this is an approximate Bayesian model

STOCHASTIC REGULARIZATION

Many have argued that SGD itself provides regularization

INITIALIZATION REGULARIZATION

The weights in a neural network are given random values initially

INITIALIZATION REGULARIZATION

The weights in a neural network are given random values initially

There is an entire literature on the best way to do this initialization

INITIALIZATION REGULARIZATION

The weights in a neural network are given random values initially

There is an entire literature on the best way to do this initialization

- Normal

- Truncated Normal

- Uniform

- Orthogonal

- Scaled by number of connections

- etc

INITIALIZATION REGULARIZATION

Try to "bias" the model into initial configurations that are easier to train

INITIALIZATION REGULARIZATION

Try to "bias" the model into initial configurations that are easier to train

Very popular way is to do transfer learning

INITIALIZATION REGULARIZATION

Try to "bias" the model into initial configurations that are easier to train

Very popular way is to do transfer learning

Train model on auxiliary task where lots of data is available

INITIALIZATION REGULARIZATION

Try to "bias" the model into initial configurations that are easier to train

Very popular way is to do transfer learning

Train model on auxiliary task where lots of data is available

Use final weight values from previous task as initial values and "fine tune" on primary task

STRUCTURAL REGULARIZATION

However, the key advantage of neural nets is the ability to easily include properties of the data directly into the model through the network's structure

Convolutional neural networks (CNNs) are a prime example of this (Kun will discuss CNNs)

Perceptron -> MLP

With a small change, we can turn our perceptron model into a multilayer perceptron

• Instead of just one linear combination, we are going to take several, each with a different set of weights (called a hidden unit)
• Each linear combination will be followed by a nonlinear activation
• Each of these nonlinear features will be fed into the logistic regression classifier
• All of the weights are learned end-to-end via SGD

MLPs learn a set of nonlinear features directly from data

"Feature learning" is the hallmark of deep learning approachs

MULTILAYER PERCEPTRONS (MLPs)

Let's set up the following MLP with 1 hidden layer that has 3 hidden units:

X_1
X_2
Pr(y = 1 | X_1, X_2)

Each neuron in the hidden layer is going to do exactly the same thing as before.

h_1
h_2
h_3
o

MULTILAYER PERCEPTRONS (MLPs)

X_1
X_2
h_1
h_2
h_3
o

Computations are:

o = b_o + \sum^3_{j=1} w_{oj}*h_j
p
p = \frac{1}{1 + exp(-o)}
h_j = \phi(w_{1j}*X_1 + w_{2j}*X_2 + b_j)

MULTILAYER PERCEPTRONS (MLPs)

X_1
X_2
h_1
h_2
h_3
o

Computations are:

o = b_o + \sum^3_{j=1} w_{oj}*h_j
p
p = \frac{1}{1 + exp(-o)}
h_j = \phi(w_{1j}*X_1 + w_{2j}*X_2 + b_j)

Output layer weight derivatives

\frac{\partial \ell}{\partial w_{oj}} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial o}*\frac{\partial o}{\partial w_{oj}}

MULTILAYER PERCEPTRONS (MLPs)

X_1
X_2
h_1
h_2
h_3
o

Computations are:

o = b_o + \sum^3_{j=1} w_{oj}*h_j
p
p = \frac{1}{1 + exp(-o)}
h_j = \phi(w_{1j}*X_1 + w_{2j}*X_2 + b_j)

Output layer weight derivatives

\frac{\partial \ell}{\partial w_{oj}} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial o}*\frac{\partial o}{\partial w_{oj}}
= (p-y)*p*(1-p)*h_j

MULTILAYER PERCEPTRONS (MLPs)

X_1
X_2
h_1
h_2
h_3
o

Computations are:

o = b_o + \sum^3_{j=1} w_{oj}*h_j
p
p = \frac{1}{1 + exp(-o)}
h_j = \phi(w_{1j}*X_1 + w_{2j}*X_2 + b_j)
\frac{\partial \ell}{\partial w_{1j}} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial o}*\frac{\partial o}{\partial h}*\frac{\partial h}{\partial w_{1j}}

Hidden layer weight derivatives

\frac{\partial \ell}{\partial w_{oj}} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial o}*\frac{\partial o}{\partial w_{oj}}

Output layer weight derivatives

= (p-y)*p*(1-p)*h_j

MULTILAYER PERCEPTRONS (MLPs)

X_1
X_2
h_1
h_2
h_3
o

Computations are:

o = b_o + \sum^3_{j=1} w_{oj}*h_j
p
p = \frac{1}{1 + exp(-o)}
h_j = \phi(w_{1j}*X_1 + w_{2j}*X_2 + b_j)
\frac{\partial \ell}{\partial w_{1j}} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial o}*\frac{\partial o}{\partial h}*\frac{\partial h}{\partial w_{1j}}

Hidden layer weight derivatives

\frac{\partial \ell}{\partial w_{oj}} = \frac{\partial \ell}{\partial p}*\frac{\partial p}{\partial o}*\frac{\partial o}{\partial w_{oj}}

Output layer weight derivatives

= (p-y)*p*(1-p)*h_j
= (p-y)*p*(1-p)*h_j*(1-h_j)*X_1

(if we use a sigmoid activation function)

MLP Terminology

X_1
X_2
Pr(y = 1 | X_1, X_2)
h_1
h_2
h_3
o
X_1
X_2
Pr(y = 1 | X_1, X_2)
h_1
h_2
h_3
o

Forward pass = computing probability from input

MLP Terminology

X_1
X_2
Pr(y = 1 | X_1, X_2)
h_1
h_2
h_3
o

Forward pass = computing probability from input

MLP Terminology

Backward pass = computing derivatives from output

X_1
X_2
Pr(y = 1 | X_1, X_2)
h_1
h_2
h_3
o

Forward pass = computing probability from input

MLP Terminology

Backward pass = computing derivatives from output

Hidden layers are often called "dense" layers

By beamandrew

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