federica bianco PRO
astro | data science | data for good
Neural Networks 101
Input
x
y
output
Input
x
y
output
function
f(x)
Input
x
y
output
f(x)
f(x) = mx + b
b
m
m: slope
b: intercept
Input
x
y
output
f(x)
f(x) = mx + b
b
m
m: slope
b: intercept
parameters
Input
x
y
output
f(x)
f(x) = mx + b
b
m
m: slope
b: intercept
parameters
x
y
goal: find the right m and b that turn x into y
goal: find the right m and b that turn x into y
Input
x
y
output
f(x)
f(x) = mx + b
b
m
m: slope
b: intercept
parameters
x
y
learn
goal: find the right m and b that turn x into y
goal: find the right m and b that turn x into y
what is machine learning?
1
what is machine learning?
ML: any model with parameters learnt from the data
Input
x
y
output
f(x) = mx + b
m = 0.4 and b=0
m: slope
b: intercept
parameters
x
L2 = (y_{1,p} - y_{1,t})^2 + (y_{2,p} - y_{2,t})^2 + (y_{3,p} - y_{3,t})^2
y_{3,p}
y_{3,t}
let's try
goal: learn the right m and b that turn x into y
f(x)
m
L2
-1.4
-.5
.6
1.5
2.4
L2 = (y_{1,p} - y_{1,t})^2 + (y_{2,p} - y_{2,t})^2 + (y_{3,p} - y_{3,t})^2
1 ~\mathrm{if} ~\sum_{i=1}^Nw_ix_i \geq\theta ~\mathrm{else}~ 0
w1
w1
w2
w2
The perceptron algorithm : 1958, Frank Rosenblatt
1958
Perceptron
The perceptron algorithm : 1958, Frank Rosenblatt
.
.
.
x_1
x_2
x_N
+b
output
weights
w_i
bias
b
linear regression:
w_2
w_1
w_N
1 ~\mathrm{if} ~\sum_{i=1}^Nw_ix_i \geq\theta ~\mathrm{else}~ 0
1958
Perceptron
y= \begin{cases}
1~ if~ \sum_i(x_i w_i) + b ~>=~Z\\
0 ~if~ \sum_i(x_i w_i) + b ~<~Z
\end{cases}
.
.
.
x_1
x_2
x_N
+b
f
w_2
w_1
w_N
output
f
activation function
weights
w_i
bias
b
y ~= f(~\sum_i w_ix_i ~+~ b)
The perceptron algorithm : 1958, Frank Rosenblatt
Perceptrons are linear classifiers: makes its predictions based on a linear predictor function
combining a set of weights (=parameters) with the feature vector.
Perceptron
The perceptron algorithm : 1958, Frank Rosenblatt
+b
f
w_2
w_1
w_N
output
f
activation function
weights
w_i
bias
b
sigmoid
f
\sigma = \frac{1}{1 + e^{-z}}
.
.
.
x_1
x_2
x_N
y ~= f(~\sum_i w_ix_i ~+~ b)
Perceptrons are linear classifiers: makes its predictions based on a linear predictor function
combining a set of weights (=parameters) with the feature vector.
Perceptron
ANN examples of activation function
The perceptron algorithm : 1958, Frank Rosenblatt
Perceptron
The Navy revealed the embryo of an electronic computer today that it expects will be able to walk, talk, see, write, reproduce itself and be conscious of its existence.
The embryo - the Weather Buerau's $2,000,000 "704" computer - learned to differentiate between left and right after 50 attempts in the Navy demonstration
NEW NAVY DEVICE LEARNS BY DOING; Psychologist Shows Embryo of Computer Designed to Read and Grow Wiser
July 8, 1958
Input
x
y
output
f(x)
x
y
A Neural Network is a kind of function that maps input to output
multilayer perceptron
x_2
x_3
output
x_1
layer of perceptrons
b_1
b_2
b_3
b_4
b
multilayer perceptron
x_2
x_3
output
input layer
hidden layer
output layer
1970: multilayer perceptron architecture
x_1
Fully connected: all nodes go to all nodes of the next layer.
b_1
b_2
b_3
b_4
Perceptrons by Marvin Minsky and Seymour Papert 1969
multilayer perceptron
x_2
x_3
output
x_1
layer of perceptrons
b_1
b_2
b_3
b_4
b
w_{11}
w_{12}
w_{13}
w_{14}
multilayer perceptron
x_2
x_3
output
x_1
layer of perceptrons
b_1
b_2
b_3
b_4
b
w_{21}
w_{22}
w_{23}
w_{24}
multilayer perceptron
layer of perceptrons
x_2
x_3
output
x_1
layer of perceptrons
b_1
b_2
b_3
b_4
b
w_{31}
w_{32}
w_{33}
w_{34}
multilayer perceptron
x_2
x_3
output
Fully connected: all nodes go to all nodes of the next layer.
layer of perceptrons
x_1
w_{11}x_1 + w_{12}x_2 + w_{13}x_3 + w_{14}x_4 + b1
multilayer perceptron
x_2
x_3
output
Fully connected: all nodes go to all nodes of the next layer.
layer of perceptrons
w_{11}x_1 + w_{12}x_2 + w_{13}x_3 + b1
w_{21}x_1 + w_{22}x_2 + w_{23}x_3 + b2
w_{31}x_1 + w_{32}x_2 + w_{33}x_3 + b3
w_{41}x_1 + w_{42}x_2 + w_{43}x_3 + b4
x_1
w: weight
sets the sensitivity of a neuron
b: bias:
up-down weights a neuron
learned parameters
multilayer perceptron
x_2
x_3
output
Fully connected: all nodes go to all nodes of the next layer.
layer of perceptrons
f(w_{11}x_1 + w_{12}x_2 + w_{13}x_3 + b1)
f(w_{21}x_1 + w_{22}x_2 + w_{23}x_3 + b1)
f(w_{31}x_1 + w_{32}x_2 + w_{33}x_3 + b1)
f(w_{41}x_1 + w_{42}x_2 + w_{43}x_3 + b1)
x_1
w: weight
sets the sensitivity of a neuron
b: bias:
up-down weights a neuron
f: activation function:
turns neurons on-off
DNN:
hyperparameters of DNN
3
EXERCISE
output
how many parameters?
input layer
hidden layer
output layer
hidden layer
EXERCISE
output
how many parameters?
input layer
hidden layer
output layer
hidden layer
output
input layer
hidden layer
output layer
hidden layer
35
(3x4)+4
(4x3)+3
how many parameters?
EXERCISE
(3)+1
output
input layer
hidden layer
output layer
hidden layer
- number of layers- 1
- number of neurons/layer-
- activation function/layer-
- layer connectivity-
- optimization metric - 1
- optimization method - 1
- parameters in optimization- M
N_l
N_l ^ {~??}
how many hyperparameters?
EXERCISE
GREEN: architecture hyperparameters
RED: training hyperparameters
N_l
EXERCISE
Lots of parameters and lots of hyperparameters! What to choose?
cheatsheet
-
architecture - wide networks tend to overfit, deep networks are hard to train
- number of epochs - the sweet spot is when learning slows down, but before you start overfitting... it may take DAYS! jumps may indicate bad initial choices (like in all gradient descent)
- loss function - needs to be appropriate to the task, e.g. classification vs regression
-
activation functions - needs to be consistent with the loss function
- optimization scheme - needs to be appropriate to the task and data
- learning rate in optimization - balance speed and accuracy
- batch size - smaller batch size is faster but leads to overtraining
An article that compars various DNNs
An article that compars various DNNs
accuracy comparison
An article that compars various DNNs
batch size
Lots of parameters and lots of hyperparameters! What to choose?
cheatsheet
- architecture - wide networks tend to overfit, deep networks are hard to train
-
number of epochs - the sweet spot is when learning slows down, but before you start overfitting... it may take DAYS! jumps may indicate bad initial choices
-
loss function - needs to be appropriate to the task, e.g. classification vs regression
-
activation functions - needs to be consistent with the loss function
- optimization scheme - needs to be appropriate to the task and data
- learning rate in optimization - balance speed and accuracy
- batch size - smaller batch size is faster but leads to overtraining
What should I choose for the loss function and how does that relate to the activation functiom and optimization?
Lots of parameters and lots of hyperparameters! What to choose?
Lots of parameters and lots of hyperparameters! What to choose?
cheatsheet
always check your loss function! it should go down smoothly and flatten out at the end of the training.
not flat? you are still learning!
too flat? you are overfitting...
loss (gallery of horrors)
jumps are not unlikely (and not necessarily a problem) if your activations are discontinuous (e.g. relu)
when you use validation you are introducing regularizations (e.g. dropout) so the loss can be smaller than for the training set
loss and learning rate (not that the appropriate learning rate depends on the chosen optimization scheme!)
Building a DNN
with keras and tensorflow
autoencoder for image recontstruction
What should I choose for the loss function and how does that relate to the activation functiom and optimization?
| loss | good for | activation last layer | size last layer |
|---|---|---|---|
| mean_squared_error | regression | linear | one node |
| mean_absolute_error | regression | linear | one node |
| mean_squared_logarithmit_error | regression | linear | one node |
| binary_crossentropy | binary classification | sigmoid | one node |
| categorical_crossentropy | multiclass classification | sigmoid | N nodes |
| Kullback_Divergence | multiclass classification, probabilistic inerpretation | sigmoid | N nodes |
On the interpretability of DNNs
resources
Neural Network and Deep Learning
an excellent and free book on NN and DL
http://neuralnetworksanddeeplearning.com/index.html
History of NN
https://cs.stanford.edu/people/eroberts/courses/soco/projects/neural-networks/History/history2.html
NN are a vast topics and we only have 2 weeks!
Some FREE references!
michael nielsen
better pedagogical approach, more basic, more clear
ian goodfellow
mathematical approach, more advanced, unfinished
michael nielsen
better pedagogical approach, more basic, more clear
An article that compars various DNNs
accuracy comparison
ANN101
By federica bianco
