_AI_Stanford_Super_#DeepLearning_Cheat_Sheet!_😊🙃😀🙃😊.pdf

CS 230 – Deep Learning Shervine Amidi & Afshine Amidi
Super VIP Cheatsheet: Deep Learning
Afshine Amidi and Shervine Amidi
November 25, 2018
Contents
1 Convolutional Neural Networks 2
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Types of layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Filter hyperparameters . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Tuning hyperparameters . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Commonly used activation functions . . . . . . . . . . . . . . . . . . . 3
1.6 Object detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6.1 Face verification and recognition . . . . . . . . . . . . . . . . . 5
1.6.2 Neural style transfer . . . . . . . . . . . . . . . . . . . . . . . 5
1.6.3 Architectures using computational tricks . . . . . . . . . . . . 6
2 Recurrent Neural Networks 7
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Handling long term dependencies . . . . . . . . . . . . . . . . . . . . 8
2.3 Learning word representation . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Motivation and notations . . . . . . . . . . . . . . . . . . . 9
2.3.2 Word embeddings . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Comparing words . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Language model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Machine translation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.7 Attention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Deep Learning Tips and Tricks 11
3.1 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Training a neural network . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.2 Finding optimal weights . . . . . . . . . . . . . . . . . . . . . 12
3.3 Parameter tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.1 Weights initialization . . . . . . . . . . . . . . . . . . . . . . 12
3.3.2 Optimizing convergence . . . . . . . . . . . . . . . . . . . . . 12
3.4 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.5 Good practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1 Convolutional Neural Networks
1.1 Overview
r Architecture of a traditional CNN – Convolutional neural networks, also known as CNNs,
are a specific type of neural networks that are generally composed of the following layers:
The convolution layer and the pooling layer can be fine-tuned with respect to hyperparameters
that are described in the next sections.
1.2 Types of layer
r Convolutional layer (CONV) – The convolution layer (CONV) uses filters that perform
convolution operations as it is scanning the input I with respect to its dimensions. Its hyperpa-
rameters include the filter size F and stride S. The resulting output O is called feature map or
activation map.
Remark: the convolution step can be generalized to the 1D and 3D cases as well.
r Pooling (POOL) – The pooling layer (POOL) is a downsampling operation, typically applied
after a convolution layer, which does some spatial invariance. In particular, max and average
pooling are special kinds of pooling where the maximum and average value is taken, respectively.
Stanford University 1 Winter 2019

Max pooling Average pooling
Purpose
Each pooling operation selects the
maximum value of the current view
Each pooling operation averages
the values of the current view
Illustration
Comments
- Preserves detected features
- Most commonly used
- Downsamples feature map
- Used in LeNet
r Fully Connected (FC) – The fully connected layer (FC) operates on a flattened input where
each input is connected to all neurons. If present, FC layers are usually found towards the end
of CNN architectures and can be used to optimize objectives such as class scores.
1.3 Filter hyperparameters
The convolution layer contains filters for which it is important to know the meaning behind its
hyperparameters.
r Dimensions of a filter – A filter of size F × F applied to an input containing C channels is
a F × F × C volume that performs convolutions on an input of size I × I × C and produces an
output feature map (also called activation map) of size O × O × 1.
Remark: the application of K filters of size F × F results in an output feature map of size
O × O × K.
r Stride – For a convolutional or a pooling operation, the stride S denotes the number of pixels
by which the window moves after each operation.
r Zero-padding – Zero-padding denotes the process of adding P zeroes to each side of the
boundaries of the input. This value can either be manually specified or automatically set through
one of the three modes detailed below:
Valid Same Full
Value P = 0
Pstart =
j
Sd I
S
e−I+F −S
2
k
Pend =
l
Sd I
S
e−I+F −S
2
m
Pstart ∈ [[0,F − 1]]
Pend = F − 1
Illustration
Purpose
- No padding
- Drops last
convolution if
dimensions do not
match
- Padding such that feature
map size has size
l
I
S
m
- Output size is
mathematically convenient
- Also called ’half’ padding
- Maximum padding
such that end
convolutions are
applied on the limits
of the input
- Filter ’sees’ the input
end-to-end
1.4 Tuning hyperparameters
r Parameter compatibility in convolution layer – By noting I the length of the input
volume size, F the length of the filter, P the amount of zero padding, S the stride, then the
output size O of the feature map along that dimension is given by:
O =
I − F + Pstart + Pend
S
+ 1
Remark: often times, Pstart = Pend , P, in which case we can replace Pstart + Pend by 2P in
the formula above.

r Understanding the complexity of the model – In order to assess the complexity of a
model, it is often useful to determine the number of parameters that its architecture will have.
In a given layer of a convolutional neural network, it is done as follows:
CONV POOL FC
Illustration
Input size I × I × C I × I × C Nin
Output size O × O × K O × O × C Nout
Number of
parameters
(F × F × C + 1) · K 0 (Nin + 1) × Nout
Remarks
- One bias parameter
per filter
- In most cases, S < F
- A common choice
for K is 2C
- Pooling operation
done channel-wise
- In most cases, S = F
- Input is flattened
- One bias parameter
per neuron
- The number of FC
neurons is free of
structural constraints
r Receptive field – The receptive field at layer k is the area denoted Rk × Rk of the input
that each pixel of the k-th activation map can ’see’. By calling Fj the filter size of layer j and
Si the stride value of layer i and with the convention S0 = 1, the receptive field at layer k can
be computed with the formula:
Rk = 1 +
k
X
j=1
(Fj − 1)
j−1
Y
i=0
Si
In the example below, we have F1 = F2 = 3 and S1 = S2 = 1, which gives R2 = 1+2 · 1+2 · 1 =
5.
1.5 Commonly used activation functions
r Rectified Linear Unit – The rectified linear unit layer (ReLU) is an activation function g
that is used on all elements of the volume. It aims at introducing non-linearities to the network.
Its variants are summarized in the table below:
ReLU Leaky ReLU ELU
g(z) = max(0,z)
g(z) = max(z,z)
with 1
g(z) = max(α(ez − 1),z)
with α 1
Non-linearity complexities
biologically interpretable
Addresses dying ReLU
issue for negative values
Differentiable everywhere
r Softmax – The softmax step can be seen as a generalized logistic function that takes as input
a vector of scores x ∈ Rn and outputs a vector of output probability p ∈ Rn through a softmax
function at the end of the architecture. It is defined as follows:
p =
p1
.
.
.
pn

where pi =
exi
n
X
j=1
exj
1.6 Object detection
r Types of models – There are 3 main types of object recognition algorithms, for which the
nature of what is predicted is different. They are described in the table below:
Image classification
Classification
w. localization
Detection
- Classifies a picture
- Predicts probability
of object
- Detects object in a picture
- Predicts probability of
object and where it is
located
- Detects up to several objects
in a picture
- Predicts probabilities of objects
and where they are located
Traditional CNN Simplified YOLO, R-CNN YOLO, R-CNN
r Detection – In the context of object detection, different methods are used depending on
whether we just want to locate the object or detect a more complex shape in the image. The
two main ones are summed up in the table below:

CS 230 – Deep Learning Shervine Amidi Afshine Amidi
Bounding box detection Landmark detection
Detects the part of the image where
the object is located
- Detects a shape or characteristics of
an object (e.g. eyes)
- More granular
Box of center (bx,by), height bh
and width bw
Reference points (l1x,l1y), ...,(lnx,lny)
r Intersection over Union – Intersection over Union, also known as IoU, is a function that
quantifies how correctly positioned a predicted bounding box Bp is over the actual bounding
box Ba. It is defined as:
IoU(Bp,Ba) =
Bp ∩ Ba
Bp ∪ Ba
Remark: we always have IoU ∈ [0,1]. By convention, a predicted bounding box Bp is considered
as being reasonably good if IoU(Bp,Ba) 0.5.
r Anchor boxes – Anchor boxing is a technique used to predict overlapping bounding boxes.
In practice, the network is allowed to predict more than one box simultaneously, where each box
prediction is constrained to have a given set of geometrical properties. For instance, the first
prediction can potentially be a rectangular box of a given form, while the second will be another
rectangular box of a different geometrical form.
r Non-max suppression – The non-max suppression technique aims at removing duplicate
overlapping bounding boxes of a same object by selecting the most representative ones. After
having removed all boxes having a probability prediction lower than 0.6, the following steps are
repeated while there are boxes remaining:
• Step 1: Pick the box with the largest prediction probability.
• Step 2: Discard any box having an IoU 0.5 with the previous box.
r YOLO – You Only Look Once (YOLO) is an object detection algorithm that performs the
following steps:
• Step 1: Divide the input image into a G × G grid.
• Step 2: For each grid cell, run a CNN that predicts y of the following form:
y =

pc,bx,by,bh,bw,c1,c2,...,cp
| {z }
repeated k times
,...
T
∈ RG×G×k×(5+p)
where pc is the probability of detecting an object, bx,by,bh,bw are the properties of the
detected bouding box, c1,...,cp is a one-hot representation of which of the p classes were
detected, and k is the number of anchor boxes.
• Step 3: Run the non-max suppression algorithm to remove any potential duplicate over-
lapping bounding boxes.
Remark: when pc = 0, then the network does not detect any object. In that case, the corre-
sponding predictions bx, ..., cp have to be ignored.
r R-CNN – Region with Convolutional Neural Networks (R-CNN) is an object detection algo-
rithm that first segments the image to find potential relevant bounding boxes and then run the
detection algorithm to find most probable objects in those bounding boxes.
Remark: although the original algorithm is computationally expensive and slow, newer archi-
tectures enabled the algorithm to run faster, such as Fast R-CNN and Faster R-CNN.

1.6.1 Face verification and recognition
r Types of models – Two main types of model are summed up in table below:
Face verification Face recognition
- Is this the correct person?
- One-to-one lookup
- Is this one of the K persons in the database?
- One-to-many lookup
r One Shot Learning – One Shot Learning is a face verification algorithm that uses a limited
training set to learn a similarity function that quantifies how different two given images are. The
similarity function applied to two images is often noted d(image 1, image 2).
r Siamese Network – Siamese Networks aim at learning how to encode images to then quantify
how different two images are. For a given input image x(i), the encoded output is often noted
as f(x(i)).
r Triplet loss – The triplet loss ` is a loss function computed on the embedding representation
of a triplet of images A (anchor), P (positive) and N (negative). The anchor and the positive
example belong to a same class, while the negative example to another one. By calling α ∈ R+
the margin parameter, this loss is defined as follows:
`(A,P,N) = max (d(A,P) − d(A,N) + α,0)
1.6.2 Neural style transfer
r Motivation – The goal of neural style transfer is to generate an image G based on a given
content C and a given style S.
r Activation – In a given layer l, the activation is noted a[l] and is of dimensions nH ×nw ×nc
r Content cost function – The content cost function Jcontent(C,G) is used to determine how
the generated image G differs from the original content image C. It is defined as follows:
Jcontent(C,G) =
1
2
||a[l](C)
− a[l](G)
||2
r Style matrix – The style matrix G[l] of a given layer l is a Gram matrix where each of its
elements G
[l]
kk0 quantifies how correlated the channels k and k0 are. It is defined with respect to
activations a[l] as follows:
G
[l]
kk0 =
n
[l]
H
X
i=1
n
[l]
w
X
j=1
a
[l]
ijk
a
[l]
ijk0
Remark: the style matrix for the style image and the generated image are noted G[l](S) and
G[l](G) respectively.
r Style cost function – The style cost function Jstyle(S,G) is used to determine how the
generated image G differs from the style S. It is defined as follows:
J
[l]
style(S,G) =
1
(2nH nwnc)2
||G[l](S)
− G[l](G)
||2
F =
1
(2nH nwnc)2
nc
X
k,k0=1

G
[l](S)
kk0 − G
[l](G)
kk0
2
r Overall cost function – The overall cost function is defined as being a combination of the
content and style cost functions, weighted by parameters α,β, as follows:
J(G) = αJcontent(C,G) + βJstyle(S,G)
Remark: a higher value of α will make the model care more about the content while a higher
value of β will make it care more about the style.
1.6.3 Architectures using computational tricks
r Generative Adversarial Network – Generative adversarial networks, also known as GANs,
are composed of a generative and a discriminative model, where the generative model aims at
generating the most truthful output that will be fed into the discriminative which aims at
differentiating the generated and true image.

Remark: use cases using variants of GANs include text to image, music generation and syn-
thesis.
r ResNet – The Residual Network architecture (also called ResNet) uses residual blocks with a
high number of layers meant to decrease the training error. The residual block has the following
characterizing equation:
a[l+2]
= g(a[l]
+ z[l+2]
)
r Inception Network – This architecture uses inception modules and aims at giving a try
at different convolutions in order to increase its performance. In particular, it uses the 1 × 1
convolution trick to lower the burden of computation.
? ? ?
2 Recurrent Neural Networks
2.1 Overview
r Architecture of a traditional RNN – Recurrent neural networks, also known as RNNs,
are a class of neural networks that allow previous outputs to be used as inputs while having
hidden states. They are typically as follows:
For each timestep t, the activation at and the output yt are expressed as follows:
at
= g1(Waaat−1
+ Waxxt
+ ba) and yt
= g2(Wyaat
+ by)
where Wax, Waa, Wya, ba, by are coefficients that are shared temporally and g1, g2 activation
functions
The pros and cons of a typical RNN architecture are summed up in the table below:
Advantages Drawbacks
- Possibility of processing input of any length
- Model size not increasing with size of input
- Computation takes into account
historical information
- Weights are shared across time
- Computation being slow
- Difficulty of accessing information
from a long time ago
- Cannot consider any future input
for the current state
r Applications of RNNs – RNN models are mostly used in the fields of natural language
processing and speech recognition. The different applications are summed up in the table below:

Type of RNN Illustration Example
One-to-one
Tx = Ty = 1
Traditional neural network
One-to-many
Tx = 1, Ty 1
Music generation
Many-to-one
Tx 1, Ty = 1
Sentiment classification
Many-to-many
Tx = Ty
Name entity recognition
Many-to-many
Tx 6= Ty
Machine translation
r Loss function – In the case of a recurrent neural network, the loss function L of all time
steps is defined based on the loss at every time step as follows:
L(b
y,y) =
Ty
X
t=1
L(b
yt
,yt
)
r Backpropagation through time – Backpropagation is done at each point in time. At
timestep T, the derivative of the loss L with respect to weight matrix W is expressed as follows:
∂L(T )
∂W
=
T
X
t=1
∂L(T )
∂W
(t)
2.2 Handling long term dependencies
r Commonly used activation functions – The most common activation functions used in
RNN modules are described below:
Sigmoid Tanh RELU
g(z) =
1
1 + e−z
g(z) =
ez − e−z
ez + e−z
g(z) = max(0,z)
r Vanishing/exploding gradient – The vanishing and exploding gradient phenomena are
often encountered in the context of RNNs. The reason why they happen is that it is difficult
to capture long term dependencies because of multiplicative gradient that can be exponentially
decreasing/increasing with respect to the number of layers.
r Gradient clipping – It is a technique used to cope with the exploding gradient problem
sometimes encountered when performing backpropagation. By capping the maximum value for
the gradient, this phenomenon is controlled in practice.
r Types of gates – In order to remedy the vanishing gradient problem, specific gates are used
in some types of RNNs and usually have a well-defined purpose. They are usually noted Γ and
are equal to:
Γ = σ(Wxt
+ Uat−1
+ b)
where W, U, b are coefficients specific to the gate and σ is the sigmoid function. The main ones
are summed up in the table below:

Type of gate Role Used in
Update gate Γu How much past should matter now? GRU, LSTM
Relevance gate Γr Drop previous information? GRU, LSTM
Forget gate Γf Erase a cell or not? LSTM
Output gate Γo How much to reveal of a cell? LSTM
r GRU/LSTM – Gated Recurrent Unit (GRU) and Long Short-Term Memory units (LSTM)
deal with the vanishing gradient problem encountered by traditional RNNs, with LSTM being
a generalization of GRU. Below is a table summing up the characterizing equations of each
architecture:
Gated Recurrent Unit
(GRU)
Long Short-Term Memory
(LSTM)
c̃t tanh(Wc[Γr ? at−1,xt] + bc) tanh(Wc[Γr ? at−1,xt] + bc)
ct Γu ? c̃t + (1 − Γu) ? ct−1 Γu ? c̃t + Γf ? ct−1
at ct Γo ? ct
Dependencies
Remark: the sign ? denotes the element-wise multiplication between two vectors.
r Variants of RNNs – The table below sums up the other commonly used RNN architectures:
Bidirectional
(BRNN)
Deep
(DRNN)
2.3 Learning word representation
In this section, we note V the vocabulary and |V | its size.
2.3.1 Motivation and notations
r Representation techniques – The two main ways of representing words are summed up in
the table below:
1-hot representation Word embedding
- Noted ow
- Naive approach, no similarity information
- Noted ew
- Takes into account words similarity
r Embedding matrix – For a given word w, the embedding matrix E is a matrix that maps
its 1-hot representation ow to its embedding ew as follows:
ew = Eow
Remark: learning the embedding matrix can be done using target/context likelihood models.
2.3.2 Word embeddings
r Word2vec – Word2vec is a framework aimed at learning word embeddings by estimating the
likelihood that a given word is surrounded by other words. Popular models include skip-gram,
negative sampling and CBOW.
r Skip-gram – The skip-gram word2vec model is a supervised learning task that learns word
embeddings by assessing the likelihood of any given target word t happening with a context
word c. By noting θt a parameter associated with t, the probability P(t|c) is given by:

P(t|c) =
exp(θT
t ec)
|V |
X
j=1
exp(θT
j ec)
Remark: summing over the whole vocabulary in the denominator of the softmax part makes
this model computationally expensive. CBOW is another word2vec model using the surrounding
words to predict a given word.
r Negative sampling – It is a set of binary classifiers using logistic regressions that aim at
assessing how a given context and a given target words are likely to appear simultaneously, with
the models being trained on sets of k negative examples and 1 positive example. Given a context
word c and a target word t, the prediction is expressed by:
P(y = 1|c,t) = σ(θT
t ec)
Remark: this method is less computationally expensive than the skip-gram model.
r GloVe – The GloVe model, short for global vectors for word representation, is a word em-
bedding technique that uses a co-occurence matrix X where each Xi,j denotes the number of
times that a target i occurred with a context j. Its cost function J is as follows:
J(θ) =
1
2
|V |
X
i,j=1
f(Xij)(θT
i ej + bi + b0
j − log(Xij))2
here f is a weighting function such that Xi,j = 0 =⇒ f(Xi,j) = 0.
Given the symmetry that e and θ play in this model, the final word embedding e
(final)
w is given
by:
e
(final)
w =
ew + θw
2
Remark: the individual components of the learned word embeddings are not necessarily inter-
pretable.
2.4 Comparing words
r Cosine similarity – The cosine similarity between words w1 and w2 is expressed as follows:
similarity =
w1 · w2
||w1|| ||w2||
= cos(θ)
Remark: θ is the angle between words w1 and w2.
r t-SNE – t-SNE (t-distributed Stochastic Neighbor Embedding) is a technique aimed at re-
ducing high-dimensional embeddings into a lower dimensional space. In practice, it is commonly
used to visualize word vectors in the 2D space.
2.5 Language model
r Overview – A language model aims at estimating the probability of a sentence P(y).
r n-gram model – This model is a naive approach aiming at quantifying the probability that
an expression appears in a corpus by counting its number of appearance in the training data.
r Perplexity – Language models are commonly assessed using the perplexity metric, also
known as PP, which can be interpreted as the inverse probability of the dataset normalized by
the number of words T. The perplexity is such that the lower, the better and is defined as
follows:
PP =
T
Y
t=1
1
P|V |
j=1
y
(t)
j · b
y
(t)
j
! 1
T
Remark: PP is commonly used in t-SNE.
2.6 Machine translation
r Overview – A machine translation model is similar to a language model except it has an
encoder network placed before. For this reason, it is sometimes referred as a conditional language
model. The goal is to find a sentence y such that:
y = arg max
y1,...,yTy
P(y1
,...,yTy
|x)
r Beam search – It is a heuristic search algorithm used in machine translation and speech
recognition to find the likeliest sentence y given an input x.
• Step 1: Find top B likely words y1
• Step 2: Compute conditional probabilities yk|x,y1,...,yk−1
• Step 3: Keep top B combinations x,y1,...,yk

Remark: if the beam width is set to 1, then this is equivalent to a naive greedy search.
r Beam width – The beam width B is a parameter for beam search. Large values of B yield
to better result but with slower performance and increased memory. Small values of B lead to
worse results but is less computationally intensive. A standard value for B is around 10.
r Length normalization – In order to improve numerical stability, beam search is usually ap-
plied on the following normalized objective, often called the normalized log-likelihood objective,
defined as:
Objective =
1
Tα
y
Ty
X
t=1
log
h
p(yt
|x,y1
, ..., yt−1
)
i
Remark: the parameter α can be seen as a softener, and its value is usually between 0.5 and 1.
r Error analysis – When obtaining a predicted translation b
y that is bad, one can wonder why
we did not get a good translation y∗ by performing the following error analysis:
Case P(y∗|x) P(b
y|x) P(y∗|x) 6 P(b
y|x)
Root cause Beam search faulty RNN faulty
Remedies Increase beam width
- Try different architecture
- Regularize
- Get more data
r Bleu score – The bilingual evaluation understudy (bleu) score quantifies how good a machine
translation is by computing a similarity score based on n-gram precision. It is defined as follows:
bleu score = exp
1
n
n
X
k=1
pk
!
where pn is the bleu score on n-gram only defined as follows:
pn =
X
n-gram∈b
y
countclip(n-gram)
X
n-gram∈b
y
count(n-gram)
Remark: a brevity penalty may be applied to short predicted translations to prevent an artificially
inflated bleu score.
2.7 Attention
r Attention model – This model allows an RNN to pay attention to specific parts of the input
that is considered as being important, which improves the performance of the resulting model
in practice. By noting αt,t0
the amount of attention that the output yt should pay to the
activation at0
and ct the context at time t, we have:
ct
=
X
t0
αt,t0

at0

with
X
t0
αt,t0

= 1
Remark: the attention scores are commonly used in image captioning and machine translation.
r Attention weight – The amount of attention that the output yt should pay to the
activation at0
is given by αt,t0
computed as follows:
αt,t0

=
exp(et,t0
)
Tx
X
t00=1
exp(et,t00

)
Remark: computation complexity is quadratic with respect to Tx.
? ? ?

3 Deep Learning Tips and Tricks
3.1 Data processing
r Data augmentation – Deep learning models usually need a lot of data to be properly trained.
It is often useful to get more data from the existing ones using data augmentation techniques.
The main ones are summed up in the table below. More precisely, given the following input
image, here are the techniques that we can apply:
Original Flip Rotation Random crop
- Image without
any modification
- Flipped with respect
to an axis for which
the meaning of the
image is preserved
- Rotation with
a slight angle
- Simulates incorrect
horizon calibration
- Random focus
on one part of
the image
- Several random
crops can be
done in a row
Color shift Noise addition Information loss Contrast change
- Nuances of RGB
is slightly changed
- Captures noise
that can occur
with light exposure
- Addition of noise
- More tolerance to
quality variation of
inputs
- Parts of image
ignored
- Mimics potential
loss of parts of image
- Luminosity changes
- Controls difference
in exposition due
to time of day
r Batch normalization – It is a step of hyperparameter γ, β that normalizes the batch {xi}.
By noting µB, σ2
B the mean and variance of that we want to correct to the batch, it is done as
follows:
xi ←− γ
xi − µB
p
σ2
B +
+ β
It is usually done after a fully connected/convolutional layer and before a non-linearity layer and
aims at allowing higher learning rates and reducing the strong dependence on initialization.
3.2 Training a neural network
3.2.1 Definitions
r Epoch – In the context of training a model, epoch is a term used to refer to one iteration
where the model sees the whole training set to update its weights.
r Mini-batch gradient descent – During the training phase, updating weights is usually not
based on the whole training set at once due to computation complexities or one data point due
to noise issues. Instead, the update step is done on mini-batches, where the number of data
points in a batch is a hyperparameter that we can tune.
r Loss function – In order to quantify how a given model performs, the loss function L is
usually used to evaluate to what extent the actual outputs y are correctly predicted by the
model outputs z.
r Cross-entropy loss – In the context of binary classification in neural networks, the cross-
entropy loss L(z,y) is commonly used and is defined as follows:
L(z,y) = −
h
y log(z) + (1 − y) log(1 − z)
i
3.2.2 Finding optimal weights
r Backpropagation – Backpropagation is a method to update the weights in the neural network
by taking into account the actual output and the desired output. The derivative with respect
to each weight w is computed using the chain rule.
Using this method, each weight is updated with the rule:
w ←− w − α
∂L(z,y)
∂w
r Updating weights – In a neural network, weights are updated as follows:
• Step 1: Take a batch of training data and perform forward propagation to compute the
loss.
• Step 2: Backpropagate the loss to get the gradient of the loss with respect to each weight.
• Step 3: Use the gradients to update the weights of the network.

3.3 Parameter tuning
3.3.1 Weights initialization
r Xavier initialization – Instead of initializing the weights in a purely random manner, Xavier
initialization enables to have initial weights that take into account characteristics that are unique
to the architecture.
r Transfer learning – Training a deep learning model requires a lot of data and more impor-
tantly a lot of time. It is often useful to take advantage of pre-trained weights on huge datasets
that took days/weeks to train, and leverage it towards our use case. Depending on how much
data we have at hand, here are the different ways to leverage this:
Training size Illustration Explanation
Small
Freezes all layers,
trains weights on softmax
Medium
Freezes most layers,
trains weights on last
layers and softmax
Large
Trains weights on layers
and softmax by initializing
weights on pre-trained ones
3.3.2 Optimizing convergence
r Learning rate – The learning rate, often noted α or sometimes η, indicates at which pace the
weights get updated. It can be fixed or adaptively changed. The current most popular method
is called Adam, which is a method that adapts the learning rate.
r Adaptive learning rates – Letting the learning rate vary when training a model can reduce
the training time and improve the numerical optimal solution. While Adam optimizer is the
most commonly used technique, others can also be useful. They are summed up in the table
below:
Method Explanation Update of w Update of b
Momentum
- Dampens oscillations
- Improvement to SGD
- 2 parameters to tune
w − αvdw b − αvdb
RMSprop
- Root Mean Square propagation
- Speeds up learning algorithm
by controlling oscillations
w − α
dw
√
sdw
b ←− b − α
db
√
sdb
Adam
- Adaptive Moment estimation
- Most popular method
- 4 parameters to tune
w − α
vdw
√
sdw +
b ←− b − α
vdb
√
sdb +
Remark: other methods include Adadelta, Adagrad and SGD.
3.4 Regularization
r Dropout – Dropout is a technique used in neural networks to prevent overfitting the training
data by dropping out neurons with probability p 0. It forces the model to avoid relying too
much on particular sets of features.
Remark: most deep learning frameworks parametrize dropout through the ’keep’ parameter 1−p.
r Weight regularization – In order to make sure that the weights are not too large and that
the model is not overfitting the training set, regularization techniques are usually performed on
the model weights. The main ones are summed up in the table below:
LASSO Ridge Elastic Net
- Shrinks coefficients to 0
- Good for variable selection
Makes coefficients smaller
Tradeoff between variable
selection and small coefficients
... + λ||θ||1
λ ∈ R
... + λ||θ||2
2
λ ∈ R
... + λ
h
(1 − α)||θ||1 + α||θ||2
2
i
λ ∈ R,α ∈ [0,1]

r Early stopping – This regularization technique stops the training process as soon as the
validation loss reaches a plateau or starts to increase.
3.5 Good practices
r Overfitting small batch – When debugging a model, it is often useful to make quick tests
to see if there is any major issue with the architecture of the model itself. In particular, in order
to make sure that the model can be properly trained, a mini-batch is passed inside the network
to see if it can overfit on it. If it cannot, it means that the model is either too complex or not
complex enough to even overfit on a small batch, let alone a normal-sized training set.
r Gradient checking – Gradient checking is a method used during the implementation of
the backward pass of a neural network. It compares the value of the analytical gradient to the
numerical gradient at given points and plays the role of a sanity-check for correctness.
Numerical gradient Analytical gradient
Formula
df
dx
(x) ≈
f(x + h) − f(x − h)
2h
df
dx
(x) = f0
(x)
Comments
- Expensive; loss has to be
computed two times per dimension
- Used to verify correctness
of analytical implementation
-Trade-off in choosing h
not too small (numerical instability)
nor too large (poor gradient approx.)
- ’Exact’ result
- Direct computation
- Used in the final implementation
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