Neural networks. Overview

Oleksandr Baiev, PhD

Senior Engineer

Samsung R&D Institute Ukraine

Neural networks. Overview

• Common principles

– Structure

– Learning

• Shallow and Deep NN

• Additional methods

– Conventional

– Voodoo

Neural networks. Overview

• Common principles

– Structure

– Learning

• Shallow and Deep NN

• Additional methods

– Conventional

– Voodoo

Canonical/Typical tasks

Solutions in general

𝑥𝑗 = 𝑥1, 𝑥2, 𝑥3, 𝑥4, … , 𝑥𝑖 , … 𝑗 ∈ 𝑋

𝑦𝑗 = 𝑦1, 𝑦2, … , 𝑦𝑘 , … 𝑗 ∈ 𝑌

𝐹: 𝑋 → 𝑌Classification

𝑦1 = 1,0,0𝑦2 = 0,0,1𝑦3 = 0,1,0𝑦4 = 0,1,0

Index of sample in dataset

sample of class “0”

sample of class “2”

sample of class “2”

sample of class “1”

Regression

𝑦1 = 0.3𝑦2 = 0.2𝑦3 = 1.0𝑦4 = 0.65

What is artificial Neural Networks?Is it biology?

Simulation of biological neural networks (synapses, axons, chains, layers, etc.) is a good abstraction for understanding topology.

Bio NN is only inspiration and illustration. Nothing more!

What is artificial Neural Networks?Let’s imagine black box!

Finputs

params

outputs

General form:𝑜𝑢𝑡𝑝𝑢𝑡𝑠 = 𝐹 𝑖𝑛𝑝𝑢𝑡𝑠, 𝑝𝑎𝑟𝑎𝑚𝑠

Steps:1) choose “form” of F 2) find params

What is artificial Neural Networks?It’s a simple math!

free parameters

activation function

𝑠𝑖 =

𝑗=1

𝑛

𝑤𝑖𝑗𝑥𝑗 + 𝑏𝑖

𝑦𝑖 = 𝑓 𝑠𝑖

Output of i-th neuron:

What is artificial Neural Networks?It’s a simple math!

activation: 𝑦 = 𝑓 𝑤𝑥 + 𝑏 = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(𝑤𝑥 + 𝑏)

What is artificial Neural Networks?It’s a simple math!

activation: 𝑦 = 𝑓 𝑤𝑥 + 𝑏 = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(𝑤𝑥 + 𝑏)

What is artificial Neural Networks?It’s a simple math!

activation: 𝑦 = 𝑓 𝑤𝑥 + 𝑏 = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(𝑤𝑥 + 𝑏)

What is artificial Neural Networks?It’s a simple math!

activation: 𝑦 = 𝑓 𝑤𝑥 + 𝑏 = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(𝑤𝑥 + 𝑏)

What is artificial Neural Networks?It’s a simple math!

activation: 𝑦 = 𝑓 𝑤𝑥 + 𝑏 = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(𝑤𝑥 + 𝑏)

What is artificial Neural Networks?It’s a simple math!

n inputsm neurons in hidden layer

𝑠𝑖 =

𝑗=1

𝑛

𝑤𝑖𝑗𝑥𝑗 + 𝑏𝑖

𝑦𝑖 = 𝑓 𝑠𝑖

Output of i-th neuron:

Output of k-th layer:

1) 𝑆𝑘 = 𝑊𝑘𝑋𝑘 + 𝐵𝑘 =

=

𝑤11 𝑤12 ⋯ 𝑤1𝑛𝑤21 𝑤21 ⋯ 𝑤21

⋯ ⋯ ⋯ ⋯𝑤𝑚1 𝑤𝑚2 ⋯ 𝑤𝑚𝑛 𝑘

𝑥1𝑥2𝑥3⋮𝑥𝑛 𝑘

+

𝑏1𝑏2𝑏3⋮𝑏𝑛 𝑘

2) 𝑌𝑘 = 𝑓𝑘 𝑆𝑘

apply element-wise

Kolmagorov & Arnold function superposition

Form of F:

Neural networks. Overview

• Common principles

– Structure

– Learning

• Shallow and Deep NN

• Additional methods

– Conventional

– Voodoo

How to find parametersW and B?

Supervised learning:

Training set (pairs of variables and responses):𝑋; 𝑌 𝑖 , 𝑖 = 1. . 𝑁

Find: 𝑊∗, 𝐵∗ = 𝑎𝑟𝑔𝑚𝑖𝑛𝑊,𝐵

𝐿 𝐹 𝑋 , 𝑌

Cost function (loss, error):

logloss: L 𝐹 𝑋 , 𝑌 =1

𝑁 𝑖=1𝑁 𝑗=1

𝑀 𝑦𝑖.𝑗 log 𝑓𝑖,𝑗

rmse: L 𝐹 𝑋 , 𝑌 =1

𝑁 𝑖=1𝑁 𝐹 𝑋𝑖 − 𝑌𝑖 2

“1” if in i-th sample is class j else “0”

previously scaled:

𝑓𝑖,𝑗 = 𝑓𝑖,𝑗 𝑗 𝑓𝑖,𝑗

Just an examples. Cost function depend on problem (classification, regression) and domain knowledge

Training or optimization algorithm

So, we have model cost 𝐿 (or error of prediction)

And we want to update weights in order to minimize 𝑳:

𝑤∗ = 𝑤 + 𝛼Δ𝑤

In accordance to gradient descent: Δ𝑤 = −𝛻𝐿

It’s clear for network with only one layer (we have predicted outputs and targets, so can evaluate 𝐿).

But how to find 𝜟𝒘 for hidden layers?

Meet “Error Back Propagation”

Find Δ𝑤 for each layer from the last to the first as influence of weights to cost:

∆𝑤𝑖,𝑗 =𝜕𝐿

𝜕𝑤𝑖,𝑗

and:

𝜕𝐿

𝜕𝑤𝑖,𝑗

=𝜕𝐿

𝜕𝑓𝑗

𝜕𝑓𝑗

𝜕𝑠𝑗

𝜕𝑠𝑗

𝜕𝑤𝑖,𝑗

Error Back PropagationDetails

𝜕𝐿

𝜕𝑤𝑖,𝑗

=𝜕𝐿

𝜕𝑓𝑗

𝜕𝑓𝑗

𝜕𝑠𝑗

𝜕𝑠𝑗

𝜕𝑤𝑖,𝑗

𝛿𝑗 =𝜕𝐿

𝜕𝑓𝑗

𝜕𝑓𝑗

𝜕𝑠𝑗

𝛿𝑗 = 𝐿′ 𝐹 𝑋 , 𝑌 𝑓′ 𝑠𝑗 , 𝑜𝑢𝑡𝑝𝑢𝑡 𝑙𝑎𝑦𝑒𝑟

𝑙 ∈ 𝑛𝑒𝑥𝑡 𝑙𝑎𝑦𝑒𝑟 𝛿𝑙𝑤 𝑗,𝑙 𝑓′ 𝑠𝑗 , ℎ𝑖𝑑𝑑𝑒𝑛 𝑙𝑎𝑦𝑒𝑟𝑠

∆𝑤𝑖,𝑗 = −𝛼𝛿𝑗 𝑥𝑖

Gradient Descentin real life

Recall gradient descent:𝑤∗ = 𝑤 + 𝛼Δ𝑤

𝛼 is a “step” coefficient. In term of ML – learning rate.

Recall cost function:

𝐿 =1

𝑁

𝑁

…

GD modification: update 𝑤 for each sample.

Sum along all samples,And what if 𝑁 = 106 or more?

Typical: 𝛼 = 0.01. . 0.1

Gradient DescentStochastic & Minibatch

“Batch” GD (L for full set)

need a lot of memory

Stochastic GD (L for each sample)fast, but fluctuation

Minibatch GD(L for subsets)

less memory & less fluctuationsSize of minibatch depends on HW Typical: minibatch=32…256

Termination criteria

By epochs count max number of iterations along all data set

By value of gradientwhen gradient is equal to 0 than minimum, but small gradient => very slow learning

When cost didn’t change during several epochsif error is not change than training procedure is not converges

Early stoppingStop when “validation” score starts increase even when “train” score continue decreasing

Typical: epochs=50…200

Neural networks. Overview

• Common principles

– Structure

– Learning

• Shallow and Deep NN

• Additional methods

– Conventional

– Voodoo

What about “form” of F?Network topology

“Shallow” networks 1, 2 hidden layers => not enough parameters => pure separation abilities

“Deep” networks is a NN with 2..10 layers

“Very deep” networks is a NN with >10 layers

Deep learning. Problems

• Big networks => Too huge separating ability => Overfitting

• Vanishing gradient problem during training

• Complex error’s surface => Local minimum

• Curse of dimensionality => memory & computations

𝑚(𝑖−1) 𝑚(𝑖)

dim 𝑊(𝑖) = 𝑚 𝑖−1 ∗ 𝑚(𝑖)

Neural networks. Overview

• Common principles

– Structure

– Learning

• Shallow and Deep NN

• Additional methods

– Conventional

– Voodoo

Additional methodsConventional

• Momentum (prevent the variations on error surface)

∆𝑤(𝑡) = −𝛼𝛻𝐿 𝑤 𝑡 + 𝛽∆𝑤(𝑡−1)

𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚

• LR decay (make smaller steps near optimum)

𝛼(𝑡) = 𝑘𝛼(𝑡−1), 0 < 𝑘 < 1

• Weight Decay (prevent weight growing, and smooth F)

𝐿∗ = 𝐿 + 𝜆 𝑤(𝑡)

L1 or L2 regularization often used

Typical: 𝛽 = 0.9

Typical: apply LR decay (𝑘 = 0.1) each 10..100 epochs

Typical: 𝐿2 with 𝜆 = 0.0005

Neural networks. Overview

• Common principles

– Structure

– Learning

• Shallow and Deep NN

• Additional methods

– Conventional

– Voodoo

Additional methodsContemporary

Dropout/DropConnect

– ensembles of networks

– 2𝑁 networks in one: for each example hide neurons output randomly (𝑃 = 0.5)

Additional methodsContemporary

Data augmentation - more data with all available cases:

– affine transformations, flips, crop, contrast, noise, scale

– pseudo-labeling

Additional methodsContemporary

New activation function:

– Linear: 𝑦𝑖 = 𝑓 𝑠𝑖 = 𝑎𝑠𝑖– ReLU: 𝑦𝑖 = 𝑚𝑎𝑥 𝑠𝑖 , 0

– Leaky ReLU: 𝑦𝑖 = 𝑠𝑖 𝑠𝑖 > 0𝑎𝑠𝑖 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

– Maxout: 𝑦𝑖 = 𝑚𝑎𝑥 𝑠1,𝑖 , 𝑠2,𝑖 , … , 𝑠𝑘,𝑖

Typical: 𝑎 = 0.01

Typical: 𝑘 = 2. . 3

Additional methodsContemporary

Pre-training

– train layer-by-layer,

– re-train “other” network

Sources

• Jeffry Hinton Course “Neural Networks for Machine Learning” [http://www.coursera.org/course/neuralnets]

• Ian Goodfellow, Yoshua Bengio and Aaron Courville “Deep Learning” [http://www.deeplearningbook.org/]

• http://neuralnetworksanddeeplearning.com

• CS231n: Convolutional Neural Networks for Visual Recognition [http://cs231n.stanford.edu/]

• CS224d: Deep Learning for Natural Language Processing [http://cs224d.stanford.edu/]

• Schmidhuber “Deep Learning in Neural Networks: An Overview”

• kaggle.com competitions and forums