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Disclaimers

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Hint: he's not a "real" doctor

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AI? Machine Learning? Neural Networks? PHP?!

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h"ps://twi"er.com/mhbergen/status/966876932313763841

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"I thought that it might be be.er to use a familiar language to learn something unfamiliar like ML." h"ps://tech.kartenmacherei.de/why-i-used-php-to-teach-myself- machine-learning-10ed90af8996

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h"ps://www.offerzen.com/blog/how-to-build-a-content-based-recommender-system-for-your-product

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h"ps://www.theatlan.c.com/magazine/archive/2013/03/the-robot-will-see-you-now/309216/

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h"ps://www.geek.com/tech/ai-beats-human-lawyers-at-their-own-game-1732154/

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h"ps://blog.floydhub.com/turning-design-mockups-into-code-with-deep-learning/

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h"ps://www.cnet.com/news/machine-learning-algorithm-ai-nicolas-cage-movies-indiana-jones/

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Artificial Neural Networks

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Coming up next · Why are they called "Neural Networks"? · How do they "learn"? · How can I write one (with PHP)? · What the hell is "Deep Learning"?

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The Human Brain

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≈86,000,000,000 neurons

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Neuron

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1943: First Ar(ficial Neuron (McCulloch and Pitts) 1957: Perceptron (Rosenblatt)

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Learnable Parameters

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Weights { } · Represents the synaptic strenght (influence of one neuron on another). Bias { } · Ensures the output best fits the incoming signal (allows the activation function to be shifted to the left or right).

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Neuron input Neuron output

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Activation Function · Models the firing rate of the neuron (frequency of the spikes along the axon). · Goal: add non-linearity into the network.

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Activation Function: Sigmoid

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Supervised Machine Learning

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Supervised Machine Learning Given a set of inputs , learn a function mapping to some known output ; So that we can accurately predict a new output from unseen inputs.

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Adapted from h-ps://www.coursera.org/learn/introduc9on-tensorflow/lecture/PoOzi/a-primer-in-machine-learning

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Adapted from h-ps://www.coursera.org/learn/introduc9on-tensorflow/lecture/PoOzi/a-primer-in-machine-learning

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"When the activation function is non- linear, then a two-layer neural network can be proven to be an universal func-on approximator"

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Training approach · Start with random weights. · Predict based on input data . · Compare with target output . · Adjust network parameters ( and ). · Repeat until we are "close enough"

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Learning the XOR

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XOR x1 x2 y 0 0 0 0 1 1 1 0 1 1 1 0

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Build Your Own Neural Network h"ps:/ /github.com/noiselabs/byonn

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// › src/NeuralNetwork.php class NeuralNetwork { public function train(array $inputs, array $targets) { } public function predict(array $input) { } }

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// › src/NeuralNetwork.php class NeuralNetwork { public function train(array $inputs, array $targets) { } public function predict(array $input) { } }

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// › src/NeuralNetwork.php class NeuralNetwork { public function train(array $inputs, array $targets) { } public function predict(array $input) { } }

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// › examples/xor.php $inputs = [ [0, 0], // 0 [0, 1], // 1 [1, 0], // 1 [1, 1] // 0 ]; $targets = [0, 1, 1, 0]; $neuralNetwork = new NeuralNetwork();

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// › examples/xor.php $inputs = [ [0, 0], // 0 [0, 1], // 1 [1, 0], // 1 [1, 1] // 0 ]; $targets = [0, 1, 1, 0]; $neuralNetwork = new NeuralNetwork();

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Network Topology

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// › src/NeuralNetwork.php class NeuralNetwork { const INPUTS = 2; const HIDDEN_NEURONS = 2; const LAYERS = 2; const OUTPUTS = 1; // ... }

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Parameters

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// › src/Parameters.php class Parameters { /** @var array Weights */ public $w = []; /** @var array Biases */ public $b = []; /** @var array The input of the activation function */ public $z = []; /** @var array The neuron output, after applying an activation function */ public $a = []; }

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class NeuralNetwork { public function __construct() { $this->p = new Parameters(); } }

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Hidden Layer

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Output Layer

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› h#ps:/ /github.com/markrogoyski/math-php

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use MathPHP\LinearAlgebra\Matrix; use MathPHP\LinearAlgebra\MatrixFactory; $matrix = [ [1, 2, 3], [4, 5, 6], [7, 8, 9], ]; $A = MatrixFactory::create($matrix); $B = MatrixFactory::create($matrix); $A+B = $A->add($B); $AB = $A->multiply($B); $A∘B = $A->hadamardProduct($B); $C = $A->map(function($x) { return $x * 2; });

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use MathPHP\LinearAlgebra\Matrix; use MathPHP\LinearAlgebra\MatrixFactory; $matrix = [ [1, 2, 3], [4, 5, 6], [7, 8, 9], ]; $A = MatrixFactory::create($matrix); $B = MatrixFactory::create($matrix); $A+B = $A->add($B); $AB = $A->multiply($B); $A∘B = $A->hadamardProduct($B); $C = $A->map(function($x) { return $x * 2; });

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use MathPHP\LinearAlgebra\Matrix; use MathPHP\LinearAlgebra\MatrixFactory; $matrix = [ [1, 2, 3], [4, 5, 6], [7, 8, 9], ]; $A = MatrixFactory::create($matrix); $B = MatrixFactory::create($matrix); $A+B = $A->add($B); $AB = $A->multiply($B); $A∘B = $A->hadamardProduct($B); $C = $A->map(function($x) { return $x * 2; });

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use MathPHP\LinearAlgebra\Matrix; use MathPHP\LinearAlgebra\MatrixFactory; $matrix = [ [1, 2, 3], [4, 5, 6], [7, 8, 9], ]; $A = MatrixFactory::create($matrix); $B = MatrixFactory::create($matrix); $A+B = $A->add($B); $AB = $A->multiply($B); $A∘B = $A->hadamardProduct($B); $C = $A->map(function($x) { return $x * 2; });

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Vectorization?

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› h#ps:/ /github.com/phpsci/phpsci

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Activation Function

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// › src/Activation/Sigmoid.php namespace Activation; use MathPHP\LinearAlgebra\Matrix; class Sigmoid { public function compute(Matrix $m): Matrix { return $values->map(function($value) { return $this->sigmoid($value); }); } private function sigmoid($t) { return 1 / (1 + exp(-$t)); } }

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// › src/Activation/Sigmoid.php namespace Activation; use MathPHP\LinearAlgebra\Matrix; class Sigmoid { public function compute(Matrix $m): Matrix { return $values->map(function($value) { return $this->sigmoid($value); }); } private function sigmoid($t) { return 1 / (1 + exp(-$t)); } }

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// › src/Activation/Sigmoid.php namespace Activation; use MathPHP\LinearAlgebra\Matrix; class Sigmoid { public function compute(Matrix $m): Matrix { return $values->map(function($value) { return $this->sigmoid($value); }); } private function sigmoid($t) { return 1 / (1 + exp(-$t)); } }

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// › src/Activation/Sigmoid.php namespace Activation; use MathPHP\LinearAlgebra\Matrix; class Sigmoid { public function compute(Matrix $m): Matrix { return $values->map(function($value) { return $this->sigmoid($value); }); } private function sigmoid($t) { return 1 / (1 + exp(-$t)); } }

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class NeuralNetwork { public function __construct( ActivationFunction\Sigmoid $activationFunction ) { $this->activationFunction = $activationFunction; $this->p = new Parameters(); } }

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Algorithm for training initialise_weights_and_biases() # 1. while i < n_iterations or error > max_error: for m in training_examples: forward_pass() # 2. compute_cost() # 3. backpropagation() # 4. adjust_weights_and_biases() # 5.

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Algorithm for training initialise_weights_and_biases() # 1. while i < n_iterations or error > max_error: for m in training_examples: forward_pass() # 2. compute_cost() # 3. backpropagation() # 4. adjust_weights_and_biases() # 5.

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Algorithm for training initialise_weights_and_biases() # 1. while i < n_iterations or error > max_error: for m in training_examples: forward_pass() # 2. compute_cost() # 3. backpropagation() # 4. adjust_weights_and_biases() # 5.

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Algorithm for training initialise_weights_and_biases() # 1. while i < n_iterations or error > max_error: for m in training_examples: forward_pass() # 2. compute_cost() # 3. backpropagation() # 4. adjust_weights_and_biases() # 5.

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› Training 1. Initialise Parameters

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class NeuralNetwork { // ... private function initializeParameters(): void { // Hidden layer $this->p->b[1] = MatrixFactory::zero(self::HIDDEN_NEURONS, 1); $this->p->w[1] = MatrixFactory::zero(self::HIDDEN_NEURONS, self::INPUTS) $this->p->w[1]->map(function($v) { return random_int(1, 1000) / 1000; }); // Output layer $this->p->b[2] = MatrixFactory::zero(self::OUTPUTS, 1); $this->p->w[2] = MatrixFactory::zero(self::OUTPUTS, self::HIDDEN_NEURONS); $this->p->w[2]->map(function($v) { return random_int(1, 1000) / 1000; }); }

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class NeuralNetwork { // ... private function initializeParameters(): void { // Hidden layer $this->p->b[1] = MatrixFactory::zero(self::HIDDEN_NEURONS, 1); $this->p->w[1] = MatrixFactory::zero(self::HIDDEN_NEURONS, self::INPUTS) $this->p->w[1]->map(function($v) { return random_int(1, 1000) / 1000; }); // Output layer $this->p->b[2] = MatrixFactory::zero(self::OUTPUTS, 1); $this->p->w[2] = MatrixFactory::zero(self::OUTPUTS, self::HIDDEN_NEURONS); $this->p->w[2]->map(function($v) { return random_int(1, 1000) / 1000; }); }

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class NeuralNetwork { // ... private function initializeParameters(): void { // Hidden layer $this->p->b[1] = MatrixFactory::zero(self::HIDDEN_NEURONS, 1); $this->p->w[1] = MatrixFactory::zero(self::HIDDEN_NEURONS, self::INPUTS) $this->p->w[1]->map(function($v) { return random_int(1, 1000) / 1000; }); // Output layer $this->p->b[2] = MatrixFactory::zero(self::OUTPUTS, 1); $this->p->w[2] = MatrixFactory::zero(self::OUTPUTS, self::HIDDEN_NEURONS); $this->p->w[2]->map(function($v) { return random_int(1, 1000) / 1000; }); }

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class NeuralNetwork { // ... private function initializeParameters(): void { // Hidden layer $this->p->b[1] = MatrixFactory::zero(self::HIDDEN_NEURONS, 1); $this->p->w[1] = MatrixFactory::zero(self::HIDDEN_NEURONS, self::INPUTS) $this->p->w[1]->map(function($v) { return random_int(1, 1000) / 1000; }); // Output layer $this->p->b[2] = MatrixFactory::zero(self::OUTPUTS, 1); $this->p->w[2] = MatrixFactory::zero(self::OUTPUTS, self::HIDDEN_NEURONS); $this->p->w[2]->map(function($v) { return random_int(1, 1000) / 1000; }); }

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class NeuralNetwork { // ... private function initializeParameters(): void { // Hidden layer $this->p->b[1] = MatrixFactory::zero(self::HIDDEN_NEURONS, 1); $this->p->w[1] = MatrixFactory::zero(self::HIDDEN_NEURONS, self::INPUTS) $this->p->w[1]->map(function($v) { return random_int(1, 1000) / 1000; }); // Output layer $this->p->b[2] = MatrixFactory::zero(self::OUTPUTS, 1); $this->p->w[2] = MatrixFactory::zero(self::OUTPUTS, self::HIDDEN_NEURONS); $this->p->w[2]->map(function($v) { return random_int(1, 1000) / 1000; }); }

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class NeuralNetwork { public function __construct( ActivationFunction\Sigmoid $activationFunction ) { $this->activationFunction = $activationFunction; $this->p = new Parameters(); $this->initializeParameters(); } }

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class NeuralNetwork { // ... public function train(array $inputs, array $targets) { $inputs = $this->toMatrix($inputs); $targets = $this->toMatrix($targets); // ... } }

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class NeuralNetwork { // ... public function train(array $inputs, array $targets) { $inputs = $this->toMatrix($inputs); $targets = $this->toMatrix($targets); // ... } }

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class NeuralNetwork { public function train(array $inputs, array $targets) { // ... $maxTrainingIterations = 20000; $maxError = 0.001; $iteration = 0; $error = INF; }

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class NeuralNetwork { public function train(array $inputs, array $targets) { // ... $maxTrainingIterations = 20000; $maxError = 0.01; $iteration = 0; $error = INF; while ($iteration < $maxTrainingIterations && $error > $maxError) { $iteration++; $costs = []; for ($i = 0; $i < count($inputs); $i++) { // 1. doForwardPropagation() // 2. $costs = computeCost() // 3. doBackPropagation() // 4. updateParameters() } $error = array_sum($costs) / count($costs); } }

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› Training 2. Forward propagation

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Hidden Layer: Output Layer:

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Hidden Layer: Output Layer:

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Hidden Layer: Output Layer:

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class NeuralNetwork { private function doForwardPropagation(Matrix $input): array { // To ease calculations do: "Layer-0" activations = inputs $this->p->a[0] = $input; for ($l = 1; $l <= self::LAYERS; $l++) { // Z[l] = W[l]·A[l-1] + b[l] $this->p->z[$l] = $this->p->w[$l] ->multiply($this->p->a[$l-1])) ->add($this->p->b[$l]); $this->p->a[$l] = $this->activation->compute($this->p->z[$l]); } // Prediction: return $this->toArray($this->p->a[self::LAYERS]); }

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class NeuralNetwork { private function doForwardPropagation(Matrix $input): array { // To ease calculations do: "Layer-0" activations = inputs $this->p->a[0] = $input; for ($l = 1; $l <= self::LAYERS; $l++) { // Z[l] = W[l]·A[l-1] + b[l] $this->p->z[$l] = $this->p->w[$l] ->multiply($this->p->a[$l-1])) ->add($this->p->b[$l]); $this->p->a[$l] = $this->activation->compute($this->p->z[$l]); } // Prediction: return $this->toArray($this->p->a[self::LAYERS]); }

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class NeuralNetwork { private function doForwardPropagation(Matrix $input): array { // To ease calculations do: "Layer-0" activations = inputs $this->p->a[0] = $input; for ($l = 1; $l <= self::LAYERS; $l++) { // Z[l] = W[l]·A[l-1] + b[l] $this->p->z[$l] = $this->p->w[$l] ->multiply($this->p->a[$l-1])) ->add($this->p->b[$l]); $this->p->a[$l] = $this->activation->compute($this->p->z[$l]); } // Prediction: return $this->toArray($this->p->a[self::LAYERS]); }

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class NeuralNetwork { private function doForwardPropagation(Matrix $input): array { // To ease calculations do: "Layer-0" activations = inputs $this->p->a[0] = $input; for ($l = 1; $l <= self::LAYERS; $l++) { // Z[l] = W[l]·A[l-1] + b[l] $this->p->z[$l] = $this->p->w[$l] ->multiply($this->p->a[$l-1])) ->add($this->p->b[$l]); $this->p->a[$l] = $this->activation->compute($this->p->z[$l]); } // Prediction: return $this->toArray($this->p->a[self::LAYERS]); }

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class NeuralNetwork { private function doForwardPropagation(Matrix $input): array { // To ease calculations do: "Layer-0" activations = inputs $this->p->a[0] = $input; for ($l = 1; $l <= self::LAYERS; $l++) { // Z[l] = W[l]·A[l-1] + b[l] $this->p->z[$l] = $this->p->w[$l] ->multiply($this->p->a[$l-1])) ->add($this->p->b[$l]); $this->p->a[$l] = $this->activation->compute($this->p->z[$l]); } // Prediction: return $this->toArray($this->p->a[self::LAYERS]); }

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class NeuralNetwork { private function doForwardPropagation(Matrix $input): array { // To ease calculations do: "Layer-0" activations = inputs $this->p->a[0] = $input; for ($l = 1; $l <= self::LAYERS; $l++) { // Z[l] = W[l]·A[l-1] + b[l] $this->p->z[$l] = $this->p->w[$l] ->multiply($this->p->a[$l-1])) ->add($this->p->b[$l]); $this->p->a[$l] = $this->activation->compute($this->p->z[$l]); } // Prediction: return $this->toArray($this->p->a[self::LAYERS]); }

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class NeuralNetwork { public function train(array $inputs, array $targets) { // ... while ($iteration < $maxTrainingIterations && $error > $maxError) { $costs = []; for ($i = 0; $i < count($inputs); $i++) { $prediction = $this->doForwardPropagation($inputs[$i]); } } }

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› Training 3. Compute cost

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Cost function: Mean Squared Error

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// › src/CostFunction/MeanSquaredError.php namespace CostFunction; use MathPHP\LinearAlgebra\Vector; class MeanSquaredError { public function compute(Matrix $prediction, Matrix $target): float { return ($target[0] - $prediction[0]) ** 2; } }

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class NeuralNetwork { public function __construct( ActivationFunction\Sigmoid $activationFunction, CostFunction\MeanSquaredError $costFunction ) { $this->activationFunction = $activationFunction; $this->costFunction = $costFunction; $this->p = new Parameters(); }

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class NeuralNetwork { // ... private function computeCost(Matrix $prediction, Matrix $target): float { return $this->costFunction->compute($prediction, $target); }

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class NeuralNetwork { public function train(array $inputs, array $targets) { // ... while ($iteration < $maxTrainingIterations && $error > $maxError) { $costs = []; for ($i = 0; $i < count($inputs); $i++) { $prediction = $this->doForwardPropagation($inputs[$i]); $costs[$i] = $this->computeCost($prediction, $targets[$i]); } } }

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Goal: minimise the wrongness

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› Training 4. Backpropagation

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Backpropagation · A supervised learning method for multilayer feed-forward networks. · Backward propaga,on of errors using gradient descent (calculates the gradient of the error function with respect to the neural network's weights).

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Adapted from h-ps://www.khanacademy.org/math/mul:variable-calculus/mul:variable-deriva:ves/par:al-deriva:ve-and-gradient-ar:cles/ a/introduc:on-to-par:al-deriva:ves

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Output Layer: Hidden Layer:

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Output Layer: Hidden Layer:

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// › src/Parameters.php class Parameters { // ... /** * Gradient of the cost with respect to `w`. * * @var array|Matrix[] */ public $dw = []; /** * Gradient of the cost with respect to `b`. * * @var array|Matrix[] */ public $db = []; }

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Mean Squared Error derivative

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Mean Squared Error derivative

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namespace CostFunction; use MathPHP\LinearAlgebra\Matrix; use MathPHP\LinearAlgebra\Vector; class MeanSquaredError { public function compute(Matrix $prediction, Matrix $target): float { return ($target[0] - $prediction[0]) ** 2; } public function differentiate(Matrix $prediction, Matrix $target): Matrix { // ∂L = 2·(Y -Ŷ) return ($target->subtract($prediction))->scalarMultiply(2); } }

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Sigmoid derivative

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Sigmoid derivative

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namespace Activation; use MathPHP\LinearAlgebra\Matrix; class Sigmoid { public function compute(Matrix $m): Matrix { return $values->map(function($value) { return $this->sigmoid($value); }); } public function differentiate(Matrix $values): Matrix { // ∂σ = σ·(1 - σ) return $values->map(function($value) { $computedValue = $this->sigmoid(value); return $computedValue * (1 - $computedValue); }); } private function sigmoid($t) { return 1 / (1 + exp(-$t)); } }

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class NeuralNetwork { private function doBackPropagation(Matrix $target): void { $l = self::LAYERS; // Output Layer $da[$l] = $this->costFunction->differentiate($this->p->a[$l], $target); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; // Hidden Layer(s) for ($l = (self::LAYERS - 1); $l >= 1; $l--) { $da[$l] = $this->p->w[$l+1]->transpose()->multiply($dz[$l+1]); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; } }

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Output Layer: Hidden Layer:

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class NeuralNetwork { private function doBackPropagation(Matrix $target): void { $l = self::LAYERS; // Output Layer $da[$l] = $this->costFunction->differentiate($this->p->a[$l], $target); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; // Hidden Layer(s) for ($l = (self::LAYERS - 1); $l >= 1; $l--) { $da[$l] = $this->p->w[$l+1]->transpose()->multiply($dz[$l+1]); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; } }

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class NeuralNetwork { private function doBackPropagation(Matrix $target): void { $l = self::LAYERS; // Output Layer $da[$l] = $this->costFunction->differentiate($this->p->a[$l], $target); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; // Hidden Layer(s) for ($l = (self::LAYERS - 1); $l >= 1; $l--) { $da[$l] = $this->p->w[$l+1]->transpose()->multiply($dz[$l+1]); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; } }

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class NeuralNetwork { private function doBackPropagation(Matrix $target): void { $l = self::LAYERS; // Output Layer $da[$l] = $this->costFunction->differentiate($this->p->a[$l], $target); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; // Hidden Layer(s) for ($l = (self::LAYERS - 1); $l >= 1; $l--) { $da[$l] = $this->p->w[$l+1]->transpose()->multiply($dz[$l+1]); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; } }

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Output Layer: Hidden Layer:

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class NeuralNetwork { private function doBackPropagation(Matrix $target): void { $l = self::LAYERS; // Output Layer $da[$l] = $this->costFunction->differentiate($this->p->a[$l], $target); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; // Hidden Layer(s) for ($l = (self::LAYERS - 1); $l >= 1; $l--) { $da[$l] = $this->p->w[$l+1]->transpose()->multiply($dz[$l+1]); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; } }

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class NeuralNetwork { private function doBackPropagation(Matrix $target): void { $l = self::LAYERS; // Output Layer $da[$l] = $this->costFunction->differentiate($this->p->a[$l], $target); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; // Hidden Layer(s) for ($l = (self::LAYERS - 1); $l >= 1; $l--) { $da[$l] = $this->p->w[$l+1]->transpose()->multiply($dz[$l+1]); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; } }

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class NeuralNetwork { private function doBackPropagation(Matrix $target): void { $l = self::LAYERS; // Output Layer $da[$l] = $this->costFunction->differentiate($this->p->a[$l], $target); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; // Hidden Layer(s) for ($l = (self::LAYERS - 1); $l >= 1; $l--) { $da[$l] = $this->p->w[$l+1]->transpose()->multiply($dz[$l+1]); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; } }

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class NeuralNetwork { private function doBackPropagation(Matrix $target): void { $l = self::LAYERS; // Output Layer $da[$l] = $this->costFunction->differentiate($this->p->a[$l], $target); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; // Hidden Layer(s) for ($l = (self::LAYERS - 1); $l >= 1; $l--) { $da[$l] = $this->p->w[$l+1]->transpose()->multiply($dz[$l+1]); $dz[$l] = $da[$l]->hadamardProduct( $this->activations[$l]->differentiate($this->p->z[$l])); $this->p->dw[$l] = $dz[$l]->multiply($this->p->a[$l-1]->transpose()); $this->p->db[$l] = $dz[$l]; } }

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class NeuralNetwork { // ... public function train(array $inputs, array $targets) { // ... while ($iteration < $maxTrainingIterations && $error > $maxError) { $iteration++; $costs = []; for ($i = 0; $i < count($inputs); $i++) { $prediction = $this->doForwardPropagation($inputs[$i]); $costs[$i] = $this->computeCost($prediction, $targets[$i]); $this->doBackPropagation($targets[$i]); } } }

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› Training 5. Update Parameters

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Gradient Descent

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Gradient Descent

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Gradient Descent

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class NeuralNetwork { public function __construct( ActivationFunction\Sigmoid $activationFunction, CostFunction\MeanSquaredError $costFunction, float $learningRate = 0.1 ) { $this->activationFunction = $activationFunction; $this->costFunction = $costFunction; $this->learningRate = $learningRate; $this->p = new Parameters(); }

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class NeuralNetwork { // ... private function updateParameters(): void { for ($l = 1; $l <= self::LAYERS; $l++) { $this->p->w[$l] = $this->p->w[$l]->subtract( $this->p->dw[$l]->scalarMultiply($this->learningRate)); $this->p->b[$l] = $this->p->b[$l]->subtract( $this->p->db[$l]->scalarMultiply($this->learningRate)); } }

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class NeuralNetwork { // ... private function updateParameters(): void { for ($l = 1; $l <= self::LAYERS; $l++) { $this->p->w[$l] = $this->p->w[$l]->subtract( $this->p->dw[$l]->scalarMultiply($this->learningRate)); $this->p->b[$l] = $this->p->b[$l]->subtract( $this->p->db[$l]->scalarMultiply($this->learningRate)); } }

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class NeuralNetwork { // ... private function updateParameters(): void { for ($l = 1; $l <= self::LAYERS; $l++) { $this->p->w[$l] = $this->p->w[$l]->subtract( $this->p->dw[$l]->scalarMultiply($this->learningRate)); $this->p->b[$l] = $this->p->b[$l]->subtract( $this->p->db[$l]->scalarMultiply($this->learningRate)); } }

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class NeuralNetwork { // ... private function updateParameters(): void { for ($l = 1; $l <= self::LAYERS; $l++) { $this->p->w[$l] = $this->p->w[$l]->subtract( $this->p->dw[$l]->scalarMultiply($this->learningRate)); $this->p->b[$l] = $this->p->b[$l]->subtract( $this->p->db[$l]->scalarMultiply($this->learningRate)); } }

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class NeuralNetwork { // ... public function train(array $inputs, array $targets) { // ... while ($iteration < $maxTrainingIterations && $error > $maxError) { $iteration++; $costs = []; for ($i = 0; $i < count($inputs); $i++) { $prediction = $this->doForwardPropagation($inputs[$i]); $costs[$i] = $this->computeCost($prediction, $targets[$i]); $this->doBackPropagation($targets[$i]); $this->updateParameters(); } } }

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class NeuralNetwork { // ... public function train(array $inputs, array $targets) { // ... while ($iteration < $maxTrainingIterations && $error > $maxError) { $iteration++; $costs = []; for ($i = 0; $i < count($inputs); $i++) { $prediction = $this->doForwardPropagation($inputs[$i]); $costs[$i] = $this->computeCost($prediction, $targets[$i]); $this->doBackPropagation($targets[$i]); $this->updateParameters(); } $error = array_sum($costs) / count($costs); } }

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class NeuralNetwork { public function train(array $inputs, array $targets) { $inputs = $this->toMatrix($inputs); $targets = $this->toMatrix($targets); $maxTrainingIterations = 20000; $maxError = 0.001; $iteration = 0; $error = INF; while ($iteration < $maxTrainingIterations && $error > $maxError) { $iteration++; $costs = []; for ($i = 0; $i < count($inputs); $i++) { $prediction = $this->doForwardPropagation($inputs[$i]); $costs[$i] = $this->computeCost($prediction, $targets[$i]); $this->doBackPropagation($targets[$i]); $this->updateParameters(); } $error = array_sum($costs) / count($costs); } }

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Algorithm for training initialise_weights_and_biases() # 1. while i < n_iterations or error > max_error: for m in training_examples: forward_pass() # 2. compute_cost() # 3. backpropagation() # 4. adjust_weights_and_biases() # 5.

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$ php examples/xor.php Training for 20000 epochs or until the cost falls below 0.001... * Epoch: 1000, Error: 0.229587 * Epoch: 2000, Error: 0.062260 * Epoch: 3000, Error: 0.009333 * Epoch: 4000, Error: 0.004388 * Epoch: 5000, Error: 0.002788 * Epoch: 6000, Error: 0.002020 * Epoch: 7000, Error: 0.001575 * Epoch: 8000, Error: 0.001286 * Epoch: 9000, Error: 0.001084 * Epoch: 9500, Error: 0.001005 Predicting... * Input: [0, 0] Prediction: 0.0341 Target: 0 * Input: [0, 1] Prediction: 0.9697 Target: 1 * Input: [1, 0] Prediction: 0.9698 Target: 1 * Input: [1, 1] Prediction: 0.0317 Target: 0

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$ php examples/xor.php Training for 20000 epochs or until the cost falls below 0.001... * Epoch: 1000, Error: 0.229587 * Epoch: 2000, Error: 0.062260 * Epoch: 3000, Error: 0.009333 * Epoch: 4000, Error: 0.004388 * Epoch: 5000, Error: 0.002788 * Epoch: 6000, Error: 0.002020 * Epoch: 7000, Error: 0.001575 * Epoch: 8000, Error: 0.001286 * Epoch: 9000, Error: 0.001084 * Epoch: 9500, Error: 0.001005 Predicting... * Input: [0, 0] Prediction: 0.0341 Target: 0 * Input: [0, 1] Prediction: 0.9697 Target: 1 * Input: [1, 0] Prediction: 0.9698 Target: 1 * Input: [1, 1] Prediction: 0.0317 Target: 0

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$ php examples/xor.php Training for 20000 epochs or until the cost falls below 0.001... * Epoch: 1000, Error: 0.229587 * Epoch: 2000, Error: 0.062260 * Epoch: 3000, Error: 0.009333 * Epoch: 4000, Error: 0.004388 * Epoch: 5000, Error: 0.002788 * Epoch: 6000, Error: 0.002020 * Epoch: 7000, Error: 0.001575 * Epoch: 8000, Error: 0.001286 * Epoch: 9000, Error: 0.001084 * Epoch: 9500, Error: 0.001005 Predicting... * Input: [0, 0] Prediction: 0.0341 Target: 0 * Input: [0, 1] Prediction: 0.9697 Target: 1 * Input: [1, 0] Prediction: 0.9698 Target: 1 * Input: [1, 1] Prediction: 0.0317 Target: 0

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$ php examples/xor.php Training for 20000 epochs or until the cost falls below 0.001... * Epoch: 1000, Error: 0.229587 * Epoch: 2000, Error: 0.062260 * Epoch: 3000, Error: 0.009333 * Epoch: 4000, Error: 0.004388 * Epoch: 5000, Error: 0.002788 * Epoch: 6000, Error: 0.002020 * Epoch: 7000, Error: 0.001575 * Epoch: 8000, Error: 0.001286 * Epoch: 9000, Error: 0.001084 * Epoch: 9500, Error: 0.001005 Predicting... * Input: [0, 0] Prediction: 0.0341 Target: 0 * Input: [0, 1] Prediction: 0.9697 Target: 1 * Input: [1, 0] Prediction: 0.9698 Target: 1 * Input: [1, 1] Prediction: 0.0317 Target: 0

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// › examples/xor.php $inputs = [ [0, 0], // 0 [0, 1], // 1 [1, 0], // 1 [1, 1] // 0 ]; $targets = [0, 1, 1, 0]; $neuralNetwork = new NeuralNetwork( new Activation\Sigmoid(), new CostFunction\MeanSquaredError() ); $neuralNetwork->train($inputs, $targets); echo "Predicting...\n"; echo sprintf("* Input: [0,0] Prediction: %.2f Target: 0\n", $neuralNetwork->predict([0,0]])); echo sprintf("* Input: [0,1] Prediction: %.2f Target: 1\n", $neuralNetwork->predict([0,1]])); echo sprintf("* Input: [1,0] Prediction: %.2f Target: 1\n", $neuralNetwork->predict([1,0]])); echo sprintf("* Input: [1,1] Prediction: %.2f Target: 0\n", $neuralNetwork->predict([1,1]]));

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class NeuralNetwork { // ... public function predict(array $input): array { return $this->doForwardPropagation($this->toMatrix($input)); }

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$ php examples/xor.php Training for 20000 epochs or until the cost falls below 0.001... * Epoch: 1000, Error: 0.229587 * Epoch: 2000, Error: 0.062260 * Epoch: 3000, Error: 0.009333 * Epoch: 4000, Error: 0.004388 * Epoch: 5000, Error: 0.002788 * Epoch: 6000, Error: 0.002020 * Epoch: 7000, Error: 0.001575 * Epoch: 8000, Error: 0.001286 * Epoch: 9000, Error: 0.001084 * Epoch: 9500, Error: 0.001005 Predicting... * Input: [0, 0] Prediction: 0.0341 Target: 0 * Input: [0, 1] Prediction: 0.9697 Target: 1 * Input: [1, 0] Prediction: 0.9698 Target: 1 * Input: [1, 1] Prediction: 0.0317 Target: 0

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$ php examples/xor.php Training for 20000 epochs or until the cost falls below 0.001... * Epoch: 1000, Error: 0.229587 * Epoch: 2000, Error: 0.062260 * Epoch: 3000, Error: 0.009333 * Epoch: 4000, Error: 0.004388 * Epoch: 5000, Error: 0.002788 * Epoch: 6000, Error: 0.002020 * Epoch: 7000, Error: 0.001575 * Epoch: 8000, Error: 0.001286 * Epoch: 9000, Error: 0.001084 * Epoch: 9500, Error: 0.001005 Predicting... * Input: [0, 0] Prediction: 0.0341 Target: 0 * Input: [0, 1] Prediction: 0.9697 Target: 1 * Input: [1, 0] Prediction: 0.9698 Target: 1 * Input: [1, 1] Prediction: 0.0317 Target: 0

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But the fun doesn't stop here! · Pick a more creative example · Experiment with different activation and cost functions, and learning rates (send me a PR?) · Try a a different topology (more hidden layers?) · Compete at Kaggle!

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Artificial Neural Networks Recap · Inspired by the human brain. · Today we saw a feed-forward network (there are other architectures). · Uses Supervised Machine Learning (learn from examples). · Error is minimised using Backpropagation and Gradient Descent. · Can approximate any function*.

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Getting deeper...

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"Deep Learning"

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Neural Networks and Machine Learning resources · https://en.wikipedia.org/wiki/Artificial_neural_network · https://www.coursera.org/learn/machine-learning · https://www.coursera.org/specializations/deep-learning · https://www.cs.toronto.edu/~hinton/coursera_lectures.html · https://developers.google.com/machine-learning/crash-course/ · http://neuralnetworksanddeeplearning.com/

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h"ps://youtu.be/fXOsFF95i5

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The End

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Vielen Dank! › h#p:/ /bit.ly/2KCD1dx-byonn-ipc19