Michelle Fullwood - A gentle introduction to deep learning with TensorFlow

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A GENTLE INTRODUCTION TO DEEP LEARNING WITH TENSORFLOW Michelle Fullwood @michelleful Slides: michelleful.github.io/PyCon2017

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PREREQUISITES Knowledge of concepts of supervised ML Familiarity with linear and logistic regression

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TARGET (Deep) Feed-forward neural networks How they're constructed Why they work How to train and optimize them Image source: Fjodor van Veen (2016) Neural Network Zoo

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DEEP LEARNING LEARNING CURVE

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DEEP LEARNING LEARNING CURVE

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DEEP LEARNING LEARNING CURVE

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DEEP LEARNING LEARNING CURVE

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DEEP LEARNING LEARNING CURVE

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Traditional machine learning Deep learning

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TENSORFLOW Popular deep learning toolkit From Google Brain, Apache-licensed Python API, makes calls to C++ back- end Works on CPUs and GPUs

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LINEAR REGRESSION FROM SCRATCH

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LINEAR REGRESSION

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INPUTS

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INPUTS X_train = np.array([ [1250, 350, 3], [1700, 900, 6], [1400, 600, 3] ]) Y_train = np.array([345000, 580000, 360000])

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MODEL Multiply each feature by a weight and add them up. Add an intercept to get our nal estimate.

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MODEL

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MODEL - PARAMETERS weights = np.array([300, -10, -1]) intercept = -26497

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MODEL - OPERATIONS

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MODEL - OPERATIONS def model(X, weights, intercept): return X.dot(weights) + intercept Y_hat = model(X_train, weights, intercept)

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MODEL - COST FUNCTION

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MODEL - COST FUNCTION

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MODEL - COST FUNCTION

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COST FUNCTION def cost(Y_hat, Y): return np.sum((Y_hat - Y)**2)

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OPTIMIZATION Hold X and Y constant. Adjust parameters to minimize cost.

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OPTIMIZATION

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TRIAL AND ERROR Image source: Wikimedia Commons

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OPTIMIZATION

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OPTIMIZATION

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OPTIMIZATION - GRADIENT CALCULATION Goal: = + + + b y ^ w 0 x 0 w 1 x 1 w 2 x 2 ϵ = (y − y ^) 2 , ∂ϵ ∂w i ∂ϵ ∂b

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OPTIMIZATION - GRADIENT CALCULATION Chain rule: = ∂ϵ ∂w i dϵ dy ^ ∂y ^ ∂w i

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OPTIMIZATION - GRADIENT CALCULATION = + + + b y ^ w 0 x 0 w 1 x 1 w 2 x 2 = ∂y ^ ∂w 0 x 0

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OPTIMIZATION - GRADIENT CALCULATION ϵ = (y − y ^) 2 = dϵ dy ^ −2(y − ) y ^

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OPTIMIZATION - GRADIENT CALCULATION = ∂y ^ ∂w 0 x 0 = −2(y − ) dϵ dy ^ y ^ = −2(y − ) ∂ϵ ∂w 0 y ^ x 0

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OPTIMIZATION - GRADIENT CALCULATION = + + + b ⋅ 1 y ^ w 0 x 0 w 1 x 1 w 2 x 2 = −2(y − ) ⋅ 1 ∂ϵ ∂b y ^

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OPTIMIZATION - GRADIENT CALCULATION delta_y = y - y_hat gradient_weights = -2 * delta_y * weights gradient_intercept = -2 * delta_y * 1

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OPTIMIZATION - PARAMETER UPDATE weights = weights - gradient_weights intercept = intercept - gradient_intercept

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OPTIMIZATION - OVERSHOOT

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OPTIMIZATION - UNDERSHOOT

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OPTIMIZATION - PARAMETER UPDATE learning_rate = 0.05 weights = weights - \ learning_rate * gradient_weights intercept = intercept - \ learning_rate * gradient_intercept

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TRAINING def training_round(x, y, weights, intercept, alpha=learning_rate): # calculate our estimate y_hat = model(x, weights, intercept) # calculate error delta_y = y - y_hat # calculate gradients gradient_weights = -2 * delta_y * weights gradient_intercept = -2 * delta_y # update parameters weights = weights - alpha * gradient_weights intercept = intercept - alpha * gradient_intercept return weights, intercept

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TRAINING NUM_EPOCHS = 100 def train(X, Y): # initialize parameters weights = np.random.randn(3) intercept = 0 # training rounds for i in range(NUM_EPOCHS): for (x, y) in zip(X, Y): weights, intercept = training_round(x, y, weights, intercept)

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TESTING def test(X_test, Y_test, weights, intercept): Y_predicted = model(X_test, weights, intercept) error = cost(Y_predicted, Y_test) return np.sqrt(np.mean(error)) >>> test(X_test, Y_test, final_weights, final_intercept) 6052.79

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Uh, wasn't this supposed to be a talk about neural networks? Why are we talking about linear regression?

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SURPRISE! YOU'VE ALREADY MADE A NEURAL NETWORK!

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LINEAR REGRESSION = SIMPLEST NEURAL NETWORK

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ONCE MORE, WITH TENSORFLOW

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Inputs Model - Parameters Model - Operations Cost function Optimization Train Test

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INPUTS → PLACEHOLDERS import tensorflow as tf X = tf.placeholder(tf.float32, [None, 3]) Y = tf.placeholder(tf.float32, [None, 1])

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PARAMETERS → VARIABLES # create tf.Variable(s) W = tf.get_variable("weights", [3, 1], initializer=tf.random_normal_initializer()) b = tf.get_variable("intercept", [1], initializer=tf.constant_initializer(0))

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OPERATIONS Y_hat = tf.matmul(X, W) + b

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COST FUNCTION cost = tf.reduce_mean(tf.square(Y_hat - Y))

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OPTIMIZATION learning_rate = 0.05 optimizer = tf.train.GradientDescentOptimizer (learning_rate).minimize(cost)

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TRAINING with tf.Session() as sess: # initialize variables sess.run(tf.global_variables_initializer()) # train for _ in range(NUM_EPOCHS): for (X_batch, Y_batch) in get_minibatches( X_train, Y_train, BATCH_SIZE): sess.run(optimizer, feed_dict={ X: X_batch, Y: Y_batch })

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# Placeholders X = tf.placeholder(tf.float32, [None, 3]) Y = tf.placeholder(tf.float32, [None, 1]) # Parameters/Variables W = tf.get_variable("weights", [3, 1], initializer=tf.random_normal_initializer()) b = tf.get_variable("intercept", [1], initializer=tf.constant_initializer(0)) # Operations Y_hat = tf.matmul(X, W) + b # Cost function cost = tf.reduce_mean(tf.square(Y_hat - Y)) # Optimization optimizer = tf.train.GradientDescentOptimizer (learning_rate).minimize(cost) # ------------------------------------------------ # Train with tf.Session() as sess: # initialize variables sess.run(tf.global_variables_initializer()) # run training rounds for _ in range(NUM_EPOCHS): for X_batch, Y_batch in get_minibatches( X_train, Y_train, BATCH_SIZE): sess.run(optimizer, feed_dict={X: X_batch, Y: Y_batch})

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COMPUTATION GRAPH

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COMPUTATION GRAPH

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FORWARD PROPAGATION

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FORWARD PROPAGATION

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FORWARD PROPAGATION

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FORWARD PROPAGATION

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FORWARD PROPAGATION

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FORWARD PROPAGATION def training_round(x, y, weights, intercept, alpha=learning_rate): # calculate our estimate y_hat = model(x, weights, intercept) # calculate error delta_y = y - y_hat # calculate gradients gradient_weights = -2 * delta_y * weights gradient_intercept = -2 * delta_y # update parameters weights = weights - alpha * gradient_weights intercept = intercept - alpha * gradient_intercept return weights, intercept

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BACKPROPAGATION

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BACKPROPAGATION

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BACKPROPAGATION

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BACKPROPAGATION

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BACKPROPAGATION

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BACKPROPAGATION

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BACKPROPAGATION

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BACKPROPAGATION

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BACKPROPAGATION

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BACKPROPAGATION

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BACKPROPAGATION

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BACKPROPAGATION def training_round(x, y, weights, intercept, alpha=learning_rate): # calculate our estimate y_hat = model(x, weights, intercept) # calculate error delta_y = y - y_hat # calculate gradients gradient_weights = -2 * delta_y * weights gradient_intercept = -2 * delta_y # update parameters weights = weights - alpha * gradient_weights intercept = intercept - alpha * gradient_intercept return weights, intercept

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VARIABLE UPDATE

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VARIABLE UPDATE

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VARIABLE UPDATE

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VARIABLE UPDATE def training_round(x, y, weights, intercept, alpha=learning_rate): # calculate our estimate y_hat = model(x, weights, intercept) # calculate error delta_y = y - y_hat # calculate gradients gradient_weights = -2 * delta_y * weights gradient_intercept = -2 * delta_y # update parameters weights = weights - alpha * gradient_weights intercept = intercept - alpha * gradient_intercept return weights, intercept

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NUMPY → TENSORFLOW sess.run(optimizer, feed_dict={ X: X_batch, Y: Y_batch })

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TESTING with tf.Session() as sess: # train # ... (code from above) # test Y_predicted = sess.run(model, feed_dict = {X: X_test}) squared_error = tf.reduce_mean( tf.square(Y_test, Y_predicted)) >>> np.sqrt(squared_error) 5967.39

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LOGISTIC REGRESSION

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PROBLEM

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BINARY CLASSIFICATION

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BINARY LOGISTIC REGRESSION - MODEL Take a weighted sum of the features and add a bias term to get the logit. Convert the logit to a probability via the logistic-sigmoid function.

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BINARY LOGISTIC REGRESSION - MODEL

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LOGISTIC-SIGMOID FUNCTION f(x) = e x 1+ex

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CLASSIFICATION WITH LOGISTIC REGRESSION Image generated with playground.tensor ow.org

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MODEL

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SOFTMAX Z = np.sum(np.exp(logits))

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MODEL

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PLACEHOLDERS # X = vector length 784 (= 28 x 28 pixels) # Y = one-hot vectors # digit 0 = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0] X = tf.placeholder(tf.float32, [None, 28*28]) Y = tf.placeholder(tf.float32, [None, 10])

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VARIABLES # Parameters/Variables W = tf.get_variable("weights", [784, 10], initializer=tf.random_normal_initializer()) b = tf.get_variable("bias", [10], initializer=tf.constant_initializer(0))

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OPERATIONS Y_logits = tf.matmul(X, W) + b

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COST FUNCTION cost = tf.reduce_mean( tf.nn.softmax_cross_entropy_with_logits( logits=Y_logits, labels=Y))

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COST FUNCTION Cross Entropy H( ) = − log( ) y ^ ∑ i y i y ^ i

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OPTIMIZATION learning_rate = 0.05 optimizer = tf.train.GradientDescentOptimizer (learning_rate).minimize(cost)

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TRAINING with tf.Session() as sess: sess.run(tf.global_variables_initializer()) for i in range(NUM_EPOCHS): for (X_batch, Y_batch) in get_minibatches( X_train, Y_train, BATCH_SIZE): sess.run(optimizer, feed_dict={X: X_batch, Y: Y_batch})

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TESTING predict = tf.argmax(Y_logits, 1) with tf.Session() as sess: # training code from above predictions = sess.run(predict, feed_dict={X: X_test}) accuracy = tf.reduce_mean(np.mean( np.argmax(Y_test, axis=1) == predictions) >>> accuracy 0.925

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DEFICIENCIES OF LINEAR MODELS Image generated with playground.tensor ow.org

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DEFICIENCIES OF LINEAR MODELS Image generated with playground.tensor ow.org

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LET'S GO DEEPER!

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ADDING ANOTHER LAYER

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ADDING ANOTHER LAYER - VARIABLES HIDDEN_NODES = 128 W1 = tf.get_variable("weights1", [784, HIDDEN_NODES], initializer=tf.random_normal_initializer()) b1 = tf.get_variable("bias1", [HIDDEN_NODES], initializer=tf.constant_initializer(0)) W2 = tf.get_variable("weights2", [HIDDEN_NODES, 10], initializer=tf.random_normal_initializer()) b2 = tf.get_variable("bias2", [10], initializer=tf.constant_initializer(0))

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ADDING ANOTHER LAYER - OPERATIONS hidden = tf.matmul(X, W1) + b1 y_logits = tf.matmul(hidden, W2) + b2

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RESULTS # hidden layers Train accuracy Test accuracy 0 93.0 92.5 1 89.2 88.8

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IS DEEP LEARNING JUST HYPE? (Well, it's a little bit over-hyped...)

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PROBLEM A linear transformation of a linear transformation is still a linear transformation! We need to add non-linearity to the system.

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ADDING NON-LINEARITY

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ADDING NON-LINEARITY

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NON-LINEAR ACTIVATION FUNCTIONS

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ADDING NON-LINEARITY

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OPERATIONS hidden = tf.nn.relu(tf.matmul(X, W1) + b1) y_logits = tf.matmul(hidden, W2) + b2

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RESULTS # hidden layers Train accuracy Test accuracy 0 93.0 92.5 1 97.9 95.2

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WHAT THE HIDDEN LAYER BOUGHT US Image generated with playground.tensor ow.org

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WHAT THE HIDDEN LAYER BOUGHT US Image generated with playground.tensor ow.org

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ADDING HIDDEN NEURONS 2 hidden neurons Image generated with ConvNetJS by Andrej Karpathy

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ADDING HIDDEN NEURONS 3 hidden neurons Image generated with ConvNetJS by Andrej Karpathy

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ADDING HIDDEN NEURONS 4 hidden neurons Image generated with ConvNetJS by Andrej Karpathy

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ADDING HIDDEN NEURONS 5 hidden neurons Image generated with ConvNetJS by Andrej Karpathy

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ADDING HIDDEN NEURONS Image generated with ConvNetJS by Andrej Karpathy

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ADDING HIDDEN NEURONS Image generated with ConvNetJS by Andrej Karpathy

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UNIVERSAL APPROXIMATION THEOREM A feedforward network with a single hidden layer containing a nite number of neurons can approximate (basically) any interesting function

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ARE WE DEEP LEARNING YET? No!

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OPERATIONS hidden_1 = tf.nn.relu(tf.matmul(X, W1) + b1) hidden_2 = tf.nn.relu(tf.matmul(hidden_1, W2) + b2) y_logits = tf.matmul(hidden_2, W3) + b3

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WHY GO DEEP? 3 reasons: Deeper networks are more powerful

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MORE POWERFUL

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WHY GO DEEP? 3 reasons: Deeper networks are more powerful Narrower networks are less prone to over tting