deep learning fundamentals - cross entropy

Deep Learning FundamentalsApril 13, 2021

[email protected]

http://cross-entropy.net/ml530/Deep_Learning_1.pdf

mailto:[email protected]

http://cross-entropy.net/ml530/Deep_Learning_1.pdf

Agenda for Tonight

• Homework Review

• [DLP] Part I: Fundamentals of Deep Learning1. What is Deep Learning?

2. Before We Begin: the Mathematical Building Blocks of Neural Networks

3. Getting Started with Neural Networks

4. Fundamentals of Machine Learning

https://twitter.com/DeepLearningAI_/status/1310595139933548546?s=20

https://twitter.com/DeepLearningAI_/status/1310595139933548546?s=20

Deep Learning with Python

The cover of our text book is captioned “Habit of a Persian Lady in 1568”, from Thomas Jeffreys’ book, “A Collection of the Dresses of Different Nations”

[DLP] Chapter 1: What is Deep Learning?

1. Artificial Intelligence, Machine Learning, and Deep Learninga. Artificial Intelligenceb. Machine Learningc. Learning Representations from

Datad. The “Deep” in Deep Learninge. Understanding How Deep

Learning Works, in Three Figuresf. What Deep Learning Has

Achieved So Farg. Don’t Believe the Short-Term

Hypeh. The Promise of AI

2. Before Deep Learning: a Brief History of Machine Learninga. Probabilistic Modelingb. Early Neural Networksc. Kernel Methodsd. Decision Trees, Random Forests,

and Gradient Boosting Machinese. Back to Neural Networksf. What Makes Deep Learning

Differentg. The Modern Machine Learning

Landscape

[DLP] Chapter 1: What is Deep Learning?

3. Why Deep Learning? Why Now?a. Hardware

b. Data

c. Algorithms

d. A New Wave of Investment

e. The Democratization of Deep Learning

f. Will it Last?

• This chapter covers• High-level definitions of

fundamental concepts

• Timeline of the development of machine learning

• Key factors behind deep learning’s rising popularity and future potential

Artificial Intelligence

• Concise definition: the effort to automate intellectual tasks normally performed by humans

• Initial take: expert rules• Fine for chess

• Difficult to develop rules for image classification, speech recognition, or language translation


Relationships Between AI, ML, and DL

• Expert Rules

• Linear Regression

• Logistic Regression

• Random Forests

• Gradient Boosting

• Multi-Layer Perceptron Network

• Convolutional Neural Networks

• Recurrent Neural Networks


Expert Rules Example

% Data: fruit(X) :- attributes(Y)

fruit(banana) :- colour(yellow), shape(crescent).

fruit(apple) :- (colour(green); colour(red)), shape(sphere), stem(yes).

fruit(lemon) :- colour(yellow), (shape(sphere);shape('tapered sphere')), acidic(yes).

fruit(lime) :- colour(green), shape(sphere), acidic(yes).

fruit(pear) :- colour(green), shape('tapered sphere').

fruit(plum) :- colour(purple), shape(sphere), stone(yes).

fruit(grape) :- (colour(purple);colour(green)), shape(sphere).

fruit(orange) :- colour(orange), shape(sphere).

fruit(satsuma) :- colour(orange), shape('flat sphere').

fruit(peach) :- colour(peach).

fruit(rhubarb) :- (colour(red); colour(green)), shape(stick).

fruit(cherry) :- colour(red), shape(sphere), stem(yes), stone(yes).

What is the value for colour?

[red, orange, yellow, green, purple, peach]

green

What is the value for shape?

[sphere, crescent, tapered sphere, flat sphere, stick]

stick

The fruit is rhubarb

http://www.paulbrownmagic.com/blog/simple_prolog_expert


http://www.paulbrownmagic.com/blog/simple_prolog_expert

Machine Learning

• Ada Lovelace, 1843: “The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform.”

• Alan Turing, 1950: quoting Ada Lovelace, while pondering whether general-purpose computers could be capable of learning

trained versus programmed


Learning Representations from Data

• Need three things (for supervised learning)• Input data points: structured data, image files, sound files, text documents

• Examples of expected output

• Way to measure whether the algorithm is doing a good job

• Input representation examples• Image as Red, Green, and Blue picture element (pixel) values

• Image as Hue, Saturation, and Value

https://en.wikipedia.org/wiki/HSL_and_HSV


https://en.wikipedia.org/wiki/HSL_and_HSV

Example Data

• The inputs are the coordinates of our points

• The expected outputs are the colors of the points

• A way to measure whether our algorithm is doing a good job could be the percentage of points that are being classified correctly (accuracy)


New Representation

import numpy as np

translation = np.array([ -2, -2 ])

theta = 0.25 * np.pi

Rotation = np.array([[ np.cos(theta), - np.sin(theta) ], [np.sin(theta), np.cos(theta) ]])

np.dot(Input + translation, Rotation)


Deep Neural Network for Digit Classification

• Successive layers of increasingly meaningful representations

• Alternative names for deep learning• Layered representations learning

• Hierarchical representations learning


Deep Representations Learned by aDigit-Classification Model


How Deep Learning Works: Part 1 of 3

Neural Network Parameterized by its Weights



Loss Function Measures Quality of Network’s Output



Loss Score Used as Feedback Signal to Adjust Weights


Deep Learning Achievements

• Near-human-level image classification

• Near-human-level speech recognition

• Near-human-level handwriting transcription

• Improved machine translation

• Improved text-to-speech conversion

• Digital assistants such as Google Now and Amazon Alexa

• Near-human-level autonomous driving

• Improved ad targeting, as used by Google, Baidu, and Bing

• Improved search results on the web

• Ability to answer natural-language questions

• Superhuman Go playing


Deep Learning Hype

• Although some world-changing applications like autonomous cars are already within reach, many more are likely to remain elusive for a long time, such as believable dialogue systems, human-level machine translation across arbitrary languages, and human-level natural language understanding

• Previous AI “Winters”:1. XOR (eXclusive Or): my perception is the inability of perceptron to solve this

problem cast a shadow on AI (though this was understood at the time)2. By the early 90s, rule-based systems had proven expensive to maintain

difficult to scale, and limited in scope

• We are currently in the intense optimism phase of a new cycle


Promise of AI

• Most of the research findings of deep learning aren’t yet applied to the full range of problems they can solve across industries• “Your doctor doesn’t use AI, and neither does your accountant” [I thought I

uploaded an image of a document during tax time]

• “Back in 1995, it would have been difficult to believe in the future impact of the internet”

• “In a not-so-distant future, AI will be your assistant; it will answer your questions, help educate your kids, and watch over your health. It will deliver your groceries to your door and drive you from point A to point B.”

• Don’t believe the short-term hype, but do believe in the long-term vision


History: Probabilistic Modeling

• Naïve Bayes

• Logistic Regression

Machine Learning

Early Neural Networks

• 1950s: Perceptron

• 1980s: Backpropagation

• Late 1980s: Yann LeCun’s work on MNIST

Machine Learning

History: Kernel Method

• Example kernel method for classification

Machine Learning

History: Decision Tree Ensembles

• Parameters that are learned are questions about the data“Is feature2 in the data greater than 3.5?”

Machine Learning

History: Back to Neural Networks

• AlexNet was not the first fast GPU-implementation of a CNN to win an image recognition contest. A CNN on GPU by K. Chellapilla et al. (2006) was 4 times faster than an equivalent implementation on CPU.[6] A deep CNN of Dan Ciresan et al. (2011) at IDSIA was already 60 times faster[7] and achieved superhuman performance in August 2011.[8] Between May 15, 2011 and September 10, 2012, their CNN won no less than four image competitions.[9][10] They also significantly improved on the best performance in the literature for multiple image databases.[11]

• According to the AlexNet paper,[5] Ciresan's earlier net is "somewhat similar." Both were originally written with CUDA to run with GPU support. In fact, both are actually just variants of the CNN designs introduced by Yann LeCun et al. (1989)

https://en.wikipedia.org/wiki/AlexNet

Machine Learning

https://en.wikipedia.org/wiki/AlexNet#cite_note-6

https://en.wikipedia.org/wiki/IDSIA

https://en.wikipedia.org/wiki/AlexNet#cite_note-flexible-7



https://en.wikipedia.org/wiki/AlexNet#cite_note-schdeepscholar-10

https://en.wikipedia.org/wiki/Database

https://en.wikipedia.org/wiki/AlexNet#cite_note-mcdns-11

https://en.wikipedia.org/wiki/AlexNet#cite_note-:0-5

https://en.wikipedia.org/wiki/CUDA

https://en.wikipedia.org/wiki/GPU

https://en.wikipedia.org/wiki/Yann_LeCun

https://en.wikipedia.org/wiki/AlexNet

Reasons for Deep Learning Success

• Incremental layer-by-layer way in which increasingly complex representations are developed

• Fact that these intermediate incremental representations are learned jointly

Machine Learning

Why Deep Learning? Why Now?

• Three technical forces driving advances in machine learning• Hardware

• Datasets and benchmarks

• Algorithmic advances

• Following a scientific revolution, progress generally follows a sigmoid curve: it starts with a period of fast progress, which generally stabilizes as researchers hit hard limitations, and then, further improvements become incremental

Deep Learning

Models to Try

• Deep Learning should be viewed as another tool in the toolbox

• There are many possible machine learning methods to apply

• Suggestion is to try …• Linear models

• Tree-based ensembles; e.g. random forests and gradient boosting

• Deep learning; e.g. feedforward, convolutional, and recurrent networks

• Gradient boosting and deep learning have won a lot of Kaggle competitions

Deep Learning

[DLP] Chapter 2: Before We Begin, the Mathematical Building Blocks of Neural Networks

1. A first look at a neural network

2. Data representations for neural networks

3. The gears of neural networks: tensor operations

4. The engine of neural networks: gradient-based optimization

5. Looking back at our first example

6. Chapter summary

MNIST Sample Digits

• The Modified (Segmented) National Institutes of Standards and Technology (MNIST) data set is part of the history of Deep Learning

• Yann LeCun (Bell Labs at the time) used this data to learn convolution filters in 1989

http://yann.lecun.com/exdb/mnist/

First Look

http://yann.lecun.com/exdb/mnist/

Loading the MNIST Data

First Look

Network Architecture and Compilation

First Look

Preprocessing the Data

First Look

Network Training and Evaluation

First Look

Scalars (0D Tensors)

Another name for a number

Data Representations

shape = ()

Vectors (1D Tensors)

Another name for a one-dimensional array of numbers


shape = (5,)

Matrices (2D Tensors)

Another name for a two-dimensional array of numbers


shape = (3, 5)

3D Tensors and Higher-Dimensional Tensors

Packing 2D tensors into an array creates a 3D tensor


shape = (3, 3, 5)

Key Attributes of a Tensor

• Number of axes: sometimes called the rank; sometimes called the number of dimensions [the number of indices for specifying a cell]

• Shape: a list consisting of sizes for the axes of the tensor

• Data type• int32: typically used for word indices and class indices

• uint8: typically used for pixel values

• float32: typically used for numeric features


Displaying a Digit

plt.imshow(Image.open(“filename.png”)) works too


Manipulating Tensors in Numpy


The Notion of Data Batches

The first axis is considered to be the batch axis


Real-World Examples of Data Tensors

• Vector data: 2D tensors of shape (samples, features)

• Timeseries data or sequence data: 3D tensors of shape (samples, timesteps, features)

• Images: 4D tensor of shape (samples, height, width, channels) or (samples, channels, height, width)

• Video: 5D tensor of shape (samples, frames, height, width, channels) or (samples, frames, channels, height, width)


Examples of Vector Data


Examples of Timeseries Data or Sequence Data

Mel Frequency Cepstral Coefficient (MFCC) representation of audio clips fits here as well


Image Data

• Tensorflow uses “channels last” format; while the no-longer-maintained Theano used “channels first” format


Video Data


The Gears of Neural Networks: Tensor Operations

Tensor Operations

Element-wise Operations

Tensor Operations

Broadcasting

• Broadcasting is used to add bias values to the product of an input matrix and a weight matrix

(samples, features) x (features, neurons) + (neurons)

= (samples, neurons) + neurons

• Broadcasting consists of two steps

Tensor Operations

Example: Adding a Vector to a Matrix

Tensor Operations

Tensor Dot: Part 1 of 3

Tensor Operations


Tensor Operations

Tensor Reshaping: Part 1 of 2

Tensor Operations

Tensor Reshaping: Part 2 of 2

Tensor Operations

Geometric Interpretation of Tensor OperationsThis example illustrates something similar to what happens during a weight update operation based on momentum

Tensor Operations

A Geometric Interpretation of Deep Learning

Uncrumpling a complicated manifold of data (think about the XOR problem)

Tensor Operations

What’s a Derivative?

• A derivative quantifies the change in a function’s value as an input changes

• Produces the change needed for the path of steepest ascent

Gradient-Based Optimization

Stochastic Gradient Descent: Part 1 of 3



Momentum can help us avoid local minima


Chaining Derivatives: the Backpropagation AlgorithmWe take derivatives for the operations used as part of forward propagation (inference) to update weights

Book says:

Alternative version of chain rule:

𝜕𝑙𝑜𝑠𝑠

𝜕𝑤𝑒𝑖𝑔ℎ𝑡−1=

𝜕𝑙𝑜𝑠𝑠

𝜕𝑎𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛−1

𝜕𝑎𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛−1𝜕𝑝𝑟𝑜𝑑𝑢𝑐𝑡−1

𝜕𝑝𝑟𝑜𝑑𝑢𝑐𝑡−1𝜕𝑤𝑒𝑖𝑔ℎ𝑡−1


Network Review

Looking Back

Chapter Summary

Activation Functions

[DLP] Chapter 3: Getting Started with Neural Networks1. Anatomy of a Neural Network

2. Introduction to Keras

3. Setting Up a Deep Learning Workstation

4. Classifying Movie Reviews: a Binary Classification Example

5. Classifying Newswires: a MultiClass Classification Example

6. Predicting House Prices: a Regression Example

7. Chapter Summary

Neural Networks

Training involves …

Anatomy of a Neural Network

Relationship between the Network, Layers, Loss Function, and Optimizer


Layers as the Lego Bricks of Deep Learning ☺

The first layer requires an input_shape parameter [the dimensions of a single observation], while additional layers do not require this parameter


Networks of Layers

• A deep learning model is a directed, acyclic graph of layers

• Most common instance is a “linear” (simple, sequential) stack of layers

• Other common instances include …• Two-branch networks; e.g. question goes down one branch and text passage

goes down another (or maybe multi-modal input, for example an image and a text description)

• Multi-head networks; e.g. we have one output predict whether a news article discusses “politics” and another output predict whether a news article discusses “health”

• Inception blocks; e.g. we want to use a few different convolution (input filtering) approaches in parallel


Loss Function and Optimizers

• Loss functions for this class include: cross entropy, mean squared error, mean absolute error, content loss, style loss, total variation loss, Kullback Leibler loss, temporal difference loss, actor loss, and critic loss

• Optimization functions for the class include: Stochastic Gradient Descent (SGD), Root Mean Squared (Gradient) Propagation (RMSProp), and Adaptive Moments (AdaM: RMSProp + Momentum)


Keras Features

Introduction to Keras

Google Search Interest for Deep Learning Frameworks


Deep Learning Software and Hardware Stack

• Nvidia Graphics Processing Units (GPUs) and Google Tensor Processing Units (TPUs) support efficient deep learning

• Nvidia’s Common Unified Device Architecture (CUDA) Application Programming Interface (API) and the CUDA Deep Neural Network (DNN) library provide an interface to Nvidia GPUs

• Eigen library implements the Basic Linear Algebra Subprograms (BLAS) specification, allowing tensor manipulation on Central Processing Units (CPUs)

Theano and CNTK are no longer maintained


Typical Keras Workflow


Network Definition:Sequential Model versus the Functional APISame model with both methods …


Model Configuration and Training


Two Options for Getting Keras Running

Setting Up a Deep Learning Workstation

Loading the Internet Movie DataBase (IMDB) Sentiment Analysis Datanum_words is the size of the vocabulary

Classifying Movie Reviews

Decoding a Document


Turning Lists of Integers into Tensors

Note: if more than one value in a row is one, we should refer to this as a multi-hot encoding

• one-hot encoding for identifying a class in a dense target vector

• multi-hot encoding to identify the tokens present in a document in a dense input vector


Encoding the Integer Sequences into a Binary Matrix


Architecture Decisions for Simple Feedforward Network• How many layers to use

• How many hidden units to choose for each layer

• Which activation functions to use• Do *not* forget to include activation functions: unexplained suboptimality

will ensue


Common Activation Functions

Rectified Linear Unit (ReLU): max(x,0)

[no saturation issue]

Sigmoid: 1/(1+exp(-x))

[usually used for output layer]


IMDB Network Architecture

• Features flow from bottom to top• An output is called a “head”

• Two hidden layers and an output layer with weights• It’s a deep neural network

• We’ll get to more than one hundred layers soon enough


Model Definition


Parameters and Outputs for a Dense Layer

• Parameters• (Number of Inputs from Previous Layer + 1) * (Number of “Units”)

• + 1 for bias weights: one for each “unit”

• We used to refer to “units” as neurons• The names have been changed to protect the innocent? Our approach was inspired by

neuroscience, but our brains aren’t using RMSProp ☺

• These are the same weight vectors we’ve come to know and love: projecting inputs to a new representation, one feature at a time [the number of “units” is the number of new features for the new representation]

• Output Shape• (Batch Size) x (Number of “Units”)


Why Are Activation Functions Necessary?

Try omitting activation functions from 1) the output layer and 2) hidden layers … so you can recognize this issue later


Compiling the Model


Setting Aside a Validation Set


Training the Model


Plotting the Training and Validation Loss


Where Do We Start Overfitting?


Plotting the Training and Validation Accuracy


Retraining the Model from Scratch

Why are we “retraining the model from scratch”?


Generating Predictions on New Data


Ideas for Experiments


Wrapping Up the IMDB Example


Loading the Reuters Dataset

Classifying Newswires

Decoding Newswires Back to Text


Preparing the Document Matrices


Preparing the Target Matrices


Defining the Model


Notes About the Architecture


Validating the Approach


Retraining a Model from Scratch


Why are we “retraining a model from scratch”?

Comparing to Random[and a Majority Classifier]

Nota bene (note well): 813 of the 2,356 test examples belonged to class 3

The accuracy of a majority classifier is 36.2%


Generating Predictions for New Data


Dense Versus Sparse Labels


Model With an Information Bottleneck

71% accuracy: an 8% absolute drop


Further Experiments


Wrapping Up


Loading the Boston Housing Dataset

1970s home prices in thousands of dollars

Predicting House Prices

Brief Discussion of Bias

• Boston Housing dataset has been used by many popular textbooks

• The data explicitly offers a race-related variable for modeling• Avoid using proxy variables that lead to discrimination based on race, gender,

religion, etc

• Example: don’t ask about gender if all you really want to know is whether the candidate can lift X pounds


http://lib.stat.cmu.edu/datasets/boston

http://lib.stat.cmu.edu/datasets/boston

Normalizing the Data


Model Definition


3-Fold Cross-Validation


4-Fold Cross-Validation Implementation


Cross-Validation Loop


Cross-Validation Results


Alternative Implementation [saved history]


Plotting the Average Mean Absolute Error (MAE)


Visualization Suggestions


Smoothing the Curve

The smoothed_points expression should look familiar: RMSProp (0.9 for last squared gradient) and AdaM (0.999 for last gradient)


Plotting the Smoothed MAE


Training the Final Model


Wrapping Up


Chapter Summary

[DLP] Chapter 4: Fundamentals of Machine Learning1. Four Branches of Machine Learning

2. Evaluating Machine Learning Models

3. Data Preprocessing, Feature Engineering, and Feature Learning

4. Overfitting and Underfitting

5. The Universal Workflow of Machine Learning

Supervised Learning Examples

Four Branches of Machine Learning

Unsupervised Learning

• Dimensionality Reduction

• Clustering


Self-Supervised Learning

• Learning Without Human Annotated Labels

• Autoencoders

• Trying to predict the next word given previous words

• Trying to predict the next frame given previous frames


Reinforcement Learning

• Google Deep Mind used reinforcement learning to create a model to play Atari games

• AlphaGo was created to play Go

• Occasional rewards

• Examples of possible applications include: self-driving cars, robotics, resource management, and education


From Yann LeCun …

https://t.co/2LSb622114



Also from Yann LeCun …




Classification and Regression Glossary


Simple Hold-out Validation Split

Evaluating Machine Learning Models

Hold-out Validation Implementation[note the concatenation]


K-Fold Cross-Validation

Used for smaller data sets• If K is too small, we’ll experience high bias (underfitting)

• If K is too large, we’ll experience high variance (overfitting)


K-Fold Cross Validation Implementation


Iterated K-Fold Cross-Validation with Shuffling

• history = []

• for i in range(iterationCount):• shuffle(data)

• history.append(crossValidation(data, K = k))

• Requires building iterationCount * K + 1 models


Things to Keep in Mind


Value Normalization

• Dividing by 255 was an example of min-max normalization:

• value = (value – min(value)) / (max(value) – min(value))

• The max pixel value was 255 and the min pixel value was 0

• Alternatively, you can use center-and-scale normalization:

[-1,1] is fine too

Consider removing outliers

Data Preprocessing, Feature Engineering, and Feature Learning

Missing Values

• “In general, with neural networks, it’s safe to input missing values as 0, with the condition that zero isn’t a meaningful value”

• It’s possible to add indicator variables: 1 if missing; 0 otherwise

• If you expect missing values at test time, be sure to train with missing values:• We train like we deploy and deploy like we train


Feature Engineering Example

Three different inputs for the “What time is it?” model …

Why no radius on the polar coordinates?


Feature Engineering

• Does this mean you don’t have to worry about feature engineering as long as you’re using deep neural networks?

• No …


Original versus Lower Capacity Model

Original Model:16 units for each hidden layer

Lower Capacity Model: 4 units for each hidden layer

Overfitting and Underfitting

Original versus Lower Capacity Model

Smaller network starts overfitting later and it’s performance degrades more slowly


Original versus Higher Capacity Model:Validation Data

Validation Loss Noisierfor Higher Capacity Model(512 versus 16 units for each hidden layer)


Original versus Higher Capacity Model:Training DataMore capacity gives a model the ability to more quickly model the training data, but it also makes it susceptible to overfitting


Regularization [for Smaller Weights]


Example for Effect of Weight Regularization

• Note: the goal of weight regularization is to improvegeneralization performance ☺

• Use “metric” rather than “loss” for comparing generalization performance; e.g. regularized crossentropy can be used for the loss function with crossentropyused for the evaluation metric [this allows us to use the same evaluation function when comparing performance of models on validation data]


Additional Weight Regularizers for Keras


Adding Dropout(dropoutRate)


Adding Dropout to the IMDB Network


Recap of Most Common Ways to Prevent Overfitting


Define the Problem

Universal Workflow of Machine Learning

Hypothesis


Choosing a Measure of Success

• Accuracy

• Precision and Recall

• Area Under the Receiver Operating Characteristic (ROC) Curve (AUC)

• Maximize Recall subject to a constraint on the False Positive Rate?

• Mean Average Precision


Deciding on an Evaluation Protocol


Preparing Your Data


Key Choices for Your First Iteration


Choosing the Last-Layer Activation and Loss Function


How Big Should the Model Be?

Developing a model that overfits …


Regularizing the Model


Tuning the Model

We call these hyperparameters to distinguish them from the parameters of the model; i.e. the weights.

Note: We tune against validation data. Much like the “private leaderboard”, we only get one look at test perf.


Chapter Summary

plot_model() and tensorboard

# To use plot_model() to generate ".png" file:

# $ sudo apt install python-pydot

# $ pip install pydot

# $ pip install graphviz

# To review tensorboard output:

# Start the tensorboard server ...

# $ tensorboard --logdir=logs --bind_all

# Use browser to navigate to tensorboard server ...

# http://host:6006/

# Tensorboard reference: https://github.com/tensorflow/tensorboard

from tensorflow.keras.datasets import mnist

from tensorflow.keras.models import Sequential

from tensorflow.keras.layers import Dense, Dropout

from tensorflow.keras.utils import plot_model

from tensorflow.keras.callbacks import TensorBoard

(x_train, y_train), (x_test, y_test) = mnist.load_data()

x_train = x_train.reshape(60000, 784).astype("float32") / 255

x_test = x_test.reshape(10000, 784).astype("float32") / 255

model = Sequential()

model.add(Dense(512, activation="relu", input_shape=(784,), name = "hidden0"))

model.add(Dropout(0.2, name = "dropout0"))

model.add(Dense(512, activation="relu", name = "hidden1"))

model.add(Dropout(0.2, name = "dropout1"))

model.add(Dense(10, activation="softmax", name = "output"))

model.summary()

plot_model(model, to_file = "model.png", show_shapes = True)

model.compile(loss = "sparse_categorical_crossentropy",

optimizer = "rmsprop",

metrics =[ "accuracy" ])

history = model.fit(x_train, y_train,

batch_size = 128, epochs = 20,

validation_split = 0.1,

callbacks = [ TensorBoard(log_dir = "logs", histogram_freq = 5) ])

score = model.evaluate(x_test, y_test)

print(f"Test loss: {score[0]}\tTest accuracy: {score[1]}")

Bonus Material

https://github.com/tensorflow/tensorboard

TensorBoard Scalars

TensorBoard Graphs

TensorBoard DistributionsFrom top to bottom the lines represent: [maximum, 93%, 84%, 69%, 50%, 31%, 16%, 7%, minimum]

TensorBoard Histograms

deep learning fundamentals - cross entropy

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