Numerical optimization
by Patrick R. Nicolas, Arun Manivannan, Pascal Bugnion
Scala: Guide for Data Science Professionals
- Scala: Guide for Data Science Professionals
- Table of Contents
- Scala: Guide for Data Science Professionals
- Scala: Guide for Data Science Professionals
- Credits
- Preface
- 1. Module 1
- 1. Scala and Data Science
- 2. Manipulating Data with Breeze
- Code examples
- Installing Breeze
- Getting help on Breeze
- Basic Breeze data types
- Vectors
- Dense and sparse vectors and the vector trait
- Matrices
- Building vectors and matrices
- Advanced indexing and slicing
- Mutating vectors and matrices
- Matrix multiplication, transposition, and the orientation of vectors
- Data preprocessing and feature engineering
- Breeze – function optimization
- Numerical derivatives
- Regularization
- An example – logistic regression
- Towards re-usable code
- Alternatives to Breeze
- Summary
- References
- 3. Plotting with breeze-viz
- 4. Parallel Collections and Futures
- 5. Scala and SQL through JDBC
- Interacting with JDBC
- First steps with JDBC
- JDBC summary
- Functional wrappers for JDBC
- Safer JDBC connections with the loan pattern
- Enriching JDBC statements with the "pimp my library" pattern
- Wrapping result sets in a stream
- Looser coupling with type classes
- Creating a data access layer
- Summary
- References
- 6. Slick – A Functional Interface for SQL
- 7. Web APIs
- 8. Scala and MongoDB
- 9. Concurrency with Akka
- GitHub follower graph
- Actors as people
- Hello world with Akka
- Case classes as messages
- Actor construction
- Anatomy of an actor
- Follower network crawler
- Fetcher actors
- Routing
- Message passing between actors
- Queue control and the pull pattern
- Accessing the sender of a message
- Stateful actors
- Follower network crawler
- Fault tolerance
- Custom supervisor strategies
- Life-cycle hooks
- What we have not talked about
- Summary
- References
- 10. Distributed Batch Processing with Spark
- 11. Spark SQL and DataFrames
- DataFrames – a whirlwind introduction
- Aggregation operations
- Joining DataFrames together
- Custom functions on DataFrames
- DataFrame immutability and persistence
- SQL statements on DataFrames
- Complex data types – arrays, maps, and structs
- Interacting with data sources
- Standalone programs
- Summary
- References
- 12. Distributed Machine Learning with MLlib
- 13. Web APIs with Play
- Client-server applications
- Introduction to web frameworks
- Model-View-Controller architecture
- Single page applications
- Building an application
- The Play framework
- Dynamic routing
- Actions
- Interacting with JSON
- Querying external APIs and consuming JSON
- Creating APIs with Play: a summary
- Rest APIs: best practice
- Summary
- References
- 14. Visualization with D3 and the Play Framework
- A. Pattern Matching and Extractors
- II. Module 2
- 1. Getting Started with Breeze
- Introduction
- Getting Breeze – the linear algebra library
- Working with vectors
- Getting ready
- How to do it...
- Creating vectors
- Constructing a vector from values
- Creating a vector out of a function
- Creating a vector of linearly spaced values
- Creating a vector with values in a specific range
- Creating an entire vector with a single value
- Slicing a sub-vector from a bigger vector
- Creating a Breeze Vector from a Scala Vector
- Vector arithmetic
- Scalar operations
- Calculating the dot product of two vectors
- Creating a new vector by adding two vectors together
- Appending vectors and converting a vector of one type to another
- Concatenating two vectors
- Standard deviation
- Find the largest value in a vector
- Finding the sum, square root and log of all the values in the vector
- Working with matrices
- Vectors and matrices with randomly distributed values
- How it works...
- Creating vectors with uniformly distributed random values
- Creating vectors with normally distributed random values
- Creating vectors with random values that have a Poisson distribution
- Creating a matrix with uniformly random values
- Creating a matrix with normally distributed random values
- Creating a matrix with random values that has a Poisson distribution
- How it works...
- Reading and writing CSV files
- 2. Getting Started with Apache Spark DataFrames
- Introduction
- Getting Apache Spark
- Creating a DataFrame from CSV
- Manipulating DataFrames
- Creating a DataFrame from Scala case classes
- 3. Loading and Preparing Data – DataFrame
- 4. Data Visualization
- 5. Learning from Data
- Introduction
- Supervised and unsupervised learning
- Gradient descent
- Predicting continuous values using linear regression
- Binary classification using LogisticRegression and SVM
- Binary classification using LogisticRegression with Pipeline API
- How to do it...
- Importing and splitting data as test and training sets
- Construct the participants of the Pipeline
- Preparing a pipeline and training a model
- Predicting against test data
- Evaluating a model without cross-validation
- Constructing parameters for cross-validation
- Constructing cross-validator and fit the best model
- Evaluating the model with cross-validation
- How to do it...
- Clustering using K-means
- Feature reduction using principal component analysis
- How to do it...
- Dimensionality reduction of data for supervised learning
- Mean-normalizing the training data
- Extracting the principal components
- Preparing the labeled data
- Preparing the test data
- Classify and evaluate the metrics
- Dimensionality reduction of data for unsupervised learning
- Mean-normalizing the training data
- Extracting the principal components
- Arriving at the number of components
- Evaluating the metrics
- How to do it...
- 6. Scaling Up
- 7. Going Further
- 1. Getting Started with Breeze
- III. Module 3
- 1. Getting Started
- 2. Hello World!
- 3. Data Preprocessing
- 4. Unsupervised Learning
- 5. Naïve Bayes Classifiers
- 6. Regression and Regularization
- 7. Sequential Data Models
- 8. Kernel Models and Support Vector Machines
- 9. Artificial Neural Networks
- Feed-forward neural networks (FFNN)
- The multilayer perceptron (MLP)
- The activation function
- The network architecture
- Software design
- Model definition
- Training cycle/epoch
- Training strategies and classification
- Evaluation
- Benefits and limitations
- Summary
- 10. Genetic Algorithms
- 11. Reinforcement Learning
- 12. Scalable Frameworks
- B. Basic Concepts
- C. Bibliography
- Index
This section briefly introduces the different optimization algorithms that can be applied to minimize the loss function, with or without a penalty term. These algorithms are described in greater detail in the Summary of optimization techniques section in Appendix A, Basic Concepts.
First, let's define the least squares problem. The minimization of the loss function consists of nullifying the first order derivatives, which in turn generates a system of D equations (also known as gradient equations), D being the number of regression weights (parameters). The weights are iteratively computed by solving the system of equations using a numerical optimization algorithm.
Note
The definition of the least squares-based loss function is as follows:
The generation of gradient equations with a Jacobian J matrix (refer to the Jacobian and Hessian matrices section in Appendix A, Basic Concepts) after minimization of the loss function L is described as follows:
Iterative approximation using the Taylor series is described as follows:
Normal equations using the matrix notation and the Jacobian 
matrix is described as follows:
The logistic regression is a nonlinear function. Therefore, it requires the nonlinear minimization of the sum of least squares. The optimization algorithms for the nonlinear least squares problems can be divided into the following two categories:
- Newton (or 2nd order techniques): These algorithms calculate the second order derivatives (the Hessian matrix) to compute the regression weights that nullify the gradient. The two most common algorithms in this category are the Gauss-Newton and the Levenberg-Marquardt methods (refer to the Nonlinear least squares minimization section in Appendix A, Basic Concepts). Both algorithms are included in the Apache Commons Math library.
- Quasi-Newton (or 1st order techniques): First order algorithms do not compute but estimate the second order derivatives of the least squares residuals from the Jacobian matrix. These methods can minimize any real-valued functions, not just the least squares summation. This category of algorithms includes the Davidon-Fletcher-Powell and the Broyden-Fletcher-Goldfarb-Shannon methods (refer to the Quasi-Newton algorithms section in Appendix A, Basic Concepts).
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