TFOCS for Spark: a Community Port of TFOCS for Apache Spark

This package is an implementation of the TFOCS convex solver for Apache Spark.

The original Matlab TFOCS library provides building blocks to construct efficient solvers for convex problems. TFOCS for Spark implements a useful subset of this functionality, in Scala, and is designed to operate on distributed data using the Spark cluster computing framework. TFOCS for Spark includes support for:

• Convex optimization using Nesterov's accelerated method (Auslender and Teboulle variant)
• Adaptive step size using backtracking Lipschitz estimation
• Automatic acceleration restart using the gradient test
• Linear operator structure optimizations
• Smoothed Conic Dual (SCD) formulation solver, with continuation support
• Smoothed linear program solver
• Multiple data distribution patterns. (Currently support is only implemented for RDD[Vector] row matrices.)

The name "TFOCS" is being used with permission from the original TFOCS developers, who are not involved in the development of this package and hence not responsible for the support. To report issues or request features about TFOCS for Spark, please use our GitHub issues page.

LASSO Example

Solve the l1 regularized least squares problem 0.5 * ||A * x' - b||_2^2 + lambda * ||x||_1 (lasso linear regression):

import org.apache.spark.mllib.linalg.{ DenseVector, Vectors }
import org.apache.spark.mllib.optimization.tfocs.SolverL1RLS

// Design matrix
val A = sc.parallelize(Array(
  Vectors.dense(0.61, 0.98, 0.32),
  Vectors.dense(0.10, 0.22, 0.92),
  Vectors.dense(0.79, 0.02, 0.20)), 2)

// Observations
val b = sc.parallelize(Array(3.69, 3.36, 1.59), 2).glom.map(new DenseVector(_))

// Regularization term
val lambda = 0.1

SolverL1RLS.run(A, b, lambda)

Alternatively, the above optimization may be performed using the TFOCS optimizer directly rather than via the SolverL1RLS helper:

import org.apache.spark.mllib.optimization.tfocs.fs.dvector.double._
import org.apache.spark.mllib.optimization.tfocs.fs.vector.double._
import org.apache.spark.mllib.optimization.tfocs.fs.vector.dvector._
import org.apache.spark.mllib.optimization.tfocs.TFOCS
import org.apache.spark.mllib.optimization.tfocs.vs.dvector._
import org.apache.spark.mllib.optimization.tfocs.vs.vector._

// Initial x vector
val x0 = Vectors.zeros(3).toDense

TFOCS.optimize(new SmoothQuad(b), new LinopMatrix(A), new ProxL1(lambda), x0)

Linear Program Example

To solve the smoothed standard form linear program:

minimize c' * x + 0.5 * mu * ||x - x0||_2^2
s.t.     A' * x == b' and x >= 0

import org.apache.spark.mllib.linalg.{ DenseVector, Vectors }
import org.apache.spark.mllib.optimization.tfocs.SolverSLP

// Constraint matrix
val A = sc.parallelize(Array(
  Vectors.sparse(3, Seq((0, 0.88))),
  Vectors.sparse(3, Seq((1, 0.63))),
  Vectors.sparse(3, Seq((0, 0.29), (2, 0.18)))), 2)

// Constraint vector
var b = new DenseVector(Array(9.50, 6.84, 5.09))

// Objective vector
val c = sc.parallelize(Array(1.0, 2.0, 3.0), 2).glom.map(new DenseVector(_))

// Smoothing parameter
val mu = 1e-2

SolverSLP.run(c, A, b, mu)

Solvers

• SolverL1RLS A solver for lasso problems.
• SolverSLP A solver for smoothed standard form linear programs.
• TFOCS A general purpose convex solver.
• TFOCS_SCD A solver for problems using the TFOCS Smooth Conic Dual formulation.

Software Architecture Overview

The primary types used in the TFOCS for Spark library are as follows:

• DenseVector A wrapper around Array[Double] with support for vector operations. (Imported from org.apache.spark.mllib.linalg)

• DVector A distributed vector, stored as an RDD[DenseVector], where each partition comprises a single DenseVector containing a slice of the complete distributed vector. More information is available in org.apache.spark.mllib.optimization.tfocs.VectorSpace.

• DMatrix A distributed matrix, stored as an RDD[Vector], where each (possibly sparse) Vector represents a row of the matrix. More information is available in org.apache.spark.mllib.optimization.tfocs.VectorSpace.

The primary abstractions of the TFOCS for Spark library are as follows:

• VectorSpace A basic vector space interface with support for computing linear combinations and dot products. This abstraction supports local computation as well as distributed computation using implementations based on different data distribution models.

• LinearOperator An interface for performing a linear mapping from one vector space to another.

• SmoothFunction An interface for evaluating a smooth function and computing its gradient.

• ProxCapableFunction An interface for evaluating a function and computing the minimizing value of its proximity operator.

The following naming conventions are used in this library:

• To the extent possible, classes and functions are given the same name as the corresponding implementation in Matlab TFOCS.

• VectorSpace implementations are placed in the vs namespace. For example the VectorSpace for DVector vectors is named vs.dvector.

• Function implementations (implementations of LinearOperator, SmoothFunction, and ProxCapableFunction) are placed in the fs (function space) namespace, and are specifically named according to their input and output types. For example, functions with input type DVector and output type Double are placed in the fs.dvector.double namespace.

TODOs

• Block matrix cluster distribution pattern.
• Block matrix sparse storage format.
• Efficient computation on sparse vectors (not just sparse matrices).
• Arbitrary vector space support in TFOCS_SCD.
• Additional objective functions.