Regression

A Julia package for regression analysis.

This package is based on EmpiricalRisks, and provides a set of algorithms to perform regression analysis.

This package supports all regression problems that can be formulated as regularized empirical risk minimization, as

In particular, it supports:

• Linear regression
• Ridge regression
• LASSO
• Logistic regression
• Multinomial Logistic regression
• Problems with customized loss and regularizers

The package also provides a variety of solvers

• Analytical solution (for linear & ridge regression)
• Gradient descent
• BFGS
• L-BFGS
• Proximal gradient descent (recommended for LASSO & sparse regression)
• Accelerated gradient descent (experimental)
• Accelerated proximal gradient descent (experimental)

The package provides a high-level interface to simplify typical use.

Example:

The following script shows how one can use this package to perform logistic regression:

d = 3      # sample dimension
n = 1000   # number of samples

# prepare data
w = randn(d+1)    # generate the weight vector
X = randn(d, n)   # generate input features
y = sign(X'w[1:d] + w[d+1] + 0.2 * randn(n))  # generate (noisy) response

# perform estimation
ret = Regression.solve(
    logisticreg(X, y; bias=1.0),   # construct a logistic regression problem
    reg=SqrL2Reg(1.0e-2),          # apply squared L2 regularization
    options=Options(verbosity=:iter, grtol=1.0e-6 * n))  # set options

# extract results
w_e = ret.sol

The high-level interface involves two parts: problem construction and problem solving.

The package provide several functions to construct regression problems:

• UnivariateRegression(loss, X, Y, bias)

  Construct a univariate regression problem, where the both arguments to the loss function are scalars.

  params	descriptions
  loss	the loss function, which should be an instance of UnivariateLoss.
  X	a matrix of inputs (as columns)
  y	a vector of corresponding outputs
  bias	The bias term

  Let d be the length of each input. When bias is zero, the parameter w is a vector of length d, and the prediction is given by w'x. When bias is non-zero, the parameter w is a vector of length d+1, and the prediction is given by w[1:d]'x + w[d+1].

• MultivariateRegression(loss, X, Y, k, bias)

  Construct a multivariate regression problem, where the prediction is a vector.

  params	descriptions
  loss	the loss function, which should be an instance of MultivariateLoss.
  X	a matrix of inputs (as columns)
  y	a matrix of corresponding outputs (as columns)
  k	The length of each prediction output
  bias	The bias term

  Let d be the length of each input. When bias is zero, the parameter W is a matrix of size (k, d), and the prediction is given by W * x. When bias is non-zero, the parameter W is a matrix of size (k, d+1), and the prediction is given by W[:, 1:d] * x + W[:,d+1].

The package also provides convenience functions to construct common problems:

• linearreg(X, Y[; bias=0])

  Construct a linear regression problem.

  When Y is a vector, it is a univariate regression problem, when Y is a matrix, it is a multivariate regression problem.

  Note that each column of X corresponds to a sample. The same applies to Y when Y is a matrix.

• logisticreg(X, y[; bias=0])

  Construct a logistic regression problem.

• mlogisticreg(X, y, k[; bias=0])

  Construct a multinomial logistic regression problem.

  Here, X is a sample matrix, y is a vector of class labels (values in 1:k), and k be the number of classes.

With a constructed problem, you can solve the problem with the solve function.

Note: The solve function is not exported (in order to avoid confliction with other optimization packages). You should write Regression.solve when calling this function.

• Regression.solve(pb[; ...])

  Solve the regression problem pb, which can be constructed using the construction functions above.

  This function allows the users to supply the following keyword arguments:

  params	description
  reg	The regularizer. (See documentation on regularizers for details.)
  init	The initial guess of the parameters. (If omitted, we use all-zeros as initial guess by default)
  solver	The chosen solver (see below for details). The default is BFGS()
  options	The options to control the solving procedure (see below for details)
  callback	The callback function, which will be invoked at each iteration. in the following way: callback(t, theta, v, g), where t is the iteration number, theta is the solution at current step, v is the current objective value, and g is the current gradient. Default is no_op, which does nothing.

• Regression.Options(...)

  Construct an option struct to control the solving procedure.

  It accepts the following keyword arguments:

  params	description
  maxiter	The maximum number of iterations (default = 200)
  ftol	Tolerance of function value changes (default = 1.0e-6)
  xtol	Tolerance of solution change (default = 1.0e-8)
  grtol	Tolerance of the gradient norm (default = 1.0e-8)
  armijo	The Armijo coefficient in line search
  beta	The back tracking ratio in line search
  verbosity	The level of display, which is a symbol, whose value can be :none, :final, or :iter. (default = :none)

As mentioned, the package implements a variety of solvers, one can construct a solver using the following functions:

GD()       # Gradient descent
BFGS()     # BFGS Quasi-Newton method
LBFGS(m)   # L-BFGS method (with history size m)
ProxGD()   # Proximal gradient descent (suitable for sparse learning, etc)

# the following solver remains in experimental status
AGD()      # Accelerated gradient descent
ProxAGD()  # Accelerated proximal gradient descent

Those who care more on performance can directly call the Regression.solve! function, as follows:

# Note: solve! will update the solution θ inplace
function solve!{T<:FloatingPoint}(
    solver::DescentSolver,  # the chosen solver
    f::Functional{T},       # the objective functional
    θ::Array{T},            # the solution (which would be updated inplace)
    options::Options,       # options to control the procedure
    callback::Function)     # callback function

# Here, the functional f can be constructed using the following functions:

# empirical risk minimization
f = RiskFun(rmodel, X, Y)   # rmodel is the risk model

# regularized empirical risk minimization
f = RegRiskFun(rmodel, reg, X, Y)   # rmodel is the risk model, reg is the regularizer

Note that for linear regression and ridge regression, there exists analytic solution. The package also provides functions that directly compute the analytic solution to these problems, using linear algebraic methods.

• llsq(X, Y; ...)

  Solve a linear least square problem.

  This function allows keyword arguments as follows:

  params	descriptions
  trans	If trans == true, it minimizes ``
  bias	The bias term, namely the value to be augmented to the inputs. Default = 0, which indicates no augmentation
  method	A symbol to indicate the matrix factorization method to be used, whose value can be qrlq, orth, or svd. Default = qrlq.

• ridgereg(X, Y, r; ...)

  Solve a ridge regression problem analytically.

  This function allows keyword arguments as follows:

  params	descriptions
  trans	If trans == true, it minimizes ``
  bias	The bias term, namely the value to be augmented to the inputs. Default = 0, which indicates no augmentation