ReHLine

ReHLine is designed to be a computationally efficient and practically useful software package for large-scale empirical risk minimization (ERM) problems.

The ReHLine solver has four appealing "linear properties":

• It applies to any convex piecewise linear-quadratic loss function, including the hinge loss, the check loss, the Huber loss, etc.
• In addition, it supports linear equality and inequality constraints on the parameter vector.
• The optimization algorithm has a provable linear convergence rate.
• The per-iteration computational complexity is linear in the sample size.

We develop both Python and R interfaces to the ReHLine solver, and the core algorithm is implemented in efficient C++ code. This repository provides the rehline R package.

📝 Formulation

ReHLine is designed to address the empirical regularized ReLU-ReHU minimization problem, named ReHLine optimization, of the following form:

$$ \min_{\mathbf{\beta} \in \mathbb{R}^d} \sum_{i=1}^n \sum_{l=1}^L \text{ReLU}( u_{li} \mathbf{x}_ i^\intercal \mathbf{\beta} + v_{li}) + \sum_{i=1}^n \sum_{h=1}^H {\text{ReHU}}_ {\tau_{hi}}( s_{hi} \mathbf{x}_ i^\intercal \mathbf{\beta} + t_{hi}) + \frac{1}{2} \Vert \mathbf{\beta} \Vert_2^2, \qquad \text{ s.t. } \mathbf{A} \mathbf{\beta} + \mathbf{b} \geq \mathbf{0}, $$

where $\mathbf{U} = (u_{li}),\mathbf{V} = (v_{li}) \in \mathbb{R}^{L \times n}$ and $\mathbf{S} = (s_{hi}),\mathbf{T} = (t_{hi}),\mathbf{\tau} = (\tau_{hi}) \in \mathbb{R}^{H \times n}$ are the ReLU-ReHU loss parameters, and $(\mathbf{A},\mathbf{b})$ are the constraint parameters. The ReLU and ReHU functions are defined as $\mathrm{ReLU}(z)=\max(z,0)$ and

$$ \mathrm{ReHU}_\tau(z) = \begin{cases} \ 0, & z \leq 0 \\ \ z^2/2, & 0 < z \leq \tau \\ \ \tau( z - \tau/2 ), & z > \tau \end{cases}. $$

This formulation has a wide range of applications spanning various fields, including statistics, machine learning, computational biology, and social studies. Some popular examples include SVMs with fairness constraints (FairSVM), elastic net regularized quantile regression (ElasticQR), and ridge regularized Huber minimization (RidgeHuber).

📒 Basic usage

The usage of the rehline R package is very straightforwawrd. For general ReHLine optimization problems, simply specify the corresponding matrices in the rehline() function, or set some of them to NULL if they are not used. For example:

library(rehline)
set.seed(123)
n = 500
d = 10
L = 5
H = 4
m = 3

Xmat = matrix(rnorm(n * d), n, d)
Umat = matrix(rnorm(L * n), L, n)
Vmat = matrix(rnorm(L * n), L, n)
Smat = matrix(rnorm(H * n), H, n)
Tmat = matrix(rnorm(H * n), H, n)
Tau = Inf
Amat = matrix(rnorm(m * d), m, d)
bvec = rnorm(m)

res = rehline(
    Xmat, Umat, Vmat, Smat, Tmat, Tau, Amat, bvec,
    max_iter = 1000, tol = 1e-3, verbose = 0
)
print(res$beta)
## [1] -0.032638926 -0.029052500  0.021690510  0.004606850 -0.015769712
## [6]  0.002429307  0.008920372  0.051252755 -0.023877601 -0.013258638

res = rehline(
    Xmat, Umat = NULL, Vmat = NULL,
    Smat = Smat, Tmat = Tmat, Tau = Tau,
    Amat = NULL, bvec = NULL,
    max_iter = 1000, tol = 1e-3, verbose = 0
)
print(res$beta)
## [1] -0.051807571 -0.049215902 -0.052786551 -0.002927573 -0.040757982
## [6]  0.043229483 -0.039403712 -0.007757519 -0.025120729 -0.021587389

The rehline package also provides some functions to directly fit models on data sets. For example, the code below fits an SVM using the svm() function, and compares the result with the LiblineaR package.

library(rehline)
library(LiblineaR)

# Simulate a data set
set.seed(123)
n = 5000
p = 100
x1 = matrix(rnorm(n / 2 * p, -0.25, 0.1), n / 2)
x2 = matrix(rnorm(n / 2 * p, 0.25, 0.1), n / 2)
x = rbind(x1, x2)
beta = 0.1 * rnorm(p)
prob = plogis(c(x %*% beta))
y = 2 * rbinom(n, 1, prob) - 1
C = 0.5

# Compute using ReHLine
system.time(
    res1 <- svm(x, y, C = C * n, max_iter = 1000,
        tol = 1e-3, verbose = 0)
)
##   user  system elapsed
##  0.010   0.001   0.011
beta1 = res1$beta
head(beta1)
## [1]  0.61322066 -0.93920265  0.07600151  0.03649106  0.42362796 -0.54425647

# Compare the result with LiblineaR
system.time(
    res2 <- LiblineaR(x, y, type = 3, cost = C, epsilon = 1e-3,
        bias = 0, verbose = FALSE)
)
##   user  system elapsed
##  0.150   0.002   0.152
beta2 = as.numeric(res2$W)
head(beta2)
## [1]  0.6155191 -0.9444753  0.0734169  0.0397688  0.4258707 -0.5430257

⌛ Benchmark (powered by benchopt)

Some existing problems of recent interest in statistics and machine learning can be solved by ReHLine, and we provide reproducible benchmark code and results at the ReHLine-benchmark repository.

Problem	Results
SVM	Result
Smoothed SVM	Result
FairSVM	Result
ElasticQR	Result
RidgeHuber	Result