Author: James Joseph Balamuta and Dirk Eddelbuettel

# Solving Linear Regression using Numeric Optimization

#### James Joseph Balamuta and Dirk Eddelbuettel

Abstract

In this vignette, we describe how to use RcppEnsmallen in a standalone C++ file.

# Overview

`RcppEnsmallen` package provides an embedded copy of the `ensmallen` C++ library of optimization functions. Optimizers contained within are state of the art and possess a high level of code quality. Each optimizer must be accessed through C++ by implementing the appropriate objective functions and, then, surfaced into R through `RcppArmadillo`. Alternatively, work has been done by Dirk Schumacher in `armacmp` to automatically create the underlying C++ code from R.

Note: Optimizers in `RcppEnsmallen` only work with `armadillo` data structures. Thus, if using `Eigen` through `RcppEigen`, please consider the `RcppNumerical` package.

## Linear Regression

Consider the Residual Sum of Squares, also known as RSS, defined as:

\[RSS\left( \beta \right) = \left( { \mathbf{y} - \mathbf{X} \beta } \right)^{\top} \left( \mathbf{y} - \mathbf{X} \beta \right)\]

The objective function we wish to minimize would be defined as:

\[f(\beta) = \rVert \mathbf{y} - \mathbf{X}\beta\lVert_2\]

The gradient is defined as:

\[\frac{\partial RSS}{\partial \beta} = -2 \mathbf{X}^{\top} \left(\mathbf{y} - \mathbf{X} \beta \right)\]

### Two-Step Implementation

When using `ensmallen` to solve this problem, we must create a C++ class that computes both the objective function value and its gradient value either together or separately under member functions named as:

- `Evaluate()`: Value of the objective function under the parameters.
- `Gradient()`: Convergence to the correct value under the given parameters.
- `EvaluateWithGradient()`: Perform both steps at the same time. (Optional)

In the Linear Regression scenario, we will define each step separately to emphasize the calculation occurring. Generally, the one step `EvaluateWithGradient()` function will be faster than the two step variant. More details on design can be found on `ensmallen` documentation page for differentiable functions.

Before writing the class, `RcppEnsmallen` requires accessing the library in a standalone C++ file with the follow include and Rcpp Attribute declarations:

    #include <RcppEnsmallen.h>
    // [[Rcpp::depends(RcppEnsmallen)]]

The overaching Linear regression class should be constructed as follows:

    #include <RcppEnsmallen.h>
    // [[Rcpp::depends(RcppEnsmallen)]]

    // Define a differentiable objective function by implementing both Evaluate()
    // and Gradient() separately.
    class LinearRegressionFunction
    {
    public:
      // Construct the object with the given the design
      // matrix and responses.
      LinearRegressionFunction(const arma::mat& X,
                               const arma::vec& y) :
      X(X), y(y) { }

      // Return the objective function for model parameters beta.
      double Evaluate(const arma::mat& beta)
      {
          return std::pow(arma::norm(y - X * beta), 2.0);
      }

      // Compute the gradient for model parameters beta
      void Gradient(const arma::mat& beta, arma::mat& g)
      {
          g = -2 * X.t() * (y - X * beta);
      }

    private:
      // The design matrix.
      const arma::mat& X;
      // The responses to each data point.
      const arma::vec& y;
    };

From there:

1. Construct a C++ function that exports into R.
2. Within the function, determine an appropriate optimizer for the problem.
3. Combine the optimizer with the linear regression class to compute the solution to the problem.

Note: Make sure to have the definition of the Linear Regression class in the same C++ file as the exported C++ function into R.

    // [[Rcpp::export]]
    arma::mat lin_reg_lbfgs(const arma::mat& X, const arma::vec& y) {

      // Construct the first objective function.
      LinearRegressionFunction lrf(X, y);

      // Create the L_BFGS optimizer with default parameters.
      // The ens::L_BFGS type can be replaced with any ensmallen optimizer that can
      // handle differentiable functions.
      ens::L_BFGS lbfgs;

      lbfgs.MaxIterations() = 10;

      // Create a starting point for our optimization randomly.
      // The model has p parameters, so the shape is p x 1.
      arma::mat beta(X.n_cols, 1, arma::fill::randn);

      // Run the optimization
      lbfgs.Optimize(lrf, beta);

      return beta;
    }

### Verifying Results

Prior to using the new optimizer in mission critical work, compare the results to methods already implemented in R. The best way to achieve this is to create an oracle model by specifying the parameters known to generate data and, then, try to recover them. Moreover, if a method is already implemented in R feel free to try to check the result equality within an appropriate tolerance threshold.

Following with this methodology, data must be generated.

    n <- 1e6
    beta <- c(-2, 1.5, 3, 8.2, 6.6)
    p <- length(beta)
    X <- cbind(1, matrix(rnorm(n), ncol = p - 1))
    y <- X %*% beta + rnorm(n / (p - 1))

Next, the optimization procedure is used to estimate the parameters of interest. Under this example, the results of the estimation can be compared to `lm()`. That said, `lm()` may have more precise results when compared against the optimizer as it is implemented with a closed-form solution to linear regression plus the computational is performed more rigorously.

    coefs_lbfgs <- lin_reg_lbfgs(X, y)
    coefs_lm <- lm.fit(X, y)$coefficients

Comparison of Estimated Coefficients
| LBFGS | LM
--- | --- | ---
Beta1 | -1.999402 | -1.999402
Beta2 | 1.501791 | 1.501791
Beta3 | 3.002372 | 3.002372
Beta4 | 8.199673 | 8.199673
Beta5 | 6.600396 | 6.600396