Abstract

Contained within are examples related to using RcppEnsmallen in everyday work alongside of creating an *R* package with it.

`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.

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

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

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

\[f(\beta) = (y - X\beta)^2\]

The gradient is defined as:

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

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.
const arma::mat& X,
LinearRegressionFunction(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)
{2 * X.t() * (y - X * beta);
g = -
}
private:
// The design matrix.
const arma::mat& X;
// The responses to each data point.
const arma::vec& y;
};
```

From there:

- Construct a
*C++*function that exports into*R*. - Within the function, determine an appropriate optimizer for the problem.
- 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]]
const arma::mat& X, const arma::vec& y) {
arma::mat lin_reg_lbfgs(
// 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;
10;
lbfgs.MaxIterations() =
// Create a starting point for our optimization randomly.
// The model has p parameters, so the shape is p x 1.
1, arma::fill::randn);
arma::mat beta(X.n_cols,
// Run the optimization
lbfgs.Optimize(lrf, beta);
return beta;
}
```

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.

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

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.

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

LBFGS | LM | |
---|---|---|

Beta1 | -1.999843 | -1.999843 |

Beta2 | 1.500141 | 1.500141 |

Beta3 | 3.002068 | 3.002068 |

Beta4 | 8.198676 | 8.198676 |

Beta5 | 6.597958 | 6.597958 |

RcppEnsmallen is best used within an *R* package. The setup for `RcppEnsmallen`

’s use mirrors that of other `Rcpp`

-based projects. In particular, the `DESCRIPTION`

file requires the `LinkingTo`

field to contain:

`LinkingTo: Rcpp, RcppArmadillo (>= 0.8.400.0.0), RcppEnsmallen (>= 0.2.10.0.1)`

Next, the `src/`

directory must contain both a `Makevars`

and `Makevars.win`

file with:

```
CXX_STD = CXX11
PKG_CXXFLAGS = $(SHLIB_OPENMP_CXXFLAGS)
PKG_LIBS = $(SHLIB_OPENMP_CXXFLAGS) $(LAPACK_LIBS) $(BLAS_LIBS) $(FLIBS)
```

The `Makevars.win`

file provides the appropriate configuration for Windows while `Makevars`

acts on Unix-alike systems like macOS, Linux, and Solaris.