---
title: Which optimization algorithm to choose?
author: Marie Laure Delignette Muller, Christophe Dutang
date: '`r Sys.Date()`'
output:
  bookdown::html_document2:
    base_format: rmarkdown::html_vignette
    fig_caption: yes
    toc: true
    number_sections: yes
link-citations: true
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{Which optimization algorithm to choose?} 
  %!\VignetteEncoding{UTF-8}
  \usepackage[utf8]{inputenc}
pkgdown:
  as_is: true
---

```{r setup, echo=FALSE, message=FALSE, warning=FALSE}
require(fitdistrplus)
require(knitr) #for kable() function
set.seed(12345)
options(digits = 3)
```


# Quick overview of main optimization methods

We present very quickly the main optimization methods.
Please refer to **Numerical Optimization (Nocedal \& Wright, 2006)** 
or **Numerical Optimization: theoretical and practical aspects 
(Bonnans, Gilbert, Lemarechal \& Sagastizabal, 2006)** for a good introduction. 
We consider the following problem $\min_x f(x)$ for $x\in\mathbb{R}^n$.

## Derivative-free optimization methods
The Nelder-Mead method is one of the most well known derivative-free methods
that use only values of $f$ to search for the minimum.
It consists in building a simplex of $n+1$ points and moving/shrinking
this simplex into the good direction.

1. set initial points $x_1, \dots, x_{n+1}$. 
2. order points such that $f(x_1)\leq f(x_2)\leq\dots\leq f(x_{n+1})$. 
3. compute $x_o$ as the centroid of $x_1, \dots, x_{n}$. 
4. Reflection:  
    + compute the reflected point $x_r = x_o + \alpha(x_o-x_{n+1})$. 
    + **if** $f(x_1)\leq f(x_r)<f(x_n)$, 
  then replace $x_{n+1}$ by $x_r$,
  go to step 2. 
    + **else** go step 5.

5. Expansion:  
    + **if** $f(x_r)<f(x_1)$, then compute the expansion point
    $x_e= x_o+\gamma(x_o-x_{n+1})$. 
    + **if** $f(x_e) <f(x_r)$, then replace $x_{n+1}$ by $x_e$,
  go to step 2. 
    + **else** $x_{n+1}$ by $x_r$, go to step 2. 
    + **else** go to step 6. 
6. Contraction:  
    + compute the contracted point $x_c = x_o + \beta(x_o-x_{n+1})$. 
    + **if** $f(x_c)<f(x_{n+1})$, 
  then replace $x_{n+1}$ by $x_c$,
  go to step 2.  
    + **else** go step 7. 
7. Reduction:  
    + for $i=2,\dots, n+1$, compute $x_i = x_1+\sigma(x_i-x_{1})$. 
    
The Nelder-Mead method is available in `optim`.
By default, in `optim`, $\alpha=1$, $\beta=1/2$, $\gamma=2$ and $\sigma=1/2$.

## Hessian-free optimization methods

For smooth non-linear function, the following method is generally used:
a local method combined with line search work on the scheme $x_{k+1} =x_k + t_k d_{k}$, where the local method will specify the direction $d_k$ and the line search will specify the step size $t_k \in \mathbb{R}$.	

### Computing the direction $d_k$
A desirable property for $d_k$ is that $d_k$ ensures a descent $f(x_{k+1}) < f(x_{k})$. 
Newton methods are such that $d_k$ minimizes a local quadratic approximation of $f$ based on a Taylor expansion, that is  $q_f(d) = f(x_k) + g(x_k)^Td +\frac{1}{2} d^T H(x_k) d$ where $g$ denotes the gradient and $H$ denotes the Hessian.

The \emph{exact Newton} consists in using the exact solution of local minimization problem $d_k = - H(x_k)^{-1} g(x_k)$.	
In practice, other methods are preferred (at least to ensure positive definiteness).
The \emph{quasi-Newton} method approximates the Hessian by a matrix $H_k$ as a function of $H_{k-1}$, $x_k$, $f(x_k)$ and then $d_k$ solves the system $H_k d = -  g(x_k)$. 
Some implementation may also directly approximate the inverse of the Hessian $W_k$ in order to compute $d_k = -W_k g(x_k)$. Using the Sherman-Morrison-Woodbury formula, we can switch between $W_k$ and $H_k$.

To determine $W_k$, first it must verify the secant equation $H_k y_k =s_k$ or $y_k=W_k s_k$ where $y_k = g_{k+1}-g_k$ and $s_k=x_{k+1}-x_k$. To define the $n(n-1)$ terms, we generally impose a symmetry and a minimum distance conditions. We say we have a rank 2 update if  $H_k = H_{k-1} + a u u^T + b v v^T$ and a rank 1 update if  $H_k = H_{k-1} + a u u^T $. Rank $n$ update is justified by the spectral decomposition theorem.


There are two rank-2 updates which are symmetric and preserve positive definiteness

  * DFP \emph{Davidon-Fletcher-Powell} minimizes $\min || H - H_k ||_F$ such that $H=H^T$:
$$ 
H_{k+1} = \left (I-\frac {y_k s_k^T} {y_k^T s_k} \right ) H_k \left (I-\frac {s_k y_k^T} {y_k^T s_k} \right )+\frac{y_k y_k^T} {y_k^T s_k}  
\Leftrightarrow
W_{k+1} = W_k +  \frac{s_k s_k^T}{y_k^{T} s_k} - \frac {W_k y_k y_k^T W_k^T} {y_k^T W_k y_k} .
$$  
  * BFGS \emph{Broyden-Fletcher-Goldfarb-Shanno} minimizes $\min || W - W_k ||_F$ such that $W=W^T$:
$$
H_{k+1} = H_k - \frac{ H_k y_k y_k^T H_k }{ y_k^T H_k y_k }  + \frac{ s_k s_k^T }{ y_k^T s_k }
\Leftrightarrow
W_{k+1} = \left (I-\frac {y_k s_k^T} {y_k^T s_k} \right )^T W_k \left (I-\frac { y_k s_k^T} {y_k^T s_k} \right )+\frac{s_k s_k^T} {y_k^T s_k} .
$$

In `R`, the so-called BFGS scheme is implemented in `optim`.


Another possible method (which is initially arised from quadratic problems)
is the nonlinear conjugate gradients.
This consists in computing directions $(d_0, \dots, d_k)$ that are conjugate
with respect to a matrix close to the true Hessian $H(x_k)$.
Directions are computed iteratively by $d_k = -g(x_k) + \beta_k d_{k-1}$ for $k>1$, once initiated by $d_1 = -g(x_1)$. 
$\beta_k$ are updated according a scheme:  

  * $\beta_k = \frac{ g_k^T g_k}{g_{k-1}^T g_{k-1} }$: Fletcher-Reeves update, 
  * $\beta_k = \frac{ g_k^T (g_k-g_{k-1} )}{g_{k-1}^T g_{k-1}}$: Polak-Ribiere update. 

There exists also three-term formula for computing direction 
$d_k = -g(x_k) + \beta_k d_{k-1}+\gamma_{k} d_t$ for $t<k$.
A possible scheme is the Beale-Sorenson update defined as
 $\beta_k = \frac{ g_k^T (g_k-g_{k-1} )}{d^T_{k-1}(g_{k}- g_{k-1})}$ 
 and $\gamma_k = \frac{ g_k^T (g_{t+1}-g_{t} )}{d^T_{t}(g_{t+1}- g_{t})}$ if $k>t+1$ otherwise $\gamma_k=0$ if $k=t$.
See Yuan (2006) for other well-known schemes such as Hestenses-Stiefel, Dixon or Conjugate-Descent.
The three updates (Fletcher-Reeves, Polak-Ribiere, Beale-Sorenson) of the (non-linear) conjugate gradient are available in `optim`.


### Computing the stepsize $t_k$

Let $\phi_k(t) = f(x_k + t d_k)$ for a given direction/iterate $(d_k, x_k)$. 
We need to find conditions to find a satisfactory stepsize $t_k$. In literature, we consider the  descent condition: $\phi_k'(0) < 0$
and the Armijo condition: $\phi_k(t) \leq \phi_k(0) + t c_1 \phi_k'(0)$ ensures a decrease of $f$.
Nocedal \& Wright (2006) presents a backtracking (or geometric) approach satisfying the Armijo condition and minimal condition, i.e. Goldstein and Price condition.

* set $t_{k,0}$ e.g. 1, $0 < \alpha < 1$, 
* **Repeat** until Armijo satisfied, 
	+ $t_{k,i+1} =  \alpha \times t_{k,i}$.
* **end Repeat**

This backtracking linesearch is available in `optim`.


## Benchmark

To simplify the benchmark of optimization methods, we create a `fitbench` function that computes
the desired estimation method for all optimization methods. 
This function is currently not exported in the package.
```{r, echo=TRUE, eval=FALSE}
fitbench <- function(data, distr, method, grad = NULL, 
                     control = list(trace = 0, REPORT = 1, maxit = 1000), 
                     lower = -Inf, upper = +Inf, ...) 
```
```{r, echo=FALSE}
fitbench <- fitdistrplus:::fitbench
```



# Numerical illustration with the beta distribution


## Log-likelihood function and its gradient for beta distribution

### Theoretical value
The density of the beta distribution is given by 
$$
f(x; \delta_1,\delta_2) = \frac{x^{\delta_1-1}(1-x)^{\delta_2-1}}{\beta(\delta_1,\delta_2)},
$$
where $\beta$ denotes the beta function, see the NIST Handbook of mathematical functions https://dlmf.nist.gov/.
We recall that $\beta(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$.
There the log-likelihood for a set of observations $(x_1,\dots,x_n)$ is
$$
\log L(\delta_1,\delta_2) = (\delta_1-1)\sum_{i=1}^n\log(x_i)+ (\delta_2-1)\sum_{i=1}^n\log(1-x_i)+ n \log(\beta(\delta_1,\delta_2))
$$
The gradient with respect to $a$ and $b$ is 
$$
\nabla \log L(\delta_1,\delta_2) = 
\left(\begin{matrix}
\sum\limits_{i=1}^n\ln(x_i) - n\psi(\delta_1)+n\psi( \delta_1+\delta_2)  \\
\sum\limits_{i=1}^n\ln(1-x_i)- n\psi(\delta_2)+n\psi( \delta_1+\delta_2)
\end{matrix}\right),
$$
where $\psi(x)=\Gamma'(x)/\Gamma(x)$ is the digamma function, 
see the NIST Handbook of mathematical functions https://dlmf.nist.gov/.

### `R` implementation
As in the `fitdistrplus` package, we minimize the opposite of the log-likelihood: 
we implement the opposite of the gradient in `grlnL`. Both the log-likelihood and its gradient
are not exported.
```{r}
lnL <- function(par, fix.arg, obs, ddistnam) 
  fitdistrplus:::loglikelihood(par, fix.arg, obs, ddistnam) 
grlnlbeta <- fitdistrplus:::grlnlbeta
```



## Random generation of a sample

```{r, fig.height=4, fig.width=4}
#(1) beta distribution
n <- 200
x <- rbeta(n, 3, 3/4)
grlnlbeta(c(3, 4), x) #test
hist(x, prob=TRUE, xlim=0:1)
lines(density(x), col="red")
curve(dbeta(x, 3, 3/4), col="green", add=TRUE)
legend("topleft", lty=1, col=c("red","green"), legend=c("empirical", "theoretical"), bty="n")
```

## Fit Beta distribution

Define control parameters.
```{r}
ctr <- list(trace=0, REPORT=1, maxit=1000)
```
Call `mledist` with the default optimization function 
(`optim` implemented in `stats` package) 
with and without the gradient for the different optimization methods.
```{r}
unconstropt <- fitbench(x, "beta", "mle", grad=grlnlbeta, lower=0)
```
In the case of constrained optimization, `mledist` permits the direct use 
of `constrOptim` function (still implemented in `stats` package)
that allow linear inequality constraints by using a logarithmic barrier.

Use a exp/log transformation of the shape parameters $\delta_1$ and $\delta_2$ 
to ensure that the shape parameters are strictly positive.
```{r}
dbeta2 <- function(x, shape1, shape2, log)
  dbeta(x, exp(shape1), exp(shape2), log=log)
#take the log of the starting values
startarg <- lapply(fitdistrplus:::startargdefault(x, "beta"), log)
#redefine the gradient for the new parametrization
grbetaexp <- function(par, obs, ...) 
    grlnlbeta(exp(par), obs) * exp(par)
    

expopt <- fitbench(x, distr="beta2", method="mle", grad=grbetaexp, start=startarg) 
#get back to original parametrization
expopt[c("fitted shape1", "fitted shape2"), ] <- exp(expopt[c("fitted shape1", "fitted shape2"), ])
```
Then we extract the values of the fitted parameters, the value of the 
corresponding log-likelihood and the number of counts to the function
to minimize and its gradient (whether it is the theoretical gradient or the
numerically approximated one).



## Results of the numerical investigation
Results are displayed in the following tables:
(1) the original parametrization without specifying the gradient (`-B` stands for bounded version),
(2) the original parametrization with the (true) gradient (`-B` stands for bounded version and `-G` for gradient),
(3) the log-transformed parametrization without specifying the gradient,
(4) the log-transformed parametrization with the (true) gradient (`-G` stands for gradient).

```{r, results='asis', echo=FALSE}
kable(unconstropt[, grep("G-", colnames(unconstropt), invert=TRUE)], digits=3,
      caption="Unconstrained optimization with approximated gradient")
```

```{r, results='asis', echo=FALSE}
kable(unconstropt[, grep("G-", colnames(unconstropt))], digits=3,
      caption="Unconstrained optimization with true gradient")
```

```{r, results='asis', echo=FALSE}
kable(expopt[, grep("G-", colnames(expopt), invert=TRUE)], digits=3,
      caption="Exponential trick optimization with approximated gradient")
```

```{r, results='asis', echo=FALSE}
kable(expopt[, grep("G-", colnames(expopt))], digits=3,
      caption="Exponential trick optimization with true gradient")
```


Using `llsurface`, we plot the log-likehood surface around the true value (green) and the fitted parameters (red).
```{r, fig.width=4, fig.height=4}
llsurface(min.arg=c(0.1, 0.1), max.arg=c(7, 3), xlim=c(.1,7), 
          plot.arg=c("shape1", "shape2"), nlev=25,
          lseq=50, data=x, distr="beta", back.col = FALSE)
points(unconstropt[1,"BFGS"], unconstropt[2,"BFGS"], pch="+", col="red")
points(3, 3/4, pch="x", col="green")
```

We can simulate bootstrap replicates using the `bootdist` function.
```{r, fig.width=4, fig.height=4}
b1 <- bootdist(fitdist(x, "beta", method = "mle", optim.method = "BFGS"), 
               niter = 100, parallel = "snow", ncpus = 2)
summary(b1)
plot(b1, trueval = c(3, 3/4))
```


# Numerical illustration with the negative binomial distribution


## Log-likelihood function and its gradient for negative binomial distribution

### Theoretical value
The p.m.f. of the Negative binomial distribution is given by 
$$
f(x; m,p) = \frac{\Gamma(x+m)}{\Gamma(m)x!} p^m (1-p)^x,
$$
where $\Gamma$ denotes the beta function, see the NIST Handbook of mathematical functions https://dlmf.nist.gov/.
There exists an alternative representation where $\mu=m (1-p)/p$ or equivalently $p=m/(m+\mu)$.
Thus, the log-likelihood for a set of observations $(x_1,\dots,x_n)$ is
$$
\log L(m,p) = 
\sum_{i=1}^{n} \log\Gamma(x_i+m)
-n\log\Gamma(m)
-\sum_{i=1}^{n} \log(x_i!)
+ mn\log(p)
+\sum_{i=1}^{n} {x_i}\log(1-p)
$$
The gradient with respect to $m$ and $p$ is 
$$
\nabla \log L(m,p) = 
\left(\begin{matrix}
\sum_{i=1}^{n} \psi(x_i+m)
-n \psi(m)
+ n\log(p)
\\
 mn/p
-\sum_{i=1}^{n} {x_i}/(1-p)
\end{matrix}\right),
$$
where $\psi(x)=\Gamma'(x)/\Gamma(x)$ is the digamma function, 
see the NIST Handbook of mathematical functions https://dlmf.nist.gov/.

### `R` implementation
As in the `fitdistrplus` package, we minimize the opposite of the log-likelihood: we implement the opposite of the gradient in `grlnL`.
```{r}
grlnlNB <- function(x, obs, ...)
{
  m <- x[1]
  p <- x[2]
  n <- length(obs)
  c(sum(psigamma(obs+m)) - n*psigamma(m) + n*log(p),
    m*n/p - sum(obs)/(1-p))
}
```





## Random generation of a sample

```{r, fig.height=4, fig.width=4}
#(2) negative binomial distribution
n <- 200
trueval <- c("size"=10, "prob"=3/4, "mu"=10/3)
x <- rnbinom(n, trueval["size"], trueval["prob"])

hist(x, prob=TRUE, ylim=c(0, .3), xlim=c(0, 10))
lines(density(x), col="red")
points(min(x):max(x), dnbinom(min(x):max(x), trueval["size"], trueval["prob"]), 
       col = "green")
legend("topright", lty = 1, col = c("red", "green"), 
       legend = c("empirical", "theoretical"), bty="n")
```

## Fit a negative binomial distribution

Define control parameters and make the benchmark.
```{r}
ctr <- list(trace = 0, REPORT = 1, maxit = 1000)
unconstropt <- fitbench(x, "nbinom", "mle", grad = grlnlNB, lower = 0)
unconstropt <- rbind(unconstropt, 
                     "fitted prob" = unconstropt["fitted mu", ] / (1 + unconstropt["fitted mu", ]))
```
In the case of constrained optimization, `mledist` permits the direct use 
of `constrOptim` function (still implemented in `stats` package)
that allow linear inequality constraints by using a logarithmic barrier.

Use a exp/log transformation of the shape parameters $\delta_1$ and $\delta_2$ 
to ensure that the shape parameters are strictly positive.
```{r}
dnbinom2 <- function(x, size, prob, log)
  dnbinom(x, exp(size), 1 / (1 + exp(-prob)), log = log)
# transform starting values
startarg <- fitdistrplus:::startargdefault(x, "nbinom")
startarg$mu <- startarg$size / (startarg$size + startarg$mu)
startarg <- list(size = log(startarg[[1]]), 
                 prob = log(startarg[[2]] / (1 - startarg[[2]])))

# redefine the gradient for the new parametrization
Trans <- function(x)
  c(exp(x[1]), plogis(x[2]))
grNBexp <- function(par, obs, ...) 
    grlnlNB(Trans(par), obs) * c(exp(par[1]), plogis(x[2])*(1-plogis(x[2])))

expopt <- fitbench(x, distr="nbinom2", method="mle", grad=grNBexp, start=startarg) 
# get back to original parametrization
expopt[c("fitted size", "fitted prob"), ] <- 
  apply(expopt[c("fitted size", "fitted prob"), ], 2, Trans)
```
Then we extract the values of the fitted parameters, the value of the 
corresponding log-likelihood and the number of counts to the function
to minimize and its gradient (whether it is the theoretical gradient or the
numerically approximated one).


## Results of the numerical investigation
Results are displayed in the following tables:
(1) the original parametrization without specifying the gradient (`-B` stands for bounded version),
(2) the original parametrization with the (true) gradient (`-B` stands for bounded version and `-G` for gradient),
(3) the log-transformed parametrization without specifying the gradient,
(4) the log-transformed parametrization with the (true) gradient (`-G` stands for gradient).

```{r, echo=FALSE}
kable(unconstropt[, grep("G-", colnames(unconstropt), invert=TRUE)], digits=3,
      caption="Unconstrained optimization with approximated gradient")
```

```{r, results='asis', echo=FALSE}
kable(unconstropt[, grep("G-", colnames(unconstropt))], digits=3,
      caption="Unconstrained optimization with true gradient")
```

```{r, results='asis', echo=FALSE}
kable(expopt[, grep("G-", colnames(expopt), invert=TRUE)], digits=3,
      caption="Exponential trick optimization with approximated gradient")
```
```{r, results='asis', echo=FALSE}
kable(expopt[, grep("G-", colnames(expopt))], digits=3,
      caption="Exponential trick optimization with true gradient")
```



Using `llsurface`, we plot the log-likehood surface around the true value (green) and the fitted parameters (red).
```{r, fig.width=4, fig.height=4}
llsurface(min.arg = c(5, 0.3), max.arg = c(15, 1), xlim=c(5, 15),
          plot.arg = c("size", "prob"), nlev = 25,
          lseq = 50, data = x, distr = "nbinom", back.col = FALSE)
points(unconstropt["fitted size", "BFGS"], unconstropt["fitted prob", "BFGS"], 
       pch = "+", col = "red")
points(trueval["size"], trueval["prob"], pch = "x", col = "green")
```

We can simulate bootstrap replicates using the `bootdist` function.
```{r, fig.width=4, fig.height=4}
b1 <- bootdist(fitdist(x, "nbinom", method = "mle", optim.method = "BFGS"), 
               niter = 100, parallel = "snow", ncpus = 2)
summary(b1)
plot(b1, trueval=trueval[c("size", "mu")]) 
```



# Conclusion

Based on the two previous examples, we observe that all methods converge to the same
point. This is reassuring.  
However, the number of function evaluations (and the gradient evaluations) is
very different from a method to another.
Furthermore, specifying the true gradient of the log-likelihood does not
help at all the fitting procedure and generally slows down the convergence.
Generally, the best method is the standard BFGS method or the BFGS method
with the exponential transformation of the parameters.
Since the exponential function is differentiable, the asymptotic properties are
still preserved (by the Delta method) but for finite-sample this may produce a small bias.