Which optimization algorithm to choose?

Marie Laure Delignette Muller, Christophe Dutang

2024-10-26

1. 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 $x\in\mathbb{R}^n$.

1.1. 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 $\sigma=1/2$.

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 $t_k \in \mathbb{R}$.

1.2.1. 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 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 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 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 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.

1.2.2. 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.

1.3. 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.

fitbench <- function(data, distr, method, grad = NULL, 
                     control = list(trace = 0, REPORT = 1, maxit = 1000), 
                     lower = -Inf, upper = +Inf, ...) 

2. Numerical illustration with the beta distribution

2.1. Log-likelihood function and its gradient for beta distribution

2.1.1. 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/.

2.1.2. 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.

lnL <- function(par, fix.arg, obs, ddistnam) 
  fitdistrplus:::loglikelihood(par, fix.arg, obs, ddistnam) 
grlnlbeta <- fitdistrplus:::grlnlbeta

2.2. Random generation of a sample

#(1) beta distribution
n <- 200
x <- rbeta(n, 3, 3/4)
grlnlbeta(c(3, 4), x) #test

## [1] -133  317

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")

2.3 Fit Beta distribution

Define control parameters.

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.

unconstropt <- fitbench(x, "beta", "mle", grad=grlnlbeta, lower=0)

##     BFGS       NM     CGFR     CGPR     CGBS L-BFGS-B     NM-B   G-BFGS 
##       14       14       14       14       14       14       14       14 
##   G-CGFR   G-CGPR   G-CGBS G-BFGS-B   G-NM-B G-CGFR-B G-CGPR-B G-CGBS-B 
##       14       14       14       14       14       14       14       14

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.

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) 

##   BFGS     NM   CGFR   CGPR   CGBS G-BFGS G-CGFR G-CGPR G-CGBS 
##     14     14     14     14     14     14     14     14     14

#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).

2.4. 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).

Unconstrained optimization with approximated gradient

	BFGS	NM	CGFR	CGPR	CGBS	L-BFGS-B	NM-B
fitted shape1	2.665	2.664	2.665	2.665	2.665	2.665	2.665
fitted shape2	0.731	0.731	0.731	0.731	0.731	0.731	0.731
fitted loglik	114.165	114.165	114.165	114.165	114.165	114.165	114.165
func. eval. nb.	15.000	47.000	191.000	221.000	235.000	8.000	92.000
grad. eval. nb.	11.000	NA	95.000	115.000	171.000	8.000	NA
time (sec)	0.006	0.004	0.032	0.038	0.051	0.003	0.011

Unconstrained optimization with true gradient

	G-BFGS	G-CGFR	G-CGPR	G-CGBS	G-BFGS-B	G-NM-B	G-CGFR-B	G-CGPR-B	G-CGBS-B
fitted shape1	2.665	2.665	2.665	2.665	2.665	2.665	2.665	2.665	2.665
fitted shape2	0.731	0.731	0.731	0.731	0.731	0.731	0.731	0.731	0.731
fitted loglik	114.165	114.165	114.165	114.165	114.165	114.165	114.165	114.165	114.165
func. eval. nb.	22.000	249.000	225.000	138.000	40.000	92.000	471.000	403.000	255.000
grad. eval. nb.	5.000	71.000	69.000	43.000	6.000	NA	96.000	104.000	58.000
time (sec)	0.009	0.076	0.072	0.046	0.019	0.020	0.131	0.135	0.078

Exponential trick optimization with approximated gradient

	BFGS	NM	CGFR	CGPR	CGBS
fitted shape1	2.665	2.664	2.665	2.665	2.665
fitted shape2	0.731	0.731	0.731	0.731	0.731
fitted loglik	114.165	114.165	114.165	114.165	114.165
func. eval. nb.	8.000	41.000	37.000	49.000	47.000
grad. eval. nb.	5.000	NA	19.000	39.000	33.000
time (sec)	0.013	0.003	0.008	0.012	0.011

Exponential trick optimization with true gradient

	G-BFGS	G-CGFR	G-CGPR	G-CGBS
fitted shape1	2.665	2.665	2.665	2.665
fitted shape2	0.731	0.731	0.731	0.731
fitted loglik	114.165	114.165	114.165	114.165
func. eval. nb.	21.000	175.000	146.000	112.000
grad. eval. nb.	5.000	39.000	47.000	35.000
time (sec)	0.010	0.046	0.051	0.039

Using llsurface, we plot the log-likehood surface around the true value (green) and the fitted parameters (red).

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.

b1 <- bootdist(fitdist(x, "beta", method = "mle", optim.method = "BFGS"), 
               niter = 100, parallel = "snow", ncpus = 2)
summary(b1)

## Parametric bootstrap medians and 95% percentile CI 
##        Median  2.5% 97.5%
## shape1   2.73 2.272 3.283
## shape2   0.75 0.652 0.888

plot(b1, trueval = c(3, 3/4))

3. Numerical illustration with the negative binomial distribution

3.1. Log-likelihood function and its gradient for negative binomial distribution

3.1.1. 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/.

3.1.2. R implementation

As in the fitdistrplus package, we minimize the opposite of the log-likelihood: we implement the opposite of the gradient in grlnL.

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))
}

3.2. Random generation of a sample

#(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")

3.3. Fit a negative binomial distribution

Define control parameters and make the benchmark.

ctr <- list(trace = 0, REPORT = 1, maxit = 1000)
unconstropt <- fitbench(x, "nbinom", "mle", grad = grlnlNB, lower = 0)

##     BFGS       NM     CGFR     CGPR     CGBS L-BFGS-B     NM-B   G-BFGS 
##       14       14       14       14       14       14       14       14 
##   G-CGFR   G-CGPR   G-CGBS G-BFGS-B   G-NM-B G-CGFR-B G-CGPR-B G-CGBS-B 
##       14       14       14       14       14       14       14       14

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.

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) 

##   BFGS     NM   CGFR   CGPR   CGBS G-BFGS G-CGFR G-CGPR G-CGBS 
##     14     14     14     14     14     14     14     14     14

# 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).

3.4. 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).

Unconstrained optimization with approximated gradient

	BFGS	NM	CGFR	CGPR	CGBS	L-BFGS-B	NM-B
fitted size	57.333	62.504	57.337	57.335	57.335	57.333	58.446
fitted mu	3.440	3.440	3.440	3.440	3.440	3.440	3.440
fitted loglik	-402.675	-402.674	-402.675	-402.675	-402.675	-402.675	-402.675
func. eval. nb.	2.000	39.000	2001.000	1001.000	1001.000	2.000	0.000
grad. eval. nb.	1.000	NA	1001.000	1001.000	1001.000	2.000	NA
time (sec)	0.001	0.003	0.213	0.166	0.167	0.001	0.003
fitted prob	0.775	0.775	0.775	0.775	0.775	0.775	0.775

Unconstrained optimization with true gradient

	G-BFGS	G-CGFR	G-CGPR	G-CGBS	G-BFGS-B	G-NM-B	G-CGFR-B	G-CGPR-B	G-CGBS-B
fitted size	57.333	57.333	57.333	57.333	57.333	58.446	57.333	57.333	57.333
fitted mu	3.440	3.440	3.440	3.440	3.440	3.440	3.440	3.440	3.440
fitted loglik	-402.675	-402.675	-402.675	-402.675	-402.675	-402.675	-402.675	-402.675	-402.675
func. eval. nb.	28.000	28.000	28.000	28.000	0.000	0.000	0.000	0.000	0.000
grad. eval. nb.	1.000	1.000	1.000	1.000	NA	NA	NA	NA	NA
time (sec)	0.008	0.001	0.001	0.002	0.002	0.003	0.002	0.002	0.002
fitted prob	0.775	0.775	0.775	0.775	0.775	0.775	0.775	0.775	0.775

Exponential trick optimization with approximated gradient

	BFGS	NM	CGFR	CGPR	CGBS
fitted size	57.335	62.320	58.595	60.173	60.875
fitted prob	0.943	0.948	0.945	0.946	0.947
fitted loglik	-402.675	-402.674	-402.675	-402.674	-402.674
func. eval. nb.	3.000	45.000	2501.000	2273.000	2272.000
grad. eval. nb.	1.000	NA	1001.000	1001.000	1001.000
time (sec)	0.004	0.002	0.229	0.221	0.225

Exponential trick optimization with true gradient

	G-BFGS	G-CGFR	G-CGPR	G-CGBS
fitted size	57.333	57.333	57.333	57.333
fitted prob	0.943	0.943	0.943	0.943
fitted loglik	-402.675	-402.675	-402.675	-402.675
func. eval. nb.	18.000	44.000	39.000	39.000
grad. eval. nb.	1.000	3.000	3.000	3.000
time (sec)	0.006	0.002	0.002	0.002

Using llsurface, we plot the log-likehood surface around the true value (green) and the fitted parameters (red).

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.

b1 <- bootdist(fitdist(x, "nbinom", method = "mle", optim.method = "BFGS"), 
               niter = 100, parallel = "snow", ncpus = 2)
summary(b1)

## Parametric bootstrap medians and 95% percentile CI 
##      Median  2.5% 97.5%
## size  57.33 57.33 57.33
## mu     3.46  3.24  3.72

plot(b1, trueval=trueval[c("size", "mu")]) 

4. 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.