Neden lojistik regresyon optimizasyonu için Newton yöntemini kullanmaya yinelemeli yeniden ağırlıklı en küçük kareler denir?

Neden lojistik regresyon optimizasyonu için Newton yöntemini kullanmaya yinelemeli yeniden ağırlıklı en küçük kareler denir?

Benim için net görünmüyor çünkü lojistik kayıp ve en az kar kaybı tamamen farklı şeyler.

Özet: GLM'ler, Dimitriy V. Masterov'un belirttiği gibi, bunun yerine beklenen Hessian ile Newton-Raphson olan Fisher skorlaması ile uyuyor (yani gözlemlenen bilgiler yerine Fisher bilgilerinin bir tahminini kullanıyoruz). Kanonik bağlantı fonksiyonunu kullanırsak, gözlemlenen Hessian'ın beklenen Hessian'a eşit olduğu ortaya çıkar, bu nedenle NR ve Fisher puanlaması bu durumda aynıdır. Her iki durumda da, Fisher puanlamasının aslında ağırlıklı bir en küçük kareler doğrusal modeline uyduğunu ve bu yakınsaklıktan elde edilen katsayı tahminlerinin maksimum lojistik regresyon olasılığına göre olduğunu göreceğiz. Daha önce çözülmüş bir soruna lojistik regresyon uydurmayı azaltmanın yanı sıra, lojistik regresyonumuz hakkında bilgi edinmek için nihai WLS uyumunda doğrusal regresyon teşhisi kullanabilme avantajından da yararlanıyoruz.

Bunu lojistik regresyona odaklayacağım, ancak GLM'lerde maksimum olasılık hakkında daha genel bir perspektif için, bu bölümün bundan geçen ve daha genel bir ortamda IRLS'yi türeyen bu bölümün 15.3 bölümünü öneriyorum (sanırım John Fox'un Uygulamalı Regresyon Analizi ve Genelleştirilmiş Doğrusal Modeller ).

$^*$ Sonunda bkz comments

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Olasılık ve skor fonksiyonu

GLM'mizi, şeklinde bir şey yineleyerek takacağız ;

$$b^{(m+1)} = b^{(m)} - J^{-1}_{(m)}\nabla \ell(b^{(m)})$$ burada $\ell$ , günlük olasılığından ve $J_{m}$ , gözlemlenen veya beklenen Hessian günlük olasılığı.

Bizim bağlantı işlevi, bir fonksiyonudur $g$ koşullu ortalama eşleyen $\mu_i = E(y_i | x_i)$ doğrusal öngörücümüzle, bu nedenle ortalama modelimiz$g(\mu_i) = x_i^T\beta$ . $h$ , doğrusal öngörücüyü ortalamayla eşleyen ters bağlantı fonksiyonuolsun.

Lojistik regresyon için, bağımsız gözlemleri olan bir Bernoulli olasılığımız var, bu yüzden

$$\ell(b; y) = \sum_{i=1}^n y_i\log h(x_i^T b) + (1 - y_i) \log(1 - h(x_i^Tb)).$$ Türev alma, $$\frac{\partial \ell}{\partial b_j} = \sum_{i=1}^n \frac{y_i}{h(x_i^T b)} h'(x_i^T b) x_{ij} - \frac{1 - y_i}{1 - h(x_i^T b)} h'(x_i^T b) x_{ij}$$ $$= \sum_{i=1}^n x_{ij} h'(x_i^T b) \left(\frac{y_i}{h(x_i^T b)} - \frac{1 - y_i}{1 - h(x_i^T b)} \right)$$ $$= \sum_i x_{ij} \frac{h'(x_i^T b)}{h(x_i^T b)(1 - h(x_i^T b))}(y_i - h(x_i^T b)).$$

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Standart bağlantıyı kullanma

Şimdi de kurallı bağlantı işlevini kullanıyorsanız varsayalım . Sonra g - 1 c ( x ) : = h c ( x ) = $g_c = \text{logit}$ yanih ′ c =hc⋅(1-hc), bu da ∂$g^{-1}_c(x) := h_c(x) = \frac{1}{1+e^{-x}}$$h_c' = h_c \cdot (1-h_c)$

$$\frac{\partial \ell}{\partial b_j} = \sum_i x_{ij} (y_i - h_c(x_i^T b))$$ so $$\nabla \ell (b; y) = X^T (y - \hat y).$$ Furthermore, still using $h_c$, $$\frac{\partial^2 \ell}{\partial b_k \partial b_j} = - \sum_i x_{ij} \frac{\partial}{\partial b_k} h_c(x_i^T b) = - \sum_i x_{ij}x_{ik} \left[h_c(x_i^T b) (1 - h_c(x_i^T b))\right].$$

Let

$$W = \text{diag}\left(h_c(x_1^T b)(1 - h_c(x_1^T b)), \dots, h_c(x_n^T b)(1 - h_c(x_n^T b))\right) = \text{diag}\left(\hat y_1(1 - \hat y_1), \dots, \hat y_n (1 - \hat y_n)\right).$$ $$H = -X^TWX$$ and note how this doesn't have any $y_i$ in it anymore, so $E(H) = H$ (we're viewing this as a function of $b$ so the only random thing is $y$ itself). Thus we've shown that Fisher scoring is equivalent to Newton-Raphson when we use the canonical link in logistic regression. Also by virtue of $\hat y_i \in (0,1)$ $-X^TWX$$\hat y_i$$0$$1$$0$ which can make $H$ negative semidefinite and therefore computationally singular.

Now create the working response $z = W^{-1}(y - \hat y)$ and note that

$$\nabla \ell = X^T(y - \hat y) = X^T W z.$$

All together this means that we can optimize the log likelihood by iterating

$$b^{(m+1)} = b^{(m)} + (X^T W_{(m)} X)^{-1}X^T W_{(m)} z_{(m)}$$ and $(X^T W_{(m)} X)^{-1}X^T W_{(m)} z_{(m)}$ is exactly $\hat \beta$ for a weighted least squares regression of $z_{(m)}$ on $X$.

Checking this in R:

set.seed(123)
p <- 5
n <- 500
x <- matrix(rnorm(n * p), n, p)
betas <- runif(p, -2, 2)
hc <- function(x) 1 /(1 + exp(-x)) # inverse canonical link
p.true <- hc(x %*% betas)
y <- rbinom(n, 1, p.true)

# fitting with our procedure
my_IRLS_canonical <- function(x, y, b.init, hc, tol=1e-8) {
  change <- Inf
  b.old <- b.init
  while(change > tol) {
    eta <- x %*% b.old  # linear predictor
    y.hat <- hc(eta)
    h.prime_eta <- y.hat * (1 - y.hat)
    z <- (y - y.hat) / h.prime_eta

    b.new <- b.old + lm(z ~ x - 1, weights = h.prime_eta)$coef  # WLS regression
    change <- sqrt(sum((b.new - b.old)^2))
    b.old <- b.new
  }
  b.new
}

my_IRLS_canonical(x, y, rep(1,p), hc)
# x1         x2         x3         x4         x5 
# -1.1149687  2.1897992  1.0271298  0.8702975 -1.2074851

glm(y ~ x - 1, family=binomial())$coef
# x1         x2         x3         x4         x5 
# -1.1149687  2.1897992  1.0271298  0.8702975 -1.2074851 

and they agree.

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Non-canonical link functions

Now if we're not using the canonical link we don't get the simplification of $\frac{h'}{h(1-h)} = 1$ in $\nabla \ell$ so $H$ becomes much more complicated, and we therefore see a noticeable difference by using $E(H)$ in our Fisher scoring.

Here's how this will go: we already worked out the general $\nabla \ell$ so the Hessian will be the main difficulty. We need

$$\frac{\partial^2 \ell}{\partial b_k \partial b_j} = \sum_i x_{ij} \frac{\partial}{\partial b_k}h'(x_i^T b) \left(\frac{y_i}{h(x_i^T b)} - \frac{1 - y_i}{1 - h(x_i^T b)} \right)$$ $$= \sum_i x_{ij}x_{ik} \left[h''(x_i^T b) \left(\frac{y_i}{h(x_i^T b)} - \frac{1 - y_i}{1 - h(x_i^T b)} \right) - h'(x_i^T b)^2\left(\frac{y_i}{h(x_i^T b)^2} + \frac{1-y_i}{(1-h(x_i^T b))^2} \right)\right]$$

Via the linearity of expectation all we need to do to get $E(H)$ is replace each occurrence of $y_i$ with its mean under our model which is $\mu_i=h(x_i^T\beta)$. Each term in the summand will therefore contain a factor of the form

$$h''(x_i^T b) \left(\frac{h(x_i^T \beta)}{h(x_i^T b)} - \frac{1 - h(x_i^T \beta)}{1 - h(x_i^T b)} \right) - h'(x_i^T b)^2\left(\frac{h(x_i^T \beta)}{h(x_i^T b)^2} + \frac{1-h(x_i^T \beta)}{(1-h(x_i^T b))^2} \right).$$ But to actually do our optimization we'll need to estimate each $\beta$, and at step $m$ $b^{(m)}$ is the best guess we have. This means that this will reduce to $$h''(x_i^T b) \left(\frac{h(x_i^T b)}{h(x_i^T b)} - \frac{1 - h(x_i^T b)}{1 - h(x_i^T b)} \right) - h'(x_i^T b)^2\left(\frac{h(x_i^T b)}{h(x_i^T b)^2} + \frac{1-h(x_i^T b)}{(1-h(x_i^T b))^2} \right)$$ $$= - h'(x_i^T b)^2\left(\frac{1}{h(x_i^T b)} + \frac{1}{1-h(x_i^T b)} \right)$$ $$= -\frac{h'(x_i^T b)^2}{h(x_i^T b)(1-h(x_i^T b))}.$$ This means we will use $J$ with $$J_{jk} = -\sum_i x_{ij}x_{ik} \frac{h'(x_i^T b)^2}{h(x_i^T b)(1-h(x_i^T b))}.$$

Now let

$$W^* = \text{diag}\left(\frac{h'(x_1^T b)^2}{h(x_1^T b)(1-h(x_1^T b))} ,\dots, \frac{h'(x_n^T b)^2}{h(x_n^T b)(1-h(x_n^T b))}\right)$$ and note how under the canonical link $h_c' = h_c \cdot (1-h_c)$ reduces $W^*$ to $W$ from the previous section. This lets us write $$J = -X^TW^*X$$ except this is now $\hat E(H)$ rather than necessarily being $H$ itself, so this can differ from Newton-Raphson. For all $i$ $W_{ii}^* > 0$ so aside from numerical issues $J$ will be negative definite.

We have

$$\frac{\partial \ell}{\partial b_j} = \sum_i x_{ij} \frac{h'(x_i^T b)}{h(x_i^T b)(1 - h(x_i^T b))}(y_i - h(x_i^T b))$$ so letting our new working response be $z^* = D^{-1}(y-\hat y)$ with $D=\text{diag}\left(h'(x_1^T b), \dots, h'(x_n^T b)\right)$, we have $\nabla \ell = X^TW^*z^*$.

All together we are iterating

$$b^{(m+1)} = b^{(m)} + (X^T W_{(m)}^* X)^{-1}X^T W_{(m)}^* z_{(m)}^*$$ so this is still a sequence of WLS regressions except now it's not necessarily Newton-Raphson.

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I've written it out this way to emphasize the connection to Newton-Raphson, but frequently people will factor the updates so that each new point $b^{(m+1)}$ is itself the WLS solution, rather than a WLS solution added to the current point $b^{(m)}$. If we wanted to do this, we can do the following:

$$b^{(m+1)} = b^{(m)} + (X^T W_{(m)}^* X)^{-1}X^T W_{(m)}^* z_{(m)}^*$$ $$= (X^T W_{(m)}^* X)^{-1}\left(X^T W_{(m)}^* Xb^{(m)}+ X^TW^*_{(m)}z_{(m)}^* \right)$$ $$= (X^T W_{(m)}^* X)^{-1}X^TW_{(m)}^*\left(Xb^{(m)}+ z_{(m)}^* \right)$$ so if we're going this way you'll see the working response take the form $\eta^{(m)} + D^{-1}_{(m)}(y - \hat y^{(m)})$, but it's the same thing.

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Let's confirm that this works by using it to perform a probit regression on the same simulated data as before (and this is not the canonical link, so we need this more general form of IRLS).

my_IRLS_general <- function(x, y, b.init, h, h.prime, tol=1e-8) {
  change <- Inf
  b.old <- b.init
  while(change > tol) {
    eta <- x %*% b.old  # linear predictor
    y.hat <- h(eta)
    h.prime_eta <- h.prime(eta)
    w_star <- h.prime_eta^2 / (y.hat * (1 - y.hat))
    z_star <- (y - y.hat) / h.prime_eta

    b.new <- b.old + lm(z_star ~ x - 1, weights = w_star)$coef  # WLS

    change <- sqrt(sum((b.new - b.old)^2))
    b.old <- b.new
  }
  b.new
}

# probit inverse link and derivative
h_probit <- function(x) pnorm(x, 0, 1)
h.prime_probit <- function(x) dnorm(x, 0, 1)

my_IRLS_general(x, y, rep(0,p), h_probit, h.prime_probit)
# x1         x2         x3         x4         x5 
# -0.6456508  1.2520266  0.5820856  0.4982678 -0.6768585 

glm(y~x-1, family=binomial(link="probit"))$coef
# x1         x2         x3         x4         x5 
# -0.6456490  1.2520241  0.5820835  0.4982663 -0.6768581 

and again the two agree.

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Comments on convergence

Finally, a few quick comments on convergence (I'll keep this brief as this is getting really long and I'm no expert at optimization). Even though theoretically each $J_{(m)}$ is negative definite, bad initial conditions can still prevent this algorithm from converging. In the probit example above, changing the initial conditions to b.init=rep(1,p) results in this, and that doesn't even look like a suspicious initial condition. If you step through the IRLS procedure with that initialization and these simulated data, by the second time through the loop there are some $\hat y_i$ that round to exactly $1$ and so the weights become undefined. If we're using the canonical link in the algorithm I gave we won't ever be dividing by $\hat y_i (1 - \hat y_i)$ to get undefined weights, but if we've got a situation where some $\hat y_i$ are approaching $0$ or $1$, such as in the case of perfect separation, then we'll still get non-convergence as the gradient dies without us reaching anything.