Last updated: 2021-05-28
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The idea here is to code up mr.ash as a penalized method, compute gradients etc.
The penalized version of the mr.ash problem is (see Theorem 6 in mr.ash draft) \[\min_b (1/2\sigma^2) ||y - Xb||_2 + (1/\sigma^2) \sum_j \rho_{g,s_j}(b_j) + 0.5(n-p) \log(2\pi \sigma^2)\] Where the penalty \(\rho\) depends on the prior (\(g\)) and the \(s_j^2:=\sigma^2/(x_j'x_j)\). Here I’m going to assume for simplicity that \(g\) denotes the prior on b, ie not scale the prior on b by the residual variance as in the paper.
The penalty \(\rho_{g,s_j}(b)\) is inconvenient to compute because it involves the inverse of \(S\) (the posterior mean shrinkage function) which is not analytically available to us, at least in our current state of knowledge.
A simple idea is to rewrite the problem as: \[\min_b (1/2\sigma^2) ||y - XS_{g,s}(b)||_2 + (1/\sigma^2) \rho_{g,s}(S_{g,s}(b)) + 0.5(n-p) \log(2\pi \sigma^2)\] Here \(S_{g,s}(b)\) means apply \(S\) element-wise to the vector \(b\). There is a slight abuse of notation here as \(s_j\) may vary across \(j\). To be explicit, the \(j\)th element is \(S_{g,s}(b)\) is equal to \(S_{g,s_j}(b_j)\).
Because \(S_{g,s}\) is invertible there is no loss of generality in writing the optimization this way. Furthermore, \[h_{g,s}(b):=\rho_{g,s}(S_{g,s}(b))\] is easy to compute. Indeed \[h_{g,s}(b) = -s^2 l_{g,s}(b) - 0.5 s^4 [l_{g,s}'(b)]^2\]
where \(l_{g,s}(b)\) is the marginal log-likelihood function under the normal means model with prior \(g\) and variance \(s\). That is \(l_{g,s}(b) = \log(f(b))\) where \[f(b) := \sum_k \pi_k N(b; 0, \sigma_k^2+s^2).\]
Note: Tweedie’s formula says that relates shrinkage in the normal means model \(S\) to the marginal loglikelihood \(l\), via \[S_{f,s^2}(b) = b + s^2 l'(b)\]. So the term \(s^4 [l'(b)]^2\) above is \((S(b)-b)^2\).
I wanted to give a succinct summary of the overall optimization problem to share with others.
For fixed \(\sigma^2\), and letting \(w\) represent the mixture proportions in the prior \(g\) (which we often denote \(\pi\)) we can write it as follows.
Given \(y \in R^n\) and \(X \in R^{n \times p}\), \[\min ||y - XS(b; w)||_2 + h(b,w)\] subject to: \(w \in S_K\) the simplex of dimension \(K\), and \(b \in R^p\).
\[h(b,w) = -s^2 l_{w,s}(b) - 0.5 s^4 [l_{w,s}'(b)]^2\]
Everything can be written in terms of the marginal likelihood \(f\) and its first two derivatives, so we code those up first. I have not been careful about numerical issues here.. will need to deal with those at some point.
#y,s are vectors of length n
#w, sigma are vectors of length K (w are prior mixture proportions)
# returns an n-vector of "marginal likelihoods" under mixture prior
f = function(b, s, w, sigma){
if(length(s)==1){s = rep(s,length(b))}
sigmamat <- outer(s^2, sigma^2, `+`) # n time k
llik_mat <- -0.5 * (log(sigmamat) + b^2 / sigmamat)
#llik_norms <- apply(llik_mat, 1, max)
#L_mat <- exp(llik_mat - llik_norms)
L_mat <- exp(llik_mat)
return((1/sqrt(2*pi)) * as.vector(colSums(w * t(L_mat))))
}
#returns a vector of the derivative of f evaluated at each element of y
f_deriv = function(b, s, w, sigma){
if(length(s)==1){s = rep(s,length(b))}
sigmamat <- outer(s^2, sigma^2, `+`) # n time k
llik_mat <- -(3/2) * log(sigmamat) -0.5* b^2 / sigmamat
#llik_norms <- apply(llik_mat, 1, max)
#L_mat <- exp(llik_mat - llik_norms)
L_mat <- exp(llik_mat)
return((-b/sqrt(2*pi)) * as.vector(colSums(w * t(L_mat))))
}
# returns f_deriv/b ok even if b=0
f_deriv_over_b = function(b, s, w, sigma){
if(length(s)==1){s = rep(s,length(b))}
sigmamat <- outer(s^2, sigma^2, `+`) # n time k
llik_mat <- -(3/2) * log(sigmamat) -0.5* b^2 / sigmamat
#llik_norms <- apply(llik_mat, 1, max)
#L_mat <- exp(llik_mat - llik_norms)
L_mat <- exp(llik_mat)
return((-1/sqrt(2*pi)) * as.vector(colSums(w * t(L_mat))))
}
#returns a vector of the second derivatives of f evaluated at each element of y
f_deriv2 = function(b, s, w, sigma){
if(length(s)==1){s = rep(s,length(b))}
sigmamat <- outer(s^2, sigma^2, `+`) # n time k
llik_mat <- -(5/2) * log(sigmamat) -0.5* b^2 / sigmamat
#llik_norms <- apply(llik_mat, 1, max)
#L_mat <- exp(llik_mat - llik_norms)
L_mat <- exp(llik_mat)
return((b^2/sqrt(2*pi)) * as.vector(colSums(w * t(L_mat)))+ f_deriv_over_b(b,s,w,sigma))
}
Check the derivative code numerically:
n = 100
k = 5
b = rnorm(n)
w = rep(1/k,k)
prior_grid = c(0,1,2,3,4,5)
eps=1e-5
plot((f(b+eps,1,w,prior_grid)-f(b,1,w,prior_grid))/eps, f_deriv(b,1,w,prior_grid),xlab="numerical 1st derivative", ylab="analytic 1st derivative")
abline(a=0,b=1)
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
plot((f_deriv(b+eps,1,w,prior_grid)-f_deriv(b,1,w,prior_grid))/eps, f_deriv2(b,1,w,prior_grid), xlab="numerical 2nd derivative", ylab="analytic 2nd derivative")
abline(a=0,b=1)
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
Now we have \[l(b) = \log f(b)\], \[l'(b) = f'(b)/f(b)\], \[l''(b) = (f(b)f''(b)-f'(b)^2)/f(b)^2\],
l = function(b, s, w, prior_grid){
log(f(b,s,w,prior_grid))
}
l_deriv = function(b, s, w, prior_grid){
f_deriv(b,s,w,prior_grid)/f(b,s,w,prior_grid)
}
l_deriv2 = function(b, s, w, prior_grid){
((f_deriv2(b,s,w,prior_grid)*f(b,s,w,prior_grid))-f_deriv(b,s,w,prior_grid)^2)/f(b,s,w,prior_grid)^2
}
plot((l_deriv(b+eps,1,w,prior_grid)-l_deriv(b,1,w,prior_grid))/eps, l_deriv2(b,1,w,prior_grid), xlab="numerical 2nd derivative", ylab="analytic 2nd derivative")
abline(a=0,b=1)
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
And with these in place we can compute the shrinkage function and penalty function \(h\), using \[h(b) = -l(b) - 0.5(l'(b))^2\] \[S(b) = b+ s^2l'(b)\] \[S'(b) = 1 + s^2 l''(b)\].
S = function(b, s, w, prior_grid){
return(b + s^2 * l_deriv(b,s,w,prior_grid))
}
S_deriv = function(b, s, w, prior_grid){
return(1+ s^2 * l_deriv2(b,s,w,prior_grid))
}
h = function(b,s,w,prior_grid){
return(-s^2 * l(b,s,w,prior_grid) - 0.5 * s^4 * l_deriv(b,s,w,prior_grid)^2)
}
h_deriv = function(b,s,w,prior_grid){
return(-s^2 * l_deriv(b,s,w,prior_grid) - s^4 * l_deriv(b,s,w,prior_grid) * l_deriv2(b,s,w,prior_grid))
}
plot((h(b+eps,1,w,prior_grid)-h(b,1,w,prior_grid))/eps, h_deriv(b,1,w,prior_grid), xlab="numerical 1st derivative", ylab="analytic 1st derivative")
abline(a=0,b=1)
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
One thing we might want to do is invert S. We don’t have analytic form for this, but we can use the Newton-Raphson method to solve \(S(x)=y\). The iterates would be \[x \leftarrow x - (S(x)-y)/S'(x) \]
y = seq(-10,10,length=100)
x=y
par(mfrow=c(5,5))
par(mar=rep(1.5,4))
w = rep(0.5,2)
prior_grid = c(0,5)
for(i in 1:25){
plot(S(x,1,w,prior_grid),y)
x = x-(S(x,1,w,prior_grid)-y)/S_deriv(x,1,w,prior_grid)
}
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
plot(S(x,1,w,prior_grid),y)
plot(x,y)
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
obj = function(b, y, X, residual_var, w, prior_grid){
d = colSums(X^2)
s = sqrt(residual_var/d)
r = y - X %*% S(b,s,w,prior_grid)
return(0.5*(1/residual_var)*sum(r^2) + (1/residual_var)*sum(h(b,s,w,prior_grid)))
}
obj_grad = function(b, y, X, residual_var, w, prior_grid){
d = colSums(X^2)
s = sqrt(residual_var/d)
r = y - X %*% S(b,s,w,prior_grid)
return((-1/residual_var)*(t(r) %*% X) * S_deriv(b,s,w,prior_grid) + (1/residual_var)* h_deriv(b,s,w,prior_grid))
}
Try it out on simple simulation:
n = 100
p = 20
X = matrix(rnorm(n*p),nrow=n)
norm = colSums(X^2)
X = t(t(X)/sqrt(norm))
b = rnorm(p)
y = X %*% b + rnorm(n)
(obj(b,y,X,1,w,prior_grid)-obj(b+c(eps,rep(0,p-1)),y,X,1,w,prior_grid))/eps
[1] 0.2996978
-obj_grad(b,y,X,1,w,prior_grid)[1]
[1] 0.2997033
b.cg.warm = optim(b,obj,gr=obj_grad,method="CG",y=y,X=X,residual_var=1, w=w,prior_grid=prior_grid)
b.cg.null = optim(rep(0,p),obj,gr=obj_grad,method="CG",y=y,X=X,residual_var=1, w=w,prior_grid=prior_grid)
b.bfgs.warm = optim(b,obj,gr=obj_grad,method="BFGS",y=y,X=X,residual_var=1, w=w,prior_grid=prior_grid)
b.bfgs.null = optim(rep(0,p),obj,gr=obj_grad,method="BFGS",y=y,X=X,residual_var=1, w=w,prior_grid=prior_grid)
b.cg.warm$value
[1] 94.80382
b.cg.null$value
[1] 94.80382
b.bfgs.warm$value
[1] 94.80382
b.bfgs.null$value
[1] 94.80382
plot(S(b.cg.warm$par,1,w,prior_grid),b)
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
Some comments from Mihai: BFGS stores a dense approximation to the Hessian, so won’t be good for big problems. However, “limited”-BFGS might work (L-BFGS?). CG needs preconditioning in general; BFGS does not because it is computing an approximation to the Hessian.
Thoughts from me: maybe we can compute the Hessian directly and efficiently when X is, say, the trend filtering matrix.
Another question: if we write h as a function of pi, is h(pi) convex? Is rho(pi) convex?
Here I try a really hard trend-filtering example.
set.seed(100)
sd = 1
n = 100
p = n
X = matrix(0,nrow=n,ncol=n)
for(i in 1:n){
X[i:n,i] = 1:(n-i+1)
}
btrue = rep(0,n)
btrue[40] = 8
btrue[41] = -8
Y = X %*% btrue + sd*rnorm(n)
norm = colSums(X^2)
X = t(t(X)/sqrt(norm))
btrue = btrue * sqrt(norm)
plot(Y)
lines(X %*% btrue)
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
prior_grid=c(0,10,100,1000)
w=rep(1/4,4)
b.cg.null = optim(rep(0,p),obj,gr=obj_grad,method="CG",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid,control=list(maxit=1000))
b.bfgs.null = optim(rep(0,p),obj,gr=obj_grad,method="BFGS",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid,control=list(maxit=1000))
plot(Y)
lines(X %*% S(b.cg.null$par,1,w,prior_grid))
lines(X %*% S(b.bfgs.null$par,1,w,prior_grid))
b.cg.warm = optim(btrue,obj,gr=obj_grad,method="CG",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid)
b.bfgs.warm = optim(btrue,obj,gr=obj_grad,method="BFGS",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid)
lines(X %*% S(b.cg.warm$par,1,w,prior_grid))
lines(X %*% S(b.bfgs.warm$par,1,w,prior_grid),col=2)
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
b.cg.warm$value
[1] 289.8884
b.cg.null$value
[1] 385.2665
b.bfgs.warm$value
[1] 289.8891
b.bfgs.null$value
[1] 422.1747
This case is ridge regression, so should be convex… try it out.
prior_grid=c(10,10)
w=rep(1/2,2)
b.cg.null = optim(rep(0,p),obj,gr=obj_grad,method="CG",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid,control=list(maxit=1000))
b.bfgs.null = optim(rep(0,p),obj,gr=obj_grad,method="BFGS",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid,control=list(maxit=1000))
plot(Y)
lines(X %*% S(b.cg.null$par,1,w,prior_grid))
lines(X %*% S(b.bfgs.null$par,1,w,prior_grid))
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
b.bfgs.null$value
[1] 428.7797
b.cg.null$value
[1] 428.7797
Try a prior with more overlapping components
prior_grid=seq(0,100,length=100)
w=rep(1/100,100)
b.cg.null = optim(rep(0,p),obj,gr=obj_grad,method="CG",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid,control=list(maxit=1000))
b.bfgs.null = optim(rep(0,p),obj,gr=obj_grad,method="BFGS",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid,control=list(maxit=1000))
plot(Y)
lines(X %*% S(b.cg.null$par,1,w,prior_grid))
lines(X %*% S(b.bfgs.null$par,1,w,prior_grid))
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
b.bfgs.null$value
[1] 462.9663
b.cg.null$value
[1] 472.9022
And now revert to the sparser prior
prior_grid=c(0,10,100,1000)
w=rep(1/4,4)
b.cg.null = optim(b.cg.null$par,obj,gr=obj_grad,method="CG",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid,control=list(maxit=1000))
b.bfgs.null = optim(b.bfgs.null$par,obj,gr=obj_grad,method="BFGS",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid,control=list(maxit=1000))
plot(Y)
lines(X %*% S(b.cg.null$par,1,w,prior_grid))
lines(X %*% S(b.bfgs.null$par,1,w,prior_grid))
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
b.bfgs.null$value
[1] 302.7868
b.cg.null$value
[1] 310.1688
Try initializing from bhat the “overfit” solution.
bhat = chol2inv(chol(t(X) %*% X)) %*% t(X) %*% Y
plot(X %*% bhat)
lines(Y)
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
b.bfgs.bhat = optim(bhat,obj,gr=obj_grad,method="BFGS",y=Y,X=X,residual_var=1, w=w,prior_grid=prior_grid,control=list(maxit=10000))
b.bfgs.bhat$value
[1] 293.4889
plot(Y)
lines(X %*% S(as.vector(b.bfgs.null$par),1,w,prior_grid))
lines(X %*% S(as.vector(b.bfgs.warm$par),1,w,prior_grid),col=2)
lines(X %*% S(as.vector(b.bfgs.bhat$par),1,w,prior_grid),col=3)
Version | Author | Date |
---|---|---|
800415b | Matthew Stephens | 2021-05-11 |
sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] Rcpp_1.0.6 rstudioapi_0.13 whisker_0.4 knitr_1.29
[5] magrittr_2.0.1 workflowr_1.6.2 R6_2.4.1 rlang_0.4.10
[9] stringr_1.4.0 tools_3.6.0 xfun_0.16 git2r_0.27.1
[13] htmltools_0.5.0 ellipsis_0.3.1 yaml_2.2.1 digest_0.6.27
[17] rprojroot_1.3-2 tibble_3.0.4 lifecycle_1.0.0 crayon_1.3.4
[21] later_1.1.0.1 vctrs_0.3.8 fs_1.5.0 promises_1.1.1
[25] glue_1.4.2 evaluate_0.14 rmarkdown_2.3 stringi_1.4.6
[29] compiler_3.6.0 pillar_1.4.6 backports_1.1.10 httpuv_1.5.4
[33] pkgconfig_2.0.3