mr_ash_pen

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Last updated: 2021-05-28

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Introduction

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 \((S(b)-b)^2\).

Summary of optimization problem

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 \(b \in R^p\).

\[h(b,w) = -s^2 l_{w,s}(b) - 0.5 s^4 [l_{w,s}'(b)]^2\]

Code for fundamental functions

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)

Past versions of unnamed-chunk-2-1.png

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)

Past versions of unnamed-chunk-2-2.png

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)

Past versions of unnamed-chunk-3-1.png

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)

Past versions of unnamed-chunk-4-1.png

Version	Author	Date
800415b	Matthew Stephens	2021-05-11

Inverting S by Newton-Raphson method

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)
}

Past versions of unnamed-chunk-5-1.png

Version	Author	Date
800415b	Matthew Stephens	2021-05-11

plot(S(x,1,w,prior_grid),y)
plot(x,y)

Past versions of unnamed-chunk-5-2.png

Version	Author	Date
800415b	Matthew Stephens	2021-05-11

Optimizing the objective

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)

Past versions of unnamed-chunk-8-1.png

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?

Trendfiltering

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)

Past versions of unnamed-chunk-10-1.png

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)

Past versions of unnamed-chunk-11-1.png

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))

Past versions of unnamed-chunk-12-1.png

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))

Past versions of unnamed-chunk-13-1.png

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))

Past versions of unnamed-chunk-14-1.png

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)

Past versions of unnamed-chunk-15-1.png

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)

Past versions of unnamed-chunk-15-2.png

Version	Author	Date
800415b	Matthew Stephens	2021-05-11

Session information

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