apg: Accelerated proximal gradient optimization
In jpvert/apg: Optimization with accelerated proximal gradient

Description Usage Arguments Value Examples

This function implements an accelerated proximal gradient method (Nesterov 2007, Beck and Teboulle 2009). It solves:

min_x (f(x) + h(x)), x \in R^dim_x

where f is smooth, convex and h is non-smooth, convex but simple in that we can easily evaluate the proximal operator of h.

1	apg(grad_f, prox_h, dim_x, opts)

`grad_f`	A function that computes the gradient of f : grad_f(v,opts) = df(v)/dv
`prox_h`	A function that computes the proximal operator of h : prox_h(v,t,opts) = argmin_x (th(x) + 1/2 * norm(x-v)^2)*
`dim_x`	The dimension of the unknown x
`opts`	List of parameters, both for the `apg` function and for the `grad_f` and `prox_h` functions. For `apg`, the list can contain the following fields: `X_INIT`: initial starting point (default `0`) `USE_RESTART` : use adaptive restart scheme (default `TRUE`) `MAX_ITERS` : maximum iterations before termination (default `2000`) `EPS` : tolerance for termination (default `1e-6`) `ALPHA` : step-size growth factor (default `1.01`) `BETA` : step-size shrinkage factor (default `0.5`) `QUIET` : if false writes out information every 100 iters (default `FALSE`) `GEN_PLOTS` : if true generates plots of norm of proximal gradient (default `TRUE`) `USE_GRA` : if true uses UN-accelerated proximal gradient descent (typically slower) (default `FALSE`) `STEP_SIZE` : starting step-size estimate, if not set then apg makes initial guess (default `NULL`) `FIXED_STEP_SIZE` : don't change step-size (forward or back tracking), uses initial step-size throughout, only useful if good STEP_SIZE set (default `FALSE`) In addition, `opts` can contain other fields that will be passed to the `grad_f` and `prox_h` functions. This code was borrowed and adapted from the MATLAB version of Brendan O'Donoghue available at `https://github.com/bodono/apg`

A list with x, the solution of the problem:

min_x (f(x) + h(x)), x \in R^dim_x ,

and t, the last step size.

# Solve a Lasso problem:
# min_x 1/2 norm( A%*%x - b )^2 + lambda ||x||_1
n <- 50
m <- 20
lambda <- 1
A <- matrix(rnorm(m*n), nrow=n)
b <- rnorm(n)
r <- apg(grad.quad, prox.l1, m, list(A=A, b=b, lambda=lambda) )
# This gives the same result as:
# m <- glmnet(A,b,alpha=1, standardize=FALSE,lambda=1/50,intercept=FALSE)