SQN_free {stochQN}

SQN Free-Mode Optimizer

Description

Optimizes an empirical (convex) loss function over batches of sample data. Compared to function/class 'SQN', this version lets the user do all the calculations from the outside, only interacting with the object by means of a function that returns a request type and is fed the required calculation through methods 'update_gradient' and 'update_hess_vec'.

Order in which requests are made:

========== loop ===========

* calc_grad

⁠ ⁠... (repeat calc_grad)

if 'use_grad_diff':

⁠ ⁠* calc_grad_big_batch

else:

⁠ ⁠* calc_hess_vec

===========================

After running this function, apply 'run_SQN_free' to it to get the first requested piece of information.

Usage

SQN_free(mem_size = 10, bfgs_upd_freq = 20, min_curvature = 1e-04,
  y_reg = NULL, use_grad_diff = FALSE, check_nan = TRUE,
  nthreads = -1)

Arguments

mem_size

Number of correction pairs to store for approximation of Hessian-vector products.

bfgs_upd_freq

Number of iterations (batches) after which to generate a BFGS correction pair.

min_curvature

Minimum value of (s * y) / (s * s) in order to accept a correction pair. Pass 'NULL' for no minimum.

y_reg

Regularizer for 'y' vector (gets added y_reg * s). Pass 'NULL' for no regularization.

use_grad_diff

Whether to create the correction pairs using differences between gradients instead of Hessian-vector products. These gradients are calculated on a larger batch than the regular ones (given by batch_size * bfgs_upd_freq).

check_nan

Whether to check for variables becoming NaN after each iteration, and reverting the step if they do (will also reset BFGS memory).

nthreads

Number of parallel threads to use. If set to -1, will determine the number of available threads and use all of them. Note however that not all the computations can be parallelized, and the BLAS backend might use a different number of threads.

Value

An 'SQN_free' object, which can be used through functions 'update_gradient', 'update_hess_vec', and 'run_SQN_free'

References

• Byrd, R.H., Hansen, S.L., Nocedal, J. and Singer, Y., 2016. "A stochastic quasi-Newton method for large-scale optimization." SIAM Journal on Optimization, 26(2), pp.1008-1031.

• Wright, S. and Nocedal, J., 1999. "Numerical optimization." (ch 7) Springer Science, 35(67-68), p.7.

See Also

update_gradient , update_hess_vec , run_oLBFGS_free

Examples

### Example optimizing Rosenbrock 2D function
### Note that this example is not stochastic, as the
### function is not evaluated in expectation based on
### batches of data, but rather it has a given absolute
### form that never varies.
### Warning: this optimizer is meant for convex functions
### (Rosenbrock's is not convex)
library(stochQN)


fr <- function(x) { ## Rosenbrock Banana function
	x1 <- x[1]
	x2 <- x[2]
	100 * (x2 - x1 * x1)^2 + (1 - x1)^2
}
grr <- function(x) { ## Gradient of 'fr'
	x1 <- x[1]
	x2 <- x[2]
	c(-400 * x1 * (x2 - x1 * x1) - 2 * (1 - x1),
	  200 * (x2 - x1 * x1))
}
Hvr <- function(x, v) { ## Hessian of 'fr' by vector 'v'
	x1 <- x[1]
	x2 <- x[2]
	H <- matrix(c(1200 * x1^2 - 400*x2 + 2,
			  -400 * x1, -400 * x1, 200),
			nrow = 2)
	as.vector(H %*% v)
}

### Initial values of x
x_opt = as.numeric(c(0, 2))
cat(sprintf("Initial values of x: [%.3f, %.3f]\n",
			x_opt[1], x_opt[2]))

### Will use constant step size throughout
### (not recommended)
step_size = 1e-3

### Initialize the optimizer
optimizer = SQN_free()

### Keep track of the iteration number
curr_iter <- 0

### Run a loop for severa, iterations
### (Note that some iterations might require more
###  than 1 calculation request)
for (i in 1:200) {
  req <- run_SQN_free(optimizer, x_opt, step_size)
  if (req$task == "calc_grad") {
    update_gradient(optimizer, grr(req$requested_on$req_x))
  } else if (req$task == "calc_hess_vec") {
	   update_hess_vec(optimizer,
      Hvr(req$requested_on$req_x, req$requested_on$req_vec))
  }

  ### Track progress every 10 iterations
  if (req$info$iteration_number > curr_iter) {
  	curr_iter <- req$info$iteration_number
  }
  if ((curr_iter %% 10) == 0) {
  	cat(sprintf(
  	 "Iteration %3d - Current function value: %.3f\n",
  	 req$info$iteration_number, fr(x_opt)))
  }
}
cat(sprintf("Current values of x: [%.3f, %.3f]\n",
			x_opt[1], x_opt[2]))