# Stateful Optimization

#### 2020-08-29

By “Stateful” I mean what if we could create an optimizer independently of the function it was operating on and be able to pass it around, store it, and get full control over when we pass it data to continue the optimization.

This vignette is about using mize to manually control an optimization externally. Instead of passing mize a function to be optimized from a starting point, then waiting for mize to finish and get back the finished results, you might want to tell mize to optimize for a few steps, then do something with the intermediate results: log the cost, update some parameters, test for some specific convergence criterion, checkpoint the current results, or plot the current state of the result in some custom way. Then, if there’s still more optimization to be done, pass the results back off to mize and get it to crank away for a few more iterations.

This was in fact the inspiration for creating mize in the first place: I wanted access to the sort of optimization routines that the stats::optim function provided, but the lack of control was a deal breaker. One way to try and get around the problem is to only optimize for a few iterations at a time:

rb_fg <- list(
fn = function(x) { 100 * (x[2] - x[1] * x[1]) ^ 2 + (1 - x[1]) ^ 2  },
gr = function(x) { c( -400 * x[1] * (x[2] - x[1] * x[1]) - 2 * (1 - x[1]),
200 *        (x[2] - x[1] * x[1])) })
rb0 <- c(-1.2, 1)

par <- rb0
for (batch in 1:3) {
optim_res <- stats::optim(par = par, fn = rb_fg$fn, gr = rb_fg$gr,
method = "BFGS", control = list(maxit = 10))
par <- optim_res$par message("batch ", batch, " f = ", formatC(optim_res$value))
}
#> batch 1 f = 1.367
#> batch 2 f = 0.2011
#> batch 3 f = 0.006648

but even this unsatisfactory work-around causes problems, because you are reinitializing the optimization for with each batch, and losing all the information the optimizer has. In the case of methods like BFGS and CG, this is important for their efficient use. The more control you want, the fewer iterations per batch, but that leads to behavior that approaches steepest descent.

Instead, mize lets you create a stateful optimizer, that you pass to a function, and an updated version of which is returned as part of the return value of the function. This gives you complete control over what to do in between iterations, without sacrificing any of the information the optimizer is using.

## Creating an Optimizer

To create an optimizer, use the make_mize function:

opt <- make_mize(method = "BFGS")

## Initialize the Optimizer

Before starting the optimization, the optimizer needs to be initialized using the function and starting point. Mainly this is to allow the various methods to preallocate whatever storage they make use of (matrices and vectors) according to the size of the data, as specified by the starting location.

To continue the rosenbrock example from above:

opt <- mize_init(opt = opt, par = rb0, fg = rb_fg)

### A potential simplification

If you have both the starting point and the function to optimize to hand at the point when the optimizer is created, you can provide that to make_mize and it will do the initialization for you:

opt <- make_mize(method = "BFGS", par = rb0, fg = rb_fg, max_iter = 30)

And there is no need to make a separate call to mize_init. However, normally it’s more convenient to handle configuring the optimizer earlier than when the data shows up.

## Start optimizing

Using the batch of ten iteration approach we used with optim is very similar with mize:

par <- rb0
iter <- 0
for (batch in 1:3) {
for (i in 1:10) {
mize_res <- mize_step(opt = opt, par = par, fg = rb_fg)
par <- mize_res$par opt <- mize_res$opt
}
message("batch ", batch, " f = ", formatC(mize_res$f)) } #> batch 1 f = 2.604 #> batch 2 f = 0.5568 #> batch 3 f = 0.005003 The difference here is that you have to do the iterating in batches of 10 manually yourself, remembering to increment the iteration counter and pass it to mize_step. Plus, the optimizer needs to be updated with the version that was returned from the function. ### Return value of mize_step As you can see, with the greater power of mize_step to control the iteration, comes greater responsibility. You also need to decide when to stop iterating. Apart from par and opt, there are some other components to the returned result list which might help: • f - The function value, if it was calculated at par. For the few methods which don’t do this, you can of course generate it yourself via rb_fg$fn(par).
• g - The gradient vector, if it was calculated at par. If it’s not present, then obviously there’s nothing to stop you calculating rb_fg$gr(par) yourself. • nf - The number of function evaluations carried out so far (i.e. since initialization). opt is also keeping track of this, and coordinates with mize_step, so you don’t need to manually update this yourself between steps. • ng - The number of gradient evaluations carried out so far. You should treat the optimizer, opt, as a black box and not examine its horrific innards, except to check whether opt$error is non-NULL. If it’s anything other than NULL, then this means something really bad has happened during the optimization, most likely a NaN or Inf was calculated in the gradient. This can happen with a very poorly chosen starting point, and a combination of descent method and line search which doesn’t guarantee descent, such as a very aggressive momentum scheme or more likely an adaptive learning rate technique like delta-bar-delta. Monitoring the function value or the size of the change in par between iterations can help spot an imminent divergence.

Taking all that into account, here’s a self-contained example, that removes the now un-necessary batching, does some minor error checking, and keeps track of the best parameters seen so far (although with this combination of optimizer and problem, you don’t have to worry about it):

# Create the optimizer
opt <- make_mize(method = "BFGS")

# Pretend we don't have access to the function or starting point until later
rb_fg <- list(
fn = function(x) { 100 * (x[2] - x[1] * x[1]) ^ 2 + (1 - x[1]) ^ 2  },
gr = function(x) { c( -400 * x[1] * (x[2] - x[1] * x[1]) - 2 * (1 - x[1]),
200 *        (x[2] - x[1] * x[1])) })
rb0 <- c(-1.2, 1)

# Initialize
opt <- mize_init(opt = opt, par = rb0, fg = rb_fg)

# Store the best seen parameters in case something goes wrong
par <- rb0
par_best <- par
f_best <- rb_fg$fn(par_best) for (i in 1:30) { mize_res <- mize_step(opt = opt, par = par, fg = rb_fg) par <- mize_res$par
opt <- mize_res$opt # Do whatever you want with the data at each iteration if (opt$is_terminated) {
break
}
if (mize_res$f < f_best) { f_best <- mize_res$f
par_best <- par
}
}

# optimized result is in par_best
par_best
#> [1] 0.9294066 0.8642370
f_best
#> [1] 0.005002828

## Step information

# Create optimizer and do one step of optimization as usual
opt <- make_mize(method = "BFGS", par = rb0, fg = rb_fg)
par <- rb0
mize_res <- mize_step(opt = opt, par = par, fg = rb_fg)
step_info <- mize_step_summary(mize_res$opt, mize_res$par, rb_fg, par_old = par)

# info that's already available in mize_res
step_info$f #> [1] 19.49933 step_info$ng
#> [1] 3
step_info$nf #> [1] 3 # and some extra step_info$step
#> [1] 0.02168573
step_info$alpha #> [1] 9.31247e-05 mize_step_summary takes opt, par and fg like mize_step does, but also optionally wants a par_old argument. This is the value of par from the previous iteration, from which it calculates the size of the step taken in this iteration. Information available from the return value of mize_step_summary includes: • iter The iteration number. • f The function value, if it’s available, or if you have set a convergence tolerance that requires its calculation (see below). • g2n The gradient l2 (Euclidean) norm, if grad_tol is non-NULL (see the Convergence section for more). • ginfn The gradient infinity norm, if ginf_tol is non-NULL (also see the Convergence section for more). • nf The number of function evaluations so far over the course of the optimization. • ng The number of gradient evaluations so far over the course of the optimization. • step The step size of this iteration. • alpha The size of the line search value found during the gradient descent stage. This won’t be the same as step even for optimizers that don’t use an extra momentum stage because the total step size is normally the value of alpha multiplied by the magnitude of the gradient. • mu If a momentum stage was used, the momentum coefficient. • opt The optimizer with updated function and gradient counts, if f, g2n, ginfn was calculated. In many cases, f, g2n and ginfn do not require any recalculation (or aren’t calculated), but to be on the safe side, always reassign opt to the return value from mize_step_summary. Here’s a modified version of the previous example, where we log out information from mize_step_summary. We’re only going to go for 10 iterations to avoid too much output. # Create the optimizer opt <- make_mize(method = "BFGS", par = rb0, fg = rb_fg) par <- rb0 for (i in 1:10) { par_old <- par mize_res <- mize_step(opt = opt, par = par, fg = rb_fg) par <- mize_res$par
opt <- mize_res$opt # step info step_info <- mize_step_summary(opt, par, rb_fg, par_old) opt <- step_info$opt
message(paste(
Map(function(x) { paste0(x, " = ", formatC(step_info[[x]])) },
c("iter", "f", "nf", "ng", "step")),
collapse = ", "))
}
#> iter = 1, f = 19.5, nf = 3, ng = 3, step = 0.02169
#> iter = 2, f = 11.57, nf = 4, ng = 4, step = 0.04729
#> iter = 3, f = 4.281, nf = 5, ng = 5, step = 0.09808
#> iter = 4, f = 4.144, nf = 6, ng = 6, step = 0.01427
#> iter = 5, f = 4.14, nf = 7, ng = 7, step = 0.002249
#> iter = 6, f = 4.136, nf = 8, ng = 8, step = 0.002942
#> iter = 7, f = 4.128, nf = 9, ng = 9, step = 0.00557
#> iter = 8, f = 4.114, nf = 10, ng = 10, step = 0.01061
#> iter = 9, f = 4.086, nf = 11, ng = 11, step = 0.02048
#> iter = 10, f = 2.604, nf = 14, ng = 14, step = 0.8382

## Convergence

In the example up until now we have manually looped over 30 iterations and then stopped. More sophisticated stopping criteria is available. Three changes are needed:

1. When initializing the optimizer, when passing par and fg to either make_mize or mize_init, also pass termination criteria:
opt <- make_mize(method = "BFGS", par = rb0, fg = rb_fg, max_iter = 30)
# or
opt <- make_mize(method = "BFGS")
opt <- mize_init(opt = opt, par = rb0, fg = rb_fg, max_iter = 30)
1. At the end of the loop, after calling mize_step_summary, pass the return value to the function mize_check_convergence. This returns an updated version of opt which will indicate if optimization should stop by setting the opt$is_terminated boolean flag: step_info <- mize_step_summary(opt, par, rb_fg, par_old) opt <- check_mize_convergence(step_info) Note that you don’t need to manually assign opt to the value that comes from mize_step_summary, as check_mize_convergence handles that. 1. Instead of manually looping with a for loop you can use while (!opt$is_terminated).

Once opt$is_terminated is TRUE, you can find out what caused the optimization by looking at opt$terminate$what. We were using opt$is_terminated before now, where if it was set to TRUE it meant that something awful had occurred, like infinity or NaN in a gradient. check_mize_convergence also uses this flag, but now with an expanded meaning that just indicates optimization should cease, but not necessarily because a catastrophe occurred. It’s still worth checking if opt$is_terminated was set by mize_step if anything that you do in the rest of the loop assumes that the gradient or function value is finite (e.g. comparing it to a real number in a boolean condition). Apart from just maximum number of iterations, there are a variety of options that relate to convergence. There is a separate vignette which covers these convergence options, and all the parameters mentioned there can be passed to make_mize and mize_init. Whatever options you use, setting max_iter is a good idea to avoid an infinite loop. Here’s the example repeated again, this time using check_mize_convergence to control the number of iterations, rather than a for loop: # Create the optimizer opt <- make_mize(method = "BFGS") rb_fg <- list( fn = function(x) { 100 * (x[2] - x[1] * x[1]) ^ 2 + (1 - x[1]) ^ 2 }, gr = function(x) { c( -400 * x[1] * (x[2] - x[1] * x[1]) - 2 * (1 - x[1]), 200 * (x[2] - x[1] * x[1])) }) rb0 <- c(-1.2, 1) # Initialize and set convergence criteria opt <- mize_init(opt = opt, par = rb0, fg = rb_fg, max_iter = 30) # Store the best seen parameters in case something goes wrong par <- rb0 par_best <- par f_best <- rb_fg$fn(par_best)

while (!opt$is_terminated) { mize_res <- mize_step(opt = opt, par = par, fg = rb_fg) par <- mize_res$par
opt <- mize_res$opt # Do whatever you want with the data at each iteration if (opt$is_terminated) {
break
}
if (mize_res$f < f_best) { f_best <- mize_res$f
par_best <- par
}

step_info <- mize_step_summary(opt, par, rb_fg, par_old)
# Do something with the step info if you'd like
# Check convergence
opt <- check_mize_convergence(step_info)
}

# optimized result is in par_best
par_best
#> [1] 1 1
f_best
#> [1] 0