Convergence

James Melville

2018-09-10

There are a variety of ways that the optimization can terminate, running the gamut from good (you have reached the minimum and further work is pointless) to bad (the solution diverged and an infinity or NaN turned up in a calculation).

We’ll use the 2D Rosenbrock function for the examples, which has a minimum at c(1, 1), where the function equals 0.

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)

Iteration tolerance

An obvious way for the optimization to terminate is if you run out of iterations:

res <- mize(rb0, rb_fg, max_iter = 10)
res$terminate
#> $what
#> [1] "max_iter"
#> 
#> $val
#> [1] 10
res$f
#> [1] 2.552598
res$par
#> [1] -0.5690157  0.2936478

When comparing different methods, the number of iterations is obviously less important than the amount of actual CPU time you spent. Comparing results with a fixed number of iterations is not a very good idea, because different methods may do a lot more work within an iteration than others. See the section on function and gradient tolerance below.

Function tolerance

There are two ways to specify a function tolerance, based on comparing the difference between consecutive function values. abs_tol measures absolute tolerance.

res <- mize(rb0, rb_fg, max_iter = 100, abs_tol = 1e-8)
res$terminate
#> $what
#> [1] "abs_tol"
#> 
#> $val
#> [1] 4.061025e-10
res$f
#> [1] 4.606476e-11
res$par
#> [1] 0.9999967 0.9999928

However, relative tolerance is often preferred, because it measures the change in value relative to the size of the values themselves.

res <- mize(rb0, rb_fg, max_iter = 100, rel_tol = 1e-3)
# hit relative tolerance
res$terminate
#> $what
#> [1] "abs_tol"
#> 
#> $val
#> [1] 1.06609e-08

# but stopped too early!
res$iter
#> [1] 37
res$f
#> [1] 4.521673e-10
res$par
#> [1] 0.9999972 0.9999923

In this example we stopped way too early. Even efficient methods like L-BFGS may make little progress on some iterations, so don’t be too aggressive with relative tolerance.

Gradient tolerance

Gradient tolerances measure the difference between the size of the gradient on consecutive step. grad_tol uses the 2-norm (sometimes referred to as the Euclidean norm) of the gradient to measure convergence.

res <- mize(rb0, rb_fg, abs_tol = 0, grad_tol = 1e-3)

res$terminate
#> $what
#> [1] "grad_tol"
#> 
#> $val
#> [1] 0.0009378802
res$f
#> [1] 4.521673e-10
res$par
#> [1] 0.9999972 0.9999923

This seems like a good stopping criterion because it is always zero at a minimum, even if the function isn’t. It is also used to compare different methods in Nocedal and Wright’s book. However, it has also been recognized that it is not always reliable, see for instance this paper by Nocedal and co-workers.

Other workers suggest using the infinity norm (the maximum absolute component) of the gradient vector, particularly for larger problems. For example, see this conjugate gradient paper by Hager and Zhang. To use the infinity norm, set the ginf_norm parameter.

res <- mize(rb0, rb_fg, rel_tol = NULL, abs_tol = NULL, ginf_tol = 1e-3)

res$terminate
#> $what
#> [1] "ginf_tol"
#> 
#> $val
#> [1] 0.0008377527
res$f
#> [1] 4.521673e-10
res$par
#> [1] 0.9999972 0.9999923

While the gradient norms aren’t as reliable for checking convergence, they almost never incur any overhead for checking, because the gradient that’s calculated at the end of the iteration for this purpose can nearly always be re-used for the gradient descent calculation at the beginning of the next iteration, whereas the function-based convergence requires the function to be calculated at the end of the iteration and this is not always reused, although for many line search methods it is.

Step tolerance

You can also look out for the change in par itself getting too small:

# set abs_tol to zero to stop it from triggering instead of step_tol
res <- mize(rb0, rb_fg, abs_tol = 0, step_tol = .Machine$double.eps)
res$terminate
#> $what
#> [1] "max_iter"
#> 
#> $val
#> [1] 100
res$iter
#> [1] 100
res$f
#> [1] 2.383932e-23
res$par
#> [1] 1 1

In most cases, the step tolerance should be a reasonable way to spot convergence. Some optimization methods may allow for a step size of zero for some iterations, preferring to commence the next iteration using the same initial value of par, but with different optimization settings. The step tolerance criterion knows when this sort of “restart” is being attempted, and does not triggered under these conditions.

Function and gradient count tolerance

For most problems, the time spent calculating the function and gradient values will drown out any of the house-keeping that individual methods do, so the number of function and gradient evaluations is the usual determinant of how long an optimization takes. You can therefore decide to terminate based on the number of function evaluations:

res <- mize(rb0, rb_fg, max_fn = 10)

res$terminate
#> $what
#> [1] "max_fn"
#> 
#> $val
#> [1] 10
res$nf
#> [1] 10
res$f
#> [1] 4.097612
res$par
#> [1] -1.015021  1.049581

Number of gradient evaluations:

res <- mize(rb0, rb_fg, max_gr = 10)

res$terminate
#> $what
#> [1] "max_gr"
#> 
#> $val
#> [1] 10
res$ng
#> [1] 10
res$f
#> [1] 4.097612
res$par
#> [1] -1.015021  1.049581

or both:

res <- mize(rb0, rb_fg, max_fg = 10)

res$terminate
#> $what
#> [1] "max_fg"
#> 
#> $val
#> [1] 10
res$nf
#> [1] 5
res$ng
#> [1] 5
res$f
#> [1] 4.280634
res$par
#> [1] -1.048507  1.070341

The function and gradient termination criteria are checked both between iterations and during line search. On the assumption that if you specify a maximum number of evaluations, that means these calculations are expensive, mize errs on the side of caution and will sometimes calculate fewer evaluations than you ask for, because it thinks that attempting another iteration will exceed the limit.

A minor complication with convergence checking

By default, convergence is checked at every iteration. For abs_tol and rel_tol, this means that the function needs to have been evaluated at the current value of par. A lot of optimization methods do this as part of their normal working, so it doesn’t cost very much to do the convergence check. However, not all optimization methods do. If you specify a non-NULL value for rel_tol and abs_tol and the function value isn’t available, it will be calculated. This could, for some methods, add a lot of overhead.

If this is important, then using a gradient-based tolerance will be a better choice.

mize internally uses the function value as a way to keep track of the best par found during optimization. If this isn’t available, it will use a gradient norm if that is being calculated. This is less reliable than using function values, but better than nothing. If you turn off all function and gradient tolerances then mize will be unable to return the best set of parameters found over the course of the optimization. Instead, you’ll get the last set of parameters it used.

If convergence checking at every iteration is too much of a burden, you can reduce the frequency with which it is carried out with the check_conv_every parameter:

res <- mize(rb0, rb_fg, grad_tol = 1e-3, check_conv_every = 5, verbose = TRUE)
#> 09:15:11 iter 0 f = 24.2 g2 = 232.9 nf = 1 ng = 1 step = 0 alpha = 0
#> 09:15:11 iter 5 f = 4.139 g2 = 1.773 nf = 7 ng = 7 step = 0.002565 alpha = 1
#> 09:15:11 iter 10 f = 2.553 g2 = 11.67 nf = 18 ng = 18 step = 0.04657 alpha = 0.05068
#> 09:15:11 iter 15 f = 1.37 g2 = 6.954 nf = 25 ng = 25 step = 0.0922 alpha = 0.405
#> 09:15:11 iter 20 f = 0.5142 g2 = 3.001 nf = 31 ng = 31 step = 0.1319 alpha = 1
#> 09:15:11 iter 25 f = 0.1203 g2 = 2.398 nf = 37 ng = 37 step = 0.03943 alpha = 0.9408
#> 09:15:11 iter 30 f = 0.009862 g2 = 3.333 nf = 42 ng = 42 step = 0.03554 alpha = 0.1706
#> 09:15:11 iter 35 f = 2.304e-06 g2 = 0.01537 nf = 47 ng = 47 step = 0.01136 alpha = 1
#> 09:15:11 iter 40 f = 4.147e-18 g2 = 4.386e-08 nf = 52 ng = 52 step = 6.386e-07 alpha = 1

This also has the side effect of producing less output to the console when verbose = TRUE, because log_every is set to the same value of check_conv_every by default. If you set them to different values, log_every must be an integer multiple of check_conv_every. If it’s not, it will be silently set to be equal to check_conv_every.

In many cases, however, convergence checking every iteration imposes no overhead, so this is a non-issue. The vignette that runs through the methods available in mize mentions where it might be an issue.