Map the optima of an objective with a Gaussian-mixture proxy

Fits a Gaussian-mixture proxy to the Gibbs measure \(\exp(-f(x) / T)\) of a user-supplied objective f over a bounded box, by cooling a short temperature ladder through regime-(iii) importance-sampled KLD-EM (fit_kld_em()). As the temperature falls the mixture mass concentrates on the low regions of f, so the fitted mixture is a closed-form map over the optima rather than a single point estimate. Pair it with gmm_modes() to read off the distinct optima.

Usage

from_objective(
  objective,
  lower,
  upper,
  N = NULL,
  minimise = TRUE,
  temperature = NULL,
  n_steps = 6L,
  exploration = 0.5,
  inflate = 1.8,
  is_size = 10000L,
  max_iter = 70L,
  ridge_eps = 1e-04,
  seed = NULL,
  scale = c("objective", "loglik")
)

Arguments

objective: Function taking a length-p numeric vector and returning a finite numeric scalar – the objective to minimise (or maximise; see minimise).
lower, upper: Numeric vectors of equal length p giving the box over which the optima are sought. Every upper must exceed its lower. objective is only evaluated inside the box; points outside it are treated like points where objective is not finite and receive a strongly unattractive value.
N: Number of mixture components. Default max(10L, 5L * p). Raise it for objectives with many optima or strong symmetry.
minimise: Logical. If TRUE (the default) lower objective values are better (the proxy concentrates on the minima); if FALSE the proxy concentrates on the maxima.
temperature: Optional control of the cooling ladder. NULL (the default) derives a ladder from the range r of objective over a uniform probe of the box: from r / 2 down to r / 80 when scale = "objective", and from max(r / 2, 1) down to 1 when scale = "loglik". A positive scalar sets the final (lowest) temperature; a length-2 numeric c(high, low) sets both ends explicitly. A supplied temperature is used as given, whatever scale is.
n_steps: Number of temperatures in the cooling ladder. Default 6L. When scale = "loglik", temperature is NULL and n_steps is not supplied, the ladder takes as many steps as it needs to cool by no more than the default factor of \(40^{1/5} \approx 2.1\) per step, and never fewer than 6.
exploration: Probability mass the importance proposal places on uniform exploration of the box at each step, in [0, 1]. Default 0.5. Larger values explore more (and starve no basin); smaller values exploit the basins found so far.
inflate: Factor by which the current mixture covariances are inflated when used as the exploitation part of the proposal. Default 1.8.
is_size: Importance-sample size per cooling step. Default 1e4L.
max_iter: Maximum EM iterations per cooling step. Default 70L.
ridge_eps: Ridge added to each component covariance at every M-step. Forwarded to fit_kld_em().
seed: Optional integer seed for reproducibility.
scale: What the objective's values mean, which sets the default temperature ladder. "objective" (the default) is any objective to optimise: the ladder scales with its range, so the result does not depend on the objective's units. "loglik" is a negative log-likelihood (or a log-likelihood with minimise = FALSE): the ladder ends at temperature 1, where the Gibbs measure is the likelihood itself.

Value

A gmm_fit (the fitted proxy) carrying a from_objective metadata record with the box, the temperature ladder, scale, and the effective sample size at each cooling step (ess). Pass it to gmm_modes() to extract the distinct optima.

Details

The Gibbs measure can be evaluated point-wise but not directly sampled, which is precisely the setting of regime (iii): minimising the Kullback-Leibler divergence from a Gaussian mixture to a peaked target is the rank-weighted Gaussian update at the heart of fit_kld_em(). This function is that fit, driven against a sequence of cooling Gibbs targets and warm-started from the previous fit at each step. Because a multimodal f produces a multimodal target, the components spread across the basins and recover the optima together.

Recovery is most reliable with component headroom – a number of components N comfortably larger than the number of optima you expect, so the defensive proposal can keep a component on each basin. Symmetric landscapes (where several optima are exchangeable) need the most headroom.

Dimensional scope. As with the rest of regime (iii), the importance- sampling effective sample size falls sharply with dimension; the guard is p <= 5 (recommended), p <= 10 (allowed with a warning), p > 10 (rejected).

Effective sample size. Each cooling step records its importance-sampling effective sample size in the from_objective metadata. A classed proxymix_low_ess warning is raised at most once per call, and only when the final step, the one that produced the returned fit, falls below 50; its message gives the final effective sample size, how many steps were low, and the lowest value with its step.

Likelihoods. With scale = "loglik" the objective is read as a negative log-likelihood (or, with minimise = FALSE, a log-likelihood) and the ladder ends at temperature 1. The fitted map is then a proxy for the likelihood itself, restricted to the box, which is the posterior under a uniform prior on the box.

Examples

## A bimodal 1-D objective with minima at +/- 2.
f <- function(v) (v[1]^2 - 4)^2
fit <- from_objective(f, lower = -5, upper = 5, N = 6L,
                      is_size = 2000L, n_steps = 5L, seed = 1L)
gmm_modes(fit)$modes
#>           [,1]
#> [1,] -2.059751
#> [2,]  2.009906

## The likelihood surface of a logistic-regression slope.
x <- c(-1.2, -0.6, -0.1, 0.3, 0.8, 1.4)
y <- c(0, 0, 1, 0, 1, 1)
nll <- function(b) sum(log1p(exp(b[1] * x))) - sum(y * b[1] * x)
lik <- from_objective(nll, lower = -5, upper = 10, N = 4L,
                      is_size = 2000L, seed = 1L, scale = "loglik")
lik@metadata$from_objective$temperatures
#> [1] 9.172354 5.888228 3.779971 2.426567 1.557744 1.000000