Convergence of BOSS vs Dimension of Parameter Space
Dayi Li
2025-04-30
Last updated: 2025-04-30
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Knit directory: BOSS_website/
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | 13edcf7 | david.li | 2025-04-30 | wflow_publish("analysis/dimension.Rmd") |
| html | 37dfc1f | david.li | 2025-04-30 | Build site. |
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Introduction
In this vignette, we analyze how the convergence performance of BOSS changes with the dimension of parameter space.
For simplicity, we set the objective posterior as \(d\)-dimensional multivariate Gaussian for \(d = \{1, 2, 3, 4, 5\}\) with mean vector \(\mu_d = \mathbf{0}\) and randomly generated covariance matrices \(\Sigma_d\).
library(tidyverse)
library(tikzDevice)
function_path <- "./code"
output_path <- "./output/dimension"
data_path <- "./data/dimension"
source(paste0(function_path, "/00_BOSS.R"))Generate Random Multivariate Gaussian
# objective
eval_func <- function(x, d, Sigma){
return(mvtnorm::dmvnorm(x, mean = rep(0, d), sigma = Sigma, log = T))
}
d <- 1:5
sim_id <- 1:20
# pre-allocate result container
res_list <- vector("list", length(d))
for (i in d)
res_list[[i]] <- vector("list", length(sim_id))Run BOSS
for (i in d) {
for (j in sim_id) {
success <- FALSE
attempt <- 0
while (!success) {
# each attempt: new seed -> new Sigma
seed_val <- j + attempt
set.seed(seed_val)
# regenerate A and Sigma
A <- matrix(rnorm(i^2), nrow = i)
Sigma <- crossprod(A)
# define objective with the new Sigma
obj_func <- function(x) eval_func(x, d = i, Sigma = Sigma)
lower <- rep(-4 * max(Sigma), i)
upper <- rep( 4 * max(Sigma), i)
# try BOSS on this Sigma
out <- try(
BOSS(obj_func,
criterion = "modal",
update_step = 5,
max_iter = 300,
D = i,
lower = lower,
upper = upper,
noise_var = 1e-6,
modal_iter_check = 5,
modal_check_warmup = 20,
modal_k.nn = 5,
modal_eps = 0.25,
initial_design = 5 * i,
delta = 0.01^i,
optim.n = 1,
optim.max.iter = 100),
silent = TRUE
)
if (!inherits(out, "try-error")) {
# success: save and break out of retry loop
res_list[[i]][[j]] <- out
success <- TRUE
} else {
message(sprintf(
"BOSS failed for d=%d, sim=%d (attempt %d, seed=%d). Retrying with new Sigma…",
i, j, attempt, seed_val
))
attempt <- attempt + 1
}
}
}
}
save(res_list, file = paste0(output_path, "/dimension_test.rda"))load(paste0(output_path, "/dimension_test.rda"))
dim <- rep(d, each = 20)
iter <- unlist(lapply(res_list, function(x) lapply(x, function(y) max(y$modal_result$i)))) + 5*dim
iter.data <- data.frame(d= dim, Iteration = iter) %>%
group_by(d) %>%
mutate(med = median(Iteration))
ggplot(iter.data, aes(d, Iteration)) + geom_point() + geom_line(aes(d, med))
| Version | Author | Date |
|---|---|---|
| 28db27f | david.li | 2025-04-30 |
We can see pretty clearly that the number of iterations required for convergence roughly grows exponentially, which is expected based on the existing theoretical analysis of BO convergence performance.
We also need to note that the above test only showcases the performance of BOSS when the true posterior is a multivariate Gaussian. We have observed that if there is deviation from normality, then convergence would take even longer, especially for higher dimension. Under the most extreme cases where certain parameters are almost degenerate with respect to each other, BOSS can almost entirely fail.
sessionInfo()R version 4.4.1 (2024-06-14)
Platform: aarch64-apple-darwin20
Running under: macOS 15.0
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.0
locale:
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time zone: America/Toronto
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
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[5] dplyr_1.1.4 purrr_1.0.2 readr_2.1.5 tidyr_1.3.1
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