diff --git a/DESCRIPTION b/DESCRIPTION
index beddc9c29..4e44c5439 100644
--- a/DESCRIPTION
+++ b/DESCRIPTION
@@ -1,6 +1,6 @@
 Package: ergm
-Version: 4.0-6565
-Date: 2021-07-12
+Version: 4.0-6570
+Date: 2021-07-13
 Title: Fit, Simulate and Diagnose Exponential-Family Models for Networks
 Authors@R: c(
              person("Mark S.", "Handcock", role=c("aut"), email="handcock@stat.ucla.edu"),
diff --git a/R/ergm.getMCMCsample.R b/R/ergm.getMCMCsample.R
index ba1882c1a..2f54a652f 100644
--- a/R/ergm.getMCMCsample.R
+++ b/R/ergm.getMCMCsample.R
@@ -348,8 +348,11 @@ ergm_MCMC_slave <- function(state, eta,control,verbose,..., burnin=NULL, samples
 .find_OK_burnin <- function(x, ...){
   n <- nrow(x[[1]])
   ssr <- function(b, s){
+    # b is basically the number of steps corresponding to halving of
+    # the difference in the expected value of the variable at the
+    # current MCMC draw from the ultimate expected value.
     b <- round(b)
-    a <- lm(s ~ c(seq_len(b) - 1, rep(b, n - b)))
+    a <- lm(s ~ 1 + I(2^(-seq_len(n)/b)))
     sum(sigma(a)^2)
   }
   geweke <- function(b){
@@ -362,9 +365,12 @@ ergm_MCMC_slave <- function(state, eta,control,verbose,..., burnin=NULL, samples
   xs <- x %>% map(scale) %>% map(~.[,attr(.,"scaled:scale")>0,drop=FALSE]) %>% discard(~ncol(.)==0)
   if(length(xs)==0) return(FAIL)
 
-  best <- sapply(xs, function(x) optimize(ssr, c(0, n/2), s=x, tol=1)$minimum)
+  bscl <- 10 # I.e., reduce error to about 1/2^10 of the initial value.
+
+  best <- sapply(xs, function(x) optimize(ssr, c(0, n/bscl/4), s=x)$minimum)
   if(all(is.na(best) | is.infinite(best))) return(FAIL)
 
-  best <- max(best, na.rm=TRUE)
+  best <- max(best, na.rm=TRUE) * bscl
+
   list(burnin=round(best), pval=geweke(round(best)))
 }
diff --git a/tests/testthat/test-offsets.R b/tests/testthat/test-offsets.R
index d93a8918c..eb85e7900 100644
--- a/tests/testthat/test-offsets.R
+++ b/tests/testthat/test-offsets.R
@@ -41,7 +41,7 @@ test_that("Linear ERGM with partial offsets", {
 test_that("Curved ERGM with partial offsets", {
   e4 <- ergm(samplike~edges+gwesp(0.25, fix=TRUE), control=control.ergm(seed=0,MCMLE.maxit=2))
   e4a <- ergm(samplike~edges+offset(gwesp(),c(FALSE,TRUE)), offset.coef=0.25, control=control.ergm(seed=0,MCMLE.maxit=2))
-  expect_equal(coef(e4a)[-3], coef(e4), tolerance=0.03, ignore_attr=TRUE)
+  expect_equal(coef(e4a)[-3], coef(e4), tolerance=0.06, ignore_attr=TRUE)
   expect_equal(logLik(e4a), logLik(e4), tolerance=0.01, ignore_attr=TRUE)
 })