diff --git a/DESCRIPTION b/DESCRIPTION
index df043bf3..ffc831b7 100644
--- a/DESCRIPTION
+++ b/DESCRIPTION
@@ -1,7 +1,7 @@
 Type: Package
 Package: mlr3mbo
 Title: Flexible Bayesian Optimization
-Version: 0.2.3.9000
+Version: 0.2.4.9000
 Authors@R: c(
     person("Lennart", "Schneider", , "lennart.sch@web.de", role = c("cre", "aut"),
            comment = c(ORCID = "0000-0003-4152-5308")),
@@ -66,13 +66,14 @@ Suggests:
     rpart,
     stringi,
     testthat (>= 3.0.0)
+Remotes: mlr-org/bbotk
 ByteCompile: no
 Encoding: UTF-8
 Config/testthat/edition: 3
 Config/testthat/parallel: false
 NeedsCompilation: yes
 Roxygen: list(markdown = TRUE, r6 = TRUE)
-RoxygenNote: 7.3.1
+RoxygenNote: 7.3.2
 Collate:
     'mlr_acqfunctions.R'
     'AcqFunction.R'
diff --git a/NEWS.md b/NEWS.md
index 57766688..cb1beef6 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -1,5 +1,9 @@
 # mlr3mbo (development version)
 
+# mlr3mbo 0.2.4
+
+* fix: Improve runtime of `AcqOptimizer` by setting `check_values = FALSE`.
+
 # mlr3mbo 0.2.3
 
 * compatibility: Work with new bbotk and mlr3tuning version 1.0.0.
diff --git a/R/AcqFunctionCB.R b/R/AcqFunctionCB.R
index 14a980c3..81328e05 100644
--- a/R/AcqFunctionCB.R
+++ b/R/AcqFunctionCB.R
@@ -76,8 +76,8 @@ AcqFunctionCB = R6Class("AcqFunctionCB",
       constants = list(...)
       lambda  = constants$lambda
       p = self$surrogate$predict(xdt)
-      res = p$mean - self$surrogate_max_to_min * lambda * p$se
-      data.table(acq_cb = res)
+      cb = p$mean - self$surrogate_max_to_min * lambda * p$se
+      data.table(acq_cb = cb)
     }
   )
 )
diff --git a/R/AcqFunctionEI.R b/R/AcqFunctionEI.R
index 1782b972..f03ec39b 100644
--- a/R/AcqFunctionEI.R
+++ b/R/AcqFunctionEI.R
@@ -9,6 +9,13 @@
 #' @description
 #' Expected Improvement.
 #'
+#' @section Parameters:
+#' * `"epsilon"` (`numeric(1)`)\cr
+#'   \eqn{\epsilon} value used to determine the amount of exploration.
+#'   Higher values result in the importance of improvements predicted by the posterior mean
+#'   decreasing relative to the importance of potential improvements in regions of high predictive uncertainty.
+#'   Defaults to `0` (standard Expected Improvement).
+#'
 #' @references
 #' * `r format_bib("jones_1998")`
 #'
@@ -60,9 +67,15 @@ AcqFunctionEI = R6Class("AcqFunctionEI",
     #' Creates a new instance of this [R6][R6::R6Class] class.
     #'
     #' @param surrogate (`NULL` | [SurrogateLearner]).
-    initialize = function(surrogate = NULL) {
+    #' @param epsilon (`numeric(1)`).
+    initialize = function(surrogate = NULL, epsilon = 0) {
       assert_r6(surrogate, "SurrogateLearner", null.ok = TRUE)
-      super$initialize("acq_ei", surrogate = surrogate, requires_predict_type_se = TRUE, direction = "maximize", label = "Expected Improvement", man = "mlr3mbo::mlr_acqfunctions_ei")
+      assert_number(epsilon, lower = 0, finite = TRUE)
+
+      constants = ps(epsilon = p_dbl(lower = 0, default = 0))
+      constants$values$epsilon = epsilon
+
+      super$initialize("acq_ei", constants = constants, surrogate = surrogate, requires_predict_type_se = TRUE, direction = "maximize", label = "Expected Improvement", man = "mlr3mbo::mlr_acqfunctions_ei")
     },
 
     #' @description
@@ -73,14 +86,16 @@ AcqFunctionEI = R6Class("AcqFunctionEI",
   ),
 
   private = list(
-    .fun = function(xdt) {
+    .fun = function(xdt, ...) {
       if (is.null(self$y_best)) {
         stop("$y_best is not set. Missed to call $update()?")
       }
+      constants = list(...)
+      epsilon = constants$epsilon
       p = self$surrogate$predict(xdt)
       mu = p$mean
       se = p$se
-      d = self$y_best - self$surrogate_max_to_min * mu
+      d = (self$y_best - self$surrogate_max_to_min * mu) - epsilon
       d_norm = d / se
       ei = d * pnorm(d_norm) + se * dnorm(d_norm)
       ei = ifelse(se < 1e-20, 0, ei)
diff --git a/R/AcqOptimizer.R b/R/AcqOptimizer.R
index 68d6cd84..f9f0f5d6 100644
--- a/R/AcqOptimizer.R
+++ b/R/AcqOptimizer.R
@@ -93,16 +93,21 @@ AcqOptimizer = R6Class("AcqOptimizer",
     #' @field acq_function ([AcqFunction]).
     acq_function = NULL,
 
+    #' @field callbacks (`NULL` | list of [mlr3misc::Callback]).
+    callbacks = NULL,
+
     #' @description
     #' Creates a new instance of this [R6][R6::R6Class] class.
     #'
     #' @param optimizer ([bbotk::Optimizer]).
     #' @param terminator ([bbotk::Terminator]).
     #' @param acq_function (`NULL` | [AcqFunction]).
-    initialize = function(optimizer, terminator, acq_function = NULL) {
+    #' @param callbacks (`NULL` | list of [mlr3misc::Callback])
+    initialize = function(optimizer, terminator, acq_function = NULL, callbacks = NULL) {
       self$optimizer = assert_r6(optimizer, "Optimizer")
       self$terminator = assert_r6(terminator, "Terminator")
       self$acq_function = assert_r6(acq_function, "AcqFunction", null.ok = TRUE)
+      self$callbacks = assert_callbacks(as_callbacks(callbacks))
       ps = ps(
         n_candidates = p_int(lower = 1, default = 1L),
         logging_level = p_fct(levels = c("fatal", "error", "warn", "info", "debug", "trace"), default = "warn"),
@@ -146,7 +151,7 @@ AcqOptimizer = R6Class("AcqOptimizer",
       logger$set_threshold(self$param_set$values$logging_level)
       on.exit(logger$set_threshold(old_threshold))
 
-      instance = OptimInstanceBatchSingleCrit$new(objective = self$acq_function, search_space = self$acq_function$domain, terminator = self$terminator, check_values = FALSE)
+      instance = OptimInstanceBatchSingleCrit$new(objective = self$acq_function, search_space = self$acq_function$domain, terminator = self$terminator, check_values = FALSE, callbacks = self$callbacks)
 
       # warmstart
       if (self$param_set$values$warmstart) {
diff --git a/R/sugar.R b/R/sugar.R
index ac9279c6..69c6a8b0 100644
--- a/R/sugar.R
+++ b/R/sugar.R
@@ -20,7 +20,7 @@
 #' @param cols_y (`NULL` | `character()`)\cr
 #'   Column id(s) in the [bbotk::Archive] that should be used as a target.
 #'   If a list of [mlr3::LearnerRegr] is provided as the `learner` argument and `cols_y` is
-#'   specified as well, as many column names as learners must be provided. 
+#'   specified as well, as many column names as learners must be provided.
 #'   Can also be `NULL` in which case this is automatically inferred based on the archive.
 #' @param ... (named `list()`)\cr
 #' Named arguments passed to the constructor, to be set as parameters in the
@@ -90,6 +90,8 @@ acqf = function(.key, ...) {
 #' @param acq_function (`NULL` | [AcqFunction])\cr
 #'   [AcqFunction] that is to be used.
 #'   Can also be `NULL`.
+#' @param callbacks (`NULL` | list of [mlr3misc::Callback])
+#'   Callbacks used during acquisition function optimization.
 #' @param ... (named `list()`)\cr
 #' Named arguments passed to the constructor, to be set as parameters in the
 #' [paradox::ParamSet].
@@ -101,9 +103,9 @@ acqf = function(.key, ...) {
 #' library(bbotk)
 #' acqo(opt("random_search"), trm("evals"), catch_errors = FALSE)
 #' @export
-acqo = function(optimizer, terminator, acq_function = NULL, ...) {
+acqo = function(optimizer, terminator, acq_function = NULL, callbacks = NULL, ...) {
   dots = list(...)
-  acqopt = AcqOptimizer$new(optimizer = optimizer, terminator = terminator, acq_function = acq_function)
+  acqopt = AcqOptimizer$new(optimizer = optimizer, terminator = terminator, acq_function = acq_function, callbacks = callbacks)
   acqopt$param_set$values = insert_named(acqopt$param_set$values, dots)
   acqopt
 }
diff --git a/man/AcqOptimizer.Rd b/man/AcqOptimizer.Rd
index f0f8d9b3..b4c143b1 100644
--- a/man/AcqOptimizer.Rd
+++ b/man/AcqOptimizer.Rd
@@ -94,6 +94,8 @@ if (requireNamespace("mlr3learners") &
 \item{\code{terminator}}{(\link[bbotk:Terminator]{bbotk::Terminator}).}
 
 \item{\code{acq_function}}{(\link{AcqFunction}).}
+
+\item{\code{callbacks}}{(\code{NULL} | list of \link[mlr3misc:Callback]{mlr3misc::Callback}).}
 }
 \if{html}{\out{</div>}}
 }
@@ -124,7 +126,7 @@ Set of hyperparameters.}
 \subsection{Method \code{new()}}{
 Creates a new instance of this \link[R6:R6Class]{R6} class.
 \subsection{Usage}{
-\if{html}{\out{<div class="r">}}\preformatted{AcqOptimizer$new(optimizer, terminator, acq_function = NULL)}\if{html}{\out{</div>}}
+\if{html}{\out{<div class="r">}}\preformatted{AcqOptimizer$new(optimizer, terminator, acq_function = NULL, callbacks = NULL)}\if{html}{\out{</div>}}
 }
 
 \subsection{Arguments}{
@@ -135,6 +137,8 @@ Creates a new instance of this \link[R6:R6Class]{R6} class.
 \item{\code{terminator}}{(\link[bbotk:Terminator]{bbotk::Terminator}).}
 
 \item{\code{acq_function}}{(\code{NULL} | \link{AcqFunction}).}
+
+\item{\code{callbacks}}{(\code{NULL} | list of \link[mlr3misc:Callback]{mlr3misc::Callback})}
 }
 \if{html}{\out{</div>}}
 }
diff --git a/man/acqo.Rd b/man/acqo.Rd
index 9455152e..43114a0c 100644
--- a/man/acqo.Rd
+++ b/man/acqo.Rd
@@ -4,7 +4,7 @@
 \alias{acqo}
 \title{Syntactic Sugar Acquisition Function Optimizer Construction}
 \usage{
-acqo(optimizer, terminator, acq_function = NULL, ...)
+acqo(optimizer, terminator, acq_function = NULL, callbacks = NULL, ...)
 }
 \arguments{
 \item{optimizer}{(\link[bbotk:Optimizer]{bbotk::Optimizer})\cr
@@ -17,6 +17,9 @@ acqo(optimizer, terminator, acq_function = NULL, ...)
 \link{AcqFunction} that is to be used.
 Can also be \code{NULL}.}
 
+\item{callbacks}{(\code{NULL} | list of \link[mlr3misc:Callback]{mlr3misc::Callback})
+Callbacks used during acquisition function optimization.}
+
 \item{...}{(named \code{list()})\cr
 Named arguments passed to the constructor, to be set as parameters in the
 \link[paradox:ParamSet]{paradox::ParamSet}.}
diff --git a/man/mlr_acqfunctions_ei.Rd b/man/mlr_acqfunctions_ei.Rd
index 700e72ab..4357d844 100644
--- a/man/mlr_acqfunctions_ei.Rd
+++ b/man/mlr_acqfunctions_ei.Rd
@@ -17,6 +17,17 @@ acqf("ei")
 }\if{html}{\out{</div>}}
 }
 
+\section{Parameters}{
+
+\itemize{
+\item \code{"epsilon"} (\code{numeric(1)})\cr
+\eqn{\epsilon} value used to determine the amount of exploration.
+Higher values result in the importance of improvements predicted by the posterior mean
+decreasing relative to the importance of potential improvements in regions of high predictive uncertainty.
+Defaults to \code{0} (standard Expected Improvement).
+}
+}
+
 \examples{
 if (requireNamespace("mlr3learners") &
     requireNamespace("DiceKriging") &
@@ -110,13 +121,15 @@ In the case of maximization, this already includes the necessary change of sign.
 \subsection{Method \code{new()}}{
 Creates a new instance of this \link[R6:R6Class]{R6} class.
 \subsection{Usage}{
-\if{html}{\out{<div class="r">}}\preformatted{AcqFunctionEI$new(surrogate = NULL)}\if{html}{\out{</div>}}
+\if{html}{\out{<div class="r">}}\preformatted{AcqFunctionEI$new(surrogate = NULL, epsilon = 0)}\if{html}{\out{</div>}}
 }
 
 \subsection{Arguments}{
 \if{html}{\out{<div class="arguments">}}
 \describe{
 \item{\code{surrogate}}{(\code{NULL} | \link{SurrogateLearner}).}
+
+\item{\code{epsilon}}{(\code{numeric(1)}).}
 }
 \if{html}{\out{</div>}}
 }
diff --git a/pkgdown/_pkgdown.yml b/pkgdown/_pkgdown.yml
index 55a84b5d..b72e56b5 100644
--- a/pkgdown/_pkgdown.yml
+++ b/pkgdown/_pkgdown.yml
@@ -16,8 +16,8 @@ toc:
 
 navbar:
   structure:
-    left:  [reference, news, book]
-    right: [github, mattermost, stackoverflow, rss, lightswitch]
+    left:  [reference, intro, news, book]
+    right: [search, github, mattermost, stackoverflow, rss, lightswitch]
   components:
     home: ~
     reference:
diff --git a/tests/testthat/helper.R b/tests/testthat/helper.R
index b3ea492d..aba5c6e6 100644
--- a/tests/testthat/helper.R
+++ b/tests/testthat/helper.R
@@ -39,12 +39,18 @@ PS_1D_MIXED_DEPS = PS_1D_MIXED$clone(deep = TRUE)
 PS_1D_MIXED_DEPS$add_dep("x2", on = "x4", cond = CondEqual$new(TRUE))
 
 FUN_1D_MIXED = function(xs) {
+  if (is.null(xs$x2)) {
+    xs$x2 = "a"
+  }
   list(y = (xs$x1 - switch(xs$x2, "a" = 0, "b" = 1, "c" = 2)) %% xs$x3 + (if (xs$x4) xs$x1 else pi))
 }
 OBJ_1D_MIXED = ObjectiveRFun$new(fun = FUN_1D_MIXED, domain = PS_1D_MIXED, properties = "single-crit")
 OBJ_1D_MIXED_DEPS = ObjectiveRFun$new(fun = FUN_1D_MIXED, domain = PS_1D_MIXED_DEPS, properties = "single-crit")
 
 FUN_1D_2_MIXED = function(xs) {
+  if (is.null(xs$x2)) {
+    xs$x2 = "a"
+  }
   list(y1 = (xs$x1 - switch(xs$x2, "a" = 0, "b" = 1, "c" = 2)) %% xs$x3 + (if (xs$x4) xs$x1 else pi), y2 = xs$x1)
 }
 OBJ_1D_2_MIXED = ObjectiveRFun$new(fun = FUN_1D_2_MIXED, domain = PS_1D_MIXED, codomain = FUN_1D_2_CODOMAIN, properties = "multi-crit")
diff --git a/tests/testthat/test_AcqFunctionCB.R b/tests/testthat/test_AcqFunctionCB.R
index 1749b3d2..eecaa100 100644
--- a/tests/testthat/test_AcqFunctionCB.R
+++ b/tests/testthat/test_AcqFunctionCB.R
@@ -12,6 +12,9 @@ test_that("AcqFunctionCB works", {
   expect_learner(acqf$surrogate$learner)
   expect_true(acqf$requires_predict_type_se)
 
+  expect_r6(acqf$constants, "ParamSet")
+  expect_equal(acqf$constants$ids(), "lambda")
+
   design = MAKE_DESIGN(inst)
   inst$eval_batch(design)
 
diff --git a/tests/testthat/test_AcqFunctionEHVIGH.R b/tests/testthat/test_AcqFunctionEHVIGH.R
index 801fd841..58273cca 100644
--- a/tests/testthat/test_AcqFunctionEHVIGH.R
+++ b/tests/testthat/test_AcqFunctionEHVIGH.R
@@ -15,6 +15,9 @@ test_that("AcqFunctionEHVIGH works", {
   expect_true(acqf$requires_predict_type_se)
   expect_setequal(acqf$packages, c("emoa", "fastGHQuad"))
 
+  expect_r6(acqf$constants, "ParamSet")
+  expect_equal(acqf$constants$ids(), c("k", "r"))
+
   design = MAKE_DESIGN(inst)
   inst$eval_batch(design)
 
diff --git a/tests/testthat/test_AcqFunctionEI.R b/tests/testthat/test_AcqFunctionEI.R
index 59eae9ef..dd8be11a 100644
--- a/tests/testthat/test_AcqFunctionEI.R
+++ b/tests/testthat/test_AcqFunctionEI.R
@@ -13,6 +13,9 @@ test_that("AcqFunctionEI works", {
   expect_learner(acqf$surrogate$learner)
   expect_true(acqf$requires_predict_type_se)
 
+  expect_r6(acqf$constants, "ParamSet")
+  expect_equal(acqf$constants$ids(), "epsilon")
+
   design = MAKE_DESIGN(inst)
   inst$eval_batch(design)
 
diff --git a/tests/testthat/test_AcqFunctionSmsEgo.R b/tests/testthat/test_AcqFunctionSmsEgo.R
index 020a946f..b6cdece9 100644
--- a/tests/testthat/test_AcqFunctionSmsEgo.R
+++ b/tests/testthat/test_AcqFunctionSmsEgo.R
@@ -12,6 +12,9 @@ test_that("AcqFunctionSmsEgo works", {
   expect_list(acqf$surrogate$learner, types = "Learner")
   expect_true(acqf$requires_predict_type_se)
 
+  expect_r6(acqf$constants, "ParamSet")
+  expect_equal(acqf$constants$ids(), c("lambda", "epsilon"))
+
   design = MAKE_DESIGN(inst)
   inst$eval_batch(design)
 
diff --git a/tests/testthat/test_AcqOptimizer.R b/tests/testthat/test_AcqOptimizer.R
index 213963fe..69e2ca2f 100644
--- a/tests/testthat/test_AcqOptimizer.R
+++ b/tests/testthat/test_AcqOptimizer.R
@@ -120,3 +120,32 @@ test_that("AcqOptimizer deep clone", {
   expect_true(address(acqopt1$terminator) != address(acqopt2$terminator))
 })
 
+test_that("AcqOptimizer callbacks", {
+  domain = ps(x = p_dbl(lower = 10, upper = 20, trafo = function(x) x - 15))
+  objective = ObjectiveRFunDt$new(
+    fun = function(xdt) data.table(y = xdt$x ^ 2),
+    domain = domain,
+    codomain = ps(y = p_dbl(tags = "minimize")),
+    check_values = FALSE
+  )
+  instance = MAKE_INST(objective = objective, search_space = domain, terminator = trm("evals", n_evals = 5L))
+  design = MAKE_DESIGN(instance)
+  instance$eval_batch(design)
+  callback = callback_batch("mlr3mbo.acqopt_time",
+    on_optimization_begin = function(callback, context) {
+      callback$state$begin = Sys.time()
+    },
+    on_optimization_end = function(callback, context) {
+      callback$state$end = Sys.time()
+      attr(callback$state$outer_instance, "acq_opt_runtime") = as.numeric(callback$state$end - callback$state$begin)
+    }
+  )
+  callback$state$outer_instance = instance
+  acqfun = AcqFunctionEI$new(SurrogateLearner$new(REGR_FEATURELESS, archive = instance$archive))
+  acqopt = AcqOptimizer$new(opt("random_search", batch_size = 10L), trm("evals", n_evals = 10L), acq_function = acqfun, callbacks = callback)
+  acqfun$surrogate$update()
+  acqfun$update()
+  res = acqopt$optimize()
+  expect_number(attr(instance, "acq_opt_runtime"))
+})
+