mlr3tuning Tutorial - German Credit

mlr3tuning tuning optimization nested resampling german credit data set classification

In this use case, we continue working with the German credit dataset. We work on hyperparameter tuning and apply nested resampling.

Martin Binder , Florian Pfisterer
03-11-2020

Intro

This is the second part of a serial of tutorials. The other parts of this series can be found here:

We will continue working with the German credit dataset. In Part I, we peeked into the dataset by using and comparing some learners with their default parameters. We will now see how to:

Prerequisites

First, load the packages we are going to use:

We initialize the random number generator with a fixed seed for reproducibility, and decrease the verbosity of the logger to keep the output clearly represented.

set.seed(7832)
lgr::get_logger("mlr3")$set_threshold("warn")
lgr::get_logger("bbotk")$set_threshold("warn")

We use the same Task as in Part I:

task = tsk("german_credit")

We also might want to use multiple cores to reduce long run times of tuning runs.

future::plan("multiprocess")

Evaluation

We will evaluate all hyperparameter configurations using 10-fold cross-validation. We use a fixed train-test split, i.e. the same splits for each evaluation. Otherwise, some evaluation could get unusually “hard” splits, which would make comparisons unfair.

cv10 = rsmp("cv", folds = 10)

# fix the train-test splits using the $instantiate() method
cv10$instantiate(task)

# have a look at the test set instances per fold
cv10$instance
      row_id fold
   1:     18    1
   2:     19    1
   3:     35    1
   4:     38    1
   5:     55    1
  ---            
 996:    973   10
 997:    975   10
 998:    981   10
 999:    993   10
1000:    998   10

Simple Parameter Tuning

Parameter tuning in mlr3 needs two packages:

  1. The paradox package is used for the search space definition of the hyperparameters.
  2. The mlr3tuning package is used for tuning the hyperparameters.

The packages are loaded by the mlr3verse package.

Search Space and Problem Definition

First, we need to decide what Learner we want to optimize. We will use LearnerClassifKKNN, the “kernelized” k-nearest neighbor classifier. We will use kknn as a normal kNN without weighting first (i.e., using the rectangular kernel):

knn = lrn("classif.kknn", predict_type = "prob", kernel = "rectangular")

As a next step, we decide what parameters we optimize over. Before that, though, we are interested in the parameter set on which we could tune:

knn$param_set
id class lower upper nlevels
k ParamInt 1 Inf Inf
distance ParamDbl 0 Inf Inf
kernel ParamFct NA NA 10
scale ParamLgl NA NA 2
ykernel ParamUty NA NA Inf

We first tune the k parameter (i.e. the number of nearest neighbors), between 3 to 20. Second, we tune the distance function, allowing L1 and L2 distances. To do so, we use the paradox package to define a search space (see the online vignette for a more complete introduction.

search_space = ps(
  k = p_int(3, 20),
  distance = p_int(1, 2)
)

As a next step, we define a TuningInstanceSingleCrit that represents the problem we are trying to optimize.

instance_grid = TuningInstanceSingleCrit$new(
  task = task,
  learner = knn,
  resampling = cv10,
  measure = msr("classif.ce"),
  terminator = trm("none"),
  search_space = search_space
)

After having set up a tuning instance, we can start tuning. Before that, we need a tuning strategy, though. A simple tuning method is to try all possible combinations of parameters: Grid Search. While it is very intuitive and simple, it is inefficient if the search space is large. For this simple use case, it suffices, though. We get the grid_search tuner via:

tuner_grid = tnr("grid_search", resolution = 18, batch_size = 36)

Tuning works by calling $optimize(). Note that the tuning procedure modifies our tuning instance (as usual for R6 class objects). The result can be found in the instance object. Before tuning it is empty:

instance_grid$result
NULL

Now, we tune:

tuner_grid$optimize(instance_grid)

The result is returned by $optimize() together with its performance. It can be also accessed with the $result slot:

instance_grid$result
   k distance learner_param_vals  x_domain classif.ce
1: 9        2          <list[3]> <list[2]>       0.25

We can also look at the Archive of evaluated configurations:

head(as.data.table(instance_grid$archive))
k distance classif.ce runtime_learners timestamp batch_nr
3 1 0.271 1.210 2021-06-13 15:33:43 1
3 2 0.273 0.703 2021-06-13 15:33:43 1
4 1 0.292 0.963 2021-06-13 15:33:43 1
4 2 0.279 0.759 2021-06-13 15:33:43 1
5 1 0.271 1.143 2021-06-13 15:33:43 1
5 2 0.274 0.643 2021-06-13 15:33:43 1

We plot the performances depending on the sampled k and distance:

ggplot(as.data.table(instance_grid$archive),
  aes(x = k, y = classif.ce, color = as.factor(distance))) +
  geom_line() + geom_point(size = 3)

On average, the Euclidean distance (distance = 2) seems to work better. However, there is much randomness introduced by the resampling instance. So you, the reader, may see a different result, when you run the experiment yourself and set a different random seed. For k, we find that values between 7 and 13 perform well.

Random Search and Transformation

Let’s have a look at a larger search space. For example, we could tune all available parameters and limit k to large values (50). We also now tune the distance param continuously from 1 to 3 as a double and tune distance kernel and whether we scale the features.

We may find two problems when doing so:

First, the resulting difference in performance between k = 3 and k = 4 is probably larger than the difference between k = 49 and k = 50. While 4 is 33% larger than 3, 50 is only 2 percent larger than 49. To account for this we will use a transformation function for k and optimize in log-space. We define the range for k from log(3) to log(50) and exponentiate in the transformation. Now, as k has become a double instead of an int (in the search space, before transformation), we round it in the extra_trafo.

search_space_large = ps(
  k = p_dbl(log(3), log(50)),
  distance = p_dbl(1, 3),
  kernel = p_fct(c("rectangular", "gaussian", "rank", "optimal")),
  scale = p_lgl(),
  .extra_trafo = function(x, param_set) {
    x$k = round(exp(x$k))
    x
  }
)

The second problem is that grid search may (and often will) take a long time. For instance, trying out three different values for k, distance, kernel, and the two values for scale will take 54 evaluations. Because of this, we use a different search algorithm, namely the Random Search. We need to specify in the tuning instance a termination criterion. The criterion tells the search algorithm when to stop. Here, we will terminate after 36 evaluations:

tuner_random = tnr("random_search", batch_size = 36)

instance_random = TuningInstanceSingleCrit$new(
  task = task,
  learner = knn,
  resampling = cv10,
  measure = msr("classif.ce"),
  terminator = trm("evals", n_evals = 36),
  search_space = search_space_large,
)
tuner_random$optimize(instance_random)

Like before, we can review the Archive. It includes the points before and after the transformation. The archive includes a column for each parameter the Tuner sampled on the search space (values before the transformation) and additional columns with prefix x_domain_* that refer to the parameters used by the learner (values after the transformation):

as.data.table(instance_random$archive)
k distance kernel scale classif.ce x_domain_k x_domain_distance x_domain_kernel x_domain_scale runtime_learners timestamp batch_nr
2.654531 2.921236 rectangular FALSE 0.314 14 2.921236 rectangular FALSE 0.797 2021-06-13 15:33:51 1
2.588931 1.869319 rank TRUE 0.254 13 1.869319 rank TRUE 1.240 2021-06-13 15:33:51 1
3.319396 2.425029 rectangular TRUE 0.272 28 2.425029 rectangular TRUE 1.591 2021-06-13 15:33:51 1
1.164252 1.799989 gaussian FALSE 0.364 3 1.799989 gaussian FALSE 0.549 2021-06-13 15:33:51 1
2.441257 1.650704 rank TRUE 0.253 11 1.650704 rank TRUE 1.221 2021-06-13 15:33:51 1
3.158912 2.514174 optimal FALSE 0.305 24 2.514174 optimal FALSE 0.926 2021-06-13 15:33:51 1
3.047551 1.405385 gaussian TRUE 0.257 21 1.405385 gaussian TRUE 1.353 2021-06-13 15:33:51 1
2.442353 2.422242 gaussian TRUE 0.270 12 2.422242 gaussian TRUE 1.173 2021-06-13 15:33:51 1
3.521548 1.243384 rectangular TRUE 0.271 34 1.243384 rectangular TRUE 1.421 2021-06-13 15:33:51 1
2.331159 1.490977 optimal TRUE 0.252 10 1.490977 optimal TRUE 1.121 2021-06-13 15:33:51 1
1.787328 1.286609 gaussian FALSE 0.345 6 1.286609 gaussian FALSE 0.677 2021-06-13 15:33:51 1
1.297462 1.479259 rank TRUE 0.274 4 1.479259 rank TRUE 1.256 2021-06-13 15:33:51 1
1.378451 1.117869 rectangular FALSE 0.357 4 1.117869 rectangular FALSE 0.582 2021-06-13 15:33:51 1
1.988414 2.284577 rectangular FALSE 0.313 7 2.284577 rectangular FALSE 0.642 2021-06-13 15:33:51 1
2.557743 2.752538 rank TRUE 0.267 13 2.752538 rank TRUE 1.242 2021-06-13 15:33:51 1
2.961104 2.557829 rank TRUE 0.264 19 2.557829 rank TRUE 1.105 2021-06-13 15:33:51 1
2.243193 2.594618 rectangular TRUE 0.256 9 2.594618 rectangular TRUE 1.165 2021-06-13 15:33:51 1
3.666907 1.910549 rectangular FALSE 0.292 39 1.910549 rectangular FALSE 1.069 2021-06-13 15:33:51 1
1.924639 1.820168 rank TRUE 0.256 7 1.820168 rank TRUE 1.238 2021-06-13 15:33:51 1
2.390153 2.621741 optimal FALSE 0.339 11 2.621741 optimal FALSE 0.625 2021-06-13 15:33:51 1
2.033775 2.209867 rank FALSE 0.332 8 2.209867 rank FALSE 0.636 2021-06-13 15:33:51 1
2.929778 2.309448 rank FALSE 0.302 19 2.309448 rank FALSE 0.929 2021-06-13 15:33:51 1
1.824520 1.706395 rank TRUE 0.261 6 1.706395 rank TRUE 1.124 2021-06-13 15:33:51 1
2.444957 1.540520 optimal TRUE 0.249 12 1.540520 optimal TRUE 1.258 2021-06-13 15:33:51 1
3.254559 2.985368 rank FALSE 0.302 26 2.985368 rank FALSE 1.101 2021-06-13 15:33:51 1
1.335633 2.266986 rank FALSE 0.361 4 2.266986 rank FALSE 0.619 2021-06-13 15:33:51 1
3.561251 1.426416 rank FALSE 0.296 35 1.426416 rank FALSE 0.978 2021-06-13 15:33:51 1
2.052564 1.258745 rectangular FALSE 0.324 8 1.258745 rectangular FALSE 0.919 2021-06-13 15:33:51 1
3.460303 1.956236 gaussian FALSE 0.296 32 1.956236 gaussian FALSE 1.085 2021-06-13 15:33:51 1
2.073975 2.848149 rank TRUE 0.269 8 2.848149 rank TRUE 1.135 2021-06-13 15:33:51 1
2.037658 2.197522 gaussian TRUE 0.267 8 2.197522 gaussian TRUE 1.353 2021-06-13 15:33:51 1
2.438784 2.952341 rectangular FALSE 0.308 11 2.952341 rectangular FALSE 0.754 2021-06-13 15:33:51 1
3.608733 2.463585 rank FALSE 0.294 37 2.463585 rank FALSE 1.029 2021-06-13 15:33:51 1
3.530354 1.713454 rectangular FALSE 0.297 34 1.713454 rectangular FALSE 1.101 2021-06-13 15:33:51 1
2.195813 1.862947 optimal TRUE 0.257 9 1.862947 optimal TRUE 1.071 2021-06-13 15:33:51 1
3.285535 1.296423 rank FALSE 0.300 27 1.296423 rank FALSE 0.821 2021-06-13 15:33:51 1

Let’s now investigate the performance by parameters. This is especially easy using visualization:

ggplot(as.data.table(instance_random$archive),
  aes(x = x_domain_k, y = classif.ce, color = x_domain_scale)) +
  geom_point(size = 3) + geom_line()

The previous plot suggests that scale has a strong influence on performance. For the kernel, there does not seem to be a strong influence:

ggplot(as.data.table(instance_random$archive),
  aes(x = x_domain_k, y = classif.ce, color = x_domain_kernel)) +
  geom_point(size = 3) + geom_line()

Nested Resampling

Having determined tuned configurations that seem to work well, we want to find out which performance we can expect from them. However, this may require more than this naive approach:

instance_random$result_y
classif.ce 
     0.249 
instance_grid$result_y
classif.ce 
      0.25 

The problem associated with evaluating tuned models is overtuning. The more we search, the more optimistically biased the associated performance metrics from tuning become.

There is a solution to this problem, namely Nested Resampling.

The mlr3tuning package provides an AutoTuner that acts like our tuning method but is actually a Learner. The $train() method facilitates tuning of hyperparameters on the training data, using a resampling strategy (below we use 5-fold cross-validation). Then, we actually train a model with optimal hyperparameters on the whole training data.

The AutoTuner finds the best parameters and uses them for training:

at_grid = AutoTuner$new(
  learner = knn,
  resampling = rsmp("cv", folds = 5), # we can NOT use fixed resampling here
  measure = msr("classif.ce"),
  terminator = trm("none"),
  tuner = tnr("grid_search", resolution = 18),
  search_space = search_space
)

The AutoTuner behaves just like a regular Learner. It can be used to combine the steps of hyperparameter tuning and model fitting but is especially useful for resampling and fair comparison of performance through benchmarking:

rr = resample(task, at_grid, cv10, store_models = TRUE)

We check the inner tuning results for stable hyperparameters. This means that the selected hyperparameters should not vary too much. We might observe unstable models in this example because the small data set and the low number of resampling iterations might introduce too much randomness. Usually, we aim for the selection of stable hyperparameters for all outer training sets.

iteration k distance classif.ce task_id learner_id resampling_id
1 9 2 0.2600000 german_credit classif.kknn.tuned cv
2 7 1 0.2488889 german_credit classif.kknn.tuned cv
3 12 2 0.2466667 german_credit classif.kknn.tuned cv
4 7 1 0.2622222 german_credit classif.kknn.tuned cv
5 7 1 0.2411111 german_credit classif.kknn.tuned cv
6 8 2 0.2544444 german_credit classif.kknn.tuned cv
7 9 2 0.2577778 german_credit classif.kknn.tuned cv
8 14 1 0.2611111 german_credit classif.kknn.tuned cv
9 17 1 0.2511111 german_credit classif.kknn.tuned cv
10 11 2 0.2566667 german_credit classif.kknn.tuned cv

Next, we want to compare the predictive performances estimated on the outer resampling to the inner resampling (extract_inner_tuning_results(rr)). Significantly lower predictive performances on the outer resampling indicate that the models with the optimized hyperparameters overfit the data.

rr$score()
iteration task_id learner_id resampling_id classif.ce
1 german_credit classif.kknn.tuned cv 0.26
2 german_credit classif.kknn.tuned cv 0.28
3 german_credit classif.kknn.tuned cv 0.29
4 german_credit classif.kknn.tuned cv 0.21
5 german_credit classif.kknn.tuned cv 0.31
6 german_credit classif.kknn.tuned cv 0.25
7 german_credit classif.kknn.tuned cv 0.22
8 german_credit classif.kknn.tuned cv 0.28
9 german_credit classif.kknn.tuned cv 0.22
10 german_credit classif.kknn.tuned cv 0.24

The archives of the AutoTuners allows us to inspect all evaluated hyperparameters configurations with the associated predictive performances.

iteration k distance classif.ce x_domain_k x_domain_distance runtime_learners timestamp batch_nr task_id learner_id resampling_id
1 17 2 0.2711111 17 2 0.485 2021-06-13 15:33:53 1 german_credit classif.kknn.tuned cv
1 10 1 0.2644444 10 1 0.538 2021-06-13 15:33:54 2 german_credit classif.kknn.tuned cv
1 20 1 0.2733333 20 1 0.611 2021-06-13 15:33:55 3 german_credit classif.kknn.tuned cv
1 8 2 0.2677778 8 2 0.209 2021-06-13 15:33:55 4 german_credit classif.kknn.tuned cv
1 13 1 0.2722222 13 1 0.701 2021-06-13 15:33:56 5 german_credit classif.kknn.tuned cv
1 14 1 0.2700000 14 1 0.591 2021-06-13 15:33:57 6 german_credit classif.kknn.tuned cv
1 12 1 0.2677778 12 1 0.570 2021-06-13 15:33:57 7 german_credit classif.kknn.tuned cv
1 9 1 0.2677778 9 1 0.552 2021-06-13 15:33:58 8 german_credit classif.kknn.tuned cv
1 20 2 0.2777778 20 2 0.275 2021-06-13 15:33:59 9 german_credit classif.kknn.tuned cv
1 5 2 0.2977778 5 2 0.224 2021-06-13 15:33:59 10 german_credit classif.kknn.tuned cv
1 10 2 0.2666667 10 2 0.244 2021-06-13 15:33:59 11 german_credit classif.kknn.tuned cv
1 18 2 0.2744444 18 2 0.258 2021-06-13 15:34:00 12 german_credit classif.kknn.tuned cv
1 7 1 0.2622222 7 1 0.549 2021-06-13 15:34:01 13 german_credit classif.kknn.tuned cv
1 3 2 0.2811111 3 2 0.239 2021-06-13 15:34:01 14 german_credit classif.kknn.tuned cv
1 17 1 0.2688889 17 1 0.621 2021-06-13 15:34:02 15 german_credit classif.kknn.tuned cv
1 6 2 0.2788889 6 2 0.213 2021-06-13 15:34:02 16 german_credit classif.kknn.tuned cv
1 13 2 0.2677778 13 2 0.220 2021-06-13 15:34:02 17 german_credit classif.kknn.tuned cv
1 18 1 0.2688889 18 1 0.581 2021-06-13 15:34:03 18 german_credit classif.kknn.tuned cv
1 16 2 0.2733333 16 2 0.233 2021-06-13 15:34:04 19 german_credit classif.kknn.tuned cv
1 4 1 0.3022222 4 1 0.529 2021-06-13 15:34:04 20 german_credit classif.kknn.tuned cv
1 12 2 0.2711111 12 2 0.252 2021-06-13 15:34:05 21 german_credit classif.kknn.tuned cv
1 16 1 0.2666667 16 1 0.603 2021-06-13 15:34:05 22 german_credit classif.kknn.tuned cv
1 14 2 0.2744444 14 2 0.268 2021-06-13 15:34:06 23 german_credit classif.kknn.tuned cv
1 6 1 0.2977778 6 1 0.541 2021-06-13 15:34:07 24 german_credit classif.kknn.tuned cv
1 11 2 0.2644444 11 2 0.248 2021-06-13 15:34:07 25 german_credit classif.kknn.tuned cv
1 9 2 0.2600000 9 2 0.254 2021-06-13 15:34:07 26 german_credit classif.kknn.tuned cv
1 19 2 0.2755556 19 2 0.265 2021-06-13 15:34:08 27 german_credit classif.kknn.tuned cv
1 7 2 0.2600000 7 2 0.275 2021-06-13 15:34:08 28 german_credit classif.kknn.tuned cv
1 3 1 0.2977778 3 1 0.533 2021-06-13 15:34:09 29 german_credit classif.kknn.tuned cv
1 4 2 0.2933333 4 2 0.953 2021-06-13 15:34:10 30 german_credit classif.kknn.tuned cv
1 5 1 0.2888889 5 1 0.501 2021-06-13 15:34:11 31 german_credit classif.kknn.tuned cv
1 15 1 0.2700000 15 1 0.564 2021-06-13 15:34:11 32 german_credit classif.kknn.tuned cv
1 15 2 0.2722222 15 2 0.227 2021-06-13 15:34:12 33 german_credit classif.kknn.tuned cv
1 11 1 0.2744444 11 1 0.526 2021-06-13 15:34:12 34 german_credit classif.kknn.tuned cv
1 19 1 0.2744444 19 1 0.591 2021-06-13 15:34:13 35 german_credit classif.kknn.tuned cv
1 8 1 0.2655556 8 1 0.512 2021-06-13 15:34:14 36 german_credit classif.kknn.tuned cv
2 17 1 0.2577778 17 1 0.795 2021-06-13 15:33:54 1 german_credit classif.kknn.tuned cv
2 10 2 0.2655556 10 2 0.230 2021-06-13 15:33:54 2 german_credit classif.kknn.tuned cv
2 11 2 0.2655556 11 2 0.229 2021-06-13 15:33:54 3 german_credit classif.kknn.tuned cv
2 15 1 0.2633333 15 1 0.573 2021-06-13 15:33:55 4 german_credit classif.kknn.tuned cv
2 16 2 0.2566667 16 2 0.369 2021-06-13 15:33:56 5 german_credit classif.kknn.tuned cv
2 6 1 0.2600000 6 1 0.539 2021-06-13 15:33:56 6 german_credit classif.kknn.tuned cv
2 9 1 0.2611111 9 1 0.598 2021-06-13 15:33:57 7 german_credit classif.kknn.tuned cv
2 13 2 0.2666667 13 2 0.252 2021-06-13 15:33:57 8 german_credit classif.kknn.tuned cv
2 6 2 0.2688889 6 2 0.247 2021-06-13 15:33:58 9 german_credit classif.kknn.tuned cv
2 12 1 0.2611111 12 1 0.567 2021-06-13 15:33:59 10 german_credit classif.kknn.tuned cv
2 5 1 0.2600000 5 1 0.527 2021-06-13 15:33:59 11 german_credit classif.kknn.tuned cv
2 4 1 0.2577778 4 1 0.522 2021-06-13 15:34:00 12 german_credit classif.kknn.tuned cv
2 7 2 0.2577778 7 2 0.226 2021-06-13 15:34:00 13 german_credit classif.kknn.tuned cv
2 9 2 0.2544444 9 2 0.230 2021-06-13 15:34:01 14 german_credit classif.kknn.tuned cv
2 17 2 0.2644444 17 2 0.260 2021-06-13 15:34:01 15 german_credit classif.kknn.tuned cv
2 15 2 0.2644444 15 2 0.234 2021-06-13 15:34:01 16 german_credit classif.kknn.tuned cv
2 12 2 0.2655556 12 2 0.209 2021-06-13 15:34:02 17 german_credit classif.kknn.tuned cv
2 10 1 0.2622222 10 1 0.515 2021-06-13 15:34:03 18 german_credit classif.kknn.tuned cv
2 11 1 0.2688889 11 1 0.546 2021-06-13 15:34:03 19 german_credit classif.kknn.tuned cv
2 19 1 0.2611111 19 1 0.596 2021-06-13 15:34:04 20 german_credit classif.kknn.tuned cv
2 3 1 0.2811111 3 1 0.514 2021-06-13 15:34:05 21 german_credit classif.kknn.tuned cv
2 4 2 0.2977778 4 2 0.276 2021-06-13 15:34:05 22 german_credit classif.kknn.tuned cv
2 13 1 0.2722222 13 1 0.640 2021-06-13 15:34:06 23 german_credit classif.kknn.tuned cv
2 8 1 0.2744444 8 1 0.605 2021-06-13 15:34:07 24 german_credit classif.kknn.tuned cv
2 14 2 0.2622222 14 2 0.289 2021-06-13 15:34:07 25 german_credit classif.kknn.tuned cv
2 18 1 0.2611111 18 1 0.626 2021-06-13 15:34:08 26 german_credit classif.kknn.tuned cv
2 20 1 0.2622222 20 1 0.634 2021-06-13 15:34:09 27 german_credit classif.kknn.tuned cv
2 20 2 0.2566667 20 2 0.288 2021-06-13 15:34:09 28 german_credit classif.kknn.tuned cv
2 8 2 0.2577778 8 2 0.251 2021-06-13 15:34:10 29 german_credit classif.kknn.tuned cv
2 19 2 0.2644444 19 2 0.305 2021-06-13 15:34:11 30 german_credit classif.kknn.tuned cv
2 18 2 0.2677778 18 2 0.243 2021-06-13 15:34:11 31 german_credit classif.kknn.tuned cv
2 7 1 0.2488889 7 1 0.520 2021-06-13 15:34:12 32 german_credit classif.kknn.tuned cv
2 3 2 0.2688889 3 2 0.198 2021-06-13 15:34:12 33 german_credit classif.kknn.tuned cv
2 14 1 0.2733333 14 1 0.611 2021-06-13 15:34:13 34 german_credit classif.kknn.tuned cv
2 5 2 0.2611111 5 2 0.204 2021-06-13 15:34:14 35 german_credit classif.kknn.tuned cv
2 16 1 0.2588889 16 1 0.581 2021-06-13 15:34:14 36 german_credit classif.kknn.tuned cv
3 9 1 0.2544444 9 1 0.604 2021-06-13 15:33:54 1 german_credit classif.kknn.tuned cv
3 16 1 0.2600000 16 1 0.563 2021-06-13 15:33:55 2 german_credit classif.kknn.tuned cv
3 12 1 0.2688889 12 1 0.556 2021-06-13 15:33:55 3 german_credit classif.kknn.tuned cv
3 17 1 0.2633333 17 1 0.588 2021-06-13 15:33:56 4 german_credit classif.kknn.tuned cv
3 3 2 0.2766667 3 2 0.384 2021-06-13 15:33:57 5 german_credit classif.kknn.tuned cv
3 4 2 0.2688889 4 2 0.279 2021-06-13 15:33:57 6 german_credit classif.kknn.tuned cv
3 11 1 0.2622222 11 1 0.609 2021-06-13 15:33:58 7 german_credit classif.kknn.tuned cv
3 5 2 0.2566667 5 2 0.277 2021-06-13 15:33:58 8 german_credit classif.kknn.tuned cv
3 19 2 0.2700000 19 2 0.288 2021-06-13 15:33:59 9 german_credit classif.kknn.tuned cv
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8 12 1 0.2811111 12 1 0.534 2021-06-13 15:34:13 35 german_credit classif.kknn.tuned cv
8 17 2 0.2822222 17 2 0.227 2021-06-13 15:34:13 36 german_credit classif.kknn.tuned cv
9 15 2 0.2611111 15 2 0.457 2021-06-13 15:33:53 1 german_credit classif.kknn.tuned cv
9 17 2 0.2644444 17 2 0.246 2021-06-13 15:33:54 2 german_credit classif.kknn.tuned cv
9 11 1 0.2633333 11 1 0.547 2021-06-13 15:33:54 3 german_credit classif.kknn.tuned cv
9 3 2 0.2822222 3 2 0.223 2021-06-13 15:33:55 4 german_credit classif.kknn.tuned cv
9 20 1 0.2666667 20 1 0.742 2021-06-13 15:33:56 5 german_credit classif.kknn.tuned cv
9 14 1 0.2611111 14 1 0.589 2021-06-13 15:33:56 6 german_credit classif.kknn.tuned cv
9 10 1 0.2722222 10 1 0.562 2021-06-13 15:33:57 7 german_credit classif.kknn.tuned cv
9 9 2 0.2622222 9 2 0.247 2021-06-13 15:33:57 8 german_credit classif.kknn.tuned cv
9 16 2 0.2677778 16 2 0.260 2021-06-13 15:33:58 9 german_credit classif.kknn.tuned cv
9 7 1 0.2644444 7 1 0.523 2021-06-13 15:33:59 10 german_credit classif.kknn.tuned cv
9 11 2 0.2611111 11 2 0.254 2021-06-13 15:33:59 11 german_credit classif.kknn.tuned cv
9 4 1 0.2966667 4 1 0.528 2021-06-13 15:34:00 12 german_credit classif.kknn.tuned cv
9 6 1 0.2622222 6 1 0.565 2021-06-13 15:34:00 13 german_credit classif.kknn.tuned cv
9 10 2 0.2544444 10 2 0.242 2021-06-13 15:34:01 14 german_credit classif.kknn.tuned cv
9 16 1 0.2633333 16 1 0.601 2021-06-13 15:34:01 15 german_credit classif.kknn.tuned cv
9 5 2 0.2822222 5 2 0.241 2021-06-13 15:34:02 16 german_credit classif.kknn.tuned cv
9 12 1 0.2522222 12 1 0.577 2021-06-13 15:34:03 17 german_credit classif.kknn.tuned cv
9 20 2 0.2711111 20 2 0.287 2021-06-13 15:34:03 18 german_credit classif.kknn.tuned cv
9 13 1 0.2577778 13 1 0.585 2021-06-13 15:34:04 19 german_credit classif.kknn.tuned cv
9 5 1 0.2722222 5 1 0.597 2021-06-13 15:34:05 20 german_credit classif.kknn.tuned cv
9 19 1 0.2666667 19 1 0.652 2021-06-13 15:34:05 21 german_credit classif.kknn.tuned cv
9 3 1 0.2911111 3 1 0.568 2021-06-13 15:34:06 22 german_credit classif.kknn.tuned cv
9 14 2 0.2633333 14 2 0.304 2021-06-13 15:34:07 23 german_credit classif.kknn.tuned cv
9 12 2 0.2577778 12 2 1.152 2021-06-13 15:34:08 24 german_credit classif.kknn.tuned cv
9 19 2 0.2655556 19 2 0.224 2021-06-13 15:34:08 25 german_credit classif.kknn.tuned cv
9 4 2 0.2900000 4 2 0.226 2021-06-13 15:34:09 26 german_credit classif.kknn.tuned cv
9 8 1 0.2655556 8 1 0.550 2021-06-13 15:34:09 27 german_credit classif.kknn.tuned cv
9 6 2 0.2577778 6 2 0.218 2021-06-13 15:34:10 28 german_credit classif.kknn.tuned cv
9 9 1 0.2666667 9 1 0.517 2021-06-13 15:34:10 29 german_credit classif.kknn.tuned cv
9 8 2 0.2611111 8 2 0.243 2021-06-13 15:34:11 30 german_credit classif.kknn.tuned cv
9 18 1 0.2577778 18 1 0.604 2021-06-13 15:34:12 31 german_credit classif.kknn.tuned cv
9 18 2 0.2588889 18 2 0.238 2021-06-13 15:34:12 32 german_credit classif.kknn.tuned cv
9 13 2 0.2577778 13 2 0.247 2021-06-13 15:34:12 33 german_credit classif.kknn.tuned cv
9 7 2 0.2677778 7 2 0.246 2021-06-13 15:34:13 34 german_credit classif.kknn.tuned cv
9 17 1 0.2511111 17 1 0.585 2021-06-13 15:34:13 35 german_credit classif.kknn.tuned cv
9 15 1 0.2588889 15 1 0.587 2021-06-13 15:34:14 36 german_credit classif.kknn.tuned cv
10 6 2 0.2600000 6 2 0.391 2021-06-13 15:33:53 1 german_credit classif.kknn.tuned cv
10 17 1 0.2722222 17 1 0.602 2021-06-13 15:33:54 2 german_credit classif.kknn.tuned cv
10 11 2 0.2566667 11 2 0.228 2021-06-13 15:33:54 3 german_credit classif.kknn.tuned cv
10 5 2 0.2722222 5 2 0.221 2021-06-13 15:33:55 4 german_credit classif.kknn.tuned cv
10 9 1 0.2744444 9 1 0.661 2021-06-13 15:33:55 5 german_credit classif.kknn.tuned cv
10 3 2 0.2900000 3 2 0.241 2021-06-13 15:33:56 6 german_credit classif.kknn.tuned cv
10 7 1 0.2722222 7 1 0.567 2021-06-13 15:33:57 7 german_credit classif.kknn.tuned cv
10 10 1 0.2733333 10 1 0.593 2021-06-13 15:33:57 8 german_credit classif.kknn.tuned cv
10 4 2 0.2888889 4 2 0.244 2021-06-13 15:33:58 9 german_credit classif.kknn.tuned cv
10 9 2 0.2611111 9 2 0.264 2021-06-13 15:33:58 10 german_credit classif.kknn.tuned cv
10 12 2 0.2622222 12 2 0.313 2021-06-13 15:33:59 11 german_credit classif.kknn.tuned cv
10 19 1 0.2722222 19 1 0.618 2021-06-13 15:33:59 12 german_credit classif.kknn.tuned cv
10 8 1 0.2766667 8 1 0.566 2021-06-13 15:34:00 13 german_credit classif.kknn.tuned cv
10 14 1 0.2666667 14 1 0.587 2021-06-13 15:34:01 14 german_credit classif.kknn.tuned cv
10 15 1 0.2644444 15 1 0.639 2021-06-13 15:34:02 15 german_credit classif.kknn.tuned cv
10 3 1 0.2900000 3 1 0.543 2021-06-13 15:34:02 16 german_credit classif.kknn.tuned cv
10 14 2 0.2666667 14 2 0.294 2021-06-13 15:34:03 17 german_credit classif.kknn.tuned cv
10 7 2 0.2633333 7 2 0.245 2021-06-13 15:34:03 18 german_credit classif.kknn.tuned cv
10 16 2 0.2855556 16 2 0.254 2021-06-13 15:34:04 19 german_credit classif.kknn.tuned cv
10 13 2 0.2655556 13 2 0.270 2021-06-13 15:34:04 20 german_credit classif.kknn.tuned cv
10 8 2 0.2766667 8 2 0.250 2021-06-13 15:34:05 21 german_credit classif.kknn.tuned cv
10 18 2 0.2800000 18 2 0.266 2021-06-13 15:34:05 22 german_credit classif.kknn.tuned cv
10 12 1 0.2600000 12 1 0.611 2021-06-13 15:34:06 23 german_credit classif.kknn.tuned cv
10 15 2 0.2700000 15 2 0.273 2021-06-13 15:34:06 24 german_credit classif.kknn.tuned cv
10 20 1 0.2733333 20 1 0.641 2021-06-13 15:34:07 25 german_credit classif.kknn.tuned cv
10 19 2 0.2811111 19 2 0.274 2021-06-13 15:34:07 26 german_credit classif.kknn.tuned cv
10 16 1 0.2644444 16 1 0.637 2021-06-13 15:34:08 27 german_credit classif.kknn.tuned cv
10 10 2 0.2744444 10 2 0.243 2021-06-13 15:34:09 28 german_credit classif.kknn.tuned cv
10 17 2 0.2833333 17 2 0.268 2021-06-13 15:34:09 29 german_credit classif.kknn.tuned cv
10 20 2 0.2722222 20 2 1.000 2021-06-13 15:34:10 30 german_credit classif.kknn.tuned cv
10 4 1 0.3000000 4 1 0.487 2021-06-13 15:34:11 31 german_credit classif.kknn.tuned cv
10 6 1 0.2922222 6 1 0.511 2021-06-13 15:34:11 32 german_credit classif.kknn.tuned cv
10 11 1 0.2644444 11 1 0.530 2021-06-13 15:34:12 33 german_credit classif.kknn.tuned cv
10 5 1 0.2877778 5 1 0.503 2021-06-13 15:34:13 34 german_credit classif.kknn.tuned cv
10 18 1 0.2711111 18 1 0.574 2021-06-13 15:34:13 35 german_credit classif.kknn.tuned cv
10 13 1 0.2622222 13 1 0.538 2021-06-13 15:34:14 36 german_credit classif.kknn.tuned cv

We aggregate the performances of all resampling iterations:

rr$aggregate()
classif.ce 
     0.256 

Essentially, this is the performance of a “knn with optimal hyperparameters found by grid search”. Note that at_grid is not changed since resample() creates a clone for each resampling iteration.

The trained AutoTuner objects can be accessed by using

rr$learners[[1]]
<AutoTuner:classif.kknn.tuned>
* Model: list
* Search Space:
<ParamSet>
         id    class lower upper nlevels        default value
1:        k ParamInt     3    20      18 <NoDefault[3]>      
2: distance ParamInt     1     2       2 <NoDefault[3]>      
* Packages: kknn
* Predict Type: prob
* Feature Types: logical, integer, numeric, factor, ordered
* Properties: multiclass, twoclass
rr$learners[[1]]$tuning_result
   k distance learner_param_vals  x_domain classif.ce
1: 9        2          <list[3]> <list[2]>       0.26

Appendix

Example: Tuning With A Larger Budget

It is always interesting to look at what could have been. The following dataset contains an optimization run result with 3600 evaluations – more than above by a factor of 100:

k distance kernel scale classif.ce uhash timestamp batch_nr x_domain_k x_domain_distance x_domain_kernel x_domain_scale
2.191216 2.232217 gaussian FALSE 0.312 0bac4d72-2dad-4d0f-8502-22cc27284bb2 2020-12-09 15:08:29 1 9 2.232217 gaussian FALSE
3.549142 1.058476 rank FALSE 0.296 fe8de768-e91a-4a89-9fe2-521083a24466 2020-12-09 15:08:29 1 35 1.058476 rank FALSE
2.835727 2.121690 optimal TRUE 0.251 bbeab2d5-19aa-4382-a351-d5ece72a58a2 2020-12-09 15:08:29 1 17 2.121690 optimal TRUE
1.118085 1.275450 rank FALSE 0.368 93b5f285-66b9-4ea6-a8fc-700a11ed08a2 2020-12-09 15:08:29 1 3 1.275450 rank FALSE
2.790168 2.126899 optimal FALSE 0.320 1ab6ca63-96db-414e-9932-67f23592aab9 2020-12-09 15:08:29 1 16 2.126899 optimal FALSE
2.403195 2.738691 gaussian FALSE 0.312 39724a14-6cd9-493d-9cbe-0c2528e84be5 2020-12-09 15:08:29 1 11 2.738691 gaussian FALSE
3.359239 2.814982 rank FALSE 0.297 1b536ebc-969f-4c5b-8e99-861d5dde6d44 2020-12-09 15:08:29 1 29 2.814982 rank FALSE
2.470343 1.915714 rank TRUE 0.253 aaaf5e5d-ebad-4c43-b9e5-a63b17365613 2020-12-09 15:08:29 1 12 1.915714 rank TRUE
2.453694 2.153602 rank TRUE 0.253 bdd63b2b-af1e-4d8e-b947-bf2a8319eb0c 2020-12-09 15:08:29 1 12 2.153602 rank TRUE
1.711372 2.770095 rank FALSE 0.352 59b0fc13-143e-413c-b01d-e71269b2fd9a 2020-12-09 15:08:29 1 6 2.770095 rank FALSE
2.546687 2.808791 optimal FALSE 0.333 256044e6-f9e8-4e43-931f-74a8fb990e2e 2020-12-09 15:08:29 1 13 2.808791 optimal FALSE
1.243454 2.223476 gaussian FALSE 0.365 ca87b9da-3b66-4f48-95ea-c317ecc927de 2020-12-09 15:08:29 1 3 2.223476 gaussian FALSE
2.088564 1.819201 rectangular FALSE 0.310 bf8b9d59-933e-45d8-9b42-a011c1d6113a 2020-12-09 15:08:29 1 8 1.819201 rectangular FALSE
2.176556 2.564562 rectangular FALSE 0.311 7c0987d2-d365-47d8-8695-c27535c1b691 2020-12-09 15:08:29 1 9 2.564562 rectangular FALSE
3.505381 1.513896 rectangular TRUE 0.274 846ca0a7-d91f-4986-b924-7c3c0e4c0a3d 2020-12-09 15:08:29 1 33 1.513896 rectangular TRUE
2.539055 1.446643 optimal TRUE 0.247 8ddf1a8c-ac04-4193-8cd8-81e1d43b5b52 2020-12-09 15:08:29 1 13 1.446643 optimal TRUE
2.396618 1.776370 optimal TRUE 0.254 d5ca7fd8-97df-45a9-8acc-ee61966949c1 2020-12-09 15:08:29 1 11 1.776370 optimal TRUE
2.840286 2.202370 optimal FALSE 0.317 c0dbdc2f-f41d-427b-b293-225da3dae067 2020-12-09 15:08:29 1 17 2.202370 optimal FALSE
2.923746 1.265294 gaussian FALSE 0.298 530afbda-8a1a-433b-b714-5543442284c0 2020-12-09 15:08:29 1 19 1.265294 gaussian FALSE
2.165366 1.073392 rank TRUE 0.254 8e01a620-7614-4122-9349-7ed4ff1bcb87 2020-12-09 15:08:29 1 9 1.073392 rank TRUE
1.464759 2.442776 gaussian FALSE 0.359 c06d1d4f-3463-498d-9bba-5ed54aafb2c9 2020-12-09 15:08:29 1 4 2.442776 gaussian FALSE
1.556998 1.513216 rank TRUE 0.263 aec75634-5aeb-4ad3-827f-e421b4014149 2020-12-09 15:08:29 1 5 1.513216 rank TRUE
3.561517 1.630985 rank FALSE 0.297 866b0352-5314-4162-9bec-fc49640ea305 2020-12-09 15:08:29 1 35 1.630985 rank FALSE
2.003632 1.651369 rank TRUE 0.262 126e8fca-b31b-42ca-bb81-e594174c9825 2020-12-09 15:08:29 1 7 1.651369 rank TRUE
2.360999 2.113001 gaussian FALSE 0.314 4a8fc456-b089-467a-a827-feb938a716f8 2020-12-09 15:08:29 1 11 2.113001 gaussian FALSE
2.212332 2.563552 gaussian FALSE 0.312 1f1945c0-8476-43dd-a4d0-6f931637a9f7 2020-12-09 15:08:29 1 9 2.563552 gaussian FALSE
1.955481 2.749837 rank TRUE 0.267 27b99b52-68c4-4839-8abb-461d3ec159d3 2020-12-09 15:08:29 1 7 2.749837 rank TRUE
2.541345 2.682941 optimal TRUE 0.261 b3111364-03d9-4fa1-9e76-63e2c9d6b31d 2020-12-09 15:08:29 1 13 2.682941 optimal TRUE
2.199013 1.568192 gaussian FALSE 0.315 c6d08db7-fd42-4593-a4e6-159f4a91a19c 2020-12-09 15:08:29 1 9 1.568192 gaussian FALSE
2.330442 1.301446 rectangular TRUE 0.260 59f78d40-64b2-494e-89c5-e395e9ad6a56 2020-12-09 15:08:29 1 10 1.301446 rectangular TRUE
3.823214 1.478477 rectangular TRUE 0.282 44f33a16-b3ab-4b73-8b44-5c8c0aa5fee3 2020-12-09 15:08:29 1 46 1.478477 rectangular TRUE
1.310169 2.556357 rank FALSE 0.353 6674910c-b6fc-450c-b1c3-dd377f1625d7 2020-12-09 15:08:29 1 4 2.556357 rank FALSE
3.279217 2.069843 optimal FALSE 0.302 67573bd0-b38b-4889-8faa-f5b69508e20d 2020-12-09 15:08:29 1 27 2.069843 optimal FALSE
3.444615 1.565430 gaussian TRUE 0.262 dcaf7f41-e628-4c0b-8f09-f8e3c600f86a 2020-12-09 15:08:29 1 31 1.565430 gaussian TRUE
2.011522 1.786624 optimal FALSE 0.360 10f13ea9-902c-438f-ac28-48d33cbf5d57 2020-12-09 15:08:29 1 7 1.786624 optimal FALSE
3.040159 2.678914 optimal FALSE 0.305 442e6467-1b3c-435a-bd6e-7b6dd17c75e5 2020-12-09 15:08:29 1 21 2.678914 optimal FALSE
3.548292 1.759270 rectangular TRUE 0.275 e81541f6-4c78-4803-92a3-22e3390fc997 2020-12-09 15:09:11 2 35 1.759270 rectangular TRUE
1.637473 2.946271 gaussian TRUE 0.276 498b7e06-1989-46f3-a375-49da93276fdb 2020-12-09 15:09:11 2 5 2.946271 gaussian TRUE
2.825236 1.342194 rank FALSE 0.306 ef7c658b-4db6-4f9b-98cf-251407c78dd3 2020-12-09 15:09:11 2 17 1.342194 rank FALSE
1.835782 1.679039 rank FALSE 0.352 b5188764-5fde-4fdf-b732-015069d2999e 2020-12-09 15:09:11 2 6 1.679039 rank FALSE
3.660419 2.219432 rank FALSE 0.295 fa2a274e-9452-4034-bfeb-d891e9569206 2020-12-09 15:09:11 2 39 2.219432 rank FALSE
1.984375 1.798759 gaussian TRUE 0.249 9c581aa9-e2af-4354-97c0-a3174b23ca59 2020-12-09 15:09:11 2 7 1.798759 gaussian TRUE
2.871947 1.159655 rectangular TRUE 0.264 da66e0f8-2815-43fb-adff-a67605c36e75 2020-12-09 15:09:11 2 18 1.159655 rectangular TRUE
3.181291 2.285767 rectangular TRUE 0.266 8c481744-146f-4a3b-a250-ca8dd50bb37a 2020-12-09 15:09:11 2 24 2.285767 rectangular TRUE
3.367268 2.588823 gaussian FALSE 0.296 3aad091a-7e5a-4a30-81f6-0386a270bdcc 2020-12-09 15:09:11 2 29 2.588823 gaussian FALSE
3.542795 1.641481 rectangular TRUE 0.275 38f63d5d-d174-4c1d-abf5-cb3a493d2735 2020-12-09 15:09:11 2 35 1.641481 rectangular TRUE
1.816575 2.434027 gaussian TRUE 0.273 cbd2156f-89ba-4ae5-b45a-a7d93ae209f8 2020-12-09 15:09:11 2 6 2.434027 gaussian TRUE
1.655091 1.217586 optimal TRUE 0.279 b7fa1a73-2787-4b9c-b803-cde16415ce27 2020-12-09 15:09:11 2 5 1.217586 optimal TRUE
2.425897 1.440158 rectangular TRUE 0.248 e922bfb8-3219-423d-9d76-096b38c2b8b8 2020-12-09 15:09:11 2 11 1.440158 rectangular TRUE
1.652333 2.178196 rank TRUE 0.266 2041878e-171a-4dbc-b6b2-8fd4d06edd11 2020-12-09 15:09:11 2 5 2.178196 rank TRUE
1.566852 2.568562 gaussian TRUE 0.267 5348538e-35b6-41ad-8bab-7cfebd960b14 2020-12-09 15:09:11 2 5 2.568562 gaussian TRUE
2.670714 2.838954 optimal TRUE 0.269 a2f4aecd-e259-4063-9869-e099f01cd6d8 2020-12-09 15:09:11 2 14 2.838954 optimal TRUE
2.278814 2.504997 gaussian FALSE 0.311 073f2ad7-1c40-4638-aea3-21f3081c2417 2020-12-09 15:09:11 2 10 2.504997 gaussian FALSE
2.311463 2.154947 rank FALSE 0.314 e15f4fe1-07e2-4957-9297-34e2e2a2fdfd 2020-12-09 15:09:11 2 10 2.154947 rank FALSE
2.194339 1.706755 rank TRUE 0.254 a17492dd-6b4d-4065-9bf5-bb959ac5d7fd 2020-12-09 15:09:11 2 9 1.706755 rank TRUE
3.141354 2.873359 rectangular FALSE 0.299 5d1dac8f-b1ab-4156-9ae2-237ade791989 2020-12-09 15:09:11 2 23 2.873359 rectangular FALSE
3.019601 2.007410 gaussian TRUE 0.265 458356b6-e356-4df1-8972-7d109324ef8b 2020-12-09 15:09:11 2 20 2.007410 gaussian TRUE
2.310180 1.831391 gaussian TRUE 0.248 0a1a99d9-b03b-4f08-83a9-f61067eec922 2020-12-09 15:09:11 2 10 1.831391 gaussian TRUE
3.742605 1.950280 rank TRUE 0.261 234ac80a-897d-48fb-8340-bc114fe19d58 2020-12-09 15:09:11 2 42 1.950280 rank TRUE
2.379814 2.350556 optimal TRUE 0.259 a86e0178-f9bb-46d1-bbe3-ed0623ea643a 2020-12-09 15:09:11 2 11 2.350556 optimal TRUE
3.131652 2.406607 rectangular TRUE 0.271 617c04c8-1eae-4c4b-8586-4c9b1a59bc78 2020-12-09 15:09:11 2 23 2.406607 rectangular TRUE
3.676227 2.623878 rank TRUE 0.265 24825ee5-808b-4626-8553-f6348160ce67 2020-12-09 15:09:11 2 39 2.623878 rank TRUE
3.808156 1.983243 rectangular TRUE 0.277 b0bd4a99-082f-400c-8734-6b63a85a7cd6 2020-12-09 15:09:11 2 45 1.983243 rectangular TRUE
1.623169 1.821303 optimal TRUE 0.275 b2a8bbb4-0c2a-45b2-b970-8d02f6fd6575 2020-12-09 15:09:11 2 5 1.821303 optimal TRUE
2.077934 2.276293 rank TRUE 0.266 37ea9688-2cd4-453f-a671-d5a5f4500fe7 2020-12-09 15:09:11 2 8 2.276293 rank TRUE
1.691646 2.412215 gaussian FALSE 0.344 da162bcf-aa54-49eb-b56c-645b04bbbd27 2020-12-09 15:09:11 2 5 2.412215 gaussian FALSE
1.184933 2.641253 optimal FALSE 0.400 69e5e6b4-3d25-4183-989c-4d5a3c7155e2 2020-12-09 15:09:11 2 3 2.641253 optimal FALSE
3.746593 2.989723 rectangular TRUE 0.284 cb977e6d-44a3-42df-ba7f-33ee6298ef27 2020-12-09 15:09:11 2 42 2.989723 rectangular TRUE
2.764914 2.833077 optimal TRUE 0.262 3ca6027a-f362-4016-a60e-8a9bf051d23e 2020-12-09 15:09:11 2 16 2.833077 optimal TRUE
2.665630 1.695408 rectangular TRUE 0.261 9554c797-07bf-4d9d-b211-8970884ce063 2020-12-09 15:09:11 2 14 1.695408 rectangular TRUE
3.454277 1.669336 optimal FALSE 0.295 61f0b58a-e919-46f8-8b80-3e1362281f97 2020-12-09 15:09:11 2 32 1.669336 optimal FALSE
2.157694 2.847317 rectangular TRUE 0.275 9a57deb9-df41-4a9c-8479-b43905efc197 2020-12-09 15:09:11 2 9 2.847317 rectangular TRUE
1.136740 2.231883 rectangular FALSE 0.365 bd63f984-b0d7-429c-bf3f-2318964a86e5 2020-12-09 15:09:47 3 3 2.231883 rectangular FALSE
1.478165 2.316062 gaussian FALSE 0.357 fec1ef24-ea0f-4ecb-bfa5-b76c7a91cda1 2020-12-09 15:09:47 3 4 2.316062 gaussian FALSE
2.209659 1.593207 optimal FALSE 0.344 ece4b6db-00f8-48d5-a236-c1dbe8e529b6 2020-12-09 15:09:47 3 9 1.593207 optimal FALSE
1.284004 2.617680 optimal FALSE 0.398 d320b1b3-450a-4d22-a125-12c9b6be745d 2020-12-09 15:09:47 3 4 2.617680 optimal FALSE
1.602980 2.479756 gaussian TRUE 0.273 e359a4a4-3829-40ca-9c7d-3bb09b1d4dfe 2020-12-09 15:09:47 3 5 2.479756 gaussian TRUE
3.557088 1.542132 optimal TRUE 0.255 002417a0-32d1-41b9-85fe-8e15f58b0cf1 2020-12-09 15:09:47 3 35 1.542132 optimal TRUE
3.073417 1.159363 rank TRUE 0.260 79b87a03-d5d2-472f-a8a5-e106b83c44ad 2020-12-09 15:09:47 3 22 1.159363 rank TRUE
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3.543424 1.328426 optimal TRUE 0.255 b960c2d0-fd0a-4358-b235-128089e017be 2020-12-09 15:15:01 12 35 1.328426 optimal TRUE
1.123933 1.999049 optimal TRUE 0.310 861f5c88-1a0e-4a4e-8317-64154586bb74 2020-12-09 15:15:32 13 3 1.999049 optimal TRUE
2.220366 1.312939 optimal TRUE 0.264 e47b82a9-c8bf-4b68-a50b-50be2f0ee470 2020-12-09 15:15:32 13 9 1.312939 optimal TRUE
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