Contents
⁉️ Why- 📥 Installation
- 🔖 Examples
- Intermezzo: Almost Stochastic Order - a better significance test for Deep Neural Networks
- Scenario 1: Comparing multiple runs of two models
- Scenario 2: Comparing multiple runs across datasets
- Scenario 3: Comparing sample-level scores
- Scenario 4: Comparing more than two models
- How to report results
- Sample size
- Other features
- General Recommendations & other notes
- 🎓 Cite
- 🏅 Acknowledgements
- 🧑🤝🧑 Papers using deep-significance
- 📚 Bibliography
Although Deep Learning has undergone spectacular growth in the recent decade, a large portion of experimental evidence is not supported by statistical hypothesis tests. Instead, conclusions are often drawn based on single performance scores.
This is problematic: Neural network display highly non-convex loss surfaces (Li et al., 2018) and their performance depends on the specific hyperparameters that were found, or stochastic factors like Dropout masks, making comparisons between architectures more difficult. Based on comparing only (the mean of) a few scores, we often cannot conclude that one model type or algorithm is better than another. This endangers the progress in the field, as seeming success due to random chance might lead practitioners astray.
For instance, a recent study in Natural Language Processing by Narang et al. (2021) has found that many modifications proposed to transformers do not actually improve performance. Similar issues are known to plague other fields like e.g., Reinforcement Learning (Henderson et al., 2018) and Computer Vision (Borji, 2017) as well.
To help mitigate this problem, this package supplies fully-tested re-implementations of useful functions for significance testing:
- Statistical Significance tests such as Almost Stochastic Order (del Barrio et al, 2017; Dror et al., 2019), bootstrap (Efron & Tibshirani, 1994) and permutation-randomization (Noreen, 1989).
- Bonferroni correction methods for multiplicity in datasets (Bonferroni, 1936).
- Bootstrap power analysis (Yuan & Hayashi, 2003) and other functions to determine the right sample size.
All functions are fully tested and also compatible with common deep learning data structures, such as PyTorch / Tensorflow tensors as well as NumPy and Jax arrays. For examples about the usage, consult the documentation here , the scenarios in the section Examples or the demo Jupyter notebook.
The package can simply be installed using pip
by running
pip3 install deepsig
Another option is to clone the repository and install the package locally:
git clone https://github.com/Kaleidophon/deep-significance.git
cd deep-significance
pip3 install -e .
Warning: Installed like this, imports will fail when the clones repository is moved.
tl;dr: Use aso()
to compare scores for two models. If the returned eps_min < 0.5
, A is better than B. The lower
eps_min
, the more confident the result (we recommend to check eps_min < 0.2
and record eps_min
alongside
experimental results).
In the following, we will lay out three scenarios that describe common use cases for ML practitioners and how to apply the methods implemented in this package accordingly. For an introduction into statistical hypothesis testing, please refer to resources such as this blog post for a general overview or Dror et al. (2018) for a NLP-specific point of view.
We assume that we have two sets of scores we would like to compare, and
,
for instance obtained by running two models
and
multiple times with a different random seed.
We can then define a one-sided test statistic
based on the gathered observations.
An example of such test statistics is for instance the difference in observation means. We then formulate the following null-hypothesis:
That means that we actually assume the opposite of our desired case, namely that is not better than
,
but equally as good or worse, as indicated by the value of the test statistic.
Usually, the goal becomes to reject this null hypothesis using the SST.
p-value testing is a frequentist method in the realm of SST.
It introduces the notion of data that could have been observed if we were to repeat our experiment again using
the same conditions, which we will write with superscript
in order to distinguish them from our actually
observed scores (Gelman et al., 2021).
We then define the p-value as the probability that, under the null hypothesis, the test statistic using replicated
observation is larger than or equal to the observed test statistic:
We can interpret this expression as follows: Assuming that is not better than
, the test
assumes a corresponding distribution of statistics that
is drawn from. So how does the observed test statistic
fit in here? This is what the
-value expresses: When the
probability is high,
is in line with what we expected under the
null hypothesis, so we can not reject the null hypothesis, or in other words, we \emph{cannot} conclude
to be better than
. If the probability is low, that means that the observed
is quite unlikely under the null hypothesis and that the reverse case is
more likely - i.e. that it is likely larger than - and we conclude that
is indeed better than
. Note that the
-value does not express whether the null hypothesis is true. To make our decision
about whether or not to reject the null hypothesis, we typically determine a threshold - the significance level
, often set to 0.05 - that the p-value has to fall below. However, it has been argued that a better practice
involves reporting the p-value alongside the results without a pidgeonholing of results into significant and non-significant
(Wasserstein et al., 2019).
Deep neural networks are highly non-linear models, having their performance highly dependent on hyperparameters, random seeds and other (stochastic) factors. Therefore, comparing the means of two models across several runs might not be enough to decide if a model A is better than B. In fact, even aggregating more statistics like standard deviation, minimum or maximum might not be enough to make a decision. For this reason, del Barrio et al. (2017) and Dror et al. (2019) introduced Almost Stochastic Order (ASO), a test to compare two score distributions.
It builds on the concept of stochastic order: We can compare two distributions and declare one as stochastically dominant by comparing their cumulative distribution functions:
Here, the CDF of A is given in red and in green for B. If the CDF of A is lower than B for every , we know the
algorithm A to score higher. However, in practice these cases are rarely so clear-cut (imagine e.g. two normal
distributions with the same mean but different variances).
For this reason, del Barrio et al. (2017) and Dror et al. (2019) consider the notion of almost stochastic dominance
by quantifying the extent to which stochastic order is being violated (red area):
ASO returns a value , which expresses (an upper bound to) the amount of violation of stochastic order. If
(where \tau is 0.5 or less), A is stochastically dominant over B in more cases than vice versa, then the corresponding algorithm can be declared as
superior. We can also interpret
as a confidence score. The lower it is, the more sure we can be
that A is better than B. Note: ASO does not compute p-values. Instead, the null hypothesis formulated as
If we want to be more confident about the result of ASO, we can also set the rejection threshold to be lower than 0.5
(see the discussion in this section).
Furthermore, the significance level is determined as an input argument when running ASO and actively influence
the resulting
.
In the simplest scenario, we have retrieved a set of scores from a model A and a baseline B on a dataset, stemming from various model runs with different seeds. We want to test whether our model A is better than B (higher scores = better)- We can now simply apply the ASO test:
import numpy as np
from deepsig import aso
seed = 1234
np.random.seed(seed)
# Simulate scores
N = 5 # Number of random seeds
my_model_scores = np.random.normal(loc=0.9, scale=0.8, size=N)
baseline_scores = np.random.normal(loc=0, scale=1, size=N)
min_eps = aso(my_model_scores, baseline_scores, seed=seed) # min_eps = 0.225, so A is better
Note that ASO does not make any assumptions about the distributions of the scores. This means that we can apply it to any kind of test metric, as long as a higher score indicates a better performance (to apply ASO to cases where lower scores indicate better performances, just multiple your scores by -1 before feeding them into the function). The more scores of model runs is supplied, the more reliable the test becomes, so try to collect scores from as many runs as possible to reject the null hypothesis confidently.
When comparing models across datasets, we formulate one null hypothesis per dataset. However, we have to make sure not to
fall prey to the multiple comparisons problem: In short,
the more comparisons between A and B we are conducting, the more likely gets is to reject a null-hypothesis accidentally.
That is why we have to adjust our significance threshold accordingly by dividing it by the number of comparisons,
which corresponds to the Bonferroni correction (Bonferroni et al., 1936):
import numpy as np
from deepsig import aso
seed = 1234
np.random.seed(seed)
# Simulate scores for three datasets
M = 3 # Number of datasets
N = 5 # Number of random seeds
my_model_scores_per_dataset = [np.random.normal(loc=0.3, scale=0.8, size=N) for _ in range(M)]
baseline_scores_per_dataset = [np.random.normal(loc=0, scale=1, size=N) for _ in range(M)]
# epsilon_min values with Bonferroni correction
eps_min = [aso(a, b, confidence_level=0.95, num_comparisons=M, seed=seed) for a, b in zip(my_model_scores_per_dataset, baseline_scores_per_dataset)]
# eps_min = [0.006370113450148568, 0.6534772728574852, 0.0]
In previous examples, we have assumed that we compare two algorithms A and B based on their performance per run, i.e. we run each algorithm once per random seed and obtain exactly one score on our test set. In some cases however, we would like to compare two algorithms based on scores for every point in the test set. If we only use one seed per model, then this case is equivalent to scenario 1. But what if we also want to use multiple seeds per model?
In this scenario, we can do pair-wise comparisons of the score distributions between A and B and use the Bonferroni correction accordingly:
from itertools import product
import numpy as np
from deepsig import aso
seed = 1234
np.random.seed(seed)
# Simulate scores for three datasets
M = 40 # Number of data points
N = 3 # Number of random seeds
my_model_scored_samples_per_run = [np.random.normal(loc=0.3, scale=0.8, size=M) for _ in range(N)]
baseline_scored_samples_per_run = [np.random.normal(loc=0, scale=1, size=M) for _ in range(N)]
pairs = list(product(my_model_scored_samples_per_run, baseline_scored_samples_per_run))
# epsilon_min values with Bonferroni correction
eps_min = [aso(a, b, confidence_level=0.95, num_comparisons=len(pairs), seed=seed) for a, b in pairs]
# eps_min = [0.3831678636198528, 0.07194780234194881, 0.9152792807128325, 0.5273463008857844, 0.14946944524461184, 1.0,
# 0.6099543280369378, 0.22387448804041898, 1.0]
Similarly, when comparing multiple models (now again on a per-seed basis), we can use a similar approach like in the
previous example. For instance, for three models, we can create a matrix and fill the entries
with the corresponding
values.
The package implements the function multi_aso()
exactly for this purpose. It has the same arguments as aso()
, with
a few differences. First of all, the function takes a single scores
argument, which can be a list of lists (of scores),
or a nested NumPy array or Tensorflow / PyTorch / Jax tensor or dictionary (more about that later).
Let's look at an example:
import numpy as np
from deepsig import multi_aso
seed = 1234
np.random.seed(seed)
N = 5 # Number of random seeds
M = 3 # Number of different models / algorithms
# Simulate different model scores by sampling from normal distributions with increasing means
# Here, we will sample from N(0.1, 0.8), N(0.15, 0.8), N(0.2, 0.8)
my_models_scores = np.array([np.random.normal(loc=loc, scale=0.8, size=N) for loc in np.arange(0.1, 0.1 + 0.05 * M, step=0.05)])
eps_min = multi_aso(my_models_scores, confidence_level=0.95, seed=seed)
# eps_min =
# array([[1. , 0.92621655, 1. ],
# [1. , 1. , 1. ],
# [0.82081635, 0.73048716, 1. ]])
In the example, eps_min
is now a matrix, containing the score between all pairs of models (for
the same model, it set to 1 by default). The matrix is always to be read as ASO(row, column).
The function applies the bonferroni correction for multiple comparisons by
default, but this can be turned off by using use_bonferroni=False
.
Lastly, when the scores
argument is a dictionary and the function is called with return_df=True
, the resulting matrix is
given as a pandas.DataFrame
for increased readability:
import numpy as np
from deepsig import multi_aso
seed = 1234
np.random.seed(seed)
N = 5 # Number of random seeds
M = 3 # Number of different models / algorithms
# Same setup as above, but use a dict for scores
my_models_scores = {
f"model {i+1}": np.random.normal(loc=loc, scale=0.8, size=N)
for i, loc in enumerate(np.arange(0.1, 0.1 + 0.05 * M, step=0.05))
}
# my_model_scores = {
# "model 1": array([...]),
# "model 2": array([...]),
# ...
# }
eps_min = multi_aso(my_models_scores, confidence_level=0.95, return_df=True, seed=seed)
# This is now a DataFrame!
# eps_min =
# model 1 model 2 model 3
# model 1 1.000000 0.926217 1.0
# model 2 1.000000 1.000000 1.0
# model 3 0.820816 0.730487 1.0
When ASO used, two important details have to be reported, namely the confidence level and the
score. Below lists some example snippets reporting the results of scenarios 1 and 4:
Using ASO with a confidence level $\alpha = 0.05$, we found the score distribution of algorithm A based on three
random seeds to be stochastically dominant over B ($\epsilon_\text{min} = 0$).
We compared all pairs of models based on five random seeds each using ASO with a confidence level of
$\alpha = 0.05$ (before adjusting for all pair-wise comparisons using the Bonferroni correction). Almost stochastic
dominance ($\epsilon_\text{min} < \tau$ with $\tau = 0.2$) is indicated in table X.
It can be hard to determine whether the currently collected set of scores is large enough to allow for reliable
significance testing or whether more scores are required. For this reason, deep-significance
also implements functions to aid the decision of whether to
collect more samples or not.
First of all, it contains Bootstrap power analysis (Yuan & Hayashi, 2003): Given a set of scores, it gives all of them a uniform lift to create an artificial, second sample. Then, the analysis runs repeated analyses using bootstrapped versions of both samples, comparing them with a significance test. Ideally, this should yield a significant result: If the difference between the re-sampled original and the lifted sample is non-significant, the original sample has too high of a variance. The analyses then returns the percentage of comparisons that yielded significant results. If the number is too low, more scores should be collected and added to the sample.
The result of the analysis is the statistical power: The higher the power, the smaller the risk of falling prey to a Type II error - the probability of mistakenly accepting the null hypothesis, when in fact it should actually be rejected. Usually, a power of ~ 0.8 is recommended (although that is sometimes hard to achieve in a machine learning setup).
The function can be used in the following way:
import numpy as np
from deepsig import bootstrap_power_analysis
scores = np.random.normal(loc=0, scale=20, size=5) # Create too small of a sample with high variance
power = bootstrap_power_analysis(scores, show_progress=False) # 0.081, way too low
scores2 = np.random.normal(loc=0, scale=20, size=50) # Let's collect more samples
power2 = bootstrap_power_analysis(scores2, show_progress=False) # Better power with 0.2556
By default, bootstrap_power_analysis()
uses a one-sided Welch's t-test. However, this can be modified by passing
a function to the significance_test
argument, which expects a function taking two sets of scores and returning a
p-value.
Secondly, if the Almost Stochastic Order test (ASO) is being used, there is a second function available. ASO estimates the violation ratio of two samples using bootstrapping. However, there is necessarily some uncertainty around that estimate, given that we only possess a finite number of samples. Using more samples decreases the uncertainty and makes the estimate tighter. The degree to which collecting more samples increases the tightness can be computed using the following function:
import numpy as np
from deepsig import aso_uncertainty_reduction
scores1 = np.random.normal(loc=0, scale=0.3, size=5) # First sample with five scores
scores2 = np.random.normal(loc=0.2, scale=5, size=3) # Second sample with three scores
red1 = aso_uncertainty_reduction(m_old=len(scores1), n_old=len(scores2), m_new=5, n_new=5) # 1.1547005383792515
red2 = aso_uncertainty_reduction(m_old=len(scores1), n_old=len(scores2), m_new=7, n_new=3) # 1.0583005244258363
# Adding two runs to scores1 increases tightness of estimate by 1.15
# But adding two runs to scores2 only increases tightness by 1.06! So spending two more runs on scores1 is better
Waiting for all the bootstrap iterations to finish can feel tedious, especially when doing many comparisons. Therefore,
ASO supports multithreading using joblib
via the num_jobs
argument.
from deepsig import aso
import numpy as np
from timeit import timeit
a = np.random.normal(size=1000)
b = np.random.normal(size=1000)
print(timeit(lambda: aso(a, b, num_jobs=1, show_progress=False), number=5)) # 616.2249192680001
print(timeit(lambda: aso(a, b, num_jobs=4, show_progress=False), number=5)) # 208.05637107000007
If you want to select the maximum number of jobs possible on your device, you can set num_jobs=-1
:
print(timeit(lambda: aso(a, b, num_jobs=-1, show_progress=False), number=5)) # 187.26257274800003
All tests implemented in this package also can take PyTorch / Tensorflow tensors and Jax or NumPy arrays as arguments:
from deepsig import aso
import torch
a = torch.randn(5, 1)
b = torch.randn(5, 1)
aso(a, b) # It just works!
In order to ensure replicability, both aso()
and multi_aso()
supply as seed
argument. This even works
when multiple jobs are used!
Should you be suspicious of ASO and want to revert to the good old faithful tests, this package also implements
the paired-bootstrap as well as the permutation randomization test. Note that as discussed in the next section, these
tests have less statistical power than ASO. Furthermore, a function for the Bonferroni-correction using
p-values can also be found using from deepsig import bonferroni_correction
.
import numpy as np
from deepsig import bootstrap_test, permutation_test
a = np.random.normal(loc=0.8, size=10)
b = np.random.normal(size=10)
print(permutation_test(a, b)) # 0.16183816183816183
print(bootstrap_test(a, b)) # 0.103
-
Naturally, the CDFs built from
scores_a
andscores_b
can only be approximations of the true distributions. Therefore, as many scores as possible should be collected, especially if the variance between runs is high. If only one run is available, comparing sample-wise score distributions like in scenario 3 can be an option, but comparing multiple runs will always be preferable. Ideally, scores should be obtained even using different sets of hyperparameters per model. Because this is usually infeasible in practice, Bouthilier et al. (2020) recommend to vary all other sources of variation between runs to obtain the most trustworthy estimate of the "true" performance, such as data shuffling, weight initialization etc. -
num_bootstrap_iterations
can be reduced to increase the speed ofaso()
. However, this is not recommended as the result of the test will also become less accurate. Technically,is a upper bound that becomes tighter with the number of samples and bootstrap iterations (del Barrio et al., 2017). Thus, increasing the number of jobs with
num_jobs
instead is always preferred. -
While we could declare a model stochastically dominant with
, we found this to have a comparatively high Type I error (false positives). Tests in our paper have shown that a more useful threshold that trades of Type I and Type II error between different scenarios might be
.
-
Bootstrap and permutation-randomization are all non-parametric tests, i.e. they don't make any assumptions about the distribution of our test metric. Nevertheless, they differ in their statistical power, which is defined as the probability that the null hypothesis is being rejected given that there is a difference between A and B. In other words, the more powerful a test, the less conservative it is and the more it is able to pick up on smaller difference between A and B. Therefore, if the distribution is known or found out why normality tests (like e.g. Anderson-Darling or Shapiro-Wilk), something like a parametric test like Student's or Welch's t-test is preferable to bootstrap or permutation-randomization. However, because these test are in turn less applicable in a Deep Learning setting due to the reasons elaborated on in Why?, ASO is still a better choice.
Using this package in general, please cite the following:
@inproceedings{ulmer2022deep,
title={deep-significance: Easy and Meaningful Signifcance Testing in the Age of Neural Networks},
author={Ulmer, Dennis and Hardmeier, Christian and Frellsen, Jes},
booktitle={ML Evaluation Standards Workshop at the Tenth International Conference on Learning Representations},
year={2022}
}
If you use the ASO test via aso()
or `multi_aso, please cite the original works:
@inproceedings{dror2019deep,
author = {Rotem Dror and
Segev Shlomov and
Roi Reichart},
editor = {Anna Korhonen and
David R. Traum and
Llu{\'{\i}}s M{\`{a}}rquez},
title = {Deep Dominance - How to Properly Compare Deep Neural Models},
booktitle = {Proceedings of the 57th Conference of the Association for Computational
Linguistics, {ACL} 2019, Florence, Italy, July 28- August 2, 2019,
Volume 1: Long Papers},
pages = {2773--2785},
publisher = {Association for Computational Linguistics},
year = {2019},
url = {https://doi.org/10.18653/v1/p19-1266},
doi = {10.18653/v1/p19-1266},
timestamp = {Tue, 28 Jan 2020 10:27:52 +0100},
}
@incollection{del2018optimal,
title={An optimal transportation approach for assessing almost stochastic order},
author={Del Barrio, Eustasio and Cuesta-Albertos, Juan A and Matr{\'a}n, Carlos},
booktitle={The Mathematics of the Uncertain},
pages={33--44},
year={2018},
publisher={Springer}
}
For instance, you can write
In order to compare models, we use the Almost Stochastic Order test \citep{del2018optimal, dror2019deep} as
implemented by \citet{ulmer2022deep}.
This package was created out of discussions of the NLPnorth group at the IT University Copenhagen, whose members I want to thank for their feedback. The code in this repository is in multiple places based on several of Rotem Dror's repositories, namely this, this and this one. Thanks also go out to her personally for being available to answer questions and provide feedback to the implementation and documentation of this package.
The commit message template used in this project can be found here. The inline latex equations were rendered using readme2latex.
In this last section of the readme, I would like to refer to works already using deep-significance
. Open an issue or
pull request if you would like to see your work added here!
- "From Masked Language Modeling to Translation: Non-English Auxiliary Tasks Improve Zero-shot Spoken Language Understanding" (van der Groot et al., 2021)
- "Cartography Active Learning" (Zhang & Plank, 2021)
- "SkillSpan: Hard and Soft Skill Extraction from English Job Postings" (Zhang et al., 2022a)
- "What do you mean by Relation Extraction? A Survey on Datasets and Study on Scientific Relation Classification" (Bassignana & Plank, 2022)
- "KOMPETENCER: Fine-grained Skill Classification in Danish Job Postings via Distant Supervision and Transfer Learning" (Zhang et al., 2022b)
- "Exploring Predictive Uncertainty and Calibration in NLP: A Study on the Impact of Method & Data Scarcity" (Ulmer et al., 2022)
- "DBGSL: Dynamic Brain Graph Structure Learning" (Campbell et al., 2022)
- "Cross-lingual sentiment analysis of official EU Slavic languages" (Pandurang Thakkar, 2022)
- "An Empirical Study on Cross-X Transfer for Legal Judgment Prediction" (Niklaus et al., 2022)
- "Why don't you do it right? analysing annotators' disagreement in subjective tasks" (Sandri et al., 2023)
- "Not all layers are equally as important: Every Layer Counts BERT" (Charpentier et al., 2023)
- "Semantics for vision-and-language understanding" (Falcon, 2023)
- "Rethinking Relation Classification with Graph Meaning Representations" (Zhou et al., 2023)
- "Improving Legal Rhetorical Role Labeling Through Additional Data and Efficient Exploitation of Transformer Models" (Gomes de Lima, 2023)
- "Integration and Analysis of Biological Data for Computational Drug Discovery" (Güvelinir, 2024)
- "Influence of Channel Selection and Subject’s Age on the Performance of the Single Channel EEG-Based Automatic Sleep Staging Algorithms" (Nazih et al., 2023)
- "DH-FBK at SemEval-2023 Task 10: Multi-Task Learning with Classifier Ensemble Agreement for Sexism Detection" (Leonardelli et al., 2023)
- "VideoAdviser: Video Knowledge Distillation for Multimodal Transfer Learning" (Wang et al., 2023)
- "Context-aware attention layers coupled with optimal transport domain adaptation and multimodal fusion methods for recognizing dementia from spontaneous speech" (Ilias et al., 2023)
- "Evaluating Recent Legal Rhetorical Role Labeling Approaches Supported by Transformer Encoders" (Gomes de Lima et al., 2023)
- "A Noisy Elephant in the Room: Is Your Out-of-Distribution Detector Robust to Label Noise?" (Humblot-Renaux et al., 2024)
- "Non-Fluent Synthetic Target-Language Data Improve Neural Machine Translation" (Sánchez-Cartagena et al., 2024)
- "Bootstrapping LLM-based Task-oriented Dialogue Agents via Self-talk" (Ulmer et al., 2024a)
- "Non-Exchangeable Conformal Language Generation with Nearest Neighbors" (Ulmer et al., 2024b)
- "Calibrating Large Language Models Using Their Generations Only" (Ulmer at al., 2024c)
- "More Room for Language: Investigating the Effect of Retrieval on Language Models" (Samuel et al., 2024)
- "On retrospective k-space subsampling schemes for deep MRI reconstruction" (Yiasemis et al., 2024)
- "Enhancing Turkish Coreference Resolution: Insights from deep learning, dropped pronouns, and multilingual transfer learning" (Arslan et al., 2024)
- "A multivariable sensor-agnostic framework for spatio-temporal air quality forecasting based on Deep Learning" (Prado-Rujas et al., 2024)
- "Putting Context in Context: the Impact of Discussion Structure on Text Classification" (Penzo et al., 2024)
- "Transferring Sentiment Cross-Lingually within and across Same-Family Languages" (Thakar et al., 2024)
- "Exploring Generalizable Distillation for Efficient Medical Image Segmentation" (Qi et al., 2024)
- "Explainable epidemiological thematic features for event based disease surveillance" (Menya et al., 2024)
- "Time Series Classification for Detecting Parkinson’s Disease from Wrist Motions" (Donie et al., 2024)
- "What Can You Do with a Sword? Gender Biases in Text Game Affordances" (McGuire & Tomuro, 2024)
- "Transformer-Encoders for Mathematical Answer Retrieval" (Reusch, 2024)
Del Barrio, Eustasio, Juan A. Cuesta-Albertos, and Carlos Matrán. "An optimal transportation approach for assessing almost stochastic order." The Mathematics of the Uncertain. Springer, Cham, 2018. 33-44.
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Borji, Ali. "Negative results in computer vision: A perspective." Image and Vision Computing 69 (2018): 1-8.
Bouthillier, Xavier, et al. "Accounting for variance in machine learning benchmarks." Proceedings of Machine Learning and Systems 3 (2021).
Dror, Rotem, et al. "The hitchhiker’s guide to testing statistical significance in natural language processing." Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2018.
Dror, Rotem, Shlomov, Segev, and Reichart, Roi. "Deep dominance-how to properly compare deep neural models." Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. 2019.
Efron, Bradley, and Robert J. Tibshirani. "An introduction to the bootstrap." CRC press, 1994.
Andrew Gelman, John B Carlin, Hal S Stern, David B Dunson, Aki Vehtari, Donald B Rubin, John Carlin, Hal Stern, Donald Rubin, and David Dunson. Bayesian data analysis third edition, 2021.
Henderson, Peter, et al. "Deep reinforcement learning that matters." Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 32. No. 1. 2018.
Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, Tom Goldstein. "Visualizing the Loss Landscape of Neural Nets." NeurIPS 2018: 6391-6401
Narang, Sharan, et al. "Do Transformer Modifications Transfer Across Implementations and Applications?." arXiv preprint arXiv:2102.11972 (2021).
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