## deep-significance

Easy and Better Significance Testing for Deep Neural Networks

### Why?

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 (Dror et al., 2019), bootstrap (Efron & Tibshirani, 1994) and
permutation-randomization (Noreen, 1989).
• Bonferroni correction methods for multiplicity in datasets (Bonferroni, 1936).

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 or the scenarios in the section Examples.

## Installation

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.

## Examples

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.

:warning: Testing models with only one set of hyperparameters and only one test set will be able to guarantee superiority
in all settings. See General Recommendations & other notes.

In the following, I 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.

In general, in statistical significance testing, we usually compare two algorithms and on a dataset using
some evaluation metric (we assume a higher = better). The difference between the two algorithms on the
data is then defined as

where is our test statistic. We then test the following null hypothesis:

Thus, we assume our algorithm A to be equally as good or worse than algorithm B and reject the null hypothesis if A
is better than B (what we actually would like to see). Most statistical significance tests operate using
p-values, which define the probability that under the null-hypothesis, the expected by the test is larger than or
equal to the observed difference (that is, for a one-sided test, i.e. we assume A to be better than B):

We can interpret this equation as follows: Assuming that A is not better than B, the test assumes a corresponding distribution
of differences that is drawn from. How does our actually observed difference fit in there?
This is what the p-value is expressing: If this probability is high, is in line with what we expected under
the null hypothesis, so we conclude A not to better than B. If the
probability is low, that means that 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 A is indeed better than B. Note that the p-value does not
express whether the null hypothesis is true
.

To decide when we trust A to be better than B, we set a threshold that will determine when the p-value is small enough
for us to reject the null hypothesis, this is called the significance level and it is often set to be 0.05.

### Intermezzo: Almost Stochastic Order - a better significance test for Deep Neural Networks

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, 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, 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 the amount of violation of stochastic order. If
, 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.
Furthermore, the significance level is determined as an input argument when running ASO and actively influence
the resulting .

### Scenario 1 - Comparing multiple runs of two models

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

# 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)  # min_eps = 0.0, 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.

### Scenario 2 - Comparing multiple runs across datasets

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

# 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=0.05 / M) for a, b in zip(my_model_scores_per_dataset, baseline_scores_per_dataset)]
# eps_min = [0.1565800030782686, 1, 0.0]


### Scenario 3 - Comparing sample-level scores

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

# 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=0.05 / len(pairs)) for a, b in pairs]


### Scenario 4 - Comparing more than two models

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 diagonal will naturally always be 1, but we can also restrict
ourself to only filling out one half of the matrix by making use of the following property of ASO:

Note: While an appealing shortcut, it has been observed during testing this property, due to the random element
of bootstrap iterations, might not always hold exactly - the difference between the two quantities has been seen to
amount to up to * when the scores distributions of A and B are very similar.

*This is just an empirically observed value, not a tight bound.

The corresponding code can then look something like this:

import numpy as np
from deepsig import aso

N = 5  # Number of random seeds
M = 3  # Number of different models / algorithms
num_comparisons = M * (M - 1) / 2
eps_min = np.eye(M)  # M x M matrix with ones on diagonal

# 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.random.normal(loc=loc, scale=0.8, size=N) for loc in np.arange(0.1, 0.1 + 0.05 * M, step=0.05)]

for i in range(M):
for j in range(i + 1, M):

e_min = aso(my_models_scores[i], my_models_scores[j], confidence_level=0.05 / num_comparisons)
eps_min[i, j] = e_min
eps_min[j, i] = 1 - e_min

# eps_min =
# array([[1., 1., 1.],
#        [0., 1., 1.],
#        [0., 0., 1.]])


### :newspaper: How to report results

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 ($\epsilon_\text{min} < 0.5)$ is indicated in table X.


### :sparkles: Other features

#### :rocket: For the impatient: ASO with multi-threading

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=5)
b = np.random.normal(size=5)

print(timeit(lambda: aso(a, b, num_jobs=1, show_progress=False), number=5))  # 146.6909574989986
print(timeit(lambda: aso(a, b, num_jobs=4, show_progress=False), number=5))  # 50.416724971000804


#### :electric_plug: Compatibility with PyTorch, Tensorflow, Jax & Numpy

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!


#### :game_die: Permutation and bootstrap test

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


### General recommendations & other notes

• Naturally, the CDFs built from scores_a and scores_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_samples and num_bootstrap_iterations can be reduced to increase the speed of aso(). 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.

• 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.

### :mortar_board: Cite

If you use the ASO test via aso(), please cite the original work:

@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},
}


Using this package in general, please cite the following:

@software{dennis_ulmer_2021_4638709,
author       = {Dennis Ulmer},
title        = {{deep-significance: Easy and Better Significance
Testing for Deep Neural Networks}},
month        = mar,
year         = 2021,
note         = {https://github.com/Kaleidophon/deep-significance},
publisher    = {Zenodo},
version      = {v1.0.0a},
doi          = {10.5281/zenodo.4638709},
url          = {https://doi.org/10.5281/zenodo.4638709}
}


### :medal_sports: Acknowledgements

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.

### :books: Bibliography

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.

Bonferroni, Carlo. "Teoria statistica delle classi e calcolo delle probabilita." Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commericiali di Firenze 8 (1936): 3-62.

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.

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).

Noreen, Eric W. "Computer intensive methods for hypothesis testing: An introduction." Wiley, New York (1989).

## GitHub

https://github.com/Kaleidophon/deep-significance## deep-significance

Easy and Better Significance Testing for Deep Neural Networks

### Why?

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 (Dror et al., 2019), bootstrap (Efron & Tibshirani, 1994) and
permutation-randomization (Noreen, 1989).
• Bonferroni correction methods for multiplicity in datasets (Bonferroni, 1936).

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 or the scenarios in the section Examples.

## Installation

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.

## Examples

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.

:warning: Testing models with only one set of hyperparameters and only one test set will be able to guarantee superiority
in all settings. See General Recommendations & other notes.

In the following, I 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.

In general, in statistical significance testing, we usually compare two algorithms and on a dataset using
some evaluation metric (we assume a higher = better). The difference between the two algorithms on the
data is then defined as

where is our test statistic. We then test the following null hypothesis:

Thus, we assume our algorithm A to be equally as good or worse than algorithm B and reject the null hypothesis if A
is better than B (what we actually would like to see). Most statistical significance tests operate using
p-values, which define the probability that under the null-hypothesis, the expected by the test is larger than or
equal to the observed difference (that is, for a one-sided test, i.e. we assume A to be better than B):

We can interpret this equation as follows: Assuming that A is not better than B, the test assumes a corresponding distribution
of differences that is drawn from. How does our actually observed difference fit in there?
This is what the p-value is expressing: If this probability is high, is in line with what we expected under
the null hypothesis, so we conclude A not to better than B. If the
probability is low, that means that 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 A is indeed better than B. Note that the p-value does not
express whether the null hypothesis is true
.

To decide when we trust A to be better than B, we set a threshold that will determine when the p-value is small enough
for us to reject the null hypothesis, this is called the significance level and it is often set to be 0.05.

### Intermezzo: Almost Stochastic Order - a better significance test for Deep Neural Networks

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, 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, 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 the amount of violation of stochastic order. If
, 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.
Furthermore, the significance level is determined as an input argument when running ASO and actively influence
the resulting .

### Scenario 1 - Comparing multiple runs of two models

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

# 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)  # min_eps = 0.0, 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.

### Scenario 2 - Comparing multiple runs across datasets

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

# 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=0.05 / M) for a, b in zip(my_model_scores_per_dataset, baseline_scores_per_dataset)]
# eps_min = [0.1565800030782686, 1, 0.0]


### Scenario 3 - Comparing sample-level scores

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

# 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=0.05 / len(pairs)) for a, b in pairs]


### Scenario 4 - Comparing more than two models

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 diagonal will naturally always be 1, but we can also restrict
ourself to only filling out one half of the matrix by making use of the following property of ASO:

Note: While an appealing shortcut, it has been observed during testing this property, due to the random element
of bootstrap iterations, might not always hold exactly - the difference between the two quantities has been seen to
amount to up to * when the scores distributions of A and B are very similar.

*This is just an empirically observed value, not a tight bound.

The corresponding code can then look something like this:

import numpy as np
from deepsig import aso

N = 5  # Number of random seeds
M = 3  # Number of different models / algorithms
num_comparisons = M * (M - 1) / 2
eps_min = np.eye(M)  # M x M matrix with ones on diagonal

# 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.random.normal(loc=loc, scale=0.8, size=N) for loc in np.arange(0.1, 0.1 + 0.05 * M, step=0.05)]

for i in range(M):
for j in range(i + 1, M):

e_min = aso(my_models_scores[i], my_models_scores[j], confidence_level=0.05 / num_comparisons)
eps_min[i, j] = e_min
eps_min[j, i] = 1 - e_min

# eps_min =
# array([[1., 1., 1.],
#        [0., 1., 1.],
#        [0., 0., 1.]])


### :newspaper: How to report results

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 ($\epsilon_\text{min} < 0.5)$ is indicated in table X.


### :sparkles: Other features

#### :rocket: For the impatient: ASO with multi-threading

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=5)
b = np.random.normal(size=5)

print(timeit(lambda: aso(a, b, num_jobs=1, show_progress=False), number=5))  # 146.6909574989986
print(timeit(lambda: aso(a, b, num_jobs=4, show_progress=False), number=5))  # 50.416724971000804


#### :electric_plug: Compatibility with PyTorch, Tensorflow, Jax & Numpy

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!


#### :game_die: Permutation and bootstrap test

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


### General recommendations & other notes

• Naturally, the CDFs built from scores_a and scores_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_samples and num_bootstrap_iterations can be reduced to increase the speed of aso(). 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.

• 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.

### :mortar_board: Cite

If you use the ASO test via aso(), please cite the original work:

@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},
}


Using this package in general, please cite the following:

@software{dennis_ulmer_2021_4638709,
author       = {Dennis Ulmer},
title        = {{deep-significance: Easy and Better Significance
Testing for Deep Neural Networks}},
month        = mar,
year         = 2021,
note         = {https://github.com/Kaleidophon/deep-significance},
publisher    = {Zenodo},
version      = {v1.0.0a},
doi          = {10.5281/zenodo.4638709},
url          = {https://doi.org/10.5281/zenodo.4638709}
}


### :medal_sports: Acknowledgements

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.

### :books: Bibliography

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.

Bonferroni, Carlo. "Teoria statistica delle classi e calcolo delle probabilita." Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commericiali di Firenze 8 (1936): 3-62.

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.

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).

Noreen, Eric W. "Computer intensive methods for hypothesis testing: An introduction." Wiley, New York (1989).

## GitHub

https://github.com/Kaleidophon/deep-significance