# FIESTA: Fast IdEntification of State-of-The-Art models using adaptive bandit
algorithms

Henry B. Moss
STOR-i Centre for
Doctoral Training,
Lancaster University
&Andrew Moore
School of Computing
and Communications,
Lancaster University
<EMAIL_ADDRESS>
&David S. Leslie
Dept. of Mathematics
and Statistics,
Lancaster University
&Paul Rayson
School of Computing
and Communications,
Lancaster University

###### Abstract

We present FIESTA, a model selection approach that significantly reduces the
computational resources required to reliably identify state-of-the-art
performance from large collections of candidate models. Despite being known to
produce unreliable comparisons, it is still common practice to compare model
evaluations based on single choices of random seeds. We show that reliable
model selection also requires evaluations based on multiple train-test splits
(contrary to common practice in many shared tasks). Using bandit theory from
the statistics literature, we are able to adaptively determine appropriate
numbers of data splits and random seeds used to evaluate each model, focusing
computational resources on the evaluation of promising models whilst avoiding
wasting evaluations on models with lower performance. Furthermore, our user-
friendly Python implementation produces confidence guarantees of correctly
selecting the optimal model. We evaluate our algorithms by selecting between
$8$ target-dependent sentiment analysis methods using dramatically fewer model
evaluations than current model selection approaches.

##  1 Introduction and Background

Natural Language Processing (NLP) is a field driven by empirical evaluations.
Authors are under pressure to demonstrate that their models or methods achieve
state-of-the-art performance on a particular task or dataset, which by
definition requires reliable model comparison. As models become more numerous,
require larger computational resources to train, and the performance of
competing models gets closer, the task of reliable model selection has not
only become more important, but also increasingly difficult. Without full
disclosure of model settings and data splits, it is impossible to accurately
compare methods and models.

To be able to perform meaningful model comparisons, we need to be able to
reliably evaluate models. Unfortunately, evaluating a model is a non-trivial
task and the best we can do is to produce noisy estimates of model performance
with the following two distinct sources of stochasticity:

  1. 1.

We only have access to a finite training dataset, however, evaluating a model
on its training data leads to severe over-estimates of performance. To
evaluate models without over-fitting, practitioners typically randomly
partitioning data into independent training and testing sets, producing
estimates that are random quantities with often high variability for NLP
problems Moss et al. (2018). Although methods like bootstrapping Efron and
Tibshirani (1994) and leave-one-out cross validation Kohavi (1995) can provide
deterministic estimates of performance, they require the fitting of a large
number of models and so are not computationally feasible for the complex
models and large data prevalent in NLP. Standard NLP model evaluation
strategies range from using a simple (and computationally cheap) single train-
test split, to the more sophisticated $K$-fold cross validation, CV Kohavi
(1995).

  2. 2.

The vast majority of recent NLP models are non-deterministic and so their
performance has another source of stochasticity, controlled by the choice of
random seed during training. Common sources of model instability in modern NLP
include weight initialisation, data sub-sampling for stochastic gradient
calculation, negative sampling used to train word embeddings Mikolov et al.
(2013) and feature sub-sampling for ensemble methods. In particular, the often
state-of-the-art LSTMs (and its many variants) have been shown to exhibit high
sensitivity to random seeds Reimers and Gurevych (2017).

For reliable model selection, it is crucial to take into account both sources
of variability when estimating model performance. Observing a higher score for
one model could be a consequence of a particularly non-representative train-
test split and/or random seed used to evaluate the model rather than a genuine
model improvement. This subtlety is ignored by large scale NLP competitions
such as SemEval  with evaluations based on a pre-determined train-test split.

Although more precise model evaluations can be obtained with higher
computation, calculating overly precise model evaluations is a huge waste of
computational resource. On the other hand, our evaluations need to provide
reliable conclusions (with only a small probability of selecting a sub-optimal
model). It is poorly understood how to choose an appropriate evaluation
strategy for a given model selection problem. These are task specific,
depending on model stability, the closeness in performance of competing models
and subtle properties of the data such as the representativeness of train-test
splits.

In contrast to common practice, we consider model selection as a sequential
process. Rather than using a fixed evaluation strategy for each model (which
we refer to as a non-adaptive approach), we start with a cheap evaluation of
each model on just a single train-test split, and then cleverly choose where
to allocate further computational resources based on the observed evaluations.
If we decide to further test a promising model, we calculate an additional
evaluation based on another data split and seed, observing both sources of
evaluation variability and allowing reliable assessments of performance.

To perform sequential model fitting, we borrow methods from the multi-armed-
bandit (MAB) statistical literature Lai and Robbins (1985). This field covers
problems motivated by designing optimal strategies for pulling the arms of a
bandit (also known as a slot machine) in casinos. Each arm produces rewards
from different random distributions which the user must learn by pulling arms.
In particular, model selection is equivalent to the problem of best-arm-
identification; identifying the arm with the highest mean. Although appearing
simple at a first glance, this problem is deceptively complex and has provided
motivation for efficient algorithms in a wide range of domains, including
clinical trials Villar et al. (2015) and recommendation systems Li et al.
(2010).

Although we believe that we are the first to use bandits to reduce the cost
and improve the reliability of model selection, we are not the first to use
them in NLP. Recent work in machine translation makes use of another major
part of the MAB literature, seeking to optimise the long-term performance of
translation algorithms Nguyen et al. (2017); Sokolov et al. (2016); Lawrence
et al. (2017). Within NLP, our work is most similar to Haffari et al. (2017),
who use bandits to minimise the number of data queries required to calculate
the F-scores of models. However, this work does not consider the stochasticity
of the resulting estimates or easily extend to other evaluation metrics.

The main contribution of this paper is the application of three intuitive
algorithms to model selection in NLP, alongside a user-friendly Python
implementation: FIESTA (Fast IdEntification of State-of-The-
Art)111https://github.com/apmoore1/fiesta. We can automatically identify an
optimal model from large collections of candidate models to a user-chosen
confidence level in a small number of model evaluations. We focus on three
distinct scenarios that are of interest to the NLP community. Firstly, we
consider the fixed budget (FB) model selection problem (Section 4.1), a
situation common in industry, where a fixed quota of computational resources
(or time constraints for real-time decisions) must be appropriately allocated
to identify an optimal model with the highest possible confidence. In
contrast, we also consider the fixed confidence (FC) problem (Section 4.2),
which we expect to be of more use for researchers. Here, we wish to claim with
a specified confidence level that our selected model is state-of-the-art
against a collection of competing models using the minimal amount of
computation. Finally, we also consider an extension to the FC scenario, where
a practitioner has the computational capacity to fit multiple models in
parallel. We demonstrate the effectiveness of our procedures over current
model selection approaches when identifying an optimal target-dependent
sentiment analysis model from a set of eight competing candidate models
(Section 5).

##  2 Motivating example

We now provide evidence for the need to vary both data splits and random seeds
for reliable model selection. We extend the motivating example used in the
work of Reimers and Gurevych (2017), comparing two LSTM-based Named Entity
Recognition (NER) models by Ma and Hovy (2016) and Lample et al. (2016),
differing only in character representation (via a CNN and a LSTM
respectively). We base model training on Ma and Hovy (2016), however,
following the settings of Yang et al. (2018) we use a batch size of 64, a
weight decay of $10e^{-9}$ and removed momentum. We ran each of the NER models
five times with a different random seed on 150 different train, validation,
and test splits222The original CoNLL data was split with respect to time
rather than random sub-sampling, explaining the discrepancy with previous
scores on this dataset using the same models.. Reimers and Gurevych (2017)
showed the effect of model instability between these two models, where
changing the model’s random seeds can lead to drawing different conclusions
about which model performed best. We extend this argument by showing that
different conclusions can also be drawn if we instead vary the train-test
split used for the model evaluation (Figure 1). We see that while data splits
0 and 2 correctly suggest that the LSTM is optimal, using data split 1
suggests the opposite. Therefore, it is clear that we must vary both the
random seeds and train-test splits used to evaluate our models if we want
reliable model selection.

Figure 1: The left plot shows the distribution of results when varying the
data splits and random seeds, with the dashed lines representing the quartile
values. The three right plots each represent a different single data split
over five runs on different random seeds. The lines represent a single run
result.

##  3 Problem Statement

Extending notation from Arcuri and Briand (2014), we can precisely state the
task of selecting between a collection of $N$ candidate models
$S=\\{m_{1},m_{2},..m_{N}\\}$ as finding

$m^{*}=\operatorname*{argmax}_{m\in S}\mathcal{M}(m).$ (1)

$m^{*}$ is the best model according to some chosen evaluation metric
$\mathcal{M}$ that measures the performance of that model, e.g accuracy,
F-score or AUC (for an summary of model evaluation metrics see Friedman et al.
(2001)).

As already argued, Equation (1) paints an overly simplistic picture of model
selection. In reality we only have access to noisy realisations of the true
model score $\mathcal{M}(m)$ and direct comparisons of single realisations of
random variables are unreliable. Therefore, we follow the arguments of Reimers
and Gurevych (2018) and consider a meaningful way of comparing noisy model
evaluations: namely, finding the model with largest expected performance
estimate across different train-test splits and random seeds. Defining the
mean performance of model $m$ as $\mu_{m}$, we see that the task of model
selection is equivalent to the accurate learning and comparison of these $N$
unknown means:

$m^{*}=\operatorname*{argmax}_{m\in S}\mu_{m}.$

We can now set up the sequential framework of our model selection procedure
and precisely state what we mean by reliable model selection. At each step in
our algorithm we choose a model to evaluate and sample a performance estimate
by randomly generating a data split and random seed. After collecting
evaluations, we can calculate sample means for each model, which we denote as
$\hat{\mu}_{m}$. After running our algorithm for $T$ steps, reliable model
selection corresponds to knowing how confident we should be that our chosen
model $\hat{m}_{T}=\operatorname*{argmax}\hat{\mu}_{m}$ is in fact the true
optimal model $m^{*}$, i.e. we wish to make a precise statement of the form;

$\displaystyle\mathds{P}\left(\hat{m}_{T}=m^{*}\right)\geq 1-\delta,$ (2)

where $1-\delta$ represents this confidence.

In Section 1 we motivated two distinct goals of a sequential model selection
routine, which we can now state as:

  1. 1.

Fixed budget model selection (FB): We wish to find the best model using only a
fixed budget of $T$ model evaluations. The aim is to collect the $T$
evaluations that allow us to claim (2) with the largest possible confidence
level $1-\delta$.

  2. 2.

Fixed confidence model selection (FC): We wish to find the best model to a
pre-specified confidence level. The aim is to collect the minimal number of
model evaluations that allow us to claim (2).

Although an algorithm designed to do well in one of these scenarios will
likely also do well in the other, we will see that to achieve the best
performance at either FB or FC model selection, we require subtly different
algorithms.

##  4 Algorithms

We now examine model selection from a bandit viewpoint, summarising three
bandit algorithms and relating their use to three distinct model selection
scenarios. Although the underpinning theoretical arguments for these
algorithms are beyond the scope of this work, we do highlight one point that
is relevant for model selection; that scenarios enjoying the largest
efficiency gains from moving to adaptive algorithms are those where only a
subset of arms have performance close to optimal Jamieson et al. (2013). Model
selection in NLP is often in this scenario, with only a small number of
considered models being close to state-of-the-art, and so (as we demonstrate
in Section 5) NLP has a lot to gain from using our adaptive model selection
algorithms.

###  4.1 Fixed Budget by Sequential Halving

Algorithm 1 Sequential Halving for Fixed Budget Model Selection

0: Computational Budget $T$, Set of $N$ candidate models $S$

while $|S|\neq 1$ do

Evaluate each model $m$ in $S$
$\Big{\lfloor}\frac{T}{|S|\lceil\log_{2}N\rceil}\Big{\rfloor}$ times

Update the empirical means $\hat{\mu}_{m}$

Remove $\big{\lfloor}\frac{|S|}{2}\big{\rfloor}$ models with worst
$\hat{\mu}_{m}$ from $S$

end while

return Chosen model $S$

FB best-arm identification algorithms are typically based on successively
eliminating arms until just a single (ideally) optimal arm remains Jamieson et
al. (2013); Jamieson and Nowak (2014); Audibert and Bubeck (2010). We focus on
the sequential halving (SH) algorithm of Karnin et al. (2013) (Algorithm 1).
Here we break our model selection routine into a series of
$\big{\lfloor}log_{2}N\big{\rfloor}$ rounds, each discarding the least
promising half of our candidate model set, eventually resulting in a single
remaining model. Our computational budget $T$ is split equally among the
rounds to be equally budgeted among the models remaining in that round. This
allocation strategy ensures an efficient use of resources, for example the
surviving final two models are evaluated
$2^{\big{\lfloor}log_{2}N\big{\rfloor}}-1$ times as often as the models
eliminated in the first round. An example run of the algorithm is summarised
in Table 1.

Round | Candidate Models | # Evaluations
---|---|---
1 | $S=\\{m_{1},m_{2},m_{3},m_{4}\\}$ | 2
2 | $S=\\{m_{2},m_{4}\\}$ | 4
output: | $S=\\{m_{2}\\}$ |
Table 1: An example of sequential elimination selecting between four models
with a budget of $T=16$. After two evaluations of each model, two models are
eliminated. The remaining budget is then used to reliably decide between the
remaining pair. Standard practice would evaluate each model an equal four
times, wasting computational resources on sub-optimal models.

In the bandit literature Karnin et al. (2013), this algorithm is shown to have
strong theoretical guarantees of reliably choosing the optimal arm, as long as
the reward-distributions for each arm are bounded (limited to some finite
range). This is not a restrictive assumption for NLP, as the majority of
common performance metrics are bounded, for example accuracy, recall,
precision and F-score are all constrained to lie in $\left[0,1\right]$. We
will demonstrate the effectiveness of sequential halving for model selection
in Section 5.

###  4.2 Fixed Confidence by TTTS

For fixed confidence model selection, where we wish to guarantee the selection
of an optimal arm at a given confidence level, we cannot just discard arms
that are likely to be sub-optimal without accurately estimating this
likelihood of sub-optimality. Although approaches that sequentially eliminate
arms (like our sequential halving algorithm) do exist for FC best-arm
identification Jamieson et al. (2014); Karnin et al. (2013); Audibert and
Bubeck (2010); Even-Dar et al. (2002), the best theoretical guarantees for the
FC problem come from algorithms that maintain the ability to sample any arm at
any point in the selection procedure Garivier and Kaufmann (2016); Jamieson
and Nowak (2014). Rather than seeking to eliminate half the considered models
at regular intervals of computation, a model is only evaluated until we can be
sufficiently confident that it is sub-optimal. Unfortunately, the performance
guarantees for these methods are asymptotic results (in the number of arms and
the number of arm pulls) and have little practical relevance to the (at most)
tens of arms in a model selection problem.

Our practical recommendation for FC model selection is a variant of the well-
known Bayesian sampling algorithm, Thompson sampling, known as top-two
Thompson sampling (TTTS) Russo (2016). We will see that this algorithm can
efficiently allocate computational resources to quickly find optimal models.
Furthermore, this approach provides full uncertainty estimation over the final
choice of model, providing the confidence guarantees required for FC model
selection.

Our implementation makes the assumption that the evaluations of each model
roughly follow a Gaussian distribution, with different means and variances.
Although such assumptions are common in the model evaluation literature
Reimers and Gurevych (2018) and for statistical testing in NLP Dror et al.
(2018), they could be problematic for the bounded metrics common in NLP.
Therefore we also experimented with modelling the logit transformation of our
evaluations, mapping our evaluation metric to the whole real line. However,
for our examples of Section 5 we found that this mapping provided a negligible
improvement in reliability and so was not worth including in our experimental
results. This may not be the case for other tasks or less well-behaved
evaluation metrics and so we include this functionality in the FIESTA package.

Algorithm 2 Top-Two Thompson Sampling

0: Desired Confidence $1-\delta$, Set of $N$ candidate models $S$

Initialise a uniform belief $\pi$

Evaluate each model in $S$ three times 333We enforce a minimum of three
evaluations to ensure that the t distribution in our posterior remains well-
defined

Update belief $\pi$

while $\max_{m\in S}\pi_{m}\leq 1-\delta$ do

Sample distinct $m_{1}$ and $m_{2}$ according to $\pi$

Randomly choose between $m_{1}$ and $m_{2}$

Evaluate chosen model

Update belief $\pi$

end while

return Chosen model $\operatorname*{argmax}_{m\in S}\pi_{m}$

To provide efficient model selection, we use our current believed probability
that a given model is optimal $\pi_{m}=\mathds{P}\left(m^{*}=m\right)$
(producing a distribution over the models $\pi=\\{\pi_{1},..,\pi_{N}\\}$) to
drive the allocation of computational resources. Standard Thompson sampling is
a stochastic algorithm that generates a choice of model by sampling from our
current belief $\pi$, i.e. choosing to evaluate a model with the same
probability that we believe is optimal (see Russo et al. (2018) for a concise
introduction). Although this strategy allows us to focus computation on
promising arms, it actually does so too aggressively. Once we believe that an
arm is optimal with reasonably high confidence, computation will be heavily
focused on evaluating this arm even though we need to become more confident
about the sub-optimality of competing models to improve our confidence level.
This criticism motivates our chosen algorithm TTTS (summarised in Algorithm
2), where instead of sampling a single model according to $\pi$, we sample two
distinct models. We then uniformly choose between these two models for the
next evaluation, allowing a greater exploration of the arms and much improved
rates of convergence to the desired confidence level Russo (2016). We use this
new evaluation to update our belief and continue making evaluations until we
believe that a model is optimal with a higher probability than $1-\delta$ and
terminate the algorithm. An example run of TTTS is demonstrated on a synthetic
example in Figure 2, where we simulate from $5$ Gaussian distributions with
means $\\{0.65,0.69,0.69,0.70,0.71\\}$ and standard deviation $0.01$ to mimic
accuracy measurements for a model selection problem.

Figure 2: TTTS seeking the optimal model with confidence $0.99$ from $5$
synthetic models. The background represents our evolving belief $\pi$ in the
optimal model and the lines represent the proportion of the total evaluations
made on each model. We start evaluating the models uniformly but our adaptive
algorithm quickly focuses resources on the best models.

We now explain how we calculate $\pi$ (our belief in the location of the
optimal model) using well-known results from Bayesian decision theory (see
Berger (2013) for a comprehensive coverage). As justified earlier, we assume
that the evaluations of model $m$ are independently distributed with a
Gaussian distribution $\mathcal{N}(\mu_{m},\sigma_{m}^{2})$ for some unknown
mean $\mu_{m}$ and variance $\sigma_{m}^{2}$. Although we are primarily
interested in learning $\mu_{m}$, we must also learn $\sigma_{m}^{2}$ in order
to make confidence guarantees about the optimality of our selected model.
Therefore, as well as keeping track of the sample means for the evaluations of
each model $\hat{\mu}_{m}$, we also keep track of the sample variances
$\hat{S}_{m}$ and counters $T_{m}$ of the number of times each model has been
evaluated. To facilitate inference, we choose a uniform prior for the unknown
$\mu_{m}$ and $\sigma_{m}$. Not only is this a conjugate prior for Gaussian
likelihoods, but it is also shown to encourage beneficial exploratory
behaviour when using Thompson sampling on Gaussian bandit problems Honda and
Takemura (2014) and so allows fast identification of optimal arms (or models).
After observing $T_{m}$ evaluations of each model and producing estimates
$\hat{\mu}_{m}$ and $\hat{S}_{m}$, our posterior belief for each deviation
between the true and observed model means $\mu_{m}-\hat{\mu}_{m}$ satisfies
(as derived in Honda and Takemura (2014));

$\sqrt{\frac{T_{m}(T_{m}-2)}{\hat{S}_{m}}}\left(\mu_{m}-\hat{\mu}_{m}\right)|\,\hat{\mu}_{m},\hat{S}_{m}\sim
t_{T_{m}-2},$

where $t_{d}$ is a Student’s t-distribution with $d$ degrees of freedom.

$\pi$ is then defined as the probability vector, such that $\pi_{m}$ is the
relative probability that $\mu_{m}$ is the largest according to this posterior
belief. Unfortunately, there is no closed form expression for the maximum of
$N$ t-distributions and so FIESTA uses a simple Monte-Carlo approximation
based on the sample maxima of repeated draws from our posteriors for
$\mu_{m}$. In practice this is very accurate and did not slow down our
experiments, especially in comparison to the time saved by reducing the number
of model evaluations.

###  4.3 Batch Fixed Confidence by BTS

NLP practitioners often have the computational capacity to fit models in
parallel across multiple workers, evaluating multiple models or the same model
across multiple seeds at once. Their model selection routines must therefore
provide batches of models to evaluate. Our proposed solution to FB model
selection naturally provides such batches, with each successive round of SH
producing a collection of model evaluations that can be calculated in
parallel. Unfortunately, TTTS for FC model selection successively chooses and
then waits for the evaluation of single models and so is not naturally suited
to parallelism.

Extending TTTS to batch decision making is an open problem in the MAB
literature. Therefore, we instead consider batch Thompson sampling (BTS), an
extension of standard Thompson sampling (as described in Section 4.2) to batch
sampling from the related field of Bayesian optimisation Kandasamy et al.
(2018). At each step in our selection process we take $B$ model draws
according to our current belief $\pi$ that the model is optimal, where $B$
represents our computational capacity. This is in contrast to the single draw
in standard Thompson sampling and the drawn pair in TTTS. In addition, this
approach extends to the asynchronous setting, where rather than waiting for
the whole batch of $B$ models to be evaluated before choosing the next batch,
each worker can draw a new model to evaluate according to the updated $\pi$.
This flexibility provides an additional efficiency gain for problems where the
different models have a wide range of run times.

##  5 Experiments

We now test our three algorithms on a challenging model selection task typical
of NLP, selecting between eight Target Dependent Sentiment Analysis (TDSA)
models based on their macro F1 score. We consider two variants of four re-
implementations of well-known TDSA models: ATAE Wang et al. (2016), IAN Ma et
al. (2017), TDLSTM Tang et al. (2016) (without target words in the left and
right LSTM), and a non-target-aware LSTM method used as the baseline in Tang
et al. (2016).

These methods represent state-of-the-art within TDSA, with only small
differences in performance between TDLSTM, IAN, and ATAE (see figure 3). All
the models are re-implemented in PyTorch Paszke et al. (2017) using AllenNLP
Gardner et al. (2018). To ensure the only difference between the models is
their network architecture the models use the same optimiser settings and the
same regularisation. All words are lower cased and we use the same Glove
common crawl 840B token 300 dimension word embedding Pennington et al. (2014).
We use variational Gal and Ghahramani (2016) and regular Hinton et al. (2012)
dropout for regularisation and an ADAM Kingma and Ba (2014) optimiser with
standard settings, a batch size of $32$ and use at most $100$ epochs (with
early stopping on a validation set). Many of these settings are not the same
as originally implemented, however, having the same training setup is required
for fair comparison (this explains the differences between our results and the
original implementations). To increase the difficulty of our model selection
problem, we additionally create four extra models by reducing the dimensions
of the Glove vectors to 50 and removing dropout. Although these models are
clearly not state-of-the-art, they increase the size of our candidate model
set and so provide a more complicated model selection problem (an intuition
discussed in Appendix A).

All of the TDSA experiments are conducted on the well-studied SemEval 2014
task 4 Restaurant dataset Pontiki et al. (2014) and we force train-val-test
splits to follow the same ratios as this dataset’s official train-test split.
Each individual model evaluation is then made on a randomly generated train-
test split and random seed to access both sources of evaluation variability.

Figure 3: F1 scores for our candidate TDSA models. After $500$ evaluations of
each model on different data splits and model seeds we see that the TDLSTM is
the state-of-the-art model.

###  5.1 Fixed Budget Model Selection

We use the TDSA model selection problem to test fixed budget model selection.
To thoroughly test our algorithm, we consider an additional four models based
on 200 dimensional Glove vectors, bringing the total number of models to 12.
We compare our approach of sequential halving to the standard non-adaptive
approach of splitting the available computational budget equally between the
12 candidate models. For example, we would allocate a budget of $24$ model
evaluations as evaluating each model two times and selecting the model with
the highest sample mean.

Figure 4 compares the proportion of $10,000$ runs of sequential halving that
correctly identify the optimal model with the proportion identified by the
non-adaptive approach with the same computational budget. Sequential halving
identifies the optimal model more reliably ($\approx 15\%$ more often) than
the current approach to FB model selection in NLP. Using sequential halving
with $204$ evaluations almost always ($99\%$ of runs) selects the optimal
model, whereas the non-adaptive approach is only correct $85\%$ of the time.

Figure 4: Proportion of the runs correctly selecting the optimal TDSA model using sequential halving against the standard non-adaptive approach. Sequential halving consistently identifies the optimal model at a significantly higher rate across a wide range of budgets. | # | evaluations | with | Non-Adaptive | # | evaluations | with | TTTS
---|---|---|---|---|---|---|---|---
$\delta$ | min | mean | max |  |  $\%$ correctly
---
selected
min | mean | max |  |  $\%$ correctly
---
selected
0.05 | 48 | 281 | 1552 | 100 | 27 | 130 | 518 | 100
0.1 | 40 | 206 | 1192 | 99 | 24 | 96 | 460 | 99
0.2 | 32 | 128 | 608 | 96 | 24 | 65 | 274 | 97
Table 2: Number of evaluations required to select a TDSA model at a range of
confidence levels across $500$ runs of TTTS and a standard non-adaptive
approach.

###  5.2 Fixed Confidence Model Selection

We perform fixed confidence model selection on the eight TDSA candidate models
(the full models and those based on 50 dimensional vectors). We compare TTTS
to a non-adaptive approach where all models are evaluated at each step,
irrespective of the results of earlier evaluations (the standard approach for
model selection in NLP). We run this non-adaptive approach until we reach the
required confidence level calculated using the same Bayesian framework as in
TTTS.

We run each approach $500$ times and note the number evaluations required to
get to a range of confidence levels (Table 2) alongside the proportion that
correctly identify the optimal model. TTTS requires substantially less model
evaluations (in terms of the minimum, mean and max across our runs) to reach a
given confidence level than the non-adaptive approach, achieving the same
reliability at half the cost (on average). TTTS is able to quickly identify
sub-optimal models and so can avoid wasting resources repeatedly evaluating
the whole candidate set.

| # | evaluations | with | BTS-4 | # | evaluations | with | BTS-8
---|---|---|---|---|---|---|---|---
$\delta$ | min | mean | max |  |  $\%$ correctly
---
selected
min | mean | max |  |  $\%$ correctly
---
selected
0.05 | 28 | 282 | 1392 | 100 | 88 | 315 | 1128 | 100
0.1 | 24 | 144 | 520 | 100 | 56 | 178 | 784 | 100
0.2 | 24 | 76 | 280 | 98 | 32 | 106 | 352 | 99
Table 3: Number of evaluations of required to select a TDSA model at a range
of confidence levels across $500$ runs of BTS selecting batches of 4 and 8
models.

Finally, we test our proposed approach to batch FC model selection by running
exactly the same experiment but using BTS to choose collections of four and
eight models at a time (Table 3). As expected, performance degrades as we
increase batch size, with batches of four allowing more fine grained control
over model evaluations than using batches of eight. In particular, due to the
exploitative nature of Thompson sampling, we see that selecting models to a
very high confidence (95%) requires more computation with BTS than the
standard non-adaptive approach. However, BTS does reach the other confidence
levels faster and correctly identifies the optimal model more often. However,
as TTTS performs significantly better across all confidence levels, we
emphasise the need for a less-exploitative version of BTS with adjustments
similar to those used in TTTS.

##  6 Conclusions

The aim of this paper has been to propose three algorithms for model selection
in NLP, providing efficient and reliable selection for two distinct realistic
scenarios: fixed confidence and fixed budget model selection. Crucially, our
research further calls into question the current practice in NLP evaluation as
used in the literature and international competitions such as SemEval. Our
algorithms adaptively allocate resources to evaluate promising models, basing
evaluations across multiple random seeds and train-test splits. We demonstrate
that this allows significant computational savings and improves reliability
over current model selection approaches.

Although we have demonstrated that our algorithms perform well on a complex
model selection problem typical of NLP, there is still work to be done to
create algorithms more suited to these problems. Future research directions
include making selection routines more robust to evaluation outliers, relaxing
our Gaussian assumptions and developing more effective batch strategies.

##  7 Acknowledgements

The authors are grateful to reviewers, whose comments and advice have greatly
improved this paper. The research was supported by an EPSRC Doctoral Training
Grant and the STOR-i Centre for Doctoral Training. We thank Dr Chris Jewell at
the Centre for Health Informatics, Computing, and Statistics, Lancaster
University for the loan of a NVIDIA GP100-equipped workstation for this study.

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##  8 Appendix

##  Appendix A Characterising the Difficulty of a Model Selection Problem

We briefly summarise a result from the best-arm identification literature,
providing intuition for our experiment section through a mechanism to
characterise the difficulty of a model selection problem. Intuitively, model
selection difficulty increases with the size of the set of candidate models
$N$ and as the performance of sub-optimal models approaches that of the
optimal model (and becomes harder to distinguish), i.e. as
$\mu_{m^{*}}-\mu_{m}$ gets small for some sub-optimal arm $m$. In fact, it is
well known in the MAB literature that it is exactly these two properties that
characterise the complexity of a best-arm-identification problem, confirming
our intuition for model selection. Mannor and Tsitsiklis (2004) show that the
number of arm pulls required for the identification of a best arm at a
confidence level $1-\delta$ has at least a computational complexity of
$O(H\log(1/\delta))$, where

$H=\sum_{m^{\prime}\in
S\setminus\\{m*\\}}\frac{1}{\left(\mu_{m^{*}}-\mu_{m}\right)^{2}}.$


# Cosmological Dark Matter from a Bulk Black Hole

Sylvain Fichet<EMAIL_ADDRESS>ICTP South American Institute for
Fundamental Research & IFT-UNESP, R. Dr. Bento Teobaldo Ferraz 271, São Paulo,
Brazil  Centro de Ciencias Naturais e Humanas (CCNH), Universidade Federal do
ABC, Santo Andre, 09210-580 SP, Brazil  Eugenio Megías<EMAIL_ADDRESS>Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de
Física Teórica y Computacional,
Universidad de Granada, Avenida de Fuente Nueva s/n, E-18071 Granada, Spain.
Mariano Quirós<EMAIL_ADDRESS>Institut de Física d’Altes Energies (IFAE) and
The Barcelona Institute of Science and Technology (BIST), Campus UAB, 08193
Bellaterra (Barcelona) Spain

###### Abstract

We study the cosmology of a three-brane in a specific five-dimensional scalar-
gravity (i.e. soft-wall) background, known as the linear dilaton background.
We discover that the Friedmann equation of the brane-world automatically
contains a term mimicking pressureless matter. We propose to identify this
term as dark matter. This dark matter arises as a projection of the bulk black
hole on the brane, which contributes to the brane Friedmann equation via both
the Weyl tensor and the scalar stress tensor. The nontrivial matter-like
behavior is due to an exact cancellation between the Weyl and scalar
pressures. We show that the Newtonian potential only receives a mild short-
distance correction going as inverse distance squared, ensuring compatibility
of the linear dilaton brane-world with observed 4D gravity. Our setup can be
viewed as a consistent cosmological description of the holographic theories
arising in the linear dilaton background. We also present more general scalar-
gravity models where the brane cosmology features an effective energy density
whose behavior smoothly interpolates between dark radiation, dark matter and
dark energy depending on a model parameter.

##  I Introduction

Five-dimensional (5D) gravity coupled to a scalar field has proven to be a
fecund playground, leading to a host of theoretical results and models of the
real world (see e.g. Karch  _et al._ (2006); Gursoy and Kiritsis (2008);
Gursoy  _et al._ (2008); Gubser and Nellore (2008); Falkowski and Perez-
Victoria (2008); Batell and Gherghetta (2008); Batell  _et al._ (2008); Cabrer
_et al._ (2010); von Gersdorff (2010); Cabrer  _et al._ (2011); Megías and
Quirós (2019, 2021a, 2021b)). Our focus in this letter is a specific 5D
scalar-gravity background (i.e. a soft-wall background) which is sometimes
referred to as the “linear dilaton background”. This model is known to have
peculiar thermodynamic Gursoy and Kiritsis (2008) and field theoretical
properties Cabrer  _et al._ (2010); Megías and Quirós (2019, 2021a, 2021b).
For example, all quantum fields living on the linear dilaton background have a
spectral distribution that features a gapped continuum. This feature has been
recently used in extensions of the Standard Model Csáki  _et al._ (2022a, b).

In this work we put the linear dilaton background at finite temperature and
posit a flat 3-brane moving over the background, in the spirit of brane-world
models (see e.g. Brax and van de Bruck (2003)). We discover a surprising
property: from the viewpoint of a brane observer, the local Friedmann equation
automatically contains an effective energy term that may be identified as dark
matter. This dark matter emerges as a nontrivial effect from the bulk physics
projected on the brane. It originates from a combination of the 5D Weyl tensor
and of the bulk scalar vev, as we will demonstrate further below.

To bring this result into context, we remind that there is a notorious analog
in pure Anti-de Sitter (AdS) background, that has been gradually uncovered and
studied in Shiromizu  _et al._ (2000); Binetruy  _et al._ (2000); Hebecker and
March-Russell (2001); Langlois  _et al._ (2002); Langlois and Sorbo (2003). In
pure AdS the net effect of the bulk physics projected on the brane gives rise
to radiation, which is identified as cosmological dark radiation in the
context of a brane-world. This remarkable fact is in direct connection with
the fact that the bulk black hole in AdS-Schwarzschild background corresponds
to the thermal state in the holographic CFT, perhaps one of the most
fascinating entries of the AdS/CFT correspondence Aharony  _et al._ (2000). In
our case, by performing the analogous calculation with the linear dilaton
background, we discover that the bulk black hole gives rise to dark matter.

Decades of astronomical observations point to the existence of dark matter.
Determining its nature is a pressing question in fundamental physics. While a
common hypothesis is that dark matter may be a new particle (that remains so
far elusive), our study leads to a fundamentally different viewpoint. Our
setup provides, in a sense, an origin to cosmological dark matter via a
modification of gravity. See e.g. Clifton  _et al._ (2012) for a few other
attempts to explain dark matter via modified gravity.

In this paper we present the derivation of our central result, the effective
Friedmann equation of the linear dilaton brane-world, that is shown to contain
dark matter. We present nontrivial consistency checks of this result. We also
compute the deviation to the Newtonian potential. We then outline more general
models featuring a variety of equations of state depending on a model
parameter, and discuss some conceptual points and prospects. Extra
developments and technical details are laid out in Fichet  _et al._ (2022),
which can be considered as a companion to this letter.

##  II The 5D scalar-gravity system

We consider the general scalar-gravity action in the presence of a brane,

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{5}x\sqrt{g}\left(\frac{M_{5}^{3}}{2}{}^{(5)}R-\frac{1}{2}(\partial_{M}\phi)^{2}-V(\phi)\right)$
(1) $\displaystyle-$
$\displaystyle\int_{\textrm{brane}}d^{4}x\sqrt{\bar{g}}\,(V_{b}(\phi)+\Lambda_{b})+\ldots\,$

${}^{(5)}R$ is the 5D Ricci scalar, $\phi$ is the scalar field, $M_{5}$ is the
fundamental 5D Planck scale, $\bar{g}_{\mu\nu}$ is the induced metric on the
brane, $g\equiv|\det g_{MN}|$ and $\bar{g}\equiv|\det\bar{g}_{\mu\nu}|$ are
the metrics determinants, $\Lambda_{b}$ is the brane tension, $V$ and $V_{b}$
are the bulk and brane-localized potentials for $\phi$. We assume that the
brane potential sets the scalar field vacuum expectation value (vev) to a
nonzero value $\langle\phi\rangle=v_{b}$, with $V_{b}(v_{b})=0$. The bulk
potential is explicitly given further below. The ellipses encode the Gibbons-
Hawking-York term York (1972); Gibbons and Hawking (1977) and the action of
quantum fields living on the 5D background.

The 5D metric is written in a frame suitable for brane cosmology as

$\displaystyle ds^{2}$
$\displaystyle=g_{MN}dx^{M}dx^{N}\equiv-n^{2}(r)d\tau^{2}+\frac{r^{2}}{\ell^{2}}d\mathbf{x}^{2}+b^{2}(r)dr^{2}\,.$
(2)

We allow the existence of a black hole horizon encoded in the $n(r)$ and
$b(r)$ factors, the position of the horizon being given by
$n(r_{h})=0=1/b(r_{h})$. Latin indices $(M,N,\cdots)$ refer to 5D coordinates,
Greek indices $(\mu,\nu,\cdots)$ refer to 4D coordinates.

The 3-brane is localized at the position $r=r_{b}$. Our frame (2) is
appropriate to describe cosmology as seen from the brane standpoint. The
induced metric on the brane is

$\displaystyle ds^{2}$ $\displaystyle=\bar{g}_{\mu\nu}dx^{\mu}dx^{\nu}\equiv-
dt^{2}+\frac{r_{b}^{2}}{\ell^{2}}d\mathbf{x}^{2}\,,$ (3)

where we have introduced the brane cosmic time $dt=n(r_{b})d\tau$. According
to this metric, if the brane moves along $r$ in the 5D background, the
observer perceives expansion of the four-dimensional (4D) universe with Hubble
parameter $H=\dot{r_{b}}/r_{b}$, where $\dot{r_{b}}\equiv\partial_{t}r_{b}$.
We choose that $r_{b}$ equals $\ell$ at present times, such that
$r_{b}=a(t)\ell$ where $a(t)$ is the standard scale factor. An overview of the
brane-world is shown in Fig. 1.

Figure 1: Overview of the scalar-gravity system. The brane and the horizon are
located respectively at $r=r_{b}$ and $r=r_{h}$. The scalar vacuum expectation
value is fixed by a brane potential to a value $v_{b}$ which remains constant
when $r_{b}$ varies. The scalar VEV (plain) evolves in the bulk, the
blackening factor of the metric (dotted) diverges at the horizon.

The 5D equations of motion of the system are

$\displaystyle{}^{(5)}G_{MN}=\frac{1}{M_{5}^{3}}T^{\phi}_{MN}\,,\quad\frac{1}{\sqrt{g}}\partial_{M}\left(\sqrt{g}g^{MN}\partial_{N}\phi\right)=\frac{\partial
V}{\partial\phi}\,,$ (4)

with ${}^{(5)}G_{MN}={}^{(5)}R_{MN}-\frac{1}{2}g_{MN}{}^{(5)}R$ and
$T^{\phi}_{MN}=\partial_{M}\phi\partial_{N}\phi-
g_{MN}\left[\frac{1}{2}(\partial_{A}\phi)^{2}+V(\phi)\right]$. More
explicitly, the equations of motion for the 5D background in the cosmological
frame are Megías  _et al._ (2018)

$\displaystyle\frac{n^{\prime\prime}(r)}{n(r)}-\left(\frac{n^{\prime}(r)}{n(r)}-\frac{1}{r}\right)\left(\frac{b^{\prime}(r)}{b(r)}-\frac{2}{r}\right)=0\,,$
(5)
$\displaystyle\frac{n^{\prime}(r)}{n(r)}+\frac{b^{\prime}(r)}{b(r)}-r\bar{\phi}^{\prime}(r)^{2}=0\,,$
$\displaystyle\frac{n^{\prime}(r)}{n(r)}+\frac{1}{r}+r\,b^{2}(r)\bar{V}(\bar{\phi})-\frac{r}{2}\bar{\phi}^{\prime}(r)^{2}=0\,,$
$\displaystyle\bar{\phi}^{\prime\prime}(r)+\left(\frac{n^{\prime}(r)}{n(r)}-\frac{b^{\prime}(r)}{b(r)}+\frac{3}{r}\right)\bar{\phi}^{\prime}(r)-b^{2}(r)\frac{\partial\bar{V}}{\partial\bar{\phi}}=0\,,$

with the dimensionless field $\bar{\phi}\equiv\phi/(3M_{5}^{3})^{1/2}$ and
$\bar{V}\equiv V/(3M_{5}^{3})$. Importantly, even though one of these
differential equations seems redundant, it cannot be ignored because it still
implies a nontrivial algebraic relation between the integration constants.
Also notice that the integration constants can depend on $r_{b}$ through the
boundary conditions; the brane location thus influences the 5D background.

We turn to gravity from the brane viewpoint. The effective 4D Einstein
equation seen by a brane observer is computed from the 5D Einstein equation by
projecting on the brane via the Gauss equation together with the Israel
junction condition Shiromizu  _et al._ (2000). Introducing the unit vector
$n_{M}$ normal to the brane that satisfies $n_{M}n^{M}=1$ and
$\bar{g}_{MN}=g_{MN}-n_{M}n_{N}$, the 4D Einstein equation on the brane is

$\displaystyle{}^{(4)}G_{\mu\nu}=\frac{1}{M^{2}_{\rm
Pl}}\left(T_{\mu\nu}^{b}+T^{\rm
eff}_{\mu\nu}\right)+O\left(\frac{T_{b}^{2}}{M^{6}_{5}}\right)$ (6)

with
${}^{(4)}G_{\mu\nu}={}^{(4)}R_{\mu\nu}-\frac{1}{2}\bar{g}_{\mu\nu}{}^{(4)}R$,
and $T^{b}_{\mu\nu}$ the stress tensor of brane-localized matter. The
“holographic” effective stress tensor $T^{\rm
eff}_{\mu\nu}=\tau^{W}_{\mu\nu}+\tau^{\phi}_{\mu\nu}+\tau^{\Lambda}_{\mu\nu}$
contains:

i) The projection of the 5D Weyl tensor ${}^{(5)}C^{M}{}_{NPQ}$ on the brane

$\displaystyle\frac{1}{M^{2}_{\rm
Pl}}\tau^{W}_{\mu\nu}=-{}^{(5)}C^{M}{}_{NPQ}n_{M}n^{P}\bar{g}_{\mu}{}^{N}\bar{g}_{\nu}{}^{Q}\,,$
(7)

leading to corresponding values of the energy density $\rho^{W}$ and pressure
$P^{W}$ given by

$\displaystyle\rho^{W}$
$\displaystyle=\frac{3}{2}\frac{M^{2}_{Pl}}{b^{2}(r_{b})r_{b}}\left(\frac{n^{\prime}(r_{b})}{n(r_{b})}-\frac{1}{r_{b}}\right)\,,$
$\displaystyle P^{W}$
$\displaystyle=\frac{1}{2}\frac{M^{2}_{Pl}}{b^{2}(r_{b})r_{b}}\left(\frac{n^{\prime}(r_{b})}{n(r_{b})}-\frac{1}{r_{b}}\right)\,,$
(8)

where we have made use of Eqs. (5).

ii) The projection of the bulk stress tensor

$\displaystyle\frac{1}{M^{2}_{\rm Pl}}\tau^{\phi}_{\mu\nu}$
$\displaystyle=\frac{2}{3M_{5}^{3}}\left[T^{\phi}_{MN}\bar{g}_{\mu}{}^{M}\bar{g}_{\nu}{}^{N}+\big{(}T^{\phi}_{MN}n^{M}n^{N}-\frac{1}{4}T^{\phi,M}_{M}\big{)}\bar{g}_{\mu\nu}\right]$
$\displaystyle=\frac{3}{2}\left(\frac{\bar{\phi}^{\prime}(r_{b})^{2}}{2b^{2}(r_{b})}-\bar{V}\right)\bar{g}_{\mu\nu}\,,$
(9)

leading to the values of $\rho^{\phi}$ and $P^{\phi}$, after using the EoM
(5),

$\displaystyle\rho^{\phi}$
$\displaystyle=-\frac{3}{2}\frac{M^{2}_{Pl}}{b^{2}(r_{b})r_{b}}\left(\frac{n^{\prime}(r_{b})}{n(r_{b})}+\frac{1}{r_{b}}\right)\,,$
$\displaystyle P^{\phi}$
$\displaystyle=\frac{3}{2}\frac{M^{2}_{Pl}}{b^{2}(r_{b})r_{b}}\left(\frac{n^{\prime}(r_{b})}{n(r_{b})}+\frac{1}{r_{b}}\right)\,.$
(10)

iii) The contribution from the brane tension

$\displaystyle\frac{1}{M^{2}_{\rm
Pl}}\tau^{\Lambda}_{\mu\nu}=-\frac{\Lambda_{b}^{2}}{12M_{5}^{6}}\bar{g}_{\mu\nu}\,,$
(11)

which yields the values of $\rho^{\Lambda}$ and $P^{\Lambda}$ as

$\rho^{\Lambda}=-P^{\Lambda}=\frac{M^{2}_{\rm
Pl}\Lambda_{b}^{2}}{12M_{5}^{6}}\,.$ (12)

The brane tension is ultimately tuned to set the effective 4D cosmological
constant to zero.

We work in the low-energy regime

$|T_{\mu\nu}^{b}|\ll\frac{M_{5}^{6}}{M_{\rm Pl}^{2}}$ (13)

which justifies neglecting the higher order terms in (6). This restriction
implies further simplifications below.

##  III Dark matter from the linear dilaton black hole

The linear dilaton background is defined by the bulk (super)potential

$\bar{W}(\bar{\phi})=\frac{2}{\ell}e^{\bar{\phi}},\quad\bar{V}(\bar{\phi})=-\frac{3}{2\ell^{2}}e^{2\bar{\phi}}\,.$
(14)

Solving the equations of motion (5) with the potential (14) we find for the 5D
background

$\displaystyle n(r)$ $\displaystyle=$
$\displaystyle\frac{r}{\ell}\sqrt{1-\frac{r_{h}^{3}}{r^{3}}}\,,$ (15)
$\displaystyle b(r)$ $\displaystyle=$
$\displaystyle\frac{\ell}{r_{b}}\frac{e^{-\bar{v}_{b}}}{\sqrt{1-\frac{r_{h}^{3}}{r^{3}}}}\,,$
(16) $\displaystyle\bar{\phi}(r)$ $\displaystyle=$
$\displaystyle\bar{v}_{b}-\log\left(\frac{r}{r_{b}}\right)\,,$ (17)

where $r_{h}$ (an integration constant) is the location of the black hole
horizon in the brane cosmology frame. The domain of the variable $r$ is the
interval $[0,\ell]$, where $r=0$ is the metric singularity and $r=\ell$ is the
value of the brane location today, while $0\leq r_{h}\leq r_{b}\leq\ell$.
Importantly, we can notice that a power of $3$ appears in the Schwarzschild
factors, in contrast with pure AdS5 where, instead, there would be a power of
$4$.

We then evaluate the brane effective Einstein equation by plugging the bulk
solutions into Eq. (6), and deduce the Friedmann equation. The mass scale that
naturally appears in the physical quantities is

$\eta=\frac{M_{5}^{3}}{M^{2}_{\rm Pl}}=\frac{e^{\bar{v}_{b}}}{\ell}\,.$ (18)

Using the low-energy assumption (13) which here becomes

$\rho_{b}\ll\eta^{2}M^{2}_{\rm Pl}$ (19)

we obtain the first Friedmann equation on the brane,

$3M^{2}_{\rm Pl}\left(\frac{\dot{r}_{b}}{r_{b}}\right)^{2}=\rho_{b}+\rho_{\rm
eff}+O\left(\frac{\rho_{b}^{2}}{\eta^{2}M^{2}_{\rm Pl}}\right)\,$ (20)

with

$\rho_{\rm eff}=3\eta^{2}M_{\rm Pl}^{2}\frac{r^{3}_{h}}{r^{3}_{b}}\,.$ (21)

The $\rho_{\rm eff}$ energy density term is the critical result. It is a
nontrivial effect from the bulk physics: a combination of the Weyl tensor and
of the scalar stress tensor contributions. This holographically-induced
$\rho_{\rm eff}$ scales as $r_{b}^{-3}$, therefore it behaves as a
nonrelativistic matter term in the 4D Friedmann equation. (The analogous
calculation in AdS would instead give a $r^{-4}_{b}$ scaling, i.e. radiation).

In the brane-world paradigm, we identify the Standard Model fields as brane-
localized modes that give rise to the brane energy density $\rho_{b}$. The
effective energy density $\rho_{\rm eff}$ in Eq. (20) is then naturally
identified as the dark matter energy density. In other words, the linear
dilaton brane-world automatically features dark matter.

From the expression of $\rho_{\rm eff}$ in Eq. (21), the fraction of dark
matter energy in the Universe $\Omega_{\rm DM}=\rho_{\rm DM}/\rho_{\rm crit}$
(with $\rho_{\rm crit}=3H^{2}M^{2}_{\rm Pl}$) induced by the linear dilaton
background is then

$\Omega_{\rm
DM}=\left(\frac{\eta}{H}\right)^{2}\left(\frac{r_{h}}{r_{b}}\right)^{3}\,.$
(22)

At present times we have $r_{b}=\ell$, $\Omega_{{\rm DM},0}=0.26$ and
$H_{0}=1.47\times 10^{-42}\,\textrm{GeV}$. This provides a constraint between
the model parameters given by

$r_{h}\simeq 0.64\ell\left(\frac{H_{0}}{\eta}\right)^{2/3}\,.$ (23)

As in the Standard Cosmology, this dark matter dominates the universe for
temperatures $T\lesssim 0.7\,\textrm{eV}$ and is subdominant with respect to
radiation for higher temperatures.

The origin of the $r^{-3}_{b}$ scaling is better understood as follows. The
effective energy density and pressure, which appear in the Friedmann and
continuity equations, are defined as

$\rho_{\rm eff}=\rho^{W}+\rho^{\phi}+\rho^{\Lambda},\quad P_{\rm
eff}=P^{W}+P^{\phi}+P^{\Lambda}\,,$ (24)

where $\rho^{W}$ and $P^{W}$ are given by Eq. (8), $\rho^{\phi}$ and
$P^{\phi}$ by Eq. (10), and $\rho^{\Lambda}$ and $P^{\Lambda}$, after imposing
the condition for cancellation of the cosmological constant,
$\Lambda_{b}=6\eta^{2}M^{2}_{\rm Pl}$, are given by

$\rho^{\Lambda}=-P^{\Lambda}=3\eta^{2}M^{2}_{\rm Pl}\,.$ (25)

A straightforward application of the BH solutions (15) and (16) yields

$\rho^{W}+\rho^{\phi}=-3\eta^{2}M^{2}_{\rm
Pl}\left(1-\frac{r_{h}^{3}}{r_{b}^{3}}\right)\,,$ (26)

which combined with $\rho^{\Lambda}$ from (25), yields the result which
appears in Eq. (21).

On the other hand, for the effective pressure $P_{\rm eff}$ using again Eqs.
(15) and (16) we get

$P^{W}+P^{\phi}=\frac{2M^{2}_{\rm
Pl}}{b^{2}(r_{b})r_{b}}\left(\frac{n^{\prime}(r_{b})}{n(r_{b})}+\frac{1}{r_{b}}\right)=3\eta^{2}M^{2}_{\rm
Pl}\,,$ (27)

which combined with Eq. (25) yields $P_{\rm eff}=0$, leading to the equation
of state $w_{\rm eff}=P_{\rm eff}/\rho_{\rm eff}=0$

This explains the $r^{-3}_{b}$ scaling and ensures that the 4D conservation
equation i.e. that the 4D Bianchi identity $D^{\mu}{}^{(4)}G_{\mu\nu}=0$ is
satisfied. The cancellation we report here is nontrivial, as it is unclear if
there exists a symmetry that enforces it.

Another nontrivial consistency check is at the level of the 5D conservation
equation projected on the brane, which takes the general form Tanaka and
Himemoto (2003); Langlois and Sorbo (2003)

$\dot{\rho}_{\rm eff}+4H\rho_{\rm eff}+HT_{\,\,\mu}^{{\rm
eff}\,\mu}=-2T_{MN}u^{M}n^{N}\,.$ (28)

Notice the $4H$ factor arising due to 5D spacetime. On the rhs, $n^{N}$ is the
unit vector normal to the brane and outward-pointing, and $u^{M}$ is the brane
velocity vector satisfying $u_{M}u^{M}=-1$ Tanaka and Himemoto (2003);
Langlois and Sorbo (2003). In the low-energy regime we have

$u^{M}\approx\left(\frac{1}{n},{\bm{0}},Hr_{b}\right)\,,\,\quad
n^{M}\approx\left(Hr_{b}\frac{b}{n},{\bm{0}},\frac{1}{b}\right)\,$ (29)

up to $O\left(H^{2}/\eta^{2}\right)$. Using the explicit expression of
$T_{MN}$ obtained from our scalar-gravity solutions, Eqs. (15)–(17), it turns
out that $T_{MN}u^{M}n^{N}=0$ in the low-energy regime. The calculation
involves again beautiful cancellations, and it is detailed in Fichet  _et al._
(2022). One can then easily verify that the 5D conservation equation is
satisfied by the effective energy density (21), ensuring that the framework is
fully consistent.

The low-energy regime Eq. (19) implies $H\ll\eta$ since the total energy
density is $\rho\sim H^{2}M^{2}_{\rm Pl}$; it is the only assumption made
throughout the calculations. We worked at first order in $H/\eta$. The
cancellations observed in $T_{\mu\nu}^{\rm eff}$ and in the 5D conservation
equation occur up to small $O(H^{2}/\eta^{2})$ factors.

##  IV The Newtonian potential

The Newtonian potential for the LD model at present times can be deduced from
the graviton brane-to-brane propagator $G_{\bm{2}}$ using the optical theorem
Fichet  _et al._ (2022). We find the discontinuity of this propagator to be

$\text{Disc}_{s}[G_{\bm{2}}(\sqrt{s})]=2\pi\delta(s)+\frac{\sqrt{\frac{s}{\sigma^{2}}-1}}{s}\theta\Big{(}s\geq\sigma^{2}\Big{)}\,,$
(30)

where $\sigma=3\eta/2$ is the mass gap. The $\delta$ term corresponds to the
4D graviton. The second term, which encodes the rest of the 5D graviton
fluctuations, forms a gapped continuum characteristic of the linear dilaton
background Megías and Quirós (2021a). From this discontinuity we deduce that
the Newtonian potential of the linear dilaton brane-world is

$V_{N}(R)=-\frac{m_{1}m_{2}}{M^{2}_{\rm Pl}\,R}\bigg{(}1+\Delta(R)\bigg{)}\,,$
(31)

with

$\Delta(R)\approx\begin{cases}\frac{4}{3\pi\sigma R}~{}{\rm
if}~{}R\ll\frac{1}{\sigma}\\\ O\Big{(}e^{-\sigma R}\Big{)}~{}{\rm
if}~{}R\gg\frac{1}{\sigma}\end{cases}\,.$ (32)

We see that the deviation from the Newtonian potential appears essentially
below the distance scale $1/\sigma$ corresponding to the inverse mass gap. The
deviation to the potential goes as $\propto 1/R^{2}$, unlike the AdS case,
where it goes as $1/R^{3}$. Micron-scale fifth force experiments such as
Smullin  _et al._ (2005) mildly constrain the $\sigma$ scale as $\sigma\gtrsim
10$ meV. This constraint, along with Eq. (23), translates into an upper bound
on the location of the bulk black hole horizon, $r_{h}\lesssim 2.3\times
10^{-21}\ell$.

##  V Extensions and uniqueness of the linear dilaton brane-world

In the previous sections we have seen that the bulk black hole from the LD
braneworld model characterized by the exponential potential Eq. (14) leads to
a pressureless matter term on the brane. We may wonder whether such a behavior
of $\rho_{\rm eff}$ is specific to the LD model or if it appears in other 5D
scalar-gravity solutions. In the next subsections we provide hints of
uniqueness by extending the model in two different directions. We consider a
model with an exponential superpotential (like that of the LD model) but with
a different exponent, and a model where a constant is added to the exponential
superpotential. In both cases the scalar-gravity solutions will depend on a
parameter which reproduces the LD model for particular values, but generalizes
it. These more general scalar-gravity solutions are interesting per se. We
leave an extended investigation for future work. Our focus here is mostly on
illustrating the uniqueness of the behavior of $\rho_{\rm eff}$ in the LD
model.

###  V.1 A generalized exponential potential

In this section we generalize the (super)potential of the LD model given by
Eq. (14) to

$\bar{W}(\bar{\phi})=\frac{2}{\ell}e^{\nu\bar{\phi}},\quad\bar{V}(\bar{\phi})=-\frac{4-\nu^{2}}{2\ell^{2}}e^{2\nu\bar{\phi}}\,,$
(33)

where the LD model is reproduced for the value $\nu=1$, while the AdS model is
reproduced for the value $\nu=0$.

The solution to the 5D equations of motion (5) is given by

$\displaystyle n(r)$
$\displaystyle=\frac{r}{\ell}\sqrt{1-\left(\frac{r_{h}}{r}\right)^{4-\nu^{2}}}\,,$
(34) $\displaystyle b(r)$
$\displaystyle=\left(\frac{r}{\ell}\right)^{\nu^{2}-1}\left(\frac{\ell}{r_{b}}\right)^{\nu^{2}}\frac{e^{-\nu\bar{v}_{b}}}{\sqrt{1-\left(\frac{r_{h}}{r}\right)^{4-\nu^{2}}}}\,,$
(35) $\displaystyle\bar{\phi}(r)$
$\displaystyle=\bar{v}_{b}-\nu\log\left(\frac{r}{r_{b}}\right)\,,$ (36)

and the relation between the 5D and 4D Planck scales is given by

$M_{5}^{3}=\frac{1}{2}\bar{W}_{b}M_{\rm Pl}^{2}=\eta M_{\rm
Pl}^{2},\quad\eta\equiv\frac{1}{\ell}e^{\nu\bar{v}_{b}}\,.$ (37)

After using the relation for vanishing of the cosmological constant
$\Lambda_{b}=3M_{5}^{3}\bar{W}_{b}=6\eta M_{5}^{3}$, one readily gets the
brane vacuum energy and pressure as

$\rho^{\Lambda}=-P^{\Lambda}=3\eta^{2}M_{\rm Pl}^{2}\,.$ (38)

Using the solution (34)-(35) one easily gets

$\rho_{\rm eff}=3\eta^{2}M_{\rm
Pl}^{2}\frac{r_{h}^{4-\nu^{2}}}{r_{b}^{4-\nu^{2}}},\quad P_{\rm
eff}=\eta^{2}M_{\rm
Pl}^{2}(1-\nu^{2})\frac{r_{h}^{4-\nu^{2}}}{r_{b}^{4-\nu^{2}}}\,,$ (39)

which yields an equation of state

$w_{\rm eff}=\frac{1-\nu^{2}}{3}\,.$ (40)

We can see that the dark matter behavior ($w_{\rm eff}=0$) appears only for
$\nu=1$.

Interestingly, the “holographic” effective energy density in this model
interpolates from dark radiation behavior ($w_{\rm eff}=1/3$) for $\nu=0$ to
dark energy behavior ($w_{\rm eff}=-1$) for $\nu=2$. For $0\leq\nu\leq 2$ the
solution satisfies the continuity equation automatically, cf. Eq. (28).
Finally, let us point out that the singularity at $r=0$ is a good one for
$\nu\leq 2$ Cabrer  _et al._ (2010). 111For $\nu=2$ the solution to the 5D
equations of motion (5) is $n(r)=r/\ell$,
$b(r)=c_{b}(\ell/r_{b})(r/r_{b})^{3}$ and
$\bar{\phi}(r)=\bar{v}_{b}-2\log\left(r/r_{b}\right)$, where $c_{b}$ is an
arbitrary constant. This corresponds to a solution with no black hole for
which $\rho_{\rm eff}=-P_{\rm eff}=3\eta^{2}M_{\rm
Pl}^{2}\left(1-1/(c_{b}\eta\ell)^{2}\right)+\Lambda_{4}M_{\rm Pl}^{2}$, where
we have not assumed cancellation of the cosmological constant. If one fixes
$c_{b}\eta\ell=1$, then $\rho_{\rm eff}=-P_{\rm eff}=\Lambda_{4}M_{\rm
Pl}^{2}$ consistently with the 4D Einstein equations.  Detailed investigation
is left for a future work.

###  V.2 Asymptotically AdS linear dilaton model

We can also define a slightly different model interpolating between AdS and
the linear dilaton background. The model is defined by the bulk potential

$\bar{V}(\bar{\phi})=\frac{1}{8}\bar{W}^{\prime}(\bar{\phi})^{2}-\frac{1}{2}\bar{W}(\bar{\phi})^{2}\,,$
(41)

where $\bar{W}(\bar{\phi})=2(1+e^{\bar{\phi}})/\ell$ Megías and Quirós (2021a,
b). In the brane cosmology frame, the behavior of the effective energy term
depends on the parameter
$c=\exp(-\bar{v}_{b}+e^{-\bar{v}_{b}})\equiv(\eta\ell)^{-1}$. We find that
$\rho_{\rm eff}$ behaves as in AdS in the limit $c\to\infty$, and as in the
linear dilaton background in the limit $c\to 0$, with

$\rho_{\rm eff}\simeq\begin{cases}3\eta^{2}M_{\rm
Pl}^{2}\frac{r_{h}^{3}}{r^{3}_{b}}\,\;\quad{\rm if}~{}~{}c\ll 1\\\
\frac{3}{\ell^{2}}M_{\rm Pl}^{2}\,\frac{r_{h}^{4}}{r_{b}^{4}}\quad\quad{\rm
if}~{}~{}c\gg 1\end{cases}\,.$ (42)

We can recognize the dark radiation behavior for $c\gg 1$ and the dark matter
behavior, Eq. (21), for $c\ll 1$.

We confirm all these results via numerical solving of the 5D conservation
equation (28). More details are given in Ref. Fichet  _et al._ (2022), where
we also discuss the transition region. We find that for arbitrary values of
$c$, the equation of state smoothly interpolates between matter and radiation
behavior, $\rho_{\rm eff}\propto a^{-3[1+w_{\rm eff}(c)]}$ with $P_{\rm
eff}/\rho_{\rm eff}=w_{\rm eff}(c)$. The numerical value of the equation-of-
state parameter $w_{\rm eff}(c)$ is exhibited in Fig. 2 where a continuous
transition appears between $w_{\rm eff}=0$, for dark matter, and $w_{\rm
eff}=1/3$ for dark radiation.

Figure 2: Plot of the equation-of-state parameter $w_{\textrm{eff}}\equiv
P_{\rm eff}/\rho_{\rm eff}$ as a function of $c$, within the asymptotically
AdS linear dilaton model.

In summary, we find that the asymptotic AdS/linear dilaton background, created
by the potential in Eq. (41), gives rise to a cosmological brane-world in
which the behavior of the “holographic” effective energy density can range
from dark radiation to dark matter, as controlled by the $c$ parameter.

##  VI Discussion

We now discuss a few conceptual points and relations to the literature.

#### Birth of the bulk black hole.

In a typical cosmological scenario, analogously to the AdS brane-world, the
bulk horizon is created by energy leaked from the brane into the continuum of
bulk gravitons and other bulk fields. See Hebecker and March-Russell (2001);
Langlois  _et al._ (2002); Langlois and Sorbo (2003) for a consistent analysis
in AdS, and Fichet (2022) for the rate in arbitrary background. The radiation
feeds the bulk black hole, which typically grows with time. This feeding
mechanism is efficient at early times while at late times, the radiation is
negligible hence the horizon does not evolve anymore. This corresponds to the
low-energy regime in our analysis. The process of dumping energy into the
bulk, known since Gubser (2001), is either similar or truely equivalent (via
AdS/CFT) to the process of heating up a CFT sector (see e.g. Gubser (2001);
Hebecker and March-Russell (2001); von Harling and McDonald (2012, 2012); Brax
_et al._ (2019); Hong  _et al._ (2020)).

#### What is the dark matter made of?

The dark matter arising in our linear dilaton brane-world is purely made of
the curvature of spacetime. However this curvature is the result of populating
the bulk with gravitons. Deep in the bulk these gravitons are strongly
interacting, and their net effect is the presence of the bulk horizon, which
is seen by the brane observer. Since the continuum of gravitons is involved,
our result shares, in a sense, some similarity with the proposal of “continuum
dark matter” made in Csáki  _et al._ (2022a, b); Csaki  _et al._ (2022). It is
plausible that our analysis provides the consistent framework needed to
understand cosmology in such models.

##  VII Prospects

Overall, the results reported in this letter hint at an alternative view of
dark matter which certainly deserves further investigation. We thus end with a
discussion of future directions.

#### Cosmological perturbations.

The key calculation presented in this letter shows that the linear dilaton
background could explain dark matter in the homogeneous universe. Computing
perturbations and structure formation is a task beyond the present work,
however the roadmap is clear: the study of cosmological perturbations in our
model belongs to the realm of the fluid/gravity correspondence Bhattacharyya
_et al._ (2008); Hubeny  _et al._ (2012). The dark matter of our brane-world
model amounts to a (non-conformal) “holographic fluid”, whose properties such
as viscosities need to be carefully computed and compared to observations.

#### Dark matter at galactic scales.

Our brane-world model may explain dark matter at cosmological scale, however
nothing is said about galactic scales. To understand how the dark matter
emerging in our model behaves at galactic scales we would have to compute less
symmetric solutions of the 5D scalar-gravity system, as needed to describe
e.g. halos. One should thus investigate $SO(3)$-symmetric solutions, possibly
assisted by matter sources on the brane. This is left for future
investigation.

#### Dark matter decay.

In analogy with AdS, the bulk black hole may in principle be able to decay via
Hawking radiation into the brane, see Rocha (2008, 2009) for an analysis in
AdS. Since the bulk black hole is the origin of dark matter, Hawking decay
amounts in our model to “dark matter decay”. It would be very interesting to
study this mechanism and its observational consequences, as well as its
implications for holography. We leave it as an open question to investigate.

#### Continuum signatures.

In our model the graviton is accompanied by a gapped continuum that can be
experimentally tested, as exemplified by the correction to the Newtonian
potential Eq. (32). Standard Model fields can be included in the model by
introducing 5D bulk fields and identifying the brane-localized modes as the
Standard Model fields. Analogously to the graviton, each Standard Model field
is accompanied with a gapped continuum which has generally mild coupling to
the brane. Such a setup looks typically like a dark sector Brax  _et al._
(2019). The phenomenology of continuum sectors is an active topic of
investigation, see e.g. Katz  _et al._ (2016); Csáki  _et al._ (2019); Lee
(2018); Gao  _et al._ (2020); Fichet (2020); Costantino  _et al._ (2020);
Chaffey  _et al._ (2021); Csáki  _et al._ (2022a, b); Csaki  _et al._ (2022).
The present study reinforces the motivation for such models and, in a sense,
starts to explore their cosmology.

###### Acknowledgements.

We thank Philippe Brax, Csaba Csaki and Philip Tanedo for useful discussions.
The work of SF has been supported by the São Paulo Research Foundation
(FAPESP) under grants #2011/11973, #2014/21477-2 and #2018/11721-4 and by
CAPES under grant #88887.194785. EM would like to thank the ICTP South
American Institute for Fundamental Research (SAIFR), São Paulo, Brazil, for
hospitality and partial finantial support of FAPESP Grant 2016/01343-7 from
Aug-Sep 2022 where part of this work was done. The work of EM is supported by
the project PID2020-114767GB-I00 funded by MCIN/AEI/10.13039/501100011033, by
the FEDER/Junta de Andalucía-Consejería de Economía y Conocimiento 2014-2020
Operational Programme under Grant A-FQM-178-UGR18, and by Junta de Andalucía
under Grant FQM-225. The research of EM is also supported by the Ramón y Cajal
Program of the Spanish MICIN under Grant RYC-2016-20678. The work of MQ is
partly supported by Spanish MICIN under Grant PID2020-115845GB-I00, and by the
Catalan Government under Grant 2021SGR00649. IFAE is partially funded by the
CERCA program of the Generalitat de Catalunya.

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