Leonie Koban
email: leonie.koban@cnrs.fr

Sangil Lee

Daniela S Schelski

Marie-Christine Simon

Caryn Lerman

Bernd Weber

Joseph W Kable

Hilke Plassmann

M Simonetti

J Boerth

A.-A Tholen

A Ortner

S Koehlmoos

L Winkler

A M Bernardo

M K Burke

N Caulfield

G Cooper

M Donnay

J Falcone

R Jorgensen

J Kazinka

M Luery

R Mcconnell

D Miglin

T Mukherjee

S Parthasarathi

M Price

R Schlussel

H J Sharp

D Sohn

K Spence

Terilli

A BRAIN SIGNATURE OF DELAY DISCOUNTING 1 An fMRI-based brain marker of individual differences in delay discounting

Individual differences in delay discounting-how much we discount future compared to immediate rewards-are associated with general life outcomes, psychopathology, and obesity. Here, we use machine learning on fMRI activity during an intertemporal choice task to develop a functional brain marker of these individual differences in human adults. Training and cross-validating the marker in one data set (Study 1, N = 110 male adults) resulted in a significant prediction-outcome correlation (r = 0.49), generalized to predict individual differences in a completely independent data set (Study 2, N = 145 male and female adults, r = 0.45), and predicted discounting several weeks later. Out-of-sample responses of the functional brain marker, but not discounting behavior itself, differed significantly between overweight and lean individuals in both studies, and predicted fasting state blood levels of insulin, c-peptide, and leptin in Study 1.

Significant predictive weights of the marker were found in cingulate, insula, and frontoparietal areas, among others, suggesting an interplay among regions associated with valuation, conflict processing, and cognitive control. This new functional brain marker is a step towards a generalizable brain model of individual differences in delay discounting.

Future studies can evaluate it as a potential transdiagnostic marker of altered decisionmaking in different clinical and developmental populations.

Introduction

Many decisions in life have consequences at different points in time. For example, most people need to decide whether to put part of their paycheck towards a retirement fund or spend it on something fun, like a short vacation. These trade-offs between options that are immediately rewarding and those that will be more rewarding in the long run are hard, and people differ substantially in delay discounting-the degree to which they discount future compared to immediate rewards [START_REF] Kirby | Preference Reversals Due to Myopic Discounting of Delayed Reward[END_REF]. Greater delay discounting (i.e., greater impatience or higher preference for sooner rewards) is associated with obesity, addiction, and many psychiatric conditions [START_REF] Bickel | Impulsivity and cigarette smoking: delay discounting in current, never, and ex-smokers[END_REF][START_REF] Mackillop | Delayed reward discounting and addictive behavior: a meta-analysis[END_REF][START_REF] Mole | Impulsivity in disorders of food and drug misuse[END_REF][START_REF] Amlung | Steep discounting of delayed monetary and food rewards in obesity: a meta-analysis[END_REF]. It has therefore been proposed as a potential transdiagnostic marker of psychopathology [START_REF] Amlung | Delay Discounting as a Transdiagnostic Process in Psychiatric Disorders: A Meta-analysis[END_REF][START_REF] Lempert | Can delay discounting deliver on the promise of RDoC?[END_REF] and as a risk factor for short-sighted behaviors such as unhealthy diet, smoking, and excessive alcohol and drug use [START_REF] Audrain-Mcgovern | Does delay discounting play an etiological role in smoking or is it a consequence of smoking?[END_REF][START_REF] Fernie | Multiple behavioural impulsivity tasks predict prospective alcohol involvement in adolescents[END_REF]. The goal of the present study is to identify and validate an fMRI-based brain marker of individual differences in delay discounting.

Previous findings regarding the structural and functional brain bases of individual differences in delay discounting offer a mixed picture. Several studies suggest a role for areas involved in reward processing and valuation [START_REF] Bartra | The valuation system: a coordinate-based metaanalysis of BOLD fMRI experiments examining neural correlates of subjective value[END_REF][START_REF] Cooper | Brain activity in valuation regions while thinking about the future predicts individual discount rates[END_REF], and for areas central to cognitive control [START_REF] Hare | Activity in dlPFC and its effective connectivity to vmPFC are associated with temporal discounting[END_REF]. Brain areas associated with memory and prospection have also been found to contribute to individual differences in delay discounting [START_REF] Benoit | A neural mechanism mediating the impact of episodic prospection on farsighted decisions[END_REF][START_REF] Peters | The neural mechanisms of inter-temporal decision-making: understanding variability[END_REF][START_REF] Lebreton | A critical role for the hippocampus in the valuation of imagined outcomes[END_REF].

Studies using structural and functional connectivity measures suggested a role for frontostriatal and striatal-subcortical connections (van den [START_REF] Van Den Bos | Connectivity strength of dissociable striatal tracts predict individual differences in temporal discounting[END_REF]. The structure of midbrain dopaminergic nuclei and the ventral striatum has been associated with selfreported trait impulsivity [START_REF] Macniven | Medial forebrain bundle structure is linked to human impulsivity[END_REF].

A BRAIN SIGNATURE OF DELAY DISCOUNTING 5

Individual differences in delay discounting may also emerge from the combined activity across multiple brain regions or functional networks. However, only a few studies [START_REF] Berman | Dimensionality of brain networks linked to life-long individual differences in selfcontrol[END_REF][START_REF] Li | Restingstate functional connectivity predicts impulsivity in economic decision-making[END_REF][START_REF] Pehlivanova | Diminished Cortical Thickness Is Associated with Impulsive Choice in Adolescence[END_REF] have investigated the distributed patterns associated with individual differences in delay discounting. Further, most previous studies used relatively small sample sizes to explore these individual differences, increasing the risk of both false-positive and false-negative results [START_REF] Poldrack | Scanning the horizon: towards transparent and reproducible neuroimaging research[END_REF]. Given the use of standard correlation or regression analyses that are typically not cross-validated on independent data samples, previous results are difficult to compare with each other and do not provide any formal model that could predict delay discounting in completely independent studies.

Here, we address these limitations by using a machine-learning-based 'brain model' approach [START_REF] Woo | Building better biomarkers: brain models in translational neuroimaging[END_REF][START_REF] Kragel | Representation, Pattern Information, and Brain Signatures: From Neurons to Neuroimaging[END_REF]. Brain models are trained to predict a mental process or individual variable (here, delay discounting) and can be applied to independent data [START_REF] Kragel | Representation, Pattern Information, and Brain Signatures: From Neurons to Neuroimaging[END_REF][START_REF] Scheinost | Ten simple rules for predictive modeling of individual differences in neuroimaging[END_REF]. As such, brain models go beyond reporting peak coordinates by identifying specific large-scale patterns of brain activity that can be replicated, validated, or falsified in a quantifiable way. This approach has been successfully applied to brain-based prediction of pain [START_REF] Wager | An fMRI-based neurologic signature of physical pain[END_REF], working memory [START_REF] Rosenberg | Behavioral and neural signatures of working memory in childhood[END_REF], and affective states [START_REF] Yu | A Generalizable Multivariate Brain Pattern for Interpersonal Guilt[END_REF], among others. The importance of independent validation and model generalizability has also been recognized for brain-based prediction of trait-like individual differences [START_REF] Gabrieli | Prediction as a humanitarian and pragmatic contribution from human cognitive neuroscience[END_REF][START_REF] Rosenberg | Prediction complements explanation in understanding the developing brain[END_REF][START_REF] Rosenberg | How to establish robust brain-behavior relationships without thousands of individuals[END_REF].

Here, we build on this approach to predict individual differences in delay discounting. If there is a consistent activity pattern associated with individual differences in delay discounting during intertemporal choices, then this pattern should be able to predict delay discounting in new data (hold-out subjects) and even completely independent data sets. Comparing the resulting pattern to meta-analysis-based masks A BRAIN SIGNATURE OF DELAY DISCOUNTING 6 allows us to assess the contribution of brain areas associated with valuation, cognitive control, and prospection.

Materials and methods

Overview

We used an established machine-learning algorithm, LASSO-PCR [START_REF] Tibshirani | Regression Shrinkage and Selection Via the Lasso[END_REF][START_REF] Wager | Predicting individual differences in placebo analgesia: contributions of brain activity during anticipation and pain experience[END_REF] and fMRI data from two independent studies, from different scanners, labs, and countries. Study 1 (N=110) was used for training and cross-validation of a predictive model of individual differences in delay discounting. Study 2 (existing data set from previously published study, see Kable et al., 2017, N=145) was used as an independent test data set to assess the validity and replicability of the predictive model.

Participants

For Study 1, participants were recruited in the context of a seven-week dietary intervention study at the University of Bonn in Germany (https://osf.io/rj8sw/?view_only=af9cba7f84064e61b29757f768a8d3bf). Due to the nature of this longitudinal intervention study, we recruited only male participants who further fulfilled the following inclusion criteria: age between 20 and 60 years, right-handedness, non-smoker, no excessive drug or alcohol use in the past year, no psychiatric or neurological disease, body mass index (BMI) between 20 and 34, no other chronic illness or medication, following a typical Western diet without dietary restrictions, and no MRI exclusion criteria (large tattoos, non-removable piercings, metal in the body, claustrophobia, etc.). N=116 male participants performed the intertemporal choice task in Study 1. Here, we focus on behavioral and fMRI data collected during a baseline session before the group assignment and dietary intervention (to be reported elsewhere) and use post-intervention behavioral data only for demonstrating the temporal stability of interindividual differences in delay discounting. The data of six participants had to be excluded for analysis due to the following reasons: technical problems with the scanner

(1), with the synchronization between stimulation software and scanner (3), and with the response box (1), and strong motion artifacts (>5mm) and participant quitting the task midscan (1). Therefore, 110 participants (mean age=31.7; 52 lean, 48 overweight, and 10 obese; BMIs ranging from 20.6 to 33.7) were included in the final analysis of Study 1.

There were no significant differences between lean and overweight-to-obese participants in age, education, or total brain volume (see Table 1). Data from 109 participants were available for the seven-week follow-up measurements (i.e., one participant did not return for the second session).

Study 2 was conducted in the context of a large cognitive training study at the University of Pennsylvania in the United States [START_REF] Kable | No Effect of Commercial Cognitive Training on Brain Activity, Choice Behavior, or Cognitive Performance[END_REF]. The goal was to examine whether commercial cognitive training software leads to significant changes in decision-making behaviors, including delay discounting. Participants completed two sessions of scans 10 weeks apart. As with Study 1, we focus on the baseline (preintervention) behavioral and fMRI data, and report post-intervention behavioral data only to assess the temporal (10-week) stability of interindividual differences in delay discounting. Of the 160 non-pilot participants who completed session 1, we excluded those with missing runs (N=6), frequent or significant head movement (any run with >5% of mean image displacements greater than 0.5mm; N=3), more than 3 missing trials per run for two or more runs (N=2), or lack of participant blinding (N=1, one subject expressed awareness of their experimental condition, i.e. cognitive training vs. control). Of the remaining 148, we excluded three more participants whose choice was entirely one-sided (i.e., choosing only immediate reward or delayed reward), resulting in a final sample of N=145 participants for Study 2 (88 male, 57 female, mean age=24.4; 81 lean, 39 overweight, and 25 obese; BMIs ranging from 16.5 to 40.9). There were no significant differences between lean and overweight-to-obese participants in sex, age, education, or total brain volume (see Table 1). Due to drop-out, data from 102 participants was available for the 10-week post-intervention measurement of log(k).

The study protocols were approved by the institutional review boards of Bonn University's Medical School (Study 1) and the University of Pennsylvania (Study 2). All participants provided written informed consent, and were paid for their time and participation in the study. The research reported here complies with all relevant ethical regulations.

Stimuli and task

In Study 1, participants performed 108 choices (trials) between varying amounts of smaller sooner (SS) and larger later (LL) options, presented on the left or right of the screen (position randomized; see Figure 1a). Participants were instructed that one of their choices might be paid out at the end of the experiment. Thus, participants' choices were non-hypothetical and incentive-compatible. During each trial, the two options were presented for 4s, during which participants could make their choice (left or right) by pressing the corresponding response key with their left or right index finger, respectively.

Once the choice had been made, a yellow frame highlighted the chosen option and remained on the screen for the remainder of the 4s. Intertrial intervals were jittered using an approximately geometric distribution (2-11s) SS options varied among €5, €10, and €20, and always had zero delay ('today').

LL options varied between €5 and €96.80 and had delays between 2 days and 8 months (~240 days, choice combinations are presented in Figure 1-1). Amounts and delays were chosen to allow fine-grained estimation of individual k's between 0 and 0.256. Trials were presented in randomized order.

The intertemporal choice task in Study 2 consisted of 120 trials, again with the same choice sets for all participants (see Figure 1-2). In contrast to Study 1, the SS amount was fixed at US$20. Thus, participants were presented with the LL option (with amount ranging from US$22 to US$85 and delays from 19 days to 180 days) and were instructed to press one of two keys to either accept and receive this LL offer, or to reject the LL offer and receive the SS offer ($20 today) instead. Participants were informed that one trial would be randomly chosen at the end of the experiment and their choices implemented (i.e., the chosen amount would be paid via wire transfer at the indicated time delay), resulting in incentive-compatible and mutually independent choices in each trial (as in Study 1).

Blood measures and body fat measures

In Study 1, blood samples in a fasted state were collected from participants' nondominant arm before they received a standardized breakfast. HOMA-IR (a marker of insulin resistance) was calculated as the product of fasting insulin and glucose levels, divided by 405 [START_REF] Lozano | Global and regional mortality from 235 causes of death for 20 age groups in 1990 and 2010: a systematic analysis for the Global Burden of Disease Study[END_REF]. Body weight and proportion of body fat were measured using a bioimpedance scale (Tanita Europe BV, Amsterdam, the Netherlands).

For technical reasons, this body fat measure was available for only 103 participants.

Experimental design and statistical analyses

Behavioral measures. For each participant, we calculated the proportion of SS choices (with respect to total number of non-misses) and the model-based discounting parameter k. Individual k's were log-transformed in both studies to obtain less skewed distributions of discounting parameters. This log(k)-parameter describes the steepness of discounting as modeled by the hyperbolic discounting function [START_REF] Kirby | Preference Reversals Due to Myopic Discounting of Delayed Reward[END_REF]. Higher log(k) parameters reflect steeper discounting and thus greater impatience;

A BRAIN SIGNATURE OF DELAY DISCOUNTING 10 smaller (more negative) values reflect less steep discounting and thus more patient decision-making.

In Study 1 we computed k by calculating the proportion of SS choices for all target k's (i.e., the k-value for which SS and LL options of any given choice trial should theoretically be chosen at 50% each). We then used linear interpolation to identify the individual indifference point at which the proportion of SS and LL choices was equal (50% each).

In Study 2, we fit a logit utility model on choice data via maximum likelihood estimation. The logit of the probability of choosing the delayed reward was modeled as follows:

logit&p(Y ! = delayed)0 = σ 2 LL ! 1 + kD ! -20;
Where LLt is the LL amount in trial t and Dt is the delay in trial t. ! was included as a scaling parameter that controls the relationship between utility difference scale and choice.

MRI data acquisition.

Functional and structural brain imaging data for Study 1 were acquired using a Siemens Trio 3T scanner (Erlangen, Germany) at the Life & Brain Institute, Bonn University Hospital, Germany. Functional images used a T2* weighted EPI-GRAPPA sequence (TR=2.5s, TE=30ms, flip angle=90°, FOV=192mm, acceleration factor R=2, average of 400 volumes) and covered the whole brain in 37 slices (voxel size of 2 x 2 x 3mm, 10% interslice distance). Structural images were acquired using a T1

weighted MPRAGE sequence (1mm isomorphic voxels).

For Study 2, the functional and structural imaging data were acquired with a Siemens 3T Trio scanner with a 32-channel head coil. High-resolution T1-weighted anatomical images were acquired using an MPRAGE sequence (T1=1100ms; 160 axial slices, 0.9375 x 0.9375 x 1.000mm; 192 x 256 matrix). T2*-weighted functional images were acquired using an EPI sequence with 3mm isotropic voxels, 64 x 64 matrix, TR=3s, TE=25ms, 53 axial slices (no interslice gaps), 104 volumes. B0 fieldmap images were collected for distortion correction (TR=1270ms, TE=5 and 7.46ms).

Preprocessing and basic statistical analyses of fMRI data. Preprocessing for

Study 1 was performed in SPM12 and used a standard pipeline of motion correction, slice time correction, spatial normalization to MNI space, and spatial smoothing of images using an 8mm FWHM Gaussian kernel. Preprocessing for Study 2 was performed in FSL according to the original preprocessing pipeline [START_REF] Kable | No Effect of Commercial Cognitive Training on Brain Activity, Choice Behavior, or Cognitive Performance[END_REF]. This involved the standard pipeline of motion correction, b0 map unwarping, interleaved slice time correction, spatial smoothing with FWHM 9mm Gaussian kernel, and high-pass filtering (cutoff=104s).

For Study 1, we used SPM12 to fit a general linear model (GLM) for each participant's imaging data, with choice screen onset modeled as a stick function (0s duration) as the main regressor and mean-centered parametric modulators for delay, relative LL amount (LL amount divided by SS amount), SS amount, and reaction time. Six nuisance regressors were added to control for movement artifacts. For Study 2, FSL was used to fit an otherwise similar statistical model with a choice screen onset as main regressor and mean-centered parametric modulators for delay, LL amount, and reaction time. As in Study 1, six movement regressors were added to control for head movement.

Individual contrast images were calculated for the following three regressors of interest that were available for both studies: 1) choice screen onset versus implicit baseline (hereafter referred to as "Choice contrast"), 2) parametric modulation by (relative)

LL amount ("LL Amount"), and 3) parametric modulation by delay ("Delay"). Contrast images were gray matter-masked to remove voxels that were unlikely to contain

A BRAIN SIGNATURE OF DELAY DISCOUNTING 12 meaningful BOLD signal and individually z-scored to remove differences in scale across images (and thereby make results transferable across studies and data sets).

Training and cross-validation.

Training and cross-validation were performed on data from Study 1 only (see Figure 1b). Individual differences in delay discounting may result from how participants respond to intertemporal choices overall, from how they process future rewards, and from how they process time delays. Thus, to capture a combination of functional processes that together determine delay discounting, we concatenated the contrast images for Choice, LL Amount, and Delay for each participant, resulting in a feature space that was triple the size of a single brain image. We then used LASSO-PCR (least absolute shrinkage and selection operator-principal component regression) [START_REF] Tibshirani | Regression Shrinkage and Selection Via the Lasso[END_REF]-a machine learning-based regression algorithm-to train a classifier to predict log(k) across all voxel weights of the concatenated contrast images. LASSO-PCR first performs data reduction using principal component regression, thus identifying brain regions and networks that are highly correlated with each other. It then performs the LASSO algorithm, which shrinks regression weights towards zero, thus reducing the contribution of less important and more unstable components. LASSO-PCR has been shown to be advantageous for brain images for several reasons (see [START_REF] Wager | Predicting individual differences in placebo analgesia: contributions of brain activity during anticipation and pain experience[END_REF][START_REF] Wager | An fMRI-based neurologic signature of physical pain[END_REF]: it is adequate for predictions based on thousands of voxels, it takes into account multicollinearity between voxels and brain regions, and it yields interpretable results by allowing reconstruction of voxel weight maps based on PCR results.

To assess the predictive accuracy of the classifier in new subjects, we used 10fold cross-validation. Thus, the training data was split up in 10 stratified combinations of training (90%) and test sets (10%), such that every subject's data was used for the training of the classifier in nine folds and held out in the remaining fold to independently assess the prediction-outcome correlation. Tenfold cross-validation was chosen a priori as a good compromise between maximizing the sample size in each training set and being within the range of recommended folds (between 5 and 10) [START_REF] Scheinost | Ten simple rules for predictive modeling of individual differences in neuroimaging[END_REF][START_REF] Poldrack | Establishment of Best Practices for Evidence for Prediction: A Review[END_REF]. A priori set default regularization parameters were used for all machine-learning analysis to avoid biasing the model parameters to the data and thereby generating overoptimistic accuracy scores. Permutation tests (5,000 iterations of randomly permuting the log(k) values) were used to generate null distributions and to assess the statistical significance of the prediction-outcome correlation and the mean absolute error. Out-ofsample predictions of log(k) were used for all correlational analysis (e.g., with BMI, age, blood markers).

Bootstrapping and thresholding. To identify the brain areas contributing the most reliable positive or negative weights, we performed a bootstrap analysis; 5,000 samples with replacements were taken from the paired brain and outcome data, and the LASSO-PCR was repeated for each bootstrap sample. Two-tailed, uncorrected p-values were calculated for each voxel based on the proportion of weights above or below zero [START_REF] Wager | Predicting individual differences in placebo analgesia: contributions of brain activity during anticipation and pain experience[END_REF][START_REF] Wager | An fMRI-based neurologic signature of physical pain[END_REF]. False discovery rate (FDR) correction was applied to p-values to correct for multiple comparisons across the whole feature space (three combined brain maps).

Independent test set.

Study 2 was used as an independent test set to assess the validity and generalizability of the brain pattern classifier developed based on Study 1 (i.e., the k-marker). For this purpose of testing its validity in an independent data set (and for all future use of this brain-based model), the k-marker was trained on the data of all participants of Study 1. To assess the response of the predictive marker in Study 2, we calculated the matrix dot product between the k-marker and the concatenated contrast images (Choice screen onset, LL Amount, and Delay) from each participant. The dot product reflects the pattern similarity between the classifier and each participant's set of contrast images and, in sum with the classifier's intercepts, provides a predicted value of log(k). Predictive accuracy of the marker was quantified by correlating the predicted value of log(k) with the actual log(k)'s of each participant and by calculating the mean absolute error for each prediction.

Other statistical analyses. All statistical tests were performed in Matlab, were two-tailed, and used a significance criterion of p=0.05. Statistical power calculations confirmed that the sample sizes in both studies were sufficiently powered (>80%) to detect correlations of r > 0.3 at a significance level of p=0.05 (two-sided tests).

Results

Individual differences in delay discounting

In Study 1 (Bonn University, N=110), participants chose the SS option in an average of 43.7% of all trials (median=48.1%) and had a fitted mean log(k) parameter of -5.70 (median log(k)=-5.28, corresponding to a k of 0.0051). Choice behavior was characterized by substantial individual differences, with %SS choices ranging from 5.6% to 88.8%, and log(k) ranging from -9.90 to -1.36 (see Figure 1c). Individual differences were very stable over a 7-week period (see Methods), with a test-retest reliability (correlation between baseline log(k)'s and second session) of r=0.86 (p<0.001, 95%confidence interval [CI]=[0.80, 0.90], Figure 1c).

On average, participants in Study 2 chose the SS option in 57.4% of trials (median=60.0%) and had an fitted log(k) of -4.08 (median log(k)=-3.95, corresponding to a k of 0.0193). Again, individuals varied substantially in their intertemporal preferences, with %SS ranging from 0.8% to 99.2%, and log(k) ranging from -7.08 to -2.12 (see Figure 1c). As in Study 1, these individual differences were stable over time, with a test-retest reliability between baseline and post-intervention (10 weeks later) measures of log(k) of r=0.74 (Pearson correlation, p<0.001, 95%-CI=[0.63, 0.82], see Figure 1c). Thus, our data confirm both the substantial variability in delay discounting known from previous work [START_REF] Kable | The neural correlates of subjective value during intertemporal choice[END_REF][START_REF] Pehlivanova | Diminished Cortical Thickness Is Associated with Impulsive Choice in Adolescence[END_REF] and the stability of these individual differences over time [START_REF] Kirby | One-year temporal stability of delay-discount rates[END_REF][START_REF] Anokhin | Long-term test-retest reliability of delayed reward discounting in adolescents[END_REF][START_REF] Lempert | The Malleability of Intertemporal Choice[END_REF], allowing us to investigate the neurofunctional bases of these individual differences.

Significant cross-validated prediction of delay discounting based on fMRI

Training (using LASSO-PCR) and cross-validating (10-fold) the predictive marker (termed k-marker, see Figure 2a) in Study 1 resulted in a cross-validated predictionoutcome correlation (i.e., correlation between predicted and actual log(k)) of r=0.49

(permutation test: p<0.001), a mean squared error of 2.84 (permutation test: p<0.001), and a mean absolute error for predicted log(k) of 1.32 (permutation test: p<0.001; see

Validation of the k-marker in an independent test data set (Study 2)

Brain markers of individual differences become more meaningful if they can be validated in different and completely independent data [START_REF] Kragel | Representation, Pattern Information, and Brain Signatures: From Neurons to Neuroimaging[END_REF]. The validity of the marker should not depend on study-specific parameters such as the type of scanner used for data acquisition, preprocessing software, or other aspects of the data [START_REF] Woo | Building better biomarkers: brain models in translational neuroimaging[END_REF][START_REF] Kragel | Representation, Pattern Information, and Brain Signatures: From Neurons to Neuroimaging[END_REF][START_REF] Scheinost | Ten simple rules for predictive modeling of individual differences in neuroimaging[END_REF].

We therefore tested whether the k-marker-developed and cross-validated The training and cross-validation data set (Study 1) consisted of male participants only, which might limit the validity of the k-marker in females. We therefore assessed the accuracy of the k-marker in Study 2 separately in male and female participants (see Figure 2c). In male participants (N=88), the prediction-outcome correlation was r=0 2d). This shows that variability in k-marker responses is driven largely by stable individual differences and their underlying neurophysiological processes.

Response of the k-marker differs between lean and overweight participants

Given previous findings of higher delay discounting in overweight and obese people [START_REF] Jarmolowicz | Robust relation between temporal discounting rates and body mass[END_REF][START_REF] Amlung | Steep discounting of delayed monetary and food rewards in obesity: a meta-analysis[END_REF] 3b).

Thresholded activation patterns of the k-marker

Activation patterns across the whole brain gray matter and across all three contrasts are used for prediction and cross-validation. To identify the areas that contributed the most strongly with positive or negative weights, we used a bootstrapping procedure (5,000 samples). Bootstrapped weights were thresholded at q=0.05 FDR corrected across the whole weight map of the combined feature space (see Figure 2a and Figures 2-1, 2-2, and2-3).

Our results revealed a distributed network of areas that jointly contributed to individual differences in delay discounting, including the vmPFC, ventral striatum, anterior midcingulate cortex (aMCC), hippocampus, frontoparietal, and visual areas (see Figure 2). Predictive activity patterns differed for the processes captured by the three different contrast images. Of note, some regions showed negative weights (i.e., predicted less discounting) for one contrast but positive weights (i.e., predicted more discounting) for another. For the Choice (versus implicit baseline) contrast, activity in the striatum, the anterior insula, and lateral prefrontal areas contributed positive weights for more discounting, whereas activity in visual, premotor, and motor areas contributed negative Training predictive patterns on the three contrasts separately resulted in lower predictive accuracies (see Figure 5).

Similarity of k-marker brain patterns to meta-analytic maps

We next compared the predictive maps of the k-marker with term-based metaanalytic images [START_REF] Yarkoni | Large-scale automated synthesis of human functional neuroimaging data[END_REF] for processes that may contribute to intertemporal decision-making. We computed the spatial correlation (Pearson's r) between the k-marker and meta-analytic maps for 1) affective-and value-related, 2) conflict-and cognitive control-related, and 3) memory-related terms (see Figure 6). While these spatial correlations are descriptive (c.f., [START_REF] Koban | Different brain networks mediate the effects of social and conditioned expectations on pain[END_REF], they can inform us quantitatively whether and in which direction (positive or negative) previously identified functional networks contribute to individual differences in delay discounting.

Value-and affect-related maps (especially 'affect' and 'emotion') showed consistent positive correlations (r's > 0.05) with the Choice-related pattern of the k-marker, in line with the idea that more affect-related activity during intertemporal choices leads to more impatient decisions. However, stronger engagement of affective and especially 'reward'-and 'value'-related activity for increasing LL Amount (and, to a lesser extent, for increasing delays) was associated with less discounting. This suggests that lower discounting is associated with greater sensitivity of valuation-related signals to the amount of LL rewards.

In contrast to our initial hypothesis, more activity in cognitive control-related areas was not associated with lower discounting. Instead, there were positive correlations (r's from 0.05 to 0.19) of the Choice pattern with meta-analytic maps for 'attention', 'cognitive control', 'conflict', and 'executive' (i.e., more positive weights predicting greater discounting). Further, stronger activation of control-related maps by greater LL Amount was associated with greater discounting, whereas stronger activation of control-related activity for longer delays was associated with less discounting.

Finally, we assessed the contribution of brain systems related to memory and prospection. While the term 'memory' (which also includes working-memory studies)

showed a similar pattern as control-related maps, more specific terms such as 'episodic memory', 'imagery', and 'planning' were not substantially positively or negatively correlated with any of the k-marker patterns (r's around 0.05 and smaller).

Parallel findings were obtained when testing whether activity in non-overlapping meta-analytic maps [START_REF] Yarkoni | Large-scale automated synthesis of human functional neuroimaging data[END_REF] (for value-related, cognitive control-related, or episodic memory-related activity) could separately predict log(k) (see Figure 7). Whereas

Local prediction of log(k)

Finally, we assessed whether activity patterns in smaller, more local brain areas could predict out-of-sample log(k) in Study 1 and whether such predictions would accurately transfer to Study 2. For this purpose, we used an established multi-modal cortical parcellation [START_REF] Glasser | A multi-modal parcellation of human cerebral cortex[END_REF] in combination with several other, subcortical parcellations, resulting in a total of 485 regions (publicly available at https://github.com/canlab/Neuroimaging_Pattern_Masks/tree/master/Atlases_and_parcel lations/2018_Wager_combined_atlas). We trained and cross-validated a separate classifier for each region (combining functional activity across all three contrasts). For each region we then tested whether the pattern trained on Study 1 data was predictive of individual differences in Study 2. Activity patterns that consistently predicted delay discounting in both studies were found in mid-and posterior cingulate cortex, right insula, and lateral frontal and parietal areas (see Figure 8 and Table 2), in line with the contributions of these areas in the whole-brain predictive pattern. In addition, activity patterns in amygdala, hippocampus, basal ganglia and brainstem areas (periaqueductal gray) also predicted individual differences in delay discounting in both studies.

Discussion

A major goal of neuroscience and psychiatry is to identify neuromarkers of transdiagnostic processes that are altered across different diseases or predispose individuals to such diseases [START_REF] Insel | Medicine. Brain disorders? Precisely[END_REF]. Delay discounting-how much people prefer sooner compared to future rewards-has been proposed as such a transdiagnostic process across obesity and various forms of psychopathology, especially addiction and eating disorders [START_REF] Bickel | Using crowdsourcing to compare temporal, social temporal, and probability discounting among obese and non-obese individuals[END_REF][START_REF] Amlung | Delay Discounting as a Transdiagnostic Process in Psychiatric Disorders: A Meta-analysis[END_REF][START_REF] Lempert | Can delay discounting deliver on the promise of RDoC?[END_REF]. In this paper, we advanced our understanding of the brain processes that drive variability in decision-making by identifying a distributed pattern of functional brain activity that predicts individual differences in delay discounting. We first used a cross-validation procedure to develop a novel functional brain marker of delay discounting (k-marker) based on whole-brain, gray matter-masked fMRI data (N1=110). We then validated the kmarker (trained on Study 1 data only) in an independent second fMRI data set (N2=145), sampled in a cohort with different socio-demographic characteristics, on a different fMRI scanner, and employing a different delay discounting task. Prediction-outcome correlations were 0.49 (Study 1) and 0.45 (Study 2), as large or larger than prediction of individual differences in other domains reported in previous studies [START_REF] Rosenberg | A neuromarker of sustained attention from whole-brain functional connectivity[END_REF][START_REF] Beaty | Robust prediction of individual creative ability from brain functional connectivity[END_REF][START_REF] Han | Effect sizes and test-retest reliability of the fMRI-based Neurologic Pain Signature[END_REF]. In both studies, individual differences in discounting were stable over time, and k-marker responses measured at baseline significantly predicted behavior several weeks later.

Recent findings have questioned the utility of brain imaging in predicting individual differences, especially for structural and resting-state fMRI data and for univariate, voxelwise-associations [START_REF] Marek | Reproducible brain-wide association studies require thousands of individuals[END_REF]. An important advance of the present study is that it overcomes many of the limitations of previous studies by providing an independently cross-validated and multivariate 'brain model' [START_REF] Kragel | Representation, Pattern Information, and Brain Signatures: From Neurons to Neuroimaging[END_REF] of stable individual differences in impatient decision-making, in line with recent recommendations on studying brain-based prediction of individual differences [START_REF] Rosenberg | How to establish robust brain-behavior relationships without thousands of individuals[END_REF]. As such, this brain model can be directly tested, validated, or refined in other existing or future fMRI data sets acquired during an intertemporal choice task. Its predictive performance can also be tested in clinical populations, such as patients with severe obesity, eating or substance use disorders, and other types of psychopathology.

Our results also inform the debate regarding the contributions of specific brain regions and functional networks to individual differences in delay discounting. Among the brain areas that contributed with positive and/or negative weights were the vmPFC, striatum, and other regions associated with valuation and reward [START_REF] Levy | Comparing apples and oranges: using reward-specific and reward-general subjective value representation in the brain[END_REF][START_REF] Bartra | The valuation system: a coordinate-based metaanalysis of BOLD fMRI experiments examining neural correlates of subjective value[END_REF][START_REF] Clithero | Informatic parcellation of the network involved in the computation of subjective value[END_REF]. This finding is in line with previous, univariate findings [START_REF] Cooper | Brain activity in valuation regions while thinking about the future predicts individual discount rates[END_REF][START_REF] Pehlivanova | Diminished Cortical Thickness Is Associated with Impulsive Choice in Adolescence[END_REF][START_REF] Macniven | Medial forebrain bundle structure is linked to human impulsivity[END_REF].

The present results add to this emergent picture by showing that greater sensitivity of reward-and value-related areas to the amount of the LL reward is linked to more patient decision-making.

Significant weights were found most consistently in the frontoparietal areas, midcingulate cortex, and anterior insula. Activity in these areas also allowed for significant prediction based on local activity alone (see Figure 8, ROI analysis). The dorsolateral prefrontal cortex has been theorized to implement self-control and far-sighted decision-making [START_REF] Mcclure | Separate neural systems value immediate and delayed monetary rewards[END_REF][START_REF] Hare | Activity in dlPFC and its effective connectivity to vmPFC are associated with temporal discounting[END_REF]). Yet the present results are surprising as they draw a more complex picture of these areas' contribution to delay discounting, with modulation of these areas by greater LL rewards being positively associated with discounting and modulation by delay being negatively associated with discounting. Thus, areas meta-analytically associated with cognitive control were more recruited for long delays and small LL amounts for low discounters, and for shorter delays and larger LL amounts for high discounters. These are the cases in which decisions are most difficult (closer to the indifference point) and therefore require resolution of response conflict [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF][START_REF] Kool | Neural and behavioral evidence for an intrinsic cost of self-control[END_REF][START_REF] Shenhav | The expected value of control: an integrative theory of anterior cingulate cortex function[END_REF][START_REF] Hutcherson | Evidence accumulation, not" self-control[END_REF], or integration of expected value and risk of future rewards [START_REF] Tobler | Risk-dependent reward value signal in human prefrontal cortex[END_REF].

These findings have implications for models of delay discounting and self-control in cognitive neuroscience. First, they speak against the idea of a simple dual process account of intertemporal choice and self-control [START_REF] Mcclure | Separate neural systems value immediate and delayed monetary rewards[END_REF], joining previous work that has suggested more complex neural processes at play [START_REF] Kable | The neural correlates of subjective value during intertemporal choice[END_REF][START_REF] Ballard | Dissociable neural representations of future reward magnitude and delay during temporal discounting[END_REF][START_REF] Hare | Activity in dlPFC and its effective connectivity to vmPFC are associated with temporal discounting[END_REF][START_REF] Berkman | Self-Control as Value-Based Choice[END_REF]. Second, it also speaks against the idea that more frontoparietal activity is related to higher individual levels of self-control. Instead, it suggests that for which choice options control-related areas are activated is more informative than their overall level of activation. This finding is in line with value-based choice models of self-control [START_REF] Berkman | Self-Control as Value-Based Choice[END_REF] and with recent evidence that high and low discounters differ in how much attention they allocate to amount versus delay information [START_REF] Amasino | Amount and time exert independent influences on intertemporal choice[END_REF]. It also fits with the idea that low discounters may not need 'control' to discount less [START_REF] Lempert | Can delay discounting deliver on the promise of RDoC?[END_REF], and that high discounters may employ cognitive control for different types of decisions.

In line with the importance of prospection and self-projection in intertemporal decision-making, the hippocampus and adjacent midtemporal areas have been associated with individual differences in discounting [START_REF] Benoit | A neural mechanism mediating the impact of episodic prospection on farsighted decisions[END_REF][START_REF] Peters | The neural mechanisms of inter-temporal decision-making: understanding variability[END_REF] A BRAIN SIGNATURE OF DELAY DISCOUNTING 25

Büchel, 2011; [START_REF] Lebreton | A critical role for the hippocampus in the valuation of imagined outcomes[END_REF]. The k-marker has significant weights in parahippocampal areas and in occipital areas, but the weight maps were not strongly associated with broader meta-analysis-based activation maps of episodic memory or prospection, possibly because those masks also include many areas that are involved in processes other than memory and prospection. In agreement with the k-marker results

and the literature described above, our ROI-based results showed significant local prediction of individual differences in delay discounting in hippocampus, amygdala and the memory-related anteromedial thalamus.

Our findings highlight the importance of investigating distributed brain activity patterns, confirming the notion that delay discounting depends on the interactions among different functional processes and networks in the brain. In addition to frontoparietal areas, midcingulate, and hippocampus, several other cortical and subcortical areas also allowed for cross-validated local prediction of individual differences in delay discounting, across both data sets. Consistent with the whole-brain results, these included several areas in the mid and posterior insula, which is involved in interoception [START_REF] Craig | How do you feel--now? The anterior insula and human awareness[END_REF], salience [START_REF] Bartra | The valuation system: a coordinate-based metaanalysis of BOLD fMRI experiments examining neural correlates of subjective value[END_REF], and exploration [START_REF] Zhen | To learn or to gain: neural signatures of exploration in human decision-making[END_REF]-processes which may all be involved in delay discounting. Local prediction was also found in subcortical areas associated with affect and visceromotor control, including the amygdala and the periaqueductal gray.

Previous work has related individual differences in delay discounting with obesity, substance use disorders, and psychiatric diseases [START_REF] Peters | The neural mechanisms of inter-temporal decision-making: understanding variability[END_REF][START_REF] Amlung | Delay Discounting as a Transdiagnostic Process in Psychiatric Disorders: A Meta-analysis[END_REF][START_REF] Lempert | Can delay discounting deliver on the promise of RDoC?[END_REF]. In the two samples presented here, log(k) values based on participants' choices themselves were not significantly associated with BMI or overweight. However, the two studies were not designed to include a large range of BMI or many obese participants, and obesity-related alterations in discounting might be more pronounced for food than monetary rewards. In contrast to discounting behavior, responses to the k-marker did significantly differ between lean and overweight participants and predicted out-of-sample blood markers related to glucose and fat metabolism. These findings suggest that this functional brain marker reflects variance in neurophysiology that is related to stable long-term patterns in decision-making and health. The k-marker even seemed more sensitive to individual differences than the behavioral measures it was trained on, potentially because it is a closer reflection of the neurophysiological underpinnings that drive both discounting behavior and more distal health outcomes, in line with previous evidence that brain-based can outperform behavior-based prediction [START_REF] Genevsky | When brain beats behavior: neuroforecasting crowdfunding outcomes[END_REF]. Of note, our approach is cross-sectional and remains agnostic regarding potential causal links among brain function, behavior, and body weight. While higher discounting is typically considered a causal or predisposing factor for weight gain, an alternative hypothesis is that overweight and changes in metabolism lead to changes in brain physiology and subsequent behavioral outcomes [START_REF] Cornil | Obesity and responsiveness to food marketing before and after bariatric surgery[END_REF][START_REF] Schmidt | Resting-state connectivity within the brain's reward system predicts weight loss and correlates with leptin[END_REF]. Future work can test the k-marker in larger numbers of participants with obesity and other health conditions.

In conclusion, the k-marker-a novel fMRI-based brain signature-predicts individual differences in intertemporal decision-making in neurotypical, healthy adults across different populations, scanners, and analysis pipelines. It can be quantitatively tested in any other fMRI study on delay discounting for which contrast images for Choice, LL Amount, and Delay can be computed, including in other delay discounting paradigms, such as those that involve non-monetary rewards such as food or social discounting tasks [START_REF] Jones | Social discounting[END_REF][START_REF] Strombach | Social discounting involves modulation of neural value signals by temporoparietal junction[END_REF]. Future work could test the generalizability of the k-marker in children, adolescents, the elderly, or clinical populations.

Most importantly, future work will show whether the k-marker prospectively predicts clinical

Table 2. ROI-based prediction of individual differences in log(k). This analysis used

an existing atlas of the brain, including 485 regions based on several different previous parcellations [START_REF] Diedrichsen | A probabilistic MR atlas of the human cerebellum[END_REF][START_REF] Shen | Groupwise whole-brain parcellation from resting-state fMRI data for network node identification[END_REF][START_REF] Bär | Functional connectivity and network analysis of midbrain and brainstem nuclei[END_REF][START_REF] Glasser | A multi-modal parcellation of human cerebral cortex[END_REF] regions based on a combination of studies [START_REF] Shen | Groupwise whole-brain parcellation from resting-state fMRI data for network node identification[END_REF][START_REF] Bär | Functional connectivity and network analysis of midbrain and brainstem nuclei[END_REF]

Figure 2b -

 2b Figure 2b-c for additional results and random cross-validation folds). The explained variance of the prediction compared to a hypothetical mean model (prediction R 2 ) was R 2 =0.23. Individual differences in head motion (mean absolute framewise displacement) were neither related to log(k) (Pearson correlation: r=-0.05, p=0.58) nor predicted log(k) (Pearson correlation: r=0.01, p=0.92). Statistically controlling for age, education, head

  entirely on Study 1-could predict discounting in a completely independent data set. The validation data set (Study 2) was acquired on a different scanner, in a different lab and country, and using a different participant sample and different task characteristics, and was preprocessed and analyzed using different MRI analysis software. Evaluating the performance of the k-marker in Study 2 is therefore an even stronger test than crossvalidation in Study 1 alone.For this purpose, we computed the pattern expression of the k-marker using the matrix dot product for each participant's data (contrast images for Choice, parametric modulation for LL Amount, and Delay) in Study 2. The resulting predicted log(k) values were significantly correlated with actual log(k) values (Figure2c), Pearson correlation: r=0.45, p<0.001, 95%-CI=[0.31, 0.57], mean absolute error of 1.68) demonstrating the replicability of the k-marker in a completely independent data set. For the transfer test to Study 2, prediction R 2 was -3.2, indicating that, while the k-marker was very accurate in identifying the rankings among individuals, the absolute prediction values were less accurate than a hypothetical mean model.

  Figure4). At FDR-corrected level, voxels with significant weights in both directions

  areas associated with 'cognitive control' showed significant prediction in Study 1 and transfer to Study 2, areas associated with 'value' predicted discounting in Study 1 but no significant transfer to Study 2. Areas associated with 'episodic memory' showed only marginal prediction in Study 1 and no significant transfer to Study 2.

Figure 1 .

 1 Figure 1. Experimental design, analytical approach, and discounting behavior. a)

Figure 2 .

 2 Figure 2. Weight maps and predictive accuracy of the classifier ('k-marker'). a)

Figure 3 .

 3 Figure 3. Association of k-marker response with BMI, overweight, and metabolic

Figure 4 .

 4 Figure 4. Results for an alternative model trained on the data of Study 2 and

Figure 5 .

 5 Figure 5. Results (prediction-outcome correlations) for training separate predictive models for each of the three contrasts (control analysis). To assess their distinct

Figure 6 .

 6 Figure 6. Spatial similarity of the k-marker with meta-analytic maps. In order to quantitatively compare the classifier patterns with theoretically relevant functional

Figure 7 .

 7 Figure 7. Results for training separate predictive models in three different meta-

Figure 8 .Figure 1 - 1 .

 811 Figure 8. Region of interest (ROI)-based prediction of individual differences in discounting. An established cortical parcellation (Glasser et al., 2016) together with a

Figure 1 - 2 .

 12 Figure 1-2. Combination of SS, LL amounts, and delays used in Study 2. Indifference k

Figure 2 - 1 .

 21 Figure 2-1. Significant positive and negative weights contributing to the k-marker duringChoice screen onset (FDR corrected q < 0.05 across the whole feature space, i.e., three

Figure 2 - 2 .

 22 Figure 2-2. Significant positive and negative weights contributing to the k-marker for the

Figure 2 - 3 .

 23 Figure 2-3. Significant positive and negative weights contributing to the k-marker for the parametric modulation by Delay (FDR corrected q < 0.05 across the whole feature space,

  While in Study 1 participants' height and A BRAIN SIGNATURE OF DELAY DISCOUNTING 26 weight were measured by the experimenters, these values were self-reported in Study 2, which might explain the lower associations with BMI in Study 2.

Table 1 . Person-level characteristics of lean (BMI <= 25) and overweight-to-obese (BMI > 25) participants in Study 1 and Study 2.

 1 A BRAIN SIGNATURE OF DELAY DISCOUNTING 27 status and health outcomes in conditions related to abnormal discounting, such as eating disorders, substance use, and other psychiatric disorders. Except for body mass (by definition) and k-marker responses, none of the variables differed significantly between the two groups.

	Tables

  The table below shows only brain parcels that had significant prediction in both data sets (p-values are uncorrected for multiple comparisons). Note that each parcel contains many voxels, each of which may contribute with positive and/or negative weights to delay discounting for each of the three contrast images.

			A BRAIN SIGNATURE OF DELAY DISCOUNTING 39
	a24pr Bstem Midbd	Area anterior 24 prime Brainstem midbrain	L L	0.38 0.23	0.001 0.047	0.21 0.22	0.013 0.008
			Brainstem periaqueductal
	a24pr Bstem PAG	Area anterior 24 prime gray	R R&L	0.37 0.39	0.001 0.001	0.21 0.19	0.010 al., 0.023
	2016;	p32pr CA2	Pauli Area posterior 32 prime et	al., L	2016) 0.43	0.000	available 0.20	at: 0.016
	d32 https://github.com/canlab/Neuroimaging_Pattern_Masks/tree/master/Atlases_and_parcel Area dorsal 32 L 0.36 0.002 0.24 Hippocampus Hippocampus area CA2 R&L 0.22 0.040 0.20 0.015
	lations/2018_Wager_combined_atlas). Predictive patterns were trained and cross-Amygdala CM Centromedial amygdala R&L 0.22 0.038 0.17 0.042
		d32	Area dorsal 32	R	0.28	0.016	0.34	0.000
	validated on Study 1 data only (using the same procedures as for the global classifier) and
	8BM 8BM 13l 6a then tested in Study 2. Study 1 (training and Area 8BM L 0.25 0.033 Area 8BM R 0.30 0.012 Area 13I R 0.25 0.025 Area 6 anterior R 0.40 0.001 PoI2 Posterior Insular Area 2 R 0.35 0.004	0.29 0.26 -0.16 0.30 0.21	0.000 0.001 0.048 0.000 0.009
		MI	Middle Insular Area	R	cross-validation) 0.23 0.040	Study 2 (test) 0.39 0.000
			Anterior Ventral Insular		Prediction-		Prediction-
	Area name AVI	Area description Area	Hemisphere R	outcome r 0.30	p-value 0.009	outcome r 0.43	p-value 0.000
			Temporoparietooccipital
		RSC	RetroSplenial Complex	L	0.30	0.010	0.21	0.010
	TPOJ3	Junction 3	R	0.28	0.017	0.20	0.018
		RSC	RetroSplenial Complex	R	0.23	0.049	0.21	0.013
		IP1	Area lntraParietal11	R	0.29	0.013	0.23	0.006
		23d	Area dorsal 23	L	0.23	0.042	0.17	0.043
		PFm	Area PFm Complex	R	0.25	0.028	0.24	0.003
		7Am	Medial Area 7a	L	0.23	0.045	0.22	0.007
		PoI1	Posterior Insular Area 1	R	0.24	0.041	0.27	0.001
		7Am	Medial Area 7a	R	0.30	0.009	0.28	0.001
		FOP5	Area Frontal Opercular 5	R	0.27	0.022	0.36	0.000
		7PC	Area 7PC	R	0.28	0.016	0.20	0.014
		GPi	Internal globus pallidus	R	0.23	0.046	0.23	0.005
		Area1	Area 1 Anteromedial thalamic	L	0.23	0.037	0.18	0.035
	Thal_AM	nucleus	L	0.27	0.024	0.19	0.020
		p24pr	Area posterior 24 prime	L	0.30	0.014	0.21	0.011

days 20 25 0.0014

days 20

0.0117

We received funding from the ANR (Tremplin-ERC grant "Brain Gut Decision" to HP), Campus France (Marie-Sklodowska-Curie co-fund fellowship PRESTIGE-2018-2-0023 to LK), French federal funding (program 'Investissements d'avenir', ANR-10-IAIHU-06), the Federal Ministry of Education and Research Germany (Diet-Body-Brain grant 01EA1809B to BW and 01EA1707 to MCS), the National Cancer Institute (R35CA197461 [CL] and R01CA170297 [JK and CL]), and the AE Foundation (to JK). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Data and code availability statement

Data from Study 2 is available in an online public repository (doi: 10.18112/openneuro.ds002843.v1.0.0). Deidentified data from Study 1 will be made available upon publication. The resulting classifier patterns (k-marker) and code to apply it to other datasets is available at: https://github.com/ldmk/k-marker. Code for analyses is available at https://github.com/canlab and upon request to the first author.


Social influences on pain beyond the here and now

[Commentary on Bajcar et al., in press, EJP] Leonie Koban, PhD Lyon Neuroscience Research Center (CRNL), CNRS, Inserm, Université Claude Bernard Lyon 1, France Contact: leonie.koban@cnrs.fr This journal recently published a paper by Bajcar et al., entitled "From past pain to future pain through the pain of others: Information about other people's pain ratings can alleviate our subsequent pain." This paper adds novel insight to a growing literature on social influence effects on pain, by demonstrating that social feedback about one specific painful event can change how people will report pain for another, subsequent event.

In their experiment, Bajcar and colleagues tested whether seeing the pain ratings of other people can change their participants' own memory of a recent experience of pain and whether such social information might alter how they respond to a second, subsequent pain stimulation. All participants first experienced a 1-minute heat pain stimulation and rated how much pain they felt. Following a short break, they were asked to recall their initial pain and rate its intensity again. After this first recall, participants of the experimental group, but not the participants of the control group, were shown the pain ratings of other people, which were systematically lower than participants' own initial pain ratings. Then all participants were again asked to recall their initial pain and to rate it. After a 30-minutes break, all participants experienced a second pain stimulation and rated the intensity of this second pain stimulation. The authors reasoned that, if other people's ratings can change how we remember our own pain, then participants who had seen the low pain ratings of others would provide lower recalled pain intensity ratings at the time of the second recall. However, no significant group difference was observed during the pain recall. But the groups did differ significantly in how they rated the second pain stimulation half an hour later: those participants, who had seen the lower pain ratings of other people, reported the intensity of the second pain stimulation as less intense than the participants in the control group. This indicates that even if social information does not necessarily alter the memory of past pain experience, it can shift how another, future, painful event is experienced. This work resonates with previous findings that have shown that both direct observation of other people's pain (e.g., [START_REF] Craig | Social modeling influences on sensory decision theory and psychophysiological indexes of pain[END_REF]) and more abstract social information such as presentation of others' pain ratings (e.g., [START_REF] Koban | Beyond conformity: Social influences on pain reports and physiology[END_REF] can change participants' pain reports and even physiological responses to painful stimulation (indicating a change in arousal and affective brain responses). Bajcar and colleagues' findings also stimulate several ideas for future research. First, while they could not find evidence for a social influence effect on pain memory, research has shown that other types of memories can be modulated by social information. Thus, the idea that this might also be the case for pain memories should be further pursued. In the current study's design, both pain recalls (before and after seeing other people's ratings) were close in time. Therefore, participants likely remembered their previous recall as well as their initial pain rating and therefore might have strived to be consistent. While most people are influenced by others' opinions, they will usually not readily admit or reveal this. Future studies could build on this work to test for social influence effects on pain memories by using more indirect probes of participants' pain memories (for example measuring their pain recall in a different rating modality such that the need for self-consistency is not threatened).

Second, the findings of Bajcar et al. put forward the exciting idea that social influences on pain may go beyond the effects of a specific and immediate pain event, and instead extend in time to also influence subsequent experiences of pain. In other words, what we learn about others' experiences of a specific painful event could potentially generalize to how we experience other instances of pain. This suggests that by observing other people's behavior (i.e., pain ratings) in a specific situation, we implicitly learn about broader implicit social norms (i.e., how much pain should be reported and expressed in similar circumstances). Future studies could build on these findings to test how far social influence effects might extend in time. Future studies could also test whether social influence on a specific type of pain can generalize to other types of painful experiences. For example, if we learn that other people have rated heat pain as low, will this change how we experience and rate other type of pain? Uncovering whether and how social influences on pain extend over time, space, and situations could be key to better understand why social-contextual and cultural factors exert such powerful influences on pain in clinical and applied settings.


Pierre Larrenie
email: pierre.larrenie@esiee.fr

Jean-François Bercher
email: jean-francois.bercher@esiee.fr

Olivier Venard
email: olivier.venard@esiee.fr

Iyad Lahsen-Cherif
email: lahsencherif@inpt.ac.ma

LOW COMPLEXITY APPROACHES FOR END-TO-END LATENCY PREDICTION

Keywords: KPI Prediction, Machine Learning, General Regression, SDN, Networking, Queuing Theory, GNN

. We propose several low complexity, locally implementable approaches, achieving significantly lower wall time both for training and inference, with marginally worse prediction accuracy compared to state-of-the-art global GNN solutions.

Introduction

Routing while ensuring Quality of Service (QoS) is still a great challenge in any networks. Having powerful ways to transmit data is not sufficient, we must use resources wisely. This is true for wide static networks but even more for mobile networks with dynamic topology.

The emergence of Software-Defined Networking (SDN) [START_REF] Singh | A survey on Software Defined Networking: Architecture for next generation network[END_REF][START_REF] Amin | Hybrid SDN networks: A survey of existing approaches[END_REF] has made it possible to share data more efficiently between communication layers. Services are able to provide network requirements to routers based on their nature; routers acquire data about network performance, and finally allocate resources to meet these requirements. However, acquiring overall network performance can result in high consumption of network bandwidth for signalization; that is particularly constraining for networks with limited resources like Mobile Ad-Hoc Networks (MANET).

1 Note: This paper has been accepted for publication at IEEE 13th ICCCNT 2022. ©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. We consider network for which we wish to reduce the amount of signalization and perform intelligent routing.

k = T k = T -1 k = T -2 k = 0 A AGGREGATE B C D A C A E F A • • • • • • • • • • • • • • • • • •
In order to limit signalization, a first axis is to be able to estimate some key performance indicators (KPI) from other KPIs. A second point would be to be able to perform this prediction locally, at the node level, rather than a global estimation of the network. Finally, if predictions are to be performed locally, the complexity of the algorithms will need to be low, but still preserve good prediction quality. The question we address is thus the design of efficient and low-cost algorithms for KPI prediction, implementable at the local level. We focus on end-to-end latency prediction, for which we illustrate our approaches and results on a public dataset from the recent international challenge [START_REF] Suárez-Varela | The graph neural networking challenge: A worldwide competition for education in AI/ML for networks[END_REF].

The best performances of the state-of-the-art are obtained with Graph Neural Networks (GNNs) [START_REF] Rusek | Unveiling the potential of graph neural networks for network modeling and optimization in SDN[END_REF][START_REF]The graph neural networking challenge[END_REF][START_REF] Suárez-Varela | The graph neural networking challenge: A worldwide competition for education in AI/ML for networks[END_REF]. Although this is a global method while we favor local methods, we use these performances as a benchmark.

We first propose to use standard machine learning regression methods, for which we show that a careful feature engineering and feature selection (based on queue theory and the approach in [START_REF] De Aquino Afonso | GNNet challenge 2021[END_REF]) allows to obtain near state-of-the-art performances with a very low number of parameters and very low computational cost, with the ability to operate at the link level instead of a whole-graph level. Building on that, we show that it is even possible to obtain similar performances with a single feature and curve-fitting methods.

The presentation is structured as follows. In section 2, we first recall the key concepts on GNNs and queues; present some related works in the literature, before introducing the dataset used for the validation of our proposals. In section 3, we present the different approaches proposed, starting with the choice of features for machine learning methods, followed by general curve fitting methods. We then compare in section 4 the performances of these different approaches, in terms of performance as well as in terms of learning time and inference time. Finally, we conclude, discuss the overall results and draw some perspectives.

2 Related work and dataset

Graph Neural Networks (GNNs)

GNN [START_REF] Hamilton | Graph representation learning[END_REF][START_REF] Bacciu | A gentle introduction to deep learning for graphs[END_REF] is a machine learning paradigm that handles non-euclidean data: graphs. A graph is defined as a set of nodes and edges with some properties on its nodes and its edges. The key point in GNNs is the concept of Message Passing: each node of the graph will update its state according to states of its neighborhood by sending and receiving messages transmitted along edges. By repeating this mechanism T times, a node is able to capture states of its T -hop neighborhood as shown in Figure 1.

Queue Theory

Queue Theory is a well studied domain and for most of simple queue systems, explicit equations exist [START_REF] Cooper | Introduction to queueing theory[END_REF]. Further, we will refer to queue systems by using their Kendall's notation. We often take at reference M/M/1 and M/M/1/K for their markovian property, since equations are particularly easy to handle in this case.

However, for more general queue systems such as M/G/1 and M/G/1/K, equations are getting more complex. Whereas closed formulas exist for M/G/1 queues, M/G/1/K queues require to solve an equation system with K + 1 unknowns.

Queue systems analysis focus on stable queue, i.e. when the ratio ρ = λ µ ≤ 1 where λ (resp. µ) is the expected value of the arrival rate process (resp. service time). But finite queue systems are always stable since the maximal number of pending items is always finite and are subject to loss instead. To model the drop of incoming item in the queue we use the ratio ρ e = λe µ where λ e is known as the effective arrival rate and can be determined thanks to equation [START_REF] Suárez-Varela | The graph neural networking challenge: A worldwide competition for education in AI/ML for networks[END_REF].

λ e = λ(1 -π K ) = µ(1 -π 0 ) (1)
Where π 0 (resp. π K ) in the above equation ( 1) refers to the probability to the queue at equilibrium to be empty (resp. full).

Related Work

Chua, Ward, Zhang, et al. [START_REF] Chua | Stringer: Balancing latency and resource usage in service function chain provisioning[END_REF] present an heuristic and an Mixed Integer Programming approach to optimize Service Functions Chain provisioning when using Network Functions Virtualization for a service provider. Their approach relies on minimizing a trade-off between the expected latency and infrastructures resources.

Such optimization routing flow in SDN may need additional information to be exchanged between the nodes of a network. This results in an increase of the volume of signalization, by performing some measurements such as in [START_REF] Pasca | AMPS: Application aware multipath flow routing using machine learning in SDN[END_REF]. This is not a consequent problem in unconstrained networks, i.e. static wired networks with near-infinite bandwidth but may decrease performance of wireless network with poor capacity. An interesting solution to save bandwidth would be to predict some of the KPIs from other KPIs and data exchanged globally between nodes.

In [START_REF] Poularakis | SDN-enabled tactical ad hoc networks: Extending programmable control to the edge[END_REF][START_REF] Poularakis | Flexible SDN control in tactical ad hoc networks[END_REF], authors proposed a MANETs application of SDN in the domain of tactical networks. They proposed a multi-level SDN controllers architecture to build both secure and resilient networking. While orchestrating communication efficiently under military constraints such as: high-level of dynamism, frequent network failures, resources-limited devices. The proposed architecture is a trade-off between traditional centralized architecture of SDN and a decentralized architecture to meet dynamic in-network constraints.

Jahromi, Hines, and Delanev [START_REF] Jahromi | Towards application-aware networking: Ml-based end-toend application KPI/QoE metrics characterization in SDN[END_REF] proposed a Quality of Experience (QoE) management strategy in a SDN to optimize the loading time of all the tile of a mapping application. They have shown the impact of several KPIs on their application using a Generalized Linear Model (GLM). This mechanism make the application aware of the current network state.

Promising works rely on estimating KPIs at a graph-level. Note that it is very difficult, if not impossible, to address this analytically since computer networks models a complex structure of chained interfering queues for each flow in the network.

Rusek, Suárez-Varela, Mestres, et al. [START_REF] Rusek | Unveiling the potential of graph neural networks for network modeling and optimization in SDN[END_REF] used GNNs for predicting KPIs such as latency, error-rate and jitter. They relied on the Routenet architecture of Figure 2. The idea is to model the problem as a bipartite hypergraph mapping flows to links as depicted on Figure 3. Aggregating messages in such graph may result in predicting KPIs of the network in input. The model needs to know the routing scheme, traffic and links properties. Their result is very promising and has been the subject of two ITU Challenge in 2020 and 2021 [START_REF]The graph neural networking challenge[END_REF][START_REF] Suárez-Varela | The graph neural networking challenge: A worldwide competition for education in AI/ML for networks[END_REF]. These ITU challenges have very good results since the top-3 teams are around 2% error in delay prediction in the sense of Mean-Absolute Percentage Error (MAPE).

In [START_REF] De Aquino Afonso | GNNet challenge 2021[END_REF], very promising results were obtained with a a near 1% GNN model error (in the sense of MAPE) on the test set. The model mix analytical M/M/1/K queueing theory used to create extra-features to feed GNN model. In order to satisfy the constraint of scalability proposed by the challenge, the first part of model operates at the link level.

Dataset

We use public data from the challenge [START_REF] Suárez-Varela | The graph neural networking challenge: A worldwide competition for education in AI/ML for networks[END_REF] The dataset models static networks that have run for a certain amount of time; the obtained data is a mean of the global working period. The data contains information about It is important to point out that the proposed data is not in accordance with M/M/1/K queue models since process service time depends on the size of the packet. The size of the packet for each flow follows a Binomial distribution; it can be approximated by a Normal distribution inducing a general service time.

Nevertheless, it turns out that the system does not have the behavior of a M/G/1/K queue system globally but that of a complex system with interconnected queues that cannot be easily modeled.

Hence, approximating the system locally by a mixed of a simple analytical theory (M/M/1/K) and blackbox optimization (GNNs), as was proposed in [START_REF] De Aquino Afonso | GNNet challenge 2021[END_REF], is a good approach despite the lack of explicability or interpretability and the high-computational requirements with a lot of parameters to train. We show below that it is possible to obtain comparable performances with other regression approaches.

Our approaches

The main question is to define an estimator ŷ of the occupancy y according to the various available characteristics of the system, with a joint objective of low complexity and performance. In the following, we present regression approaches based on machine learning and then approaches based on curve-fitting.

Once an estimate of occupancy is obtained, it is possible to get the latency prediction dn for a specific link n by the simple relation

dn = ŷn E(|P n |) c n
where E(|P n |) is the observed average packet size on link n and c n the capacity of this link.

Performances will be evaluated using the MAPE loss-function

L (ŷ, y) = 100% N N n=1 ŷn -y n y n (2)
which is preferred to Mean Squared Error (MSE) because of its scale-invariant property.

Feature Engineering and Machine Learning

Based on the assumption that the system may be approximated by a model whose essential features come from M/M/1/K and M/G/1/K queue theory, we took essential parameters characterizing queueing systems, such as: ρ, ρ e , π 0 , π K , etc. and built further features by applying interactions and various non-linearities (powers, log, exponential, square root). Then, we selected features in this set by a forward step-wise selection method; i.e. by adding in turn each feature to potential models and keeping the feature with best performance. Finally, we selected the model with best MAPE error. For a linear regression model, this led us to select and keep a set of 4 simple features, which interestingly enough, have simple interpretations:

         π 0 = 1-ρ 1-ρ K+1 L = ρ + π 0 k kρ k ρ e = λe λ ρ = λe µ S e = k kρ k e ( 3
)
where L is the expected number of packets in the queue according to M/M/1/K, π 0 the probability that the queue is empty according to M/M/1/K theory, ρ e the effective queue utilization, and S e the unnormalized expected value of the effective number of packet in the queue buffer. These features can be thought as a kind of data preprocessing, before applying ML algorithms, and this turns out to be a key to achieving good performances. The 4 previous features have been kept as input for all the machine learning models.

Next we considered several machine learning algorithm, fitted on the training split and performances were evaluated by test split of a public dataset [START_REF] Suárez-Varela | The graph neural networking challenge: A worldwide competition for education in AI/ML for networks[END_REF]. Algorithms that were considered are: Multi-Layer Perceptron model (MLP) with 4 layers and with ReLU activation function, Linear Regression, Gradient Boosting Regression Tree (GBRT) with an ensemble of n = 100 estimators, Random Forest of n = 100 trees and Generalized Linear Model (GLM) with Poisson family and exponential link. All results of these methods are shown in Table 1.

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Curve Regression for occupancy prediction

There is a high interdependence of the features we selected in Equation 3, since all these features can be expressed in term of ρ e . Furthermore, it is confirmed by data exploration that ρ e is the prominent feature for occupancy prediction (and in turn latency prediction), as exemplified in Figure 5.

It is then tempting to try to further simplify our features space and to try to estimate the occupancy from a non-linear transformation of the single feature ρ e , as:

ŷ = g(ρ e ) ( 4
)
where ŷ is the estimate of the occupancy y. The concerns are of course to define simple and efficient functions g, with a low number of parameters, that can model the kind of growth shown in Figure 5, and of course to check that the performance remains interesting.

We followed three approaches to design the estimator g in order to predict links occupancy and end-to-end flow latency. In all cases, the parameters of g were computed by minimizing the mean squared of the regression error.

Figure 5: Data of ITU Challenge 2021 [START_REF] Suárez-Varela | The graph neural networking challenge: A worldwide competition for education in AI/ML for networks[END_REF], ρ e vs queue occupancy. Color-scale is an indicator of points cloud density.

Exponential of polynomial

The simplest approach is to use a curve-fitting regression of the form ŷ = g(ρ e ) = e pn(ρe) [START_REF]The graph neural networking challenge[END_REF] where

p n (x) ∈ R n [x]
is a polynomial of degree n with real coefficients.

In order to find coefficients of p n one can obviously consider predicting log(y) (where y denotes the queue occupancy). Choosing an arbitrary high polynomial degree results to oscillations and increases largely computation time. However choosing a too small degree does not allow the prediction of high occupancy.

Generative polynomials

The estimator g is defined as a linear combination of simple functions (f n ):

ŷ = g(ρ e ) = n α n • f n (ρ e ) (6)
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Generative polynomial similar to M/M/1/K theory The idea here is to use a polynomial f K n that will match approximately the expression in Equation 3 of the expected number of packets in the queue L.

       f K n (x) = φ K n (x) γn φ K n (x) = x n (1-x) 1-x K+1 n ≥ 0 ∀x ∈ [0; 1[ γ n = φ K n n n+1 n ≥ 0 (7)
where K is the size of the queue 1 The sequence of (f n ) K n=0 is finite and defined in interval [0; 1[. In order to improve regression capabilities, each f K n is defined as φ K n normalized by γ n , a local maximum of φ K n in the interval [0; 1[.

Bernstein Polynomials

The previous method relies on polynomial approximation. Since the expected value L can be expressed theoretically in terms of a polynomial of degree K, we are driven to the Bernstein polynomials that form a basis in the set of polynomial in the interval [0; 1]:

f K n (x) = K n x n (1 -x) K-n (8)
The approximation of any continuous function on [0; 1[ by a Bernstein polynomial converges uniformly.

Implicit function

The idea here is to define a set of N points θ n = (a n , b n ) and approximate the underlying function by linear interpolation between those points. To obtain a good positioning of these points, we select them as the solution of the following optimization problem:

min θ L(f θ (x), y) + α N n u n × u n+1 2 u n 2 u n+1 2 s.t. u n = θ n+1 -θ 0 n a 0 = 0 a N = b N = 1 a n+1 -a n ≥ 0 θ n = (a n , b n ) T ∈ [0; 1] 2 (9)
Equation 9 includes a first term for minimizing the interpolation error, and a second term weighted by a parameter α ≥ 0, to force θ n sequence to be as aligned and as far as possible. This implies that our sampling will be refined in high curvature zone of our function. The constraint formulated makes θ n an increasing sequence along the feature axis in order to get a correct interpolation of the curve, especially when N is high enough.

Comparison and Discussion

In this section, we evaluate our methods on the data from the GNN ITU Challenge 2021, described in subsection 2.4. We compare our results to those of the challenge winners, which establish the state-of-the-art in terms of pure performance. Since the actual labeled test dataset used for the challenge was released after the end of the challenge, all evaluations are performed on this particular dataset. The Table 1 presents the characteristics of the methods, in terms of the number of input features and parameters to be learned; their performance in the sense of MAPE and MSE; and the values of the execution times, both in learning time and inference time. All results were obtained with the same computer configuration: 120 Go RAM, 1 CPU Intel i9-9920X @ 3.50 GHz with 24 cores and 2 GPUs Nvidia TITAN RTX2080 24Go.

The methods used for comparison are divided into 3 groups, the first being the set of GNN approaches. 1 In results shown Table I., we consider K = 32 in order to match the data contained in the ITU challenge dataset [START_REF] Suárez-Varela | The graph neural networking challenge: A worldwide competition for education in AI/ML for networks[END_REF]. In the second group, we used classical machine learning models with only 4 input features obtained by stepwise selection, as presented in Equation 3.

In the third group, we group curve regression models using a single well-chosen feature, namely ρ e , as presented in subsection 3.2.

As we can observe, the proposed approaches achieve a much lower computational time than the GNN approaches, both in terms of learning time and inference time; this at the cost of a marginal performance degradation.

Moreover, non-GNN approaches provide a more local solution since predictions are performed at the link level and not at the whole graph level (Models predict queues occupancy, then compute analytically delay for each link and finally aggregate along path). This would allow to use them for simple local predictions, without having to rely on the global knowledge and prediction of the network.

The consequent gain in computational time of our low-complexity approaches is that they use far fewer parameters, which reduces the amount of data needed for training. The reduction in the number of parameters and the architecture (number of operations) of the solutions explains the drop in learning and inference times.

Nevertheless, when we match the distribution as presented in Figure 5, we notice that most of our data are on a low occupancy level. In practice, some models have a kind of limited behavior when the occupation of the targeted queue is close to 100%: there is a significant over-or under-prediction. However, this behavior does not really affect the overall performance due to the low density of this scenario in our dataset and the predicted values are close enough to the targets.

Conclusion

In this paper, we considered the problem of designing efficient and low-cost algorithms for KPI prediction, implementable at the local level. We have argued and proposed several alternatives to GNNs for predicting the queue occupancy of a complex system using simple ML models with carefully chosen features or general curve-fitting methods.

At the cost of a marginal performance loss, our proposals are characterized by low complexity, significantly lower learning and inference times compared to GNNs, and the possibility of local deployment. Thus, this type of solution can be used for continuous performance monitoring.

The low complexity and structures of linear regression algorithms or curve-fitting solutions should also be suitable for adaptive formulations. These last two points are current perspectives of this work. Of course, the approaches considered here will have to be considered and adapted for other types of KPI, such as error-rate or jitter.

Figure 1 :

 1 Figure 1: GNN repeating T Message Passing mechanisms: message propagation and aggregation (inspired from [7])

Figure 2 : 3 F 2 F 1 (

 2321 Figure 2: Routenet Architecture [4]

F 1 F 2 F 3 LFigure 3 :

 1233 Figure 3: Routenet [4] paths-links hypergraph transformation applied on a simple topology graph carrying 3 flows. (a) Black circles represents communication node, double headed arrows between them denotes available symmetric communications links and dotted arrows shows flows path. (b) Circle (resp. dotted) represents links (resp. flows) entities defined in the first graph (Lij is the symmetric link between node i and node j.). Unidirectional arrows encode the relation "<flow> is carried by <link>".

Figure 4 :

 4 Figure 4: End-to-end latency distribution on train and test datasets of ITU Challenge 2021, where train and test datasets describe networks of very different sizes.[1]

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  Note: This paper has been accepted for publication at IEEE 13th ICCCNT 2022. ©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

	Approaches	Input Features	Model Parameters	MAPE	MSE	Wall Training Time	Wall Inference Time
	Routenet [4]	Topology	-	100%	(N/A)	≈12h	-
	Top-1 ITU Challenge Team (PARANA)[6]	Traffic matrix Routing Scheme	654006	1.27%	1.10e-5	≈8h	214s
	MLP Linear Regression *+ GBRT (n=100) * Random Forest (n=100) * GLM -Poisson +	π 0 L S e ρ e	291 4 4 4 4	1.91% 1.74% 1.73% 1.69% 3.68%	3.18e-5 3.20e-5 2.90-5 3.00e-5 5.09e-4	≈45min <1sec ≈1min <1sec ≈1min	8.26s 0.296s 0.867s 0.994s 0.481s
	Curve-fitting exponential (deg=3) +		4	3.94%	3.75e-4	≈1sec	0.311s
	Curve-fitting exponential (deg=8) +		9	1.70%	3.53e-5	≈5secs	0.320
	Curve-fitting M/M/1/K **+ Curve-fitting Bernstein **+	ρ e	33 33	2.04% 1.68%	4.42e-5 3.13e-5	≈3min ≈2min	3.55s 3.14s
	Sampling Optimization (N = 12, α = 0) *+		24	1.77%	3.18e-5	≈1min	0.281s
	Sampling Optimization (N = 12, α =1e-5) *		24	1.77%	3.18e-5	≈1min	0.306s

Table 1 :

 1 Results Synthesis of various models for flow latency prediction. Test dataset from[START_REF] Suárez-Varela | The graph neural networking challenge: A worldwide competition for education in AI/ML for networks[END_REF] only 500,000 samples used for training (2.25% of training dataset); ** only 5,000,000 samples used for training (22.5% of training dataset); + under-estimation/over-estimation occurs on high queue occupancy prediction

*

Note: This paper has been accepted for publication at IEEE 13th ICCCNT 2022. ©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

Finally, a last point that deserves interest is the fact that these low complexity models can be interpreted/explained either by direct inspection (visualization), or by using tools such as Shapley values [START_REF] Lundberg | A unified approach to interpreting model predictions[END_REF] which allow to interpret output values by measuring contributions of each input feature on the prediction.