Accelerated cross-section predictions with XSEC

Jeriek Van den Abeele

Tools 2020

@JeriekVda

Based on arXiv:2006.16273 with A. Buckley, A. Kvellestad, A. Raklev, P. Scott, J. V. Sparre, and I. A. Vazquez-Holm

IP2I, Lyon

Global fits address the need for a consistent comparison of BSM theories to all relevant experimental data

Challenge:

Scanning increasingly high-dimensional parameter spaces with varying phenomenology

Exploration of a combined likelihood function:

$\mathcal{L} = \mathcal{L}_\mathsf{collider} \times \mathcal{L}_\mathsf{Higgs} \times \mathcal{L}_\mathsf{DM} \times \mathcal{L}_\mathsf{EWPO} \times \mathcal{L}_\mathsf{flavour} \times \ldots$

See GAMBIT talk Friday!

Global fits address the need for a consistent comparison of BSM theories to all relevant experimental data

Challenge:

Scanning increasingly high-dimensional parameter spaces with varying phenomenology

Exploration of a combined likelihood function:

$\mathcal{L} = \mathcal{L}_\mathsf{collider} \times \mathcal{L}_\mathsf{Higgs} \times \mathcal{L}_\mathsf{DM} \times \mathcal{L}_\mathsf{EWPO} \times \mathcal{L}_\mathsf{flavour} \times \ldots$

Global fits need quick, but sufficiently accurate theory predictions

BSM scans today easily require $\sim 10^7$ samples or more

Higher-order BSM production cross-sections and theoretical uncertainties make a significant difference!

[GAMBIT, 1705.07919]

CMSSM

[hep-ph/9610490]

Existing higher-order evaluation tools are insufficient for large MSSM scans

Prospino/MadGraph: full calculation, minutes/hours per point
N(N)LL-fast: fast grid interpolation, but only degenerate squark masses

$$ pp\to\tilde g \tilde g,\ \tilde g \tilde q_i,\ \tilde q_i \tilde q_j, $$

$$\tilde q_i \tilde q_j^{*},\ \tilde b_i \tilde b_i^{*},\ \tilde t_i \tilde t_i^{*}$$

at $\mathsf{\sqrt{s}=13} $ TeV

xsec 1.0 performs smart regression for strong SUSY processes at the LHC

arXiv:2006.16273

github.com/jeriek/xsec

New Python tool for NLO cross-section predictions within seconds

A. Buckley, A. Kvellestad, A. Raklev, P. Scott, J. V. Sparre, JVDA, I. A. Vazquez-Holm

Pointwise estimates of PDF, $\alpha_s$ and scale uncertainties, in addition to the subdominant regression uncertainty

Future expansions: other processes, more CoM energies, higher-order corrections, ...

$$$

A quick sketch

Interpolation ...

Interpolation ... doesn't give prediction uncertainty!

Correlation length-scale

Estimate from data!

Correlation length-scale

Estimate from data!

Correlation length-scale

Gaussian process prediction with uncertainty

Estimate from data!

prior distribution over all functions

with the estimated smoothness

f(\vec x) \sim \mathcal{GP}\left[m(\vec x), k(\vec x, \vec x')\right]

\mathrm{E}\left[f(\vec x)\right] = m(\vec x)

\mathrm{Cov}\left[f(\vec x), f(\vec x')\right] = k(\vec x, \vec x')

The Bayesian Way: quantify beliefs with probability

k(\vec x, \vec x') = \sigma_f^2 \exp\left(-\frac{|\vec x-\vec x'|^2}{l^2}\right)

prior distribution over all functions

with the estimated smoothness

f(\vec x) \sim \mathcal{GP}\left[m(\vec x), k(\vec x, \vec x')\right]

\mathrm{E}\left[f(\vec x)\right] = m(\vec x)

\mathrm{Cov}\left[f(\vec x), f(\vec x')\right] = k(\vec x, \vec x')

posterior distribution over functions

with updated $m(\vec x)$

data

The Bayesian Way: quantify beliefs with probability

k(\vec x, \vec x') = \sigma_f^2 \exp\left(-\frac{|\vec x-\vec x'|^2}{l^2}\right)

Let's make some draws from this prior distribution

At each input point, we obtain a distribution of possible function values (prior to looking at data)

A Gaussian process sets up an infinite number of correlated Gaussians, one at each parameter point

We obtain a posterior by conditioning on known data

target function

We obtain a posterior by conditioning on known data

target function

We obtain a posterior by conditioning on known data

Make posterior predictions at new points by computing correlations to known points

Posterior predictive distributions are Gaussian too

Squared Exponential kernel

Matérn-3/2 kernel

Different kernels lead to different function spaces to marginalise over

My kingdom for a good kernel ...

GPs allow us to use probabilistic inference to learn a function from data, in an interpretable, analytical, yet non-parametric Bayesian framework.

A GP model is fully specified once the mean function, the kernel and its hyperparameters are chosen.

The probabilistic interpretation only holds under the assumption that the chosen kernel accurately describes the true correlation structure.

The choice of kernel allows for great flexibility. But once chosen, it fixes the type of functions likely under the GP prior and determines the kind of structure captured by the model, e.g., periodicity and differentiability.

[D. Duvenaud, PhD thesis]

My kingdom for a good kernel ...

[D. Duvenaud, PhD thesis]

My kingdom for a good kernel ...

[D. Duvenaud, PhD thesis]

My kingdom for a good kernel ...

For our multi-dimensional case of cross-section regression, we get good results by multiplying Matérn ($\nu = 3/2$) kernels over the different mass dimensions:

\displaystyle k_s(\vec x, \vec x') = \prod_{d=1}^D \sigma_{f_d}^2 \left( 1 + \sqrt{3} \frac{|x'_d - x_d|}{l_d} \right) \exp\left(-\sqrt{3}\frac{|x'_d - x_d|}{l_d}\right)

The different lengthscale parameters $l_d$ lead to automatic relevance determination for each feature: short-range correlations for important features over which the target function varies strongly.

This is an anisotropic, stationary kernel. It allows for functions that are less smooth than with the standard squared-exponential kernel.

[Deisenroth+, Distill.pub]

My kingdom for a good kernel ...

... with good hyperparameters

Short lengthscale, small noise

Long lengthscale, large noise

Underfitting, almost linear

Overfitting of fluctuations,

[Deisenroth+, Distill.pub]

can lead to large uncertainties!

Typically, kernel hyperparameters are estimated by maximising the (log) marginal likelihood $p( \vec y\ |\ \vec X, \vec \theta) $, aka the empirical Bayes method.

Alternative: MCMC integration over a range of $\vec \theta$.

Gradient-based optimisation can get stuck in local optima and plateaus. Multiple initialisations can help, or global optimisation methods like differential evolution.

Global optimum: somewhere in between

[Deisenroth+, unpublished]

... with good hyperparameters

Typically, kernel hyperparameters are estimated by maximising the (log) marginal likelihood $p( \vec y\ |\ \vec X, \vec \theta) $, aka the empirical Bayes method.

Alternative: MCMC integration over a range of $\vec \theta$.

Gradient-based optimisation can get stuck in local optima and plateaus. Multiple initialisations can help, or global optimisation methods like differential evolution.

The standard approach systematically underestimates prediction errors.

After accounting for the additional uncertainty from learning the hyper-parameters, the prediction error increases when far from training points.

[Wågberg+, 1606.03865]

... with good hyperparameters

Other tricks to improve the numerical stability of training:

Normalising features and target values
Factoring out known behaviour from the target values
Training on log-transformed targets, for extreme values and/or a large range, or to ensure positive predictions

... with good hyperparameters

Workflow

Generating data

Random sampling

SUSY spectrum

Cross-sections

Optimise kernel hyperparameters

Training GPs

GP predictions

Input parameters

Linear algebra

Cross-section

estimates

Compute covariances between training points

Workflow

Generating data

Random sampling

SUSY spectrum

Cross-sections

Optimise kernel hyperparameters

Training GPs

GP predictions

Input parameters

Linear algebra

Cross-section

estimates

XSEC

Compute covariances between training points

Training scales as $\mathcal{O}(n^3)$, prediction as $\mathcal{O}(n^2)$

A balancing act

Random sampling with different priors, directly in mass space

Evaluation speed

Sample coverage

Need to cover a large parameter space

Distributed Gaussian processes

[Liu+, 1806.00720]

Distributed Gaussian Processes

Divide-and-conquer approach for dealing with large datasets

Make local predictions with smaller data subsets
Compute a weighted average of these predictions

The exact weighting procedure is important, to ensure

Smooth final predictions
Valid regression uncertainties

"Generalized Robust Bayesian Committee Machine"

Sometimes, a curious problem arises: negative predictive variances!

It is due to numerical errors when computing the inverse of the covariance matrix $K$. When $K$ contains many training points, there is a good chance that some of them are similar:

\vec x_1 \sim \vec x_2 \quad \implies \quad k(\vec x_1, \vec X) \simeq k(\vec x_2, \vec X)

GP regularisation

Nearly equal columns make $K$ ill-conditioned. One or more eigenvalues $\lambda_i$ are close to zero and $K$ can no longer be inverted reliably. The number of significant digits lost is roughly the $\log_{10}$ of the condition number

\kappa = \frac{\lambda_\mathsf{max}}{\lambda_\mathsf{min}}

This becomes problematic when $\kappa \gtrsim 10^8 $. In the worst-case scenario,

\kappa = N \left(\frac{\sigma_f}{\sigma_n}\right)^2 + 1

signal-to-noise ratio

number of points

[Deisenroth+, Distill.pub]

GP regularisation

Add a white-noise component to the kernel
- Can compute minimal value that reduces $\kappa$ sufficiently
- Can model discrepancy between GP model and latent function, e.g., regarding assumptions of stationarity and differentiability

[Deisenroth+, Distill.pub]

[Mohammadi+, 1602.00853]

Avoid a large S/N ratio by adding a likelihood penalty term guiding the hyperparameter optimisation
- Nearly noiseless data is problematic ... Artificially increasing the noise level may be necessary, degrading the model a bit

Validation results

Gluino-gluino

Gluino-squark

Squark-squark

Squark-antisquark

Getting started with XSEC

github.com/jeriek/xsec

pip install xsec
xsec-download-gprocs --process_type gg

# Set directory and cache choices
xsec.init(data_dir="gprocs")

# Set center-of-mass energy (in GeV)
xsec.set_energy(13000)

# Load GP models for the specified process(es)
processes = [(1000021, 1000021)]
xsec.load_processes(processes)

# Enter dictionary with parameter values
xsec.set_parameters(
    {
        "m1000021": 1000,
        "m1000001": 500,
        "m1000002": 500,
        "m1000003": 500,
        "m1000004": 500,
        "m1000005": 500,
        "m1000006": 500,
        "m2000001": 500,
        "m2000002": 500,
        "m2000003": 500,
        "m2000004": 500,
        "m2000005": 500,
        "m2000006": 500,
        "sbotmix11": 0,
        "stopmix11": 0,
        "mean": 500,
    }
)

# Evaluate the cross-section with the given input parameters
xsec.eval_xsection()

# Finalise the evaluation procedure
xsec.finalise()

Fast estimate of SUSY (strong) production cross- sections at NLO, and uncertainties from

regression itself
renormalisation scale
PDF variation
$\alpha_s$ variation

Goal

$$ pp\to\tilde g \tilde g,\ \tilde g \tilde q_i,\ \tilde q_i \tilde q_j, $$

$$\tilde q_i \tilde q_j^{*},\ \tilde b_i \tilde b_i^{*},\ \tilde t_i \tilde t_i^{*}$$

Interface

Method

Pre-trained, distributed Gaussian processes

Python tool with command-line interface

Processes

at $\mathsf{\sqrt{s}=13}$ TeV

github.com/jeriek/xsec

arXiv:2006.16273

Tutorial on Friday!

Thank you!

Backup slides

Some linear algebra

Regression problem, with 'measurement' noise:

$y=f(\vec x) + \varepsilon, \ \varepsilon\sim \mathcal{N}(0,\sigma_n^2) \quad \rightarrow \quad $ infer $f$, given data $\mathcal{D} = \{\vec X, \vec y\}$

Assume covariance structure expressed by a kernel function, like

k(\vec x_i, \vec x_j) = \sigma_f^2\, \exp\!\left(-\frac{|\vec x_i - \vec x_j|^2}{2\mathcal{l}^2}\right) + \sigma_n^2\, \delta_{ij}

Consider the data as a sample from a multivariate Gaussian distribution

$[\vec x_1, \vec x_2, \ldots]$

$[y_1, y_2, \ldots]$

signal kernel

white-noise kernel

Some linear algebra

Regression problem, with 'measurement' noise:

$y=f(\vec x) + \varepsilon, \ \varepsilon\sim \mathcal{N}(0,\sigma_n^2) \quad \rightarrow \quad $ infer $f$, given data $\mathcal{D} = \{\vec X, \vec y\}$

Training: optimise kernel hyperparameters by maximising the marginal likelihood

p\left( \vec y\ |\ \vec X, \vec \theta \right) = \mathcal{N} \left( \vec y \ |\ m(\vec X), K_{\vec\theta} \right)

y_*\, |\, \mathcal D, \vec x_* \sim \mathcal{N}\left(\mu_*, \sigma_*^2\right)

\propto |K_{\vec\theta}|^{-\frac{1}{2}} \times \exp \left\{ -\frac{1}{2} \left( \vec y - m(\vec X) \right)^T K_{\vec\theta}^{-1} \left(\vec y - m(\vec X) \right) \right\} % %%k(\vec X, \vec X)^{-1} \left( \vec y - \mu(\vec X) \right), % k(\vec x_*, \vec x_*) - k(\vec x_*, \vec X) k(\vec X, \vec X)^{-1} k(\vec X, \vec x_*)

Posterior predictive distribution at a new point $\vec x_*$ :

with

\mu_* = m(\vec x_*) + K_* K^{-1}(\vec{y}-m(\vec X)) %%\tilde\mu(\vec x_*) = \mu(\vec x_*) + k(\vec x_*, \vec X) k(\vec X, \vec X)^{-1}\vec{y}\\ %\tilde\sigma^2(\vec x_*) = k(\vec x_*, \vec x_*) - k(\vec x_*, \vec X) k(\vec X, \vec X)^{-1} k(\vec X, \vec x_*)

Implicit integration over points not in $\vec X$

\sigma_*^2 = K_{**}-K_*K^{-1}K_*^T %%\tilde\mu(\vec x_*) = \mu(\vec x_*) + k(\vec x_*, \vec X) k(\vec X, \vec X)^{-1}\vec{y}\\ %\tilde\sigma^2(\vec x_*) = k(\vec x_*, \vec x_*) - k(\vec x_*, \vec X) k(\vec X, \vec X)^{-1} k(\vec X, \vec x_*)

[

K = k(\vec X, \vec X) \\ K_* = k(x_*, \vec X) \\ K_{**} = k(\vec x_*, \vec x_*)

Gaussian Processes 101 (non-parametric regression)

prior over functions

The covariance matrix controls smoothness.

Assume it is given by a kernel function, like

k(x_1, x_2) = \exp\!\left(-\frac{|x_2-x_1|^2}{2\mathcal{l}^2}\right).

posterior over functions

Bayesian approach to estimate $ y_* = f(x_*) $ :

y_* | \mathbf{y} \sim \mathcal{N}(K_* K^{-1}\mathbf{y},\ K_{**}-K_*K^{-1}K_*^T)

Consider the data as a sample from a multivariate Gaussian distribution.

\begin{pmatrix} \mathbf{y} \\ y_{*} \end{pmatrix} \sim \mathcal{N}{\left( \begin{pmatrix} \mathbf{0} \\ 0 \end{pmatrix} , \begin{bmatrix} K & K_{*}\\ K_{*}^T & K_{**}\\ \end{bmatrix} \right)}

K_{ij} = k(x_i, x_j) \\ K_{*i}= k(x_*,x_i) \\ K_{**} = k(x_*,x_*)

data

mean

covariance

Scale-dependence of LO/NLO

[Beenakker+, hep-ph/9610490]