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:
L=Lcollider× LHiggs× LDM× LEWPO× Lflavour×…
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:
L=Lcollider× LHiggs× LDM× LEWPO× Lflavour×…
Global fits need quick, but sufficiently accurate theory predictions
BSM scans today easily require ∼107 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→g~g~, g~q~i, q~iq~j,
q~iq~j∗, b~ib~i∗, t~it~i∗
at s=13 TeV
xsec 1.0 performs smart regression for strong SUSY processes at the LHC
- 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, αs 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
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(x)∼GP[m(x),k(x,x′)]
f(\vec x) \sim \mathcal{GP}\left[m(\vec x), k(\vec x, \vec x')\right]
E[f(x)]=m(x)
\mathrm{E}\left[f(\vec x)\right] = m(\vec x)
Cov[f(x),f(x′)]=k(x,x′)
\mathrm{Cov}\left[f(\vec x), f(\vec x')\right] = k(\vec x, \vec x')
The Bayesian Way: quantify beliefs with probability
k(x,x′)=σf2exp(−l2∣x−x′∣2)
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(x)∼GP[m(x),k(x,x′)]
f(\vec x) \sim \mathcal{GP}\left[m(\vec x), k(\vec x, \vec x')\right]
E[f(x)]=m(x)
\mathrm{E}\left[f(\vec x)\right] = m(\vec x)
Cov[f(x),f(x′)]=k(x,x′)
\mathrm{Cov}\left[f(\vec x), f(\vec x')\right] = k(\vec x, \vec x')
posterior distribution over functions
with updated m(x)
data
The Bayesian Way: quantify beliefs with probability
k(x,x′)=σf2exp(−l2∣x−x′∣2)
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
Let's make some draws from this prior distribution
Let's make some draws from this prior distribution
Let's make some draws from this prior distribution
Let's make some draws from this prior distribution
Let's make some draws from this prior distribution
Let's make some draws from this prior distribution
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)
At each input point, we obtain a distribution of possible function values (prior to looking at data)
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
We obtain a posterior by conditioning on known data
Make posterior predictions at new points by computing correlations to known points
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
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 O(n3), prediction as O(n2)
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"
Validation results
Gluino-gluino
Gluino-squark
Squark-squark
Squark-antisquark
Fast estimate of SUSY (strong) production cross- sections at NLO, and uncertainties from
- regression itself
- renormalisation scale
- PDF variation
- αs variation
Goal
pp→g~g~, g~q~i, q~iq~j,
q~iq~j∗, b~ib~i∗, t~it~i∗
Interface
Method
Pre-trained, distributed Gaussian processes
Python tool with command-line interface
Processes
at s=13 TeV
Tutorial on Friday!
Thank you!
Backup slides
Some linear algebra
Regression problem, with 'measurement' noise:
y=f(x)+ε, ε∼N(0,σn2)→ infer f, given data D={X,y}
Assume covariance structure expressed by a kernel function, like
k(xi,xj)=σf2exp(−2l2∣xi−xj∣2)+σn2δij
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
[x1,x2,…]
[y1,y2,…]
signal kernel
white-noise kernel
Some linear algebra
Regression problem, with 'measurement' noise:
y=f(x)+ε, ε∼N(0,σn2)→ infer f, given data D={X,y}
Training: optimise kernel hyperparameters by maximising the marginal likelihood
p(y ∣ X,θ)=N(y ∣ m(X),Kθ)
p\left( \vec y\ |\ \vec X, \vec \theta \right)
= \mathcal{N} \left( \vec y \ |\ m(\vec X), K_{\vec\theta} \right)
y∗∣D,x∗∼N(μ∗,σ∗2)
y_*\, |\, \mathcal D, \vec x_* \sim \mathcal{N}\left(\mu_*, \sigma_*^2\right)
∝∣Kθ∣−21×exp{−21(y−m(X))TKθ−1(y−m(X))}
\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 x∗ :
with
μ∗=m(x∗)+K∗K−1(y−m(X))
\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 X
σ∗2=K∗∗−K∗K−1K∗T
\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(X,X)K∗=k(x∗,X)K∗∗=k(x∗,x∗)
K = k(\vec X, \vec X)
\\
K_* = k(x_*, \vec X)
\\
K_{**} = k(\vec x_*, \vec x_*)
Scale-dependence of LO/NLO
[Beenakker+, hep-ph/9610490]
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
Accelerating cross-section predictions with XSEC
By jeriek
Accelerating cross-section predictions with XSEC
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