A Tutorial on Differentiable Analysis & end-to-end learning
Nathan Simpson
PyHEP, 15/09/22
Two software libraries:
A suite of differentiable operations designed to target typical HEP use cases.
A method for optimizing observables in an end-to-end way, incorporating systematics
Also: a software package that implements helper functions for this use case
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built with
https://github.com/gradhep/relaxed
https://github.com/gradhep/neos
Tangent:
how do neural networks learn at all?
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data
activation(weight*data + bias)
learnable free parameters
“architecture”
(how you combine parameters w/ data)
“parameters”
φ
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data
φi
result
feedback?
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data
φi
result
objective(result)
(= scalar representing how good our result is)
want to minimise
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data
φi
result
objective(result)
(= scalar representing how good our result is)
minimisation
update rule?
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data
φi
result
objective(result)
(= scalar representing how good our result is)
φi+1 = φi - lr * d(workflow(data, φi))/dφi
= workflow(data, φi)
trying to roll downhill in param space!
positive
gradient
negative
step
update rule: gradient descent
step size
(“learning rate”)
gradient of workflow w.r.t. current parameters
We don’t need neural networks to do this!
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(but they are often quite useful, so you’ll see some more later)
Same thing with a straight line:
Hard to say where “model” ends and “objective” begins.
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data
φi = {m,c}
result
objective(result)
φi+1 = φi - lr * d(workflow(data, φi))/dφi
= workflow(data, φi)
still works!
as long as we can calculate this gradient!
y = mx + c
e.g. for 2D data:
data on left of line = signal,
on right = background
Idea:
Using gradient descent, we can optimise any workflow parameters with respect to any goal… *if* the full workflow is differentiable.
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A typical HEP analysis workflow
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A typical HEP analysis workflow
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More abstractly: step with free parameters (e.g. event selection)
A typical HEP analysis workflow
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A typical HEP analysis workflow
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A typical HEP analysis workflow
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A typical HEP analysis workflow
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A typical HEP analysis workflow
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A typical HEP analysis workflow
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(chain rule)
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In equation form:
but wait, this is all code right?
how do we differentiate a computer program?
How might we get those gradients?
Automatic differentiation!
Quick explanation:
→ exact, efficient gradients
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Thanks to deep learning’s prominence, we have many great software libraries [JAX, PyTorch, TensorFlow] that take advantage of hardware acceleration (GPUs, TPUs)
Img source: AskPython.com
So is that it?
We code up our analysis in PyTorch and fit the model?
…not quite :)
Not all operations can be broken into differentiable primitives, because not all operations are differentiable!
Need to figure out a way to “relax” some operations to allow us to take their gradients.
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Pictured: One very discrete boi.
Every step of the workflow needs to be differentiable!
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already differentiable
not necessarily differentiable a priori
=> Let’s change that!
In one slide: Making analysis differentiable
Example: histograms [very discrete!]
*See Kyle Cranmer’s (heavily cited) paper on this: arxiv.org/abs/hep-ex/0011057
In one slide: Making analysis differentiable
Example: histograms [very discrete!]
We developed a histogram-alternative using kernel density estimates (KDEs). [used already in HEP!]*
Integrating the KDE over a set of intervals gives the notion of “bins”. => Binned KDE (bKDE)
*See Kyle Cranmer’s (heavily cited) paper on this: arxiv.org/abs/hep-ex/0011057
In one slide: Making analysis differentiable
Example: histograms [very discrete!]
We developed a histogram-alternative using kernel density estimates (KDEs). [used already in HEP!]*
Integrating the KDE over a set of intervals gives the notion of “bins”. => Binned KDE (bKDE)
Also have:
*See Kyle Cranmer’s (heavily cited) paper on this: arxiv.org/abs/hep-ex/0011057
Now time for some code!
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What makes a good observable?
Searches for new physics endeavour to maximally discriminate simulated signal data from background processes.
But is this really what we want?
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signal
background
e.g. neural network w/ 1-D output, trained to minimize binary cross-entropy
What makes a good observable?
Searches for new physics endeavour to maximally discriminate simulated signal data from background processes.
But is this really what we want?
e.g. what happens when we include systematic variations of the signal/background?
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?
“(...) sensitivity to high-level physics questions must account for systematic uncertainties, which involve a nonlinear trade-off between the typical machine learning performance metrics and the systematic uncertainty estimates. ”
Deep Learning and its applications to LHC Physics, section 3.1,
D.Guest, K.Cranmer, D.Whiteson, 2018
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(emphasis not in original text)
Can we learn to incorporate systematics?
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Idea 2:
We can directly optimise the discovery significance/CLs of our analysis this way!
-> Systematic aware [profiling]
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Oh baby it’s code time!
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That’s it!
and thanks for listening!
If you want to:
> discuss more about this in any way
> have an interesting use case
> talk about future opportunities
> send me pet images
please reach out! email: n.s@cern.ch
I’d love to hear from you :)
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one of my cats, enjoying the homely comfort of the washing machine
This work was partially supported by the Insights ITN, funded by the European Union’s Horizon 2020 research and innovation programme, call H2020-MSCA-ITN-2017, under Grant Agreement n. 765710.
Work now supported by the Swedish Science Council and Lund University directly.
(until November)
Seeing it in practice
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Taken from my tutorial repo: github.com/gradhep/differentiable-analysis-examples/
Toy example: 1-bin counting experiment
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s = 15 + φ
b = 45 − 2φ
σb = 1 + (φ/5)²
Increasing s/b ratio
Increasing uncertainty on background
Increasing φ
Learning to discover: 1-bin example
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We’re able to recover the optimal significance in our toy problem!
Intuitively, we’re trading off uncertainty and s/b ratio in order to give the best result.
for pdf viewers: optimisation with respect to significance is able to find the optimal significance accounting for uncertainty (minimum of blue curve)
Optimising a neural network observable (neos)
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for pdf viewers: neural network contours wrap around the signal blob, but also balance the background variations to minimise uncertainty.
Optimising a neural network observable (neos)
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neos gets better CLs than all other tried methods!
additional plots that show:
> cross-section uncertainty is also optimised for free
> no over/underconstraint of nuisance parameter
Optimising a neural network observable (neos)
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More fun details and context in our preprint! :)
In collaboration with Lukas Heinrich:
You can optimize anything!
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https://github.com/gradhep/relaxed
binning!
cuts!
Backup
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You want to know how it scales!
Me too!
IRIS-HEP is very interested in this, and plans to support it for the “Analysis grand challenge” on open data, but may need more personpower.
Very much open to collaboration on any use case!
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example concerns:
> batch size may need to sufficiently represent analysis (so could require lots more VRAM compared to usual approach)
> every minibatch update = one run of the analysis, so may need lots more compute (but GPUs + autodiff are very powerful!)
Discovery significance (it still works!)
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Differences between discovery p-value and CLs
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Train to optimize CLs
Train to optimize discovery p-value
Differences between discovery p-value and CLs
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Train to optimize CLs
Train to optimize discovery p-value
Differences between discovery p-value and CLs
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Train to optimize CLs
Train to optimize discovery p-value
Which bandwidth to pick?
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