Reweighting and Boosting to uniforimty in HEP
- 1. Non-trivial applications of boosting Tatiana Likhomanenko Lund, MLHEP 2016 *many slides are taken from Alex Rogozhnikov’s presentations
- 2. Boosting recapitulation 2 Boosting combines weak learners to obtain a strong one It is usually built over decision trees State-of-the-art results in many areas General-purpose implementations are used for classification and regression
- 3. Reweighting problem in HEP
- 4. Data/MC disagreement 4 Monte Carlo (MC) simulated samples are used for training and tuning a model After, trained model is applied to real data (RD) Real data and Monte Carlo have different distributions Thus, trained model is biased (and the quality is overestimated on MC samples)
- 5. Distributions reweighting 5 Reweighting in HEP is used to minimize the difference between RD and MC samples The goal of reweighting: assign weights to MC s.t. MC and RD distributions coincide Known process is used, for which RD can be obtained (MC samples are also available) MC distribution is original, RD distribution is target
- 6. Applications beyond physics 6 Introducing corrections to fight non-response bias: assigning higher weight to answers from groups with low response. See e.g. R. Kizilcec, "Reducing non-response bias with survey reweighting: Applications for online learning researchers", 2014.
- 7. Typical approach: histogram reweighting 7 variable(s) is split into bins in each bin the MC weight is multiplied by: - total weights of events in a bin for target and original distributions 1. simple and fast 2. number of variables is very limited by statistics (typically only one, two) 3. reweighting in one variable may bring disagreement in others 4. which variable is preferable for reweighting? multiplierbin = wbin, target wbin, original wbin, target, wbin, original
- 9. Typical approach: example 9 Problems arise when there are too few events in a bin This can be detected on a holdout (see the latest row) Issues: 1. few bins - rule is rough 2. many bins - rule is not reliable Reweighting rule must be checked on a holdout!
- 10. Reweighting quality 10 How to check the quality of reweighting? One dimensional case: two samples tests (Kolmogorov-Smirnov test, Mann-Whitney test, …) Two or more dimensions? Comparing 1d projections is not a way
- 11. Comparing nDim distributions using ML 11 Final goal: classifier doesn’t use data/MC disagreement information = classifier cannot discriminate data and MC Comparison of distributions shall be done using ML: train a classifier to discriminate data and MC output of the classifier is one-dimensional variable looking at the ROC curve (alternative of two sample test) on a holdout (should be 0.5 if the classifier cannot discriminate data and MC)
- 12. Density ratio estimation approach 12 We need to estimate density ratio: Classifier trained to discriminate MC and RD should reconstruct probabilities pMC(x) and pRD(x) For reweighting we can use 1. Approach is able to reweight in many variables 2. It is successfully tried in HEP, see D. Martschei et al, "Advanced event reweighting using multivariate analysis", 2012 3. There is poor reconstruction when ratio is too small / high 4. It is slower than histogram approach fRD(x) fMC(x) fRD(x) fMC(x) ⇠ pRD(x) pMC(x)
- 13. … 13 Write ML algorithm to solve directly reweighting problem Remind that in histogram approach few bins is bad, many bins is bad too. What can we do? Better idea… Split space of variables in several large regions Find this regions ‘intellectually’
- 14. Decision tree for reweighting 14 Write ML algorithm to solve directly reweighting problem: Tree splits the space of variables with orthogonal cuts (each tree leaf is a region, or bin) There are different criteria to construct a tree (MSE, Gini index, entropy, …) Find regions with the highest difference between original and target distribution
- 15. Spitting criteria 15 Finding regions with high difference between original and target distribution by maximizing symmetrized : 2 2 = X leaf (wleaf, original wleaf, target)2 wleaf, original + wleaf, target A tree leaf may be considered as ‘a bin’; - total weights of events in a leaf for target and original distributions. wleaf, original, wleaf, target
- 16. AdaBoost (Adaptive Boosting) recall 16 building of weak learners one-by-one, predictions are summed: each time increase weights of events incorrectly classified by a tree main idea: provide base estimator (weak learner) with information about which samples have higher importance wi wi exp( ↵yid(xi))), yi = ±1 D(x) = X j ↵jdj(x) d(x)
- 17. BDT reweighter 17 Many times repeat the following steps: build a shallow tree to maximize symmetrized compute predictions in leaves: reweight distributions (compare with AdaBoost): Comparison with GBDT: different tree splitting criterion different boosting procedure 2 leaf pred = log wleaf, target wleaf, original w = ( w, if event from target (RD) distribution w · epred , if event from original (MC) distribution
- 18. BDT reweighter DEMO after BDT reweightingbefore BDT reweighting 18
- 19. KS for 1d projections 19 Bins reweighter uses only 2 last variables (60 × 60 bins); BDT reweighter uses all variables
- 20. Comparing reweighting with ML 20
- 21. hep_ml library 21 Being a variation of GBDT, BDT reweighter is able to calculate feature importances. Two features used in reweighting with bins are indeed the most important.
- 22. Summary 22 1. Comparison of multidimensional distributions is ML problem 2. Reweighting of distributions is ML problem 3. Check reweighting rule on the holdout BDT reweighter uses each time few large bins (construction is done intellectually) is able to handle many variables requires less data (for the same performance) ... but slow (being ML algorithm)
- 24. Uniformity 24 Uniformity means that we have constant efficiency (FPR/TPR) against some variable. Applications: trigger system (flight time) flat signal efficiency particle identification (momentum) flat signal efficiency rare decays (mass) flat background efficiency Dalitz analysis (Dalitz variables) flat signal efficiency
- 25. Non-flatness along the mass 25 High correlation with the mass can create from pure background false peaking signal (specially if we use mass sidebands for training) Goal: FPR = const for different regions in mass FPR = background efficiency
- 26. Basic approach 26 reduce the number of features used in training leave only the set of features, which do not give enough information to reconstruct the mass of particle simple and works sometimes we have to loose information Can we modify ML to use all features, but provide uniform background efficiency (FPR)/signal efficiency (TPR) along the mass?
- 27. Gradient boosting recall 27 Gradient boosting greedily builds an ensemble of estimators by optimizing some loss function. Those could be: MSE: AdaLoss: LogLoss: Next estimator in series approximates gradient of loss in the space of functions D(x) = X j ↵jdj(x) L = X i (yi D(xi))2 L = X i e yiD(xi) , yi = ±1 L = X i log(1 + e yiD(xi) ), yi = ±1
- 28. uBoostBDT 28 Aims to get FPRregion=const Fix target efficiency, for example FPRtarget=30%, and find corresponding threshold train a tree, its decision function is increase weight for misclassified events: increase weight of background events in the regions with high FPR This way we achieve FPRregion=30% in all regions only for some threshold on training dataset d(x) wi wi exp( ↵yid(xi))), yi = ±1 wi wi exp ( (FPRregion FPRtarget))
- 29. uBoost 29 uBoost is an ensemble of uBoostBDTs, each uBoostBDT uses own FPRtarget (all possible FPRs with step of 1%) uBoostBDT returns 0 or 1 (passed or not the threshold corresponding to FPRtarget), simple averaging is used to obtain predictions. drives to uniform selection very complex training many trees estimation of threshold in uBoostBDT may be biased
- 30. Non-uniformity measure 30 difference in the efficiency can be detected by analyzing distributions uniformity = no dependence between the mass and predictions Uniform predictions Non-uniform predictions (peak in highlighted region)
- 31. Non-uniformity measure 31 Average contributions (difference between global and local distributions) from different regions in the mass: use for this Cramer-von Mises measure (integral characteristic) CvM = X region Z |Fregion(s) Fglobal(s)|2 dFglobal(s)
- 32. Minimizing non-uniformity 32 why not minimizing CvM as a loss function with GB? … because we can’t compute the gradient ROC AUC, classification accuracy are not differentiable too also, minimizing CvM doesn't encounter classification problem: the minimum of CvM is achieved i.e. on a classifier with random predictions
- 33. Flatness loss (FL) 33 Put an additional term in the loss function which will penalize for non-uniformity predictions: Flatness loss approximates non-differentiable CvM measure: L = Ladaloss + ↵LFL LFL = X region Z |Fregion(s) Fglobal(s)|2 ds @ @D(xi) LFL ⇠ 2(Fregion(s) Fglobal(s)) s=D(xi)
- 34. Rare decay analysis DEMO 34 when we train on a sideband vs MC using many features, we easily can run into problems (there exist several features which depend on the mass)
- 35. Rare decay analysis DEMO 35 all models use the same set of features for discrimination, but AdaBoost got serious dependence on the mass
- 36. PID DEMO 36 Features strongly depend on the momentum and transverse momentum. Both algorithms use the same set of features. Used MVA is a specific BDT implementation with flatness loss.
- 37. Trigger DEMO 37 Both algorithms use the same set of features. The right one is uGB+FL.
- 38. Dalitz analysis DEMO 38 The right one is uBoost algorithm. Global efficiency is set 70%
- 40. Summary 40 1. uBoost approach 2. Non-uniformity measure 3. uGB+FL approach: gradient boosting with flatness loss (FL) uBoost, uGB+FL: produce flat predictions along the set of features there is a trade off between classification quality and uniformity
- 41. Boosting summary 41 powerful general-purpose algorithm most known applications: classification, regression and ranking widely used, considered to be well-studied can be adapted to different specific scientific problems