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Learning to bin: differentiable and Bayesian optimization for multi-dimensional discriminants in high-energy physics

Johannes Erdmann, Nitish Kumar Kasaraguppe, Florian Mausolf

TL;DR

The paper tackles the problem of suboptimal binning in high-energy physics discriminants, proposing a flexible GMM-based binning where events are assigned to the bin/Gaussian component with the highest density. It introduces two optimization strategies, GATO (gradient-based with a differentiable softmax assignment) and BOBR (Bayesian optimization via a Tree-structured Parzen Estimator), to maximize the Asimov significance across bins, with penalties to enforce practical robustness. In 1D, both methods outperform equidistant binning and perform comparably; in multi-dimensional settings, GATO-GMM yields the largest gains by learning non-trivial, geometry-adapted bin boundaries, though Bayesian optimization struggles as dimensionality and parameter count grow. The work provides open-source Python plugins (gato-hep, bobr-hep) for straightforward integration into analyses, enabling analysis-aware, potentially end-to-end optimization of binning in real LHC studies.

Abstract

Categorizing events using discriminant observables is central to many high-energy physics analyses. Yet, bin boundaries are often chosen by hand. A simple, popular choice is to apply argmax projections of multi-class scores and equidistant binning of one-dimensional discriminants. We propose a binning optimization for signal significance directly in multi-dimensional discriminants. We use a Gaussian Mixture Model (GMM) to define flexible bin boundary shapes for multi-class scores, while in one dimension (binary classification) we move bin boundaries directly. On this binning model, we study two optimization strategies: a differentiable and a Bayesian optimization approach. We study two toy setups: a binary classification and a three-class problem with two signals and backgrounds. In the one-dimensional case, both approaches achieve similar gains in signal sensitivity compared to equidistant binnings for a given number of bins. In the multi-dimensional case, the GMM-based binning defines sensitive categories as well, with the differentiable approach performing best. We show that, in particular for limited separability of the signal processes, our approach outperforms argmax classification even with optimized binning in the one-dimensional projections. Both methods are released as lightweight Python plugins intended for straightforward integration into existing analyses.

Learning to bin: differentiable and Bayesian optimization for multi-dimensional discriminants in high-energy physics

TL;DR

The paper tackles the problem of suboptimal binning in high-energy physics discriminants, proposing a flexible GMM-based binning where events are assigned to the bin/Gaussian component with the highest density. It introduces two optimization strategies, GATO (gradient-based with a differentiable softmax assignment) and BOBR (Bayesian optimization via a Tree-structured Parzen Estimator), to maximize the Asimov significance across bins, with penalties to enforce practical robustness. In 1D, both methods outperform equidistant binning and perform comparably; in multi-dimensional settings, GATO-GMM yields the largest gains by learning non-trivial, geometry-adapted bin boundaries, though Bayesian optimization struggles as dimensionality and parameter count grow. The work provides open-source Python plugins (gato-hep, bobr-hep) for straightforward integration into analyses, enabling analysis-aware, potentially end-to-end optimization of binning in real LHC studies.

Abstract

Categorizing events using discriminant observables is central to many high-energy physics analyses. Yet, bin boundaries are often chosen by hand. A simple, popular choice is to apply argmax projections of multi-class scores and equidistant binning of one-dimensional discriminants. We propose a binning optimization for signal significance directly in multi-dimensional discriminants. We use a Gaussian Mixture Model (GMM) to define flexible bin boundary shapes for multi-class scores, while in one dimension (binary classification) we move bin boundaries directly. On this binning model, we study two optimization strategies: a differentiable and a Bayesian optimization approach. We study two toy setups: a binary classification and a three-class problem with two signals and backgrounds. In the one-dimensional case, both approaches achieve similar gains in signal sensitivity compared to equidistant binnings for a given number of bins. In the multi-dimensional case, the GMM-based binning defines sensitive categories as well, with the differentiable approach performing best. We show that, in particular for limited separability of the signal processes, our approach outperforms argmax classification even with optimized binning in the one-dimensional projections. Both methods are released as lightweight Python plugins intended for straightforward integration into existing analyses.
Paper Structure (11 sections, 19 equations, 5 figures, 1 table)

This paper contains 11 sections, 19 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dataset
  3. Binning model and optimization approaches
  4. GATO: gradient-descent based approach
  5. BOBR: Bayesian optimization approach
  6. Results
  7. Binary classification example
  8. Three-class example
  9. Impact of penalties on low event counts or large statistical uncertainties
  10. Conclusions
  11. Data generation details

Figures (5)

  • Figure 1: Top row: Optimization histories from a representative trial for BOBR and GATO with a sigmoid-based binning strategy. Bottom row: Five-bin distributions for the same trials.
  • Figure 2: Comparison of the significance obtained in the one-dimensional setup. Error bars are obtained from independent optimization runs, marking the 16th to 84th percentiles. The markers for the BOBR and GATO significances are shifted horizontally around the tested bin counts to enhance the visibility.
  • Figure 3: Top row: 68% and 95% density contours for the two signal processes and the background in the two-dimensional signal-score plane for the first (left) and second (right) scenario. Middle row: corresponding optimized five-bin boundaries in the same signal-score plane obtained with the GATO GMM approach. Bottom row: resulting five-bin event distributions for the two scenarios.
  • Figure 4: Comparison of the geometric mean of the two signal significances of the different methods for the two representative scenarios of the three-dimensional toy discriminant.
  • Figure 5: Expected background yield and relative background uncertainty per bin. Top row: background cross sections reduced by a factor of ten. Bottom row: total number of generated background events reduced to $10{,}000$.