Lecture notes on Machine Learning applications for global fits

This paper presents a comprehensive framework for accelerating high-energy physics global fits by employing Machine Learning surrogates, specifically Boosted Decision Trees trained via active learning, to approximate log-likelihood functions and efficiently explore parameter spaces, as demonstrated by an application to the $B^\pm \to K^\pm \nu \bar{\nu}$ anomaly and Axion-Like Particles at Belle II.

The Gist

The Big Picture: Finding a Needle in a Haystack (That's on Fire)

Imagine you are a physicist trying to find a new, invisible particle (let's call it an "Axion") that might explain a weird glitch in the universe. To do this, you have to build a giant mathematical model with dozens of dials and knobs (parameters). You need to turn these knobs to find the perfect setting where your model matches the real-world data collected by the Belle II experiment in Japan.

The Problem:
In the old days, finding the right settings was like trying to find the perfect temperature for a soufflé by baking one cake, checking it, adjusting the oven, and baking another.

• The Catch: Every time you "bake a cake" (calculate a prediction), it takes 10 seconds.
• The Scale: To find the perfect setting, you might need to bake 100,000 cakes. That's over 27 hours of non-stop baking. If you want to be thorough, it could take weeks. This is too slow.

The Solution:
This paper teaches physicists how to hire a super-fast AI sous-chef (Machine Learning) to taste the batter and guess the result instantly, so you don't have to actually bake the cake every time.

---

The Step-by-Step Recipe

1. The "Smart Taster" (Active Learning & Gaussian Processes)

You can't just throw random darts at your model to see what happens; that's inefficient. Instead, the paper suggests using a Smart Taster.

• How it works: Imagine you are exploring a dark cave. You don't walk randomly. You take a step, listen for a sound, and decide: "Should I go deeper here (Exploitation) or check that dark corner over there where I'm not sure what's happening (Exploration)?"
• The Tool: The AI uses a "Gaussian Process" to guess where the best settings are and where it is most confused. It only asks you to run the expensive 10-second calculation on the most interesting spots. This saves you from baking 99% of the cakes you don't need.

2. The "Speed-Reading Chef" (Boosted Decision Trees / XGBoost)

Once you have a small, high-quality dataset of "baked cakes" (data points), you need to teach the AI to predict the result of any new setting instantly.

• The Analogy: Think of a Decision Tree as a "20 Questions" game.
  • Question 1: Is the knob turned past 5? (Yes/No)
  • Question 2: Is the other knob red? (Yes/No)
  • Result: "Okay, if you did this, the cake will be delicious."
• Boosted Trees: One "20 Questions" game isn't smart enough. So, the AI builds a team of 500 of these simple games. They vote on the answer. This team (called XGBoost) becomes incredibly accurate at predicting the result without ever actually running the complex physics math. It turns a 10-second calculation into a microsecond guess.

3. The "Translator" (SHAP Values)

AI is often a "black box"—you put numbers in, and a number comes out, but you don't know why. Physicists hate this because they need to understand the laws of nature.

• The Analogy: Imagine the AI is a chef who says, "This cake is perfect." You ask, "Why?"
• The Tool: SHAP values act like a translator. They break down the chef's decision: "The cake is perfect because you added just enough sugar (Parameter A), but you used too much flour (Parameter B), which canceled out the bad taste."
• This lets physicists see exactly which "knobs" are driving the results and if the AI is making sense physically.

4. The "Map Maker" (MCMC Sampling)

Now that the AI can predict results instantly, the physicists want to map out the entire landscape of possibilities. They want to know not just the "best" setting, but the "good" settings and how likely they are.

• The Analogy: Imagine sending out 20 hikers (walkers) into the parameter space. They wander around, but they are smart. If they find a high mountain (a good fit), they stay there longer. If they find a swamp (a bad fit), they leave.
• The Result: After a while, the hikers form a cloud. The densest part of the cloud shows you where the "true" answer is likely hiding. This is called MCMC (Markov Chain Monte Carlo). Because the AI is so fast, the hikers can explore the whole mountain range in minutes instead of years.

---

The Real-World Test: The "Missing Energy" Mystery

The paper applies all these tools to a specific mystery: The $B^{\pm} \to K^{\pm}\nu\bar{\nu}$ anomaly.

• The Mystery: The Belle II experiment saw a B-meson decay into a Kaon and "missing energy" (neutrinos) more often than the Standard Model of physics predicts. It's a 2.7-sigma hint (a strong whisper, but not a shout) that something new is happening.
• The Suspect: A light, invisible particle called an Axion-Like Particle (ALP).
• The Challenge: The ALP has to be heavy enough to explain the data, but it also has to be "long-lived" (stable) so it doesn't decay too quickly and get caught by other experiments. It's a delicate balancing act.
• The Win: Using the ML tools described above, the authors built a model that navigated this tricky balancing act. They found a specific set of "knobs" (couplings) that explains the Belle II data while keeping the ALP stable enough to hide from other detectors.

Why This Matters

This paper isn't just about one particle; it's about a new way of doing science.

1. Speed: It turns weeks of computing into minutes.
2. Clarity: It stops the AI from being a magic black box and makes it a transparent tool that scientists can trust.
3. Future-Proofing: As experiments get bigger and data gets more complex, we can't afford to wait weeks for answers. We need these "AI sous-chefs" to help us find the next big discovery in the universe.

In short: The authors taught a computer to learn the rules of the universe so fast that it can help physicists find new particles before they run out of coffee.

Technical Summary

1. Problem Statement

In high-energy physics (HEP), global statistical fits are essential for testing theoretical models against experimental data. These fits rely on the likelihood function $L(\theta)$, which quantifies the probability of observing data given a set of model parameters $\theta$.

• The Bottleneck: Traditional minimization and scanning of the likelihood function require thousands of evaluations. In many modern scenarios (e.g., Standard Model Effective Field Theory or New Physics models), calculating the model predictions $x_i(\theta)$ involves computationally expensive processes, such as the numerical integration of Renormalization Group Equations (RGEs).
• Consequence: The computational cost renders traditional minimization prohibitive, preventing thorough exploration of the parameter space, especially when dealing with non-Gaussian observables or complex correlations.

2. Methodology

The paper proposes a comprehensive Machine Learning (ML) surrogate framework to approximate the log-likelihood function, thereby bypassing the need for repeated expensive calculations. The workflow consists of four main stages:

A. Active Learning for Data Generation

To minimize the number of expensive likelihood evaluations required to build a training dataset, the author employs Active Learning:

• Strategy: Starts with a small initial dataset (e.g., Latin Hypercube sampling). An ML algorithm iteratively selects new points to evaluate based on a trade-off between:
  • Exploitation: Sampling near the current optimum (highest likelihood).
  • Exploration: Sampling in regions of high uncertainty.
• Tool: Gaussian Processes (GP) are used as the surrogate during this phase because they provide both a prediction and an uncertainty estimate.
• Acquisition Function: The Expected Improvement (EI) is used to select the next candidate point, balancing the potential for finding a better minimum against the need to reduce model uncertainty.

B. Surrogate Modeling with Boosted Decision Trees

Once the training data is generated, a Boosted Decision Tree (BDT) ensemble, specifically XGBoost, is trained to approximate the log-likelihood (or $\chi^2$) surface.

• Architecture: An ensemble of regression trees where each tree corrects the errors of the previous ones.
• Training: Uses a loss function (Mean Absolute Error) and regularization terms to prevent overfitting.
• Optimization: Hyperparameters (learning rate, tree depth, regularization $\alpha/\lambda$, subsampling) are tuned using Optuna and early stopping based on validation set performance.
• Acceleration: Trained models can be saved as binary files or compiled into C libraries (using tl2cgen/treelite) to achieve significant speed-ups in evaluation.

C. Interpretability via SHAP Values

To ensure the "black box" nature of ML does not obscure physical insights, the framework utilizes SHAP (SHapley Additive exPlanations) values:

• Local Interpretability: Decomposes the prediction for a specific parameter point into contributions from each parameter.
• Global Interpretability: Aggregates SHAP values to determine which parameters drive the model globally.
• Interactions: Uses the shapiq library to quantify interactions between parameters (e.g., how the effect of one coupling depends on another).

D. Posterior Sampling

With the fast surrogate model in place, the framework performs Bayesian inference:

• Algorithm: Uses Markov Chain Monte Carlo (MCMC) via the emcee library (affine-invariant ensemble sampler).
• Process: The sampler explores the parameter space defined by the posterior distribution $p(\theta|data) \propto L(\theta) \times \text{Prior}(\theta)$.
• Output: Generates samples to calculate expected values, confidence intervals, and multi-dimensional contours.

3. Key Contributions

1. End-to-End ML Workflow: Provides a reproducible pipeline for global fits, integrating active learning, XGBoost regression, hyperparameter tuning, and MCMC sampling.
2. Two-Stage Modeling Strategy: Addresses the issue of large dynamic ranges in $\chi^2$ by proposing a two-stage approach:
   • A classifier discards regions where $\Delta\chi^2$ is too large (noise).
   • A regressor is trained only on the physically relevant region (low $\chi^2$), improving accuracy where it matters most.
3. Explainability Integration: Demonstrates how to use SHAP values not just for model debugging, but to decode physical parameter influences and interactions in complex New Physics models.
4. Code Repository: Provides a complete tutorial repository (Jorge-Alda/comcha_tutorial) with Python implementations using libraries like gpflow, xgboost, shap, emcee, and alpaca.

4. Application and Results

The methodology is applied to the $B^\pm \to K^\pm \nu\bar{\nu}$ anomaly observed by the Belle II experiment.

• Physics Context: The Standard Model (SM) predicts a very rare Flavor Changing Neutral Current (FCNC) decay. Belle II observed a $2.7\sigma$ excess.
• Model: The anomaly is explained by a light Axion-Like Particle (ALP) ($m \approx 2$ GeV) that mediates the decay $B^+ \to K^+ a$, followed by $a \to \nu\bar{\nu}$ (appearing as missing energy).
• Constraints: The model must satisfy:
  • The observed branching ratio excess.
  • Longevity constraints ($c\tau \geq 80$ cm) to avoid detection of decay products in Belle II and BaBar.
  • Constraints from other FCNC processes and meson oscillations.
• Results:
  • The ML surrogate successfully mapped the high-dimensional parameter space (involving ALP couplings $c_f$ and decay constant $f_a$).
  • It identified a viable region where the ALP couplings allow for the required decay rate while suppressing the ALP's decay into visible particles (via cancellations between tree-level and RGE-induced couplings).
  • The approach demonstrated that a two-stage ML model can efficiently explore the parameter space while satisfying stringent experimental constraints that would be computationally prohibitive with direct numerical integration.

5. Significance

This work represents a paradigm shift in how theoretical models are confronted with experimental data in high-energy physics:

• Computational Efficiency: It transforms global fits from being limited by the cost of likelihood evaluation to being limited only by the speed of ML inference (orders of magnitude faster).
• Scalability: The method is robust for moderate-dimensional parameter spaces ($\lesssim 50$ parameters) and handles non-differentiable likelihoods (common in RGE-based models) where gradient-based methods fail.
• Physical Insight: By integrating SHAP values, the method moves beyond simple parameter estimation to provide interpretability, allowing physicists to understand why a model fits the data (e.g., identifying specific parameter cancellations).
• Future Readiness: As datasets grow and models become more complex (e.g., full SMEFT fits), these explainable and efficient ML techniques are positioned as essential tools for discovering subtle signatures of New Physics.