[Paper Review] HEPGAME and the Simplification of Expressions

Ben Ruijl, J.A.M. Vermaseren+2 more|arXiv (Cornell University)|May 25, 2014

Numerical Methods and Algorithms21 references3 citations

TL;DR

This paper proposes SA-UCT, a modified Monte Carlo Tree Search (MCTS) algorithm with a dynamically decreasing exploration-exploitation parameter, to optimize Horner schemes for simplifying large multivariate polynomials in high-energy physics. The method reduces numerical integration time by up to 24-fold and increases the robustness of hyperparameter tuning by widening the effective range of the exploration constant by over tenfold.

ABSTRACT

Advances in high energy physics have created the need to increase computational capacity. Project HEPGAME was composed to address this challenge. One of the issues is that numerical integration of expressions of current interest have millions of terms and takes weeks to compute. We have investigated ways to simplify these expressions, using Horner schemes and common subexpression elimination. Our approach applies MCTS, a search procedure that has been successful in AI. We use it to find near-optimal Horner schemes. Although MCTS finds better solutions, this approach gives rise to two further challenges. (1) MCTS (with UCT) introduces a constant, $C_p$ that governs the balance between exploration and exploitation. This constant has to be tuned manually. (2) There should be more guided exploration at the bottom of the tree, since the current approach reduces the quality of the solution towards the end of the expression. We investigate NMCS (Nested Monte Carlo Search) to address both issues, but find that NMCS is computationally unfeasible for our problem. Then, we modify the MCTS formula by introducing a dynamic exploration-exploitation parameter $T$ that decreases linearly with the iteration number. Consequently, we provide a performance analysis. We observe that a variable $C_p$ solves our domain: it yields more exploration at the bottom and as a result the tuning problem has been simplified. The region in $C_p$ for which good values are found is increased by more than a tenfold. This result encourages us to continue our research to solve other prominent problems in High Energy Physics.

Motivation & Objective

• To address the computational bottleneck in high-energy physics where numerical integration of expressions with millions of terms takes weeks.
• To reduce the number of arithmetic operations in large multivariate polynomials through optimized Horner schemes and common subexpression elimination.
• To improve the performance and robustness of Monte Carlo Tree Search (MCTS) for expression simplification by addressing two key limitations: sensitivity to the exploration-exploitation constant $C_p$ and insufficient exploration at the tree's bottom levels.
• To develop a method that enables faster, more reliable simplification of complex expressions, making high-precision theoretical predictions computationally feasible.

Proposed method

• The paper applies MCTS with the UCT (Upper Confidence bound applied to Trees) selection criterion to search for near-optimal Horner schemes in multivariate polynomial expressions.
• It introduces SA-UCT, a variant of UCT where the exploration parameter $C_p$ decreases linearly over iterations, shifting from exploration to exploitation.
• The method uses a performance evaluation function to estimate the number of operations required by a given Horner scheme, guiding the tree search toward lower-cost solutions.
• It compares SA-UCT against standard UCT and Nested Monte Carlo Search (NMCS), evaluating their effectiveness and computational feasibility on benchmark polynomials from high-energy physics and mathematics.
• The approach is integrated into the FORM computer algebra system, with results intended for inclusion in future releases.
• The method supports both commutative and non-commutative variables, with extensions considered for handling non-commutative subexpressions and variable linear combinations.

Experimental results

Research questions

• RQ1 Can MCTS with a dynamic exploration-exploitation parameter improve the quality and robustness of Horner scheme selection for large multivariate polynomials in high-energy physics?
• RQ2 How does the dynamic adjustment of $C_p$ in SA-UCT affect the convergence and solution quality compared to fixed $C_p$ in UCT?
• RQ3 Why does NMCS fail to scale effectively in this domain, and can its limitations be overcome with algorithmic modifications?
• RQ4 To what extent does the region of effective $C_p$ values expand in SA-UCT compared to UCT, and how does this impact tuning effort?
• RQ5 Can SA-UCT be generalized to other expression simplification problems in high-energy physics, such as recurrence relation solving or non-commutative algebra?

Key findings

• SA-UCT reduces the number of operations in benchmark polynomials by up to a factor of 24 compared to naive Horner schemes.
• The effective range of $C_p$ values yielding good solutions increases by at least tenfold under SA-UCT, significantly simplifying hyperparameter tuning.
• The dynamic decrease of $C_p$ enhances exploration at the bottom of the search tree, improving solution quality in later iterations.
• NMCS, while theoretically addressing both issues, is computationally infeasible due to the high cost of the evaluation function (6 seconds per evaluation).
• The performance of SA-UCT is robust across different numbers of tree updates, with stable convergence to near-global minima as $N$ increases.
• The method enables a reduction in numerical integration time from weeks to days or even hours in real-world high-energy physics applications.