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Exploring gauge-fixing conditions with gradient-based optimization

William Detmold, Gurtej Kanwar, Yin Lin, Phiala E. Shanahan, Michael L. Wagman

Abstract

Lattice gauge fixing is required to compute gauge-variant quantities, for example those used in RI-MOM renormalization schemes or as objects of comparison for model calculations. Recently, gauge-variant quantities have also been found to be more amenable to signal-to-noise optimization using contour deformations. These applications motivate systematic parameterization and exploration of gauge-fixing schemes. This work introduces a differentiable parameterization of gauge fixing which is broad enough to cover Landau gauge, Coulomb gauge, and maximal tree gauges. The adjoint state method allows gradient-based optimization to select gauge-fixing schemes that minimize an arbitrary target loss function.

Exploring gauge-fixing conditions with gradient-based optimization

Abstract

Lattice gauge fixing is required to compute gauge-variant quantities, for example those used in RI-MOM renormalization schemes or as objects of comparison for model calculations. Recently, gauge-variant quantities have also been found to be more amenable to signal-to-noise optimization using contour deformations. These applications motivate systematic parameterization and exploration of gauge-fixing schemes. This work introduces a differentiable parameterization of gauge fixing which is broad enough to cover Landau gauge, Coulomb gauge, and maximal tree gauges. The adjoint state method allows gradient-based optimization to select gauge-fixing schemes that minimize an arbitrary target loss function.
Paper Structure (6 sections, 16 equations, 2 figures, 1 algorithm)

This paper contains 6 sections, 16 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Gauge fixing by functional minimization
  3. Gradients with respect to the gauge-fixing scheme
  4. Restricting to maximal-tree gauges
  5. Summary of the algorithm
  6. Results for a regression problem

Figures (2)

  • Figure 1: Training history for an axial (above) and randomized (below) target tree. The left plots show the training loss as a function of training updates, while the right plots show the accuracy of the learned parameters, quantified as $\sum_{x,\mu} |k^{(S)}_\mu(x) - p(x; 0, v)|$.
  • Figure 2: Snapshots of $p_\mu(x;T,v)$ at various points in the training history for an axial (above) and randomized (below) target tree, corresponding to the respective plots in Fig. \ref{['fig:random_tree_line']}.