Stochastic Trace Optimization of Parameter Dependent Matrices Based on Statistical Learning Theory
This work addresses a theoretical optimization problem in numerical linear algebra, providing incremental improvements in error bounds for stochastic trace estimation.
The paper tackles the problem of minimizing the trace of parameter-dependent matrices using a Monte Carlo estimator, deriving probabilistic bounds on the backward error with two types of bounds (epsilon nets and generic chaining) that predict small sampling amounts for matrices with small offdiagonal mass and compact parameter spaces.
We consider matrices $\boldsymbol{A}(\boldsymbolθ)\in\mathbb{R}^{m\times m}$ that depend, possibly nonlinearly, on a parameter $\boldsymbolθ$ from a compact parameter space $Θ$. We present a Monte Carlo estimator for minimizing $\text{trace}(\boldsymbol{A}(\boldsymbolθ))$ over all $\boldsymbolθ\inΘ$, and determine the sampling amount so that the backward error of the estimator is bounded with high probability. We derive two types of bounds, based on epsilon nets and on generic chaining. Both types predict a small sampling amount for matrices $\boldsymbol{A}(\boldsymbolθ)$ with small offdiagonal mass, and parameter spaces $Θ$ of small ``size.'' Dependence on the matrix dimension~$m$ is only weak or not explicit. The bounds based on epsilon nets are easier to evaluate and come with fully specified constants. In contrast, the bounds based on chaining depend on the Talagrand functionals which are difficult to evaluate, except in very special cases. Comparisons between the two types of bounds are difficult, although the literature suggests that chaining bounds can be superior.