# Smooth Nested Simulation: Bridging Cubic and Square Root Convergence Rates in High Dimensions

arXiv: https://arxiv.org/abs/2201.02958 · HTML: https://mlsanity.com/paper/2201.02958/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 80 references that are in the map.
- First posted on arXiv 9 Jan 2022.

## Builds on (most cited first, 7 of 7)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Bayesian Optimization is Superior to Random Search for Machine Learning Hyperparameter Tuning: Analysis of the Black-Box Optimization Challenge 2020](https://mlsanity.com/paper/2104.10201/) (2021)
- [Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond](https://mlsanity.com/paper/2004.11154/) (2020)
- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates](https://mlsanity.com/paper/1305.5029/) (2013)
- [Ridge Regularization: an Essential Concept in Data Science](https://mlsanity.com/paper/2006.00371/) (2020)
- [Efficient Nested Simulation Experiment Design via the Likelihood Ratio Method](https://mlsanity.com/paper/2008.13087/) (2020)

## Built on by (most cited first, 1 of 1)

- [Reliable Real-Time Value at Risk Estimation via Quantile Regression Forest with Conformal Calibration](https://mlsanity.com/paper/2602.01912/) (2026)

## Abstract (arXiv)

Nested simulation concerns estimating functionals of a conditional expectation via simulation. In this paper, we propose a new method based on kernel ridge regression to exploit the smoothness of the conditional expectation as a function of the multidimensional conditioning variable. Asymptotic analysis shows that the proposed method can effectively alleviate the curse of dimensionality on the convergence rate as the simulation budget increases, provided that the conditional expectation is sufficiently smooth. The smoothness bridges the gap between the cubic root convergence rate (that is, the optimal rate for the standard nested simulation) and the square root convergence rate (that is, the canonical rate for the standard Monte Carlo simulation). We demonstrate the performance of the proposed method via numerical examples from portfolio risk management and input uncertainty quantification.