# Estimating the minimizer and the minimum value of a regression function under passive design

arXiv: https://arxiv.org/abs/2211.16457 · HTML: https://mlsanity.com/paper/2211.16457/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 44 references that are in the map.
- First posted on arXiv 29 Nov 2022.

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

- [Highly-Smooth Zero-th Order Online Optimization Vianney Perchet](https://mlsanity.com/paper/1605.08165/) (2016)
- [Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandits](https://mlsanity.com/paper/2006.07862/) (2020)
- [Optimization of Smooth Functions with Noisy Observations: Local Minimax Rates](https://mlsanity.com/paper/1803.08586/) (2018)
- [Distributed Zero-Order Optimization under Adversarial Noise](https://mlsanity.com/paper/2102.01121/) (2021)
- [Multi-kernel Passive Stochastic Gradient Algorithms and Transfer Learning](https://mlsanity.com/paper/2008.10020/) (2020)
- [Estimation of the Global Mode of a Density: Minimaxity, Adaptation, and Computational Complexity](https://mlsanity.com/paper/2104.07870/) (2021)
- [Bayesian mode and maximum estimation and accelerated rates of contraction](https://mlsanity.com/paper/1608.03913/) (2016)

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

- [One-Shot Localisation of the Global Minimum of a Noisy One-Dimensional Function: An Iterative Neural Minimizer Compared with Set Transformers and Classical Estimators](https://mlsanity.com/paper/2604.03614/) (2026)

## Abstract (arXiv)

We propose a new method for estimating the minimizer boldsymbolx^* and the minimum value f^* of a smooth and strongly convex regression function f from the observations contaminated by random noise. Our estimator boldsymbolzn of the minimizer boldsymbolx^* is based on a version of the projected gradient descent with the gradient estimated by a regularized local polynomial algorithm. Next, we propose a two-stage procedure for estimation of the minimum value f^* of regression function f. At the first stage, we construct an accurate enough estimator of boldsymbolx^*, which can be, for example, boldsymbolzn. At the second stage, we estimate the function value at the point obtained in the first stage using a rate optimal nonparametric procedure. We derive non-asymptotic upper bounds for the quadratic risk and optimization error of boldsymbolzn, and for the risk of estimating f^*. We establish minimax lower bounds showing that, under certain choice of parameters, the proposed algorithms achieve the minimax optimal rates of convergence on the class of smooth and strongly convex functions.