# A Gradient Smoothed Functional Algorithm with Truncated Cauchy Random Perturbations for Stochastic Optimization

arXiv: https://arxiv.org/abs/2208.00290 · HTML: https://mlsanity.com/paper/2208.00290/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 29 references that are in the map.
- First posted on arXiv 30 Jul 2022.

## Builds on (most cited first, 6 of 6)

- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [(Bandit) Convex Optimization with Biased Noisy Gradient Oracles](https://mlsanity.com/paper/1609.07087/) (2016)
- [Hessian Estimation via Stein's Identity in Black-Box Problems](https://mlsanity.com/paper/2104.01317/) (2021)
- [Non-asymptotic bounds for stochastic optimization with biased noisy gradient oracles](https://mlsanity.com/paper/2002.11440/) (2020)

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

- [A Cubic-regularized Policy Newton Algorithm for Reinforcement Learning](https://mlsanity.com/paper/2304.10951/) (2023)

## Abstract (arXiv)

In this paper, we present a stochastic gradient algorithm for minimizing a smooth objective function that is an expectation over noisy cost samples, and only the latter are observed for any given parameter. Our algorithm employs a gradient estimation scheme with random perturbations, which are formed using the truncated Cauchy distribution from the delta sphere. We analyze the bias and variance of the proposed gradient estimator. Our algorithm is found to be particularly useful in the case when the objective function is non-convex, and the parameter dimension is high. From an asymptotic convergence analysis, we establish that our algorithm converges almost surely to the set of stationary points of the objective function and obtains the asymptotic convergence rate. We also show that our algorithm avoids unstable equilibria, implying convergence to local minima. Further, we perform a non-asymptotic convergence analysis of our algorithm. In particular, we establish here a non-asymptotic bound for finding an epsilon-stationary point of the non-convex objective function. Finally, we demonstrate numerically through simulations that the performance of our algorithm outperforms GSF, SPSA, and RDSA by a significant margin over a few non-convex settings and further validate its performance over convex (noisy) objectives.