A one-bit, comparison-based gradient estimator

Cai, HanQin; McKenzie, Daniel; Yin, Wotao; Zhang, Zhenliang

doi:10.1016/j.acha.2022.03.003

Cited by 7 publications

(14 citation statements)

References 13 publications

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“…In fact, HJ-Prox does not require differentiability of f . Related methods include random gradients ( 21 – 24 ), sparsity-based methods ( 25 – 27 ), derivative-free quasi-Newton methods ( 28 – 30 ), finite-difference-based methods ( 31 , 32 ), numerical quadrature-based methods ( 33 , 34 ), Bayesian methods ( 29 ), and comparison methods ( 35 ). As proximals closely relate to the gradient of Moreau envelopes, our work relates to methods that minimize Moreau envelopes (or their approximations) ( 16 , 19 , 36 – 40 ).…”

Section: Related Workmentioning

confidence: 99%

A Hamilton–Jacobi-based proximal operator

Osher

Heaton²,

Fung

2023

Proc. Natl. Acad. Sci. U.S.A.

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First-order optimization algorithms are widely used today. Two standard building blocks in these algorithms are proximal operators (proximals) and gradients. Although gradients can be computed for a wide array of functions, explicit proximal formulas are known for only limited classes of functions. We provide an algorithm, HJ-Prox, for accurately approximating such proximals. This is derived from a collection of relations between proximals, Moreau envelopes, Hamilton–Jacobi (HJ) equations, heat equations, and Monte Carlo sampling. In particular, HJ-Prox smoothly approximates the Moreau envelope and its gradient. The smoothness can be adjusted to act as a denoiser. Our approach applies even when functions are accessible only by (possibly noisy) black box samples. We show that HJ-Prox is effective numerically via several examples.

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Section: Related Workmentioning

confidence: 99%

A Hamilton–Jacobi-based proximal operator

Osher

Heaton²,

Fung

2023

Proc. Natl. Acad. Sci. U.S.A.

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“…In particular, Balasubramanian and Ghadimi (2021) studied the zeroth order Hessian estimators via the Stein's identity (Stein, 1981) and applied the estimator to cubic regularized Newton's method (Nesterov and Polyak, 2006). In addition to the above mentioned works, comparisonbased gradient estimator has also been considered by Cai et al (2022), which follows from a rich line of works in information theory (e.g., Raginsky and Rakhlin, 2011;Jamieson et al, 2012;Vershynin, 2012, 2014). Perhaps the most relevant works are (Flaxman et al, 2005) for gradient estimators, and (Wang, 2022) for Hessian estimators.…”

Section: Related Workmentioning

confidence: 99%

“…Several useful identities are stated below in Propositions 2, Proposition 3, and Proposition 4. References for the following propositions include (Nesterov and Spokoiny, 2017;Wang, 2022;Cai et al, 2022). Their proofs are included in the appendix.…”

Section: Preliminariesmentioning

confidence: 99%

“…Algorithm 2 Comparison-based gradient estimator (Cai et al, 2022) 1: Input: Number of random vectors k; Finite difference granularity: δ; Location for evaluation x ∈ R n ; Comparison oracle for the target function f :…”

Section: Comparison With Stochastic Estimatorsmentioning

confidence: 99%

“…• Comparison-based gradient estimator ∇f δ k,C (x), which estimates the normalized gradient. This estimator is defined in Algorithm 2 (Cai et al, 2022).…”

Section: Comparison With Stochastic Estimatorsmentioning

confidence: 99%

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Stochastic Zeroth Order Gradient and Hessian Estimators: Variance Reduction and Refined Bias Bounds

Feng¹,

Wang²

2022

Preprint

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We study stochastic zeroth order gradient and Hessian estimators for real-valued functions in R n . We show that, via taking finite difference along random orthogonal directions, the variance of the stochastic finite difference estimators can be significantly reduced. In particular, we design estimators for smooth functions such that, if one uses Θ (k) random directions sampled from the Stiefel's manifold St(n, k) and finite-difference granularity δ, the variance of the gradient estimator is bounded by, and the variance of the Hessian estimator is bounded by O. When k = n, the variances become negligibly small. In addition, we provide improved bias bounds for the estimators. The bias of both gradient and Hessian estimators for smooth function f is of order O δ 2 Γ , where δ is the finite-difference granularity, and Γ depends on high order derivatives of f . Our results are evidenced by empirical observations.

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