REVIEW 4 major objections 5 minor 24 references
Anisotropic Gaussian Smoothing for Gradient-based Optimization
T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read This paper shows that replacing gradients with gradients of an anisotropic Gaussian-smoothed function preserves the convergence guarantees of GD, SGD, and Adam while adding a non-local, basin-escaping search.
desk verdict An anisotropic smoothing framework that is a real generalization but whose main convergence theorems don't follow from the stated assumptions. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the $\Sigma$-Gaussian smoothed function $f_\Sigma(x)=\pi^{-d/2}\int_{\mathbb{R}^d} f(x+\Sigma u)e^{-\|u\|^2}du$ and its gradient $\nabla f_\Sigma(x)=2\pi^{-d/2}\Sigma^{-1}\int_{\mathbb{R}^d} u f(x+\Sigma u)e^{-\|u\|^2}du$, which admits a Monte Carlo estimate from function values only. The proofs rest on three inequalities: Lemma 2.1, Gaussian smoothing preserves L-smoothness, convexity, and Lipschitzness; Lemma 2.3, $|f_\Sigma(x)-f(x)|\le (Ld/4)\|\Sigma\|^2$ and $\|\nabla f_\Sigma(x)-\nabla f(x)\|\le L\|\Sigma\|^2\|\Sigma^{-1}\|((3+d)/2)^{3/2}$; and Lemma 2.5, which bounds the switching cost $|f_\Sigma(x)-f_T(x)|$ by $B(\Sigma,T)$ and is the new ingredient needed when the smoothing matrix changes direction. The switching lemma is what makes the anisotropic case genuinely different, because two anisotropic kernels cannot generally be composed into a single kernel of the same family unless their covariances are ordered.
What would settle it
Take $f(x)=\|x\|^2$ on $\mathbb{R}^2$ and run AGS-GD with $\Sigma_t=\mathrm{diag}(1,1/t)$. Every matrix is symmetric and invertible and $\|\Sigma_t\|=1$ is bounded, but $\|\Sigma_t\|^4\|\Sigma_t^{-1}\|^2=t^2$, so the smoothing term in Theorem 3.2 grows without bound; checking whether the stated boundedness condition really yields a vanishing right-hand side settles the adequacy of the paper's assumptions. A direct empirical counterpart is to run AGS-SGD under the same schedule with decreasing step sizes and record whether $\|\nabla f_{\Sigma_t}(x_t)\|$ stays away from zero.
Extended reading notes
Core claim
The central discovery is that the convergence theory of gradient-based optimization survives a non-local replacement of the gradient, provided the smoothing matrices change slowly enough. Specifically, the paper shows that when the update uses $\nabla f_{\Sigma_t}(x_{t-1})$, the smoothed gradient defined by convolution with an anisotropic Gaussian of covariance $\Sigma_t^2$, the standard descent arguments go through. For convex L-smooth functions, AGS-GD satisfies $f(x_T)-f(x_*)\le O(1/T)$ up to a smoothing cost involving $\|\Sigma_t\|^2$ and a switching cost $B(\Sigma_t,\Sigma_{t+1})$ that bounds $|f_{\Sigma_t}(x)-f_{\Sigma_{t+1}}(x)|$. For non-convex L-smooth functions, the minimum squared gradient norm along the trajectory is bounded by a similar expression involving $\|\Sigma_t\|^4\|\Sigma_t^{-1}\|^2$ plus switching costs. In the stochastic setting, AGS-SGD converges to a noisy ball whose radius is set by the step size, gradient variance, and smoothing parameters, and AGS-Adam achieves $\lim_{t\to\infty}\|\nabla f(x_t)\|^2=0$ almost surely under decay conditions on $\Sigma_t$ and the switching bounds. Unsmoothed GD, SGD, and Adam, and isotropic Gaussian smoothing, are recovered as special cases.
Load-bearing premise
Everything rests on the unstated requirement that the smoothing matrices shrink and switch fast enough for the smoothing error and switching costs to vanish at the right rates, and the paper does not prove that its experimental CMA schedule meets that requirement.
Editorial extensions
If this is right
- For convex L-smooth objectives, AGS-GD inherits the $O(1/T)$ rate of gradient descent up to a smoothing term $\frac{Ld}{4T}\sum_t\|\Sigma_t\|^2$ and a switching term built from $B(\Sigma_t,\Sigma_{t+1})$; both vanish as $T\to\infty$ when the smoothing matrices are controlled.
- For non-convex L-smooth objectives, the same framework bounds the minimum squared gradient norm along the trajectory, with the smoothing cost written in terms of $\|\Sigma_t\|^4\|\Sigma_t^{-1}\|^2$.
- AGS-SGD converges to a noisy ball whose radius is set by the learning rate, the stochastic gradient variance, and the smoothing parameters; a decreasing learning rate drives the smoothed gradient to zero.
- AGS-Adam attains $\lim_{t\to\infty}\|\nabla f(x_t)\|^2=0$ almost surely when $\|\Sigma_t\|\to 0$ and the switching bounds $B$ and $\tilde{B}$ decay fast enough, with no extra smoothing terms in the final bound.
- Because the smoothed gradient admits a Monte Carlo estimate from function values only, all three algorithms operate as zero-order methods, so the convergence guarantees carry over to derivative-free implementations.
Reading between the lines
- The paper leaves open how to choose $\Sigma_t$ so the theorems apply; enforcing $\|\Sigma_t\|^2\|\Sigma_t^{-1}\|=O(1/t)$ and $B(\Sigma_t,\Sigma_{t+1})=O(\eta_t^2)$ is a concrete sufficient schedule that would connect the theory to the CMA experiments.
- Because Lemma 2.5 says $f_H=(f_\Sigma)_T$ for $H=\sqrt{\Sigma^2+T^2}$, anisotropic smoothing can be read as a homotopy through smoothed landscapes; whether rotating the smoothing axes rather than only shrinking them helps escape basins is a testable question.
- A practical byproduct is a stopping criterion: monitor $\|\Sigma_t\|^2\|\Sigma_t^{-1}\|$ to decide when the smoothed gradient has come within a desired tolerance of the true gradient.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper introduces anisotropic Gaussian smoothing as a modification of GD, SGD, and Adam. In place of the gradient of f, the algorithms use the gradient of the Gaussian-smoothed function f_Sigma_t defined in (2), with a time-varying symmetric invertible matrix Sigma_t, and the paper presents finite-time bounds for convex and non-convex L-smooth objectives (Theorems 3.1 and 3.2), a stochastic bound for AGS-SGD (Theorem 4.1), and an almost-sure stationarity result for AGS-Adam (Theorem 5.1). The bounds are expressed with a generic switching penalty B(Sigma,T) bounding |f_Sigma(x)-f_T(x)|, and the paper claims that, as T tends to infinity, the right-hand sides tend to zero whenever the sequence (||Sigma_t||) is bounded, thereby recovering the unsmoothed GD/SGD/Adam results as Sigma_t tends to 0. The numerical section replaces nabla f_Sigma_t by the Monte Carlo estimator (7) and generates Sigma_t with a CMA-type update, testing five benchmark functions.
Significance. The generalization from isotropic (scalar or identity) smoothing to a time-varying matrix Sigma_t is natural and potentially useful: it allows direction-dependent smoothing, and the abstract formulation of the switching cost B(Sigma,T) cleanly isolates the new difficulty. The paper is written in the proof-template style of the authors' prior isotropic work [21,22]; the appendices give complete proofs of the background lemmas, and the constants in the bounds are explicit. If the convergence claims were supported by the stated assumptions, this would be a solid contribution to the smoothing and zero-order optimization literature. However, the central asymptotic claims are not merely underspecified: they are false under the stated hypotheses (see major comments), and the proof of Theorem 5.1 does not establish its conclusion. The experimental section demonstrates the behavior of a heuristic implementation (CMA plus Monte Carlo) that the theorems do not cover, so it does not compensate for the theoretical gaps.
major comments (4)
- [Section 3, after Theorem 3.2; Section 4, after Theorem 4.1] The assertion that the right-hand sides of Theorems 3.1, 3.2, and 4.1 converge to zero whenever (||Sigma_t||) is bounded is false. In Theorem 3.1, the term (Ld/(4T)) sum_{t=1}^T ||Sigma_t||^2 with the constant sequence Sigma_t = sigma I converges to Ld sigma^2 / 4 > 0 rather than to 0, and the switching term (1/T) sum_{t=1}^{T-1} t B(Sigma_{T-t}, Sigma_{T-t+1}) equals (1/T) sum_{s=1}^{T-1} (T-s) B(Sigma_s, Sigma_{s+1}); for Sigma_t alternating between two fixed matrices with different spectra, B is a positive constant c and this term grows like c(T-1)/2, so the bound diverges. Even under B(Sigma_t, Sigma_{t+1}) = O(1/t), the weighted mean is O(log T) and does not vanish. In Theorem 3.2, the second term requires (1/T) sum_{t=1}^T ||Sigma_t||^4 ||Sigma_t^{-1}||^2 to tend to 0, and the third term requires (1/T) sum_{t=0}^T B(Sigma_{t+1}, Sigma_t) to tend to 0; boundedness of (||Sigma_t||) alone gives neither, especially since ||Sigma_t^{-1}|| is uncontrolled. In Theorem 4.1, with eta_t = 1/t and B identically c > 0, the final term sum_{t=1}^T B(Sigma_t, Sigma_{t-1}) / sum_{t=1}^T eta_t behaves as c T / log T and diverges. The theorems as stated therefore do not support the advertised convergence to the minimizer in the convex case or to stationary points in the non-convex case; explicit decay conditions, such as ||Sigma_t|| tending to 0 in the relevant Cesaro sense and weighted summability of B(Sigma_t, Sigma_{t+1}) (e.g., B = o(1/t)), are missing.
- [Section 5, proof of Theorem 5.1] The final step of the proof, which states that since ||Sigma_t|| -> 0 we have lim_{t->infinity} ||nabla f(x_t)||^2 = 0 almost surely, is not justified. To pass from lim ||nabla f_{Sigma_t}(x_t)|| = 0 to lim ||nabla f(x_t)|| = 0, one must control ||nabla f_{Sigma_t}(x_t) - nabla f(x_t)||, and Lemma 2.3(b) bounds this by L ||Sigma_t||^2 ||Sigma_t^{-1}|| ((3+d)/2)^{3/2}. The assumption ||Sigma_t|| -> 0 does not imply ||Sigma_t||^2 ||Sigma_t^{-1}|| -> 0; for example, for d = 2 and Sigma_t = diag(t^{-3}, t^{-1}), one has ||Sigma_t|| = t^{-1} -> 0 but ||Sigma_t||^2 ||Sigma_t^{-1}|| = t^{-2} * t^3 = t -> infinity. The hypotheses of the theorem must be strengthened (for instance, to ||Sigma_t||^2 ||Sigma_t^{-1}|| -> 0) or an additional argument must be given; as it stands, the proof of Theorem 5.1 is incomplete.
- [Section 3, Theorems 3.1 and 3.2 and Appendix B] The statement of Theorem 3.1 is inconsistent with its proof regarding the learning rate. The stated bound is (1/(2T)) ||x_0 - x^*||^2, while the proof in Appendix B obtains (1/(2 lambda T)) ||x_0 - x^*||^2; the factor 1/lambda is dropped in the statement. In addition, the proof uses the inequality lambda (L lambda / 2 - 1) <= -lambda/2, which is valid only if lambda <= 1/L, but no step-size restriction appears in Theorem 3.1 or Theorem 3.2. The displayed proof of Theorem 3.2 also loses the 1/lambda factor, and its telescoping step bounds f_{Sigma_1}(x_1) - f_{Sigma_{T+1}}(x_{T+1}) by f(x_0) - f*, which does not follow for non-convex f because the inequality f_Sigma(x) >= f(x) from Lemma 2.1(b) is unavailable in that setting and f_{Sigma_1}(x_0) can exceed f(x_0) by up to (Ld/4) ||Sigma_1||^2.
- [Section 2, Lemma 2.5(b)] The bound in Lemma 2.5(b) is either false or unproved as written. The proof sets Sigma^2 = P D_Sigma P^T with D_Sigma = diag(sigma_1, ..., sigma_d), so sigma_i and tau_i are treated as eigenvalues of Sigma^2 and T^2; but then the estimate with the squared terms ||(D_Sigma - D_T)_+||^2 and ||(D_T - D_Sigma)_+||^2 does not follow from Lemma 2.3(a), because applying that lemma to the smoothing matrix sqrt((D_Sigma - D_T)_+) yields (Ld/4) ||(D_Sigma - D_T)_+|| without the square. If instead sigma_i and tau_i are meant to be eigenvalues of Sigma and T, the claimed bound is false: for f(x) = (1/2)||x||^2, Sigma = 2I, and T = I, the true difference is |f_Sigma(x) - f_T(x)| = 3d/4, while the claimed bound is (Ld/4)(2-1)^2 = d/4. The statement of the lemma does not define sigma_i and tau_i, and no proof in Appendix A establishes the displayed formula.
minor comments (5)
- [Section 6] The text refers to results shown in Figure 1a through Figure 1e for the five benchmark tests, but the benchmark results appear in Figure 2; the caption ordering of Figure 2 also does not match the F1-F5 enumeration in the text, and the names Powell function F18 and Rosenbrock function F6 introduce a third numbering for the same test functions.
- [Appendix B, proof of Theorem 3.1] The inequality f_{Sigma_t}(x_t) <= f_{Sigma_t}(x_{t-1}) <= f_{Sigma_{t-1}}(x_{t-1}) + B(Sigma_t, Sigma_{t+1}) should read B(Sigma_t, Sigma_{t-1}); the subsequent sum writes ||Sigma_{T+1}|| where ||Sigma_{t+1}|| is meant, and the notation f_{sigma T - t} should be f_{Sigma_{T-t}}.
- [Sections 3 and 4] The symbol lambda is used both for the step size in Algorithms 1-2 and Theorems 3.1-3.2 and for the variance bound E[||nabla f_k||^2] <= lambda in Theorem 4.1; these two roles should be separated to avoid confusion.
- [Abstract and Introduction] The abstract and introduction state that the algorithms approach global minima more effectively, but the theorems only provide conditional convergence to a minimizer in the convex case and to stationary points in the non-convex case; the global-minimum language should be qualified.
- [Section 6] The experiments implement the Monte Carlo estimator (7) and a CMA-based update for Sigma_t, whereas the theorems concern exact gradients nabla f_{Sigma_t} and require conditions on (Sigma_t) that are not verified for the CMA sequence; a sentence stating that the experiments are heuristic and not covered by the theorems would be appropriate.
Circularity Check
No circularity: the convergence theorems are derived from explicit matrix smoothing lemmas; the self-citations to prior isotropic results are proof templates, not unverified premises.
full rationale
I walked the derivation chain. AGS-GD, AGS-SGD, and AGS-Adam are defined as gradient-type methods using f_Sigma_t in place of f, and the convergence claims are proved from the preliminary lemmas in Section 2: Lemma 2.1 preserves smoothness/convexity, Lemma 2.3 bounds |f_Sigma - f| and ||grad f_Sigma - grad f||, Lemma 2.5 controls switching between different anisotropic smoothing matrices, and Corollary 2.4 converts smoothed-gradient control back to true-gradient control. The main theorems (3.1, 3.2, 4.1, 5.1) are conditional statements with explicit error terms; they do not fit parameters to data and then rename the fit as a prediction. The quantities B(Sigma,T) and ~B(Sigma,T) are introduced as generic bounds satisfying explicitly stated inequalities, not as hidden degrees of freedom that make the conclusions true by construction. The paper does cite the authors' prior isotropic papers [21,22] as proof templates, but the anisotropic technical content -- non-commuting covariance matrices, explicit ||Sigma||^2||Sigma^{-1}|| factors, and the non-trivial switching lemma 2.5 -- is proved in the present paper, so the central claim does not reduce to a self-citation chain. The main weakness is a correctness gap, not circularity: the comment after Theorems 3.1/3.2/4.1 that boundedness of ||Sigma_t|| makes the error terms vanish is not justified (the average of ||Sigma_t||^2 need not go to zero), and in Theorem 5.1 the final step from ||grad f_Sigma_t(x_t)|| to ||grad f(x_t)|| requires ||Sigma_t||^2||Sigma_t^{-1}|| -> 0 rather than merely ||Sigma_t|| -> 0. The CMA schedule in Section 6 is also not shown to satisfy the theorem hypotheses. These are unsupported implications or empirical gaps, but they are not cases where a prediction is equivalent to its input by definition or by fitted construction.
Assumptions & free parameters
free parameters (3)
- Smoothing matrix sequence {Sigma_t} =
Unspecified; CMA-updated in experiments
- Learning rates lambda and eta_t =
Unspecified
- Monte Carlo sample count N =
Unspecified
assumptions (4)
- domain assumption L-smoothness, convexity, and bounded-noise assumptions E[f_k] = f, E[grad f_k] = grad f, E[||grad f_k||^2] <= lambda
- standard math Existing Gaussian smoothing properties from Nesterov and Spokoiny [19]
- domain assumption Convergence templates from the authors' prior papers [21,22] and He et al. [9]
- standard math Fubini-Tonelli theorem and Gaussian integration identities
Cite this review
Pith. "Pith review of Anisotropic Gaussian Smoothing for Gradient-based Optimization." pith.science (2026). https://pith.science/paper/7VF733ZF
@misc{pith2026241111747,
author = {Pith},
title = {Pith review of: Anisotropic Gaussian Smoothing for Gradient-based Optimization},
year = {2026},
howpublished = {\url{https://pith.science/paper/7VF733ZF}},
note = {Machine review of arXiv:2411.11747}
}
read the original abstract
This article introduces a novel family of optimization algorithms - Anisotropic Gaussian Smoothing Gradient Descent (AGS-GD), AGS-Stochastic Gradient Descent (AGS-SGD), and AGS-Adam - that employ anisotropic Gaussian smoothing to enhance traditional gradient-based methods, including GD, SGD, and Adam. The primary goal of these approaches is to address the challenge of optimization methods becoming trapped in suboptimal local minima by replacing the standard gradient with a non-local gradient derived from averaging function values using anisotropic Gaussian smoothing. Unlike isotropic Gaussian smoothing (IGS), AGS adapts the smoothing directionality based on the properties of the underlying function, aligning better with complex loss landscapes and improving convergence. The anisotropy is computed by adjusting the covariance matrix of the Gaussian distribution, allowing for directional smoothing tailored to the gradient's behavior. This technique mitigates the impact of minor fluctuations, enabling the algorithms to approach global minima more effectively. We provide detailed convergence analyses that extend the results from both the original (unsmoothed) methods and the IGS case to the more general anisotropic smoothing, applicable to both convex and non-convex, L-smooth functions. In the stochastic setting, these algorithms converge to a noisy ball, with its size determined by the smoothing parameters. The article also outlines the theoretical benefits of anisotropic smoothing and details its practical implementation using Monte Carlo estimation, aligning with established zero-order optimization techniques.
Figures
Reference graph
Works this paper leans on
-
[1]
A restart CMA evolution strategy with in- creasing population size
Anne Auger and Nikolaus Hansen. A restart CMA evolution strategy with in- creasing population size. In 2005 IEEE Congress on Evolutionary Computation , volume 2, pages 1769–1776. IEEE, 2005
work page 2005
-
[2]
Stochastic optimization problems with nondifferentiable cost functionals
Dimitri P Bertsekas. Stochastic optimization problems with nondifferentiable cost functionals. Journal of Optimization Theory and Applications , 12(2):218–231, 1973
work page 1973
-
[3]
Andrew Blake and Andrew Zisserman. Visual reconstruction. MIT press, 1987
work page 1987
-
[4]
Randomized smoothing for stochastic optimization
John C Duchi, Peter L Bartlett, and Martin J Wainwright. Randomized smoothing for stochastic optimization. SIAM Journal on Optimization , 22(2):674–701, 2012
2012
-
[5]
The CMA evolution strategy: A comparing review
Nikolaus Hansen. The CMA evolution strategy: A comparing review. Towards a New Evolutionary Computation , pages 75–102, 2006
work page 2006
-
[6]
Evaluating the CMA evolution strategy on multimodal test functions
Nikolaus Hansen and Stefan Kern. Evaluating the CMA evolution strategy on multimodal test functions. In Parallel Problem Solving from Nature - PPSN VIII , pages 282–291, Berlin, Heidelberg, 2004. Springer
work page 2004
-
[7]
Completely derandomized self- adaptation in evolution strategies
Nikolaus Hansen and Andreas Ostermeier. Completely derandomized self- adaptation in evolution strategies. Evolutionary Computation, 9(2):159–195, 2001
work page 2001
-
[8]
On graduated optimiza- tion for stochastic non-convex problems
Elad Hazan, Kfir Yehuda Levy, and Shai Shalev-Shwartz. On graduated optimiza- tion for stochastic non-convex problems. In International conference on machine learning, pages 1833–1841. PMLR, 2016
work page 2016
Show all 24 references
-
[9]
Convergence of adam for non-convex objectives: Relaxed hyperparameters and non-ergodic case
Meixuan He, Yuqing Liang, Jinlan Liu, and Dongpo Xu. Convergence of adam for non-convex objectives: Relaxed hyperparameters and non-ergodic case. arXiv preprint arXiv 2307.11782 , 2023. 13
2023 arXiv
-
[10]
Covariance matrix adaptation for multi-objective optimization
Christian Igel, Nikolaus Hansen, and Stefan Roth. Covariance matrix adaptation for multi-objective optimization. Evolutionary Computation , 15(1):1–28, 2007
2007
-
[11]
Single loop gaussian homotopy method for non-convex optimization
Hidenori Iwakiri, Yuhang Wang, Shinji Ito, and Akiko Takeda. Single loop gaussian homotopy method for non-convex optimization. Advances in Neural Information Processing Systems, 35:7065–7076, 2022
2022
-
[12]
On large-batch training for deep learning: Generalization gap and sharp minima
Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. arXiv preprint arXiv:1609.04836 , 2016
2016 arXiv
-
[13]
A primer on zeroth-order optimization in signal processing and machine learning: Principals, recent advances, and applications
Sijia Liu, Pin-Yu Chen, Bhavya Kailkhura, Gaoyuan Zhang, Alfred O Hero III, and Pramod K Varshney. A primer on zeroth-order optimization in signal processing and machine learning: Principals, recent advances, and applications. IEEE Signal Processing Magazine, 37(5):43–54, 2020
2020
-
[14]
On the behavior of spatial critical points under gaussian blurring a folklore theorem and scale-space constraints
Marco Loog, Johannes JisseDuistermaat, and Luc MJ Florack. On the behavior of spatial critical points under gaussian blurring a folklore theorem and scale-space constraints. In Scale-Space and Morphology in Computer Vision: Third Interna- tional Conference, Scale-Space 2001 Va...
2001
-
[15]
CMA-ES for hyperparameter optimization of deep neural networks
Ilya Loshchilov and Frank Hutter. CMA-ES for hyperparameter optimization of deep neural networks. In International Conference on Learning Representations Workshop, 2017
2017
-
[16]
Optimization by Gaussian smoothing with application to geometric alignment
Hossein Mobahi. Optimization by Gaussian smoothing with application to geometric alignment. University of Illinois at Urbana-Champaign, 2012
2012
-
[17]
On the link between gaussian homotopy continuation and convex envelopes
Hossein Mobahi and John W Fisher. On the link between gaussian homotopy continuation and convex envelopes. In Energy Minimization Methods in Computer Vision and Pattern Recognition: 10th International Conference, EMMCVPR 2015, Hong Kong, China, January 13-16, 2015. Proceedings...
2015
-
[18]
Smoothing techniques for macromolecular global optimization
Jorge J More and Wu Zhijun. Smoothing techniques for macromolecular global optimization. Springer, 1996
1996
-
[19]
Nesterov and V
Y. Nesterov and V. Spokoiny. Random gradient-free minimization of convex func- tions. Foundations of Computational Mathematics , pages 1–40, 2015
2015
-
[20]
Real & Complex Analysis
Walter Rudin. Real & Complex Analysis . McGraw-Hill, 1987
1987
-
[21]
Gaussian smooth- ing gradient descent for minimizing high-dimensional non-convex functions
Andrew Starnes, Anton Dereventsov, and Clayton Webster. Gaussian smooth- ing gradient descent for minimizing high-dimensional non-convex functions. arXiv preprint arXiv 2311.00521 , 2023
2023 arXiv
-
[22]
Improved performance of stochastic gradi- ents with gaussian smoothing
Andrew Starnes and Clayton Webster. Improved performance of stochastic gradi- ents with gaussian smoothing. arXiv preprint arXiv 2311.00531 , 2024
2024 arXiv
-
[23]
Adadgs: An adaptive black-box optimization method with a nonlocal directional gaussian smoothing gradient
Hoang Tran and Guannan Zhang. Adadgs: An adaptive black-box optimization method with a nonlocal directional gaussian smoothing gradient. arXiv preprint arXiv:2011.02009, 2020
2011 arXiv
-
[24]
A Scalable Evolution Strategy with Directional Gaussian Smoothing for Blackbox Optimization
Jiaxin Zhang, Hoang Tran, Dan Lu, and Guannan Zhang. A Scalable Evolution Strategy with Directional Gaussian Smoothing for Blackbox Optimization. arXiv preprint arXiv:2002.03001, 2020. 14 Appendix A. Proofs of Background Results Proof of (3). Let v = x + Σu. Since Σ is inverti...
2002 arXiv
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