Improved Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandit

Novitskii, Vasilii; Gasnikov, Alexander

doi:10.48550/arxiv.2101.03821

Cited by 6 publications

(13 citation statements)

References 4 publications

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“…The recent work[21] obtains the same improvement, using the gradient estimator of[2]. However as we notice below that estimator is less computationally appealing.…”

mentioning

confidence: 64%

Distributed Zero-Order Optimization under Adversarial Noise

Akhavan,

Pontil,

Tsybakov

2021

Preprint

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We study the problem of distributed zero-order optimization for a class of strongly convex functions. They are formed by the average of local objectives, associated to different nodes in a prescribed network of connections. We propose a distributed zero-order projected gradient descent algorithm to solve this problem. Exchange of information within the network is permitted only between neighbouring nodes. A key feature of the algorithm is that it can query only function values, subject to a general noise model, that does not require zero mean or independent errors. We derive upper bounds for the average cumulative regret and optimization error of the algorithm which highlight the role played by a network connectivity parameter, the number of variables, the noise level, the strong convexity parameter of the global objective and certain smoothness properties of the local objectives. When the bound is specified to the standard undistributed setting, we obtain an improvement over the state-of-the-art bounds, due to the novel gradient estimation procedure proposed here. We also comment on lower bounds and observe that the dependency over certain function parameters in the bound is nearly optimal.

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“…The recent work[21] obtains the same improvement, using the gradient estimator of[2]. However as we notice below that estimator is less computationally appealing.…”

mentioning

confidence: 64%

Distributed Zero-Order Optimization under Adversarial Noise

Akhavan,

Pontil,

Tsybakov

2021

Preprint

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“…Note, that instead of finite-difference approximation approach in some applications we can use kernel approach [43,3]. The interest to this alternative has grown last time [2,39].…”

Section: Gradient-free Methodsmentioning

confidence: 99%

Accelerated gradient methods with absolute and relative noise in the gradient

Artem¹,

Gasnikov²,

Dvurechensky³

et al. 2021

Preprint

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In this article, we investigate an accelerated first-order method, namely, the method of similar triangles, which is optimal in the class of convex (strongly convex) problems with a Lipschitz gradient. The paper considers a model of additive noise in a gradient and a Euclidean prox-structure for not necessarily bounded sets. Convergence estimates are obtained in the case of strong convexity and its absence, and a stopping criterion is proposed for not strongly convex problems.

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“…Many more examples are possible, for example, as sketched in [CSV09, Section 1.2] one could consider tuning the regularization parameter in ridge regression. Comparing to for example the numerical experiment from [NG21] is less insightful as their smoothing parameter relies on the variance of the noise.…”

Section: Non-convex Optimizationmentioning

confidence: 99%

Small Errors in Random Zeroth Order Optimization are Imaginary

Jongeneel¹,

Yue²,

Kühn³

2021

Preprint

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The vast majority of zeroth order optimization methods try to imitate first order methods via some smooth approximation of the gradient. Here, the smaller the smoothing parameter, the smaller the gradient approximation error. We show that for the majority of zeroth order methods this smoothing parameter can however not be chosen arbitrarily small as numerical cancellation errors will dominate. As such, theoretical and numerical performance could differ significantly. Using classical tools from numerical differentiation we will propose a new smoothed approximation of the gradient that can be integrated into general zeroth order algorithmic frameworks. Since the proposed smoothed approximation does not suffer from cancellation error, the smoothing parameter (and hence the approximation error) can be made arbitrarily small. Sublinear convergence rates for algorithms based on our smoothed approximation are proved. Numerical experiments are also presented to demonstrate the superiority of algorithms based on the proposed approximation.

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Improved Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandit

Cited by 6 publications

References 4 publications

Distributed Zero-Order Optimization under Adversarial Noise

Distributed Zero-Order Optimization under Adversarial Noise

Accelerated gradient methods with absolute and relative noise in the gradient

Small Errors in Random Zeroth Order Optimization are Imaginary

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