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Minimisation of Quasar-Convex Functions Using Random Zeroth-Order Oracles
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Minimisation of Quasar-Convex Functions Using Random Zeroth-Order Oracles
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This paper explores the performance of a random Gaussian smoothing zeroth-order (ZO) scheme for minimising quasar-convex (QC) and strongly quasar-convex (SQC) functions in both unconstrained and constrained settings. For the unconstrained problem, we establish the ZO algorithm's convergence to a global minimum along with its complexity when applied to both QC and SQC functions. For the constrained problem, we introduce the new notion of proximal-quasar-convexity and prove analogous results to the unconstrained case. Specifically, we derive complexity bounds and prove convergence of the algorithm to a neighbourhood of a global minimum whose size can be controlled under a variance reduction scheme. Beyond the theoretical guarantees, we demonstrate the practical implications of our results on several machine learning problems where quasar-convexity naturally arises, including linear dynamical system identification and generalised linear models.
Forward citations
Cited by 4 Pith papers
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Accelerated Stochastic Zeroth-Order Quasar-Convex Optimization
A continuized zeroth-order Nesterov method achieves O(d/√ε) function-evaluation complexity for smooth quasar-convex minimization, with improved dimension dependence under a 1-norm mirror step when the solution is sparse.
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From Cursed to Competitive: Closing the ZO-FO Gap via Input-to-State Stability
Zeroth-order methods achieve the same expected convergence rate as first-order methods without extra dimension dependence by treating them as input-to-state stable systems with controllable perturbations.
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Robust Learning Meets Quasar-Convex Optimization: Inexact High-Order Proximal-Point Methods
Robust learning problems are formulated as quasar-convex optimization, and HiPPA is proposed as an inexact high-order proximal method with global and superlinear convergence guarantees.
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Mirror Descent Methods for Quasar Convex Optimization Problems With Non-Smooth Inequality Constraints
Extends mirror descent with productive/non-productive steps to quasar-convex objectives with nonsmooth constraints, but the convergence claims are only partially proven.
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