High Probability Complexity Bounds of Trust-Region Stochastic Sequential Quadratic Programming with Heavy-Tailed Noise

• URL: http://arxiv.org/abs/2503.19091v2
• Date: Sun, 06 Apr 2025 22:21:16 GMT
• Title: High Probability Complexity Bounds of Trust-Region Stochastic Sequential Quadratic Programming with Heavy-Tailed Noise
• Authors: Yuchen Fang, Javad Lavaei, Sen Na,
• Abstract summary: We propose a Trust-Region Sequential Quadratic Programming (TR-SSQP) method.<n>We establish its high-probability complexity bounds for identifying first- and second-order $epsilon$-stationary points.<n>We show that our method achieves the same high-probability first-order complexity bounds as in the light-tailed noise setting.
• Score: 23.663813244183984
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we consider nonlinear optimization problems with a stochastic objective and deterministic equality constraints. We propose a Trust-Region Stochastic Sequential Quadratic Programming (TR-SSQP) method and establish its high-probability iteration complexity bounds for identifying first- and second-order $\epsilon$-stationary points. In our algorithm, we assume that exact objective values, gradients, and Hessians are not directly accessible but can be estimated via zeroth-, first-, and second-order probabilistic oracles. Compared to existing complexity studies of SSQP methods that rely on a zeroth-order oracle with sub-exponential tail noise (i.e., light-tailed) and focus mostly on first-order stationarity, our analysis accommodates irreducible and heavy-tailed noise in the zeroth-order oracle and significantly extends the analysis to second-order stationarity. We show that under heavy-tailed noise conditions, our SSQP method achieves the same high-probability first-order iteration complexity bounds as in the light-tailed noise setting, while further exhibiting promising second-order iteration complexity bounds. Specifically, the method identifies a first-order $\epsilon$-stationary point in $\mathcal{O}(\epsilon^{-2})$ iterations and a second-order $\epsilon$-stationary point in $\mathcal{O}(\epsilon^{-3})$ iterations with high probability, provided that $\epsilon$ is lower bounded by a constant determined by the irreducible noise level in estimation. We validate our theoretical findings and evaluate the practical performance of our method on CUTEst benchmark test set.

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