提出通用随机内点法，解决大规模机器学习中的锥规划问题
Stochastic interior-point methods for smooth conic optimization with applications
- 设计四种基于不同随机梯度估计的随机内点算法
- 迭代复杂度逼近最优随机无约束优化结果
- 适用于鲁棒线性回归等大规模数据场景
锥规划在众多机器学习问题中至关重要，但针对大规模数据集的锥约束学习问题，现有实用算法多局限于特定场景，因为通用锥规划的随机算法发展仍不充分。为填补这一空白，我们提出一种通用锥规划的随机内点法（SIPM）框架，并设计了四种利用不同随机梯度估计器的新变体。在温和假设下，建立了所提SIPM的迭代复杂度，其性能在多项式对数因子范围内达到随机无约束优化的最佳已知结果。最后，在鲁棒线性回归、多任务关系学习和聚类数据流上的数值实验验证了该方法的有效性与高效性。
原文摘要 · Abstract (English)
Conic optimization plays a crucial role in many machine learning (ML) problems. However, practical algorithms for conic constrained ML problems with large datasets are often limited to specific use cases, as stochastic algorithms for general conic optimization remain underdeveloped. To fill this gap, we introduce a stochastic interior-point method (SIPM) framework for general conic optimization, along with four novel SIPM variants leveraging distinct stochastic gradient estimators. Under mild assumptions, we establish the iteration complexity of our proposed SIPMs, which, up to a polylogarithmic factor, match the best-known {results} in stochastic unconstrained optimization. Finally, our numerical experiments on robust linear regression, multi-task relationship learning, and clustering data streams demonstrate the effectiveness and efficiency of our approach.
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