通过流形识别加速正则化优化的近似二次逼近
Accelerating Inexact Successive Quadratic Approximation for Regularized Optimization Through Manifold Identification
AI总结：
本文提出ISQA+方法，通过识别流形提升正则化优化效率，实现迭代和运行时间双重超线性收敛。
摘要：
对于正则化优化问题，即最小化光滑项与促进结构解的正则化项之和，近似二次逼近（SQA）方法因其迭代的超线性收敛性而被广泛使用。然而，与光滑优化不同，它们在求解正则化子问题时运行时间较长，因为即使近似解也难以轻易计算，导致实际时间成本不令人满意。本文首先表明，对于部分光滑正则化项，尽管一般近似解无法识别使目标函数光滑的主动流形，但由常用子问题求解器生成的近似解仍能识别该流形，即使解的精度任意低。随后，我们利用这一性质提出改进的SQA方法ISQA+，在识别该流形后切换到高效的光滑优化方法。我们证明，对于广泛类别的退化解，ISQA+不仅在迭代次数上具有超线性收敛性，而且在运行时间上也具有超线性收敛性，因为每次迭代的成本是有限的。特别是，我们的超线性收敛结果适用于满足比现有文献更广泛条件的优化问题。在真实世界问题上的实验也证实，ISQA+显著提升了正则化优化的性能。
英文摘要：
For regularized optimization that minimizes the sum of a smooth term and a regularizer that promotes structured solutions, inexact proximal-Newton-type methods, or successive quadratic approximation (SQA) methods, are widely used for their superlinear convergence in terms of iterations. However, unlike the counter parts in smooth optimization, they suffer from lengthy running time in solving regularized subproblems because even approximate solutions cannot be computed easily, so their empirical time cost is not as impressive. In this work, we first show that for partly smooth regularizers, although general inexact solutions cannot identify the active manifold that makes the objective function smooth, approximate solutions generated by commonly-used subproblem solvers will identify this manifold, even with arbitrarily low solution precision. We then utilize this property to propose an improved SQA method, ISQA+, that switches to efficient smooth optimization methods after this manifold is identified. We show that for a wide class of degenerate solutions, ISQA+ possesses superlinear convergence not just only in iterations, but also in running time because the cost per iteration is bounded. In particular, our superlinear convergence result holds on problems satisfying a sharpness condition more general than that in existing literature. Experiments on real-world problems also confirm that ISQA+ greatly improves the state of the art for regularized optimization.
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