关于局部极大值的Grothendieck型不等式
A Grothendieck-type inequality for local maxima
摘要：
在优化、机器学习、信号处理等领域中，大量的问题可以通过适当的半定规划（SDP）松弛来有效地解决。不幸的是，通用SDP求解器几乎不能扩展到具有几百个变量的实例（在底层组合问题中）。另一方面，经验上已经观察到，有效的策略等于引入（非凸）秩约束，并且通过上升方法来求解所得到的平滑最优化问题。这个非凸问题一般有大量的局部极大值，因此这种成功的原因尚不清楚。本文为这种方法提供了有力的支持。对于椭圆形上最大化线性泛函的问题，我们证明了所有的局部最大值都在一个小的间隙从SDP最佳。在几个感兴趣的问题中，任意小的相对误差可以通过将秩约束$k$取为一阶来实现，而与问题的大小无关。
英文摘要：
A large number of problems in optimization, machine learning, signal processing can be effectively addressed by suitable semidefinite programming (SDP) relaxations. Unfortunately, generic SDP solvers hardly scale beyond instances with a few hundreds variables (in the underlying combinatorial problem). On the other hand, it has been observed empirically that an effective strategy amounts to introducing a (non-convex) rank constraint, and solving the resulting smooth optimization problem by ascent methods. This non-convex problem has --generically-- a large number of local maxima, and the reason for this success is therefore unclear. This paper provides rigorous support for this approach. For the problem of maximizing a linear functional over the elliptope, we prove that all local maxima are within a small gap from the SDP optimum. In several problems of interest, arbitrarily small relative error can be achieved by taking the rank constraint $k$ to be of order one, independently of the problem size.
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