论文标题
低级别的正方形总和没有伪造的本地最小值
Low-Rank Univariate Sum of Squares Has No Spurious Local Minima
论文作者
论文摘要
我们研究了将多项式$ p $分解为$ r $ quares的问题，通过最大程度地减少四次惩罚的目标$ f_p（\ mathbf {u}）= \ left \ left \ lvert \ sum_ {i = 1}^r u_i^r u_i^r u_i^2- p \ p \ p \ reir \ lerver \ lerver \ lvert \ lvert \ lvert^2 $。该目标是非convex，相当于半标准程序（SDP）的等级$ r $ burer-monteiro分解编码平方分解之和。我们表明，对于所有单变量多项式$ p $，如果$ r \ ge 2 $，则$ f_p（\ mathbf {u}）$都没有二阶的二阶关键点，这表明所有本地Optima也是全局的Optima。这与先前的工作相反，表明对于一般的SDP，除了通用条件外，$ r $大致必须是约束数量（$ p $的程度）的平方根，才能没有虚假的二阶关键点。我们的证明使用计算代数几何形状中的工具，可以解释为使用一阶和二阶必需条件构建证书。我们还表明，通过基于圆上的采样相同点的采样标准，可以使用快速傅立叶变换在几乎线性的时间内计算梯度$ \ nabla f_p $。在实验上，我们证明了该方法使用一阶优化算法（例如L-BFGS）具有非常快速的收敛性，并具有接近线性的比例为百万度多项式。
We study the problem of decomposing a polynomial $p$ into a sum of $r$ squares by minimizing a quadratically penalized objective $f_p(\mathbf{u}) = \left\lVert \sum_{i=1}^r u_i^2 - p\right\lVert^2$. This objective is nonconvex and is equivalent to the rank-$r$ Burer-Monteiro factorization of a semidefinite program (SDP) encoding the sum of squares decomposition. We show that for all univariate polynomials $p$, if $r \ge 2$ then $f_p(\mathbf{u})$ has no spurious second-order critical points, showing that all local optima are also global optima. This is in contrast to previous work showing that for general SDPs, in addition to genericity conditions, $r$ has to be roughly the square root of the number of constraints (the degree of $p$) for there to be no spurious second-order critical points. Our proof uses tools from computational algebraic geometry and can be interpreted as constructing a certificate using the first- and second-order necessary conditions. We also show that by choosing a norm based on sampling equally-spaced points on the circle, the gradient $\nabla f_p$ can be computed in nearly linear time using fast Fourier transforms. Experimentally we demonstrate that this method has very fast convergence using first-order optimization algorithms such as L-BFGS, with near-linear scaling to million-degree polynomials.
