We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both the algorithms and their analyses build extensively on existing work in the Euclidean case. The basic operation consists in running the Euclidean accelerated gradient descent method (appropriately safe-guarded against non-convexity) in the current tangent space, then moving back to the manifold and repeating. This requires lifting the cost function from the manifold to the tangent space, which can be done for example through the Riemannian exponential map. For this approach to succeed, the lifted cost function (called the pullback) must retain certain Lipschitz properties. As a contribution of independent interest, we prove precise claims to that effect, with explicit constants. Those claims are affected by the Riemannian curvature of the manifold, which in turn affects the worst-case complexity bounds for our optimization algorithms.

我们描述了黎曼流形上的第一个梯度方法，以在非凸情况下实现加速。在对成本函数的黎曼梯度和 Hessian 的 Lipschitz 假设下，这些方法比常规梯度下降更快地找到近似一阶临界点。随机版本还可以找到近似的二阶临界点。算法及其分析都广泛建立在欧几里得案例中的现有工作之上。基本操作包括在当前切线空间中运行欧几里得加速梯度下降法（适当地防止非凸性），然后移回流形并重复。这需要将成本函数从流形提升到切线空间，这可以通过例如黎曼指数映射来完成。为了使这种方法成功，提升的成本函数（称为回调）必须保留某些 Lipschitz 属性。作为独立利益的贡献，我们用明确的常数证明了这种效果的精确主张。这些说法受到流形的黎曼曲率的影响，这反过来又会影响我们优化算法的最坏情况复杂性界限。