We introduce a new online algorithm for expected log-likelihood maximization in situations where the objective function is multi-modal and/or has saddle points, that we term G-PFSO. The key element underpinning G-PFSO is a probability distribution which (a) is shown to concentrate on the target parameter value as the sample size increases and (b) can be efficiently estimated by means of a standard particle filter algorithm. This distribution depends on a learning rate, where the faster the learning rate the quicker it concentrates on the desired element of the search space, but the less likely G-PFSO is to escape from a local optimum of the objective function. In order to achieve a fast convergence rate with a slow learning rate, G-PFSO exploits the acceleration property of averaging, well-known in the stochastic gradient literature. Considering several challenging estimation problems, the numerical experiments show that, with high probability, G-PFSO successfully finds the highest mode of the objective function and converges to its global maximizer at the optimal rate. While the focus of this work is expected log-likelihood maximization, the proposed methodology and its theory apply more generally for optimizing a function defined through an expectation.
翻译:在目标功能为多模式和/或具有马鞍点的情况下,我们为预期日志最大化引入一种新的在线算法,即我们称为G-PFSO。G-PFSO的关键要素是概率分布,(a) 显示随着抽样规模的增加,集中关注目标参数值,(b) 可以通过标准粒子过滤算法有效估算。这种分布取决于学习率,即学习速度越快,学习速度越快,它就越能集中到所希望的搜索空间,但G-PFSO越不可能从目标功能的当地最佳功能中逃脱。为了在学习速度缓慢的情况下实现快速趋同率,G-PFSO利用平均加速特性,这是在随机梯度文献中广为人知的。考虑到一些具有挑战性的估算问题,数字实验表明,G-PFSO在极有可能找到目标功能的最高模式,并且以最佳速度与全球最大化相趋同。虽然这项工作的重点是预期的日志最大化,但拟议的方法和理论更普遍地适用于通过预期来优化一个确定的职能。