Abstract
Convergence of a stochastic process is an intrinsic property quite relevant for its successful practical for example for the function optimization problem. Lyapunov functions are widely used as tools to prove convergence of optimization procedures. However, identifying a Lyapunov function for a specific stochastic process is a difficult and creative task. This work aims to provide a geometric explanation to convergence results and to state and identify conditions for the convergence of not exclusively optimization methods but any stochastic process. Basically, we relate the expected directions set of a stochastic process with the half-space of a conservative vector field, concepts defined along the text. After some reasonable conditions, it is possible to assure convergence when the expected direction resembles enough to some vector field. We translate two existent and useful convergence results into convergence of processes that resemble to particular conservative vector fields. This geometric point of view could make it easier to identify Lyapunov functions for new stochastic processes which we would like to prove its convergence.
Highlights
The stochastic natural gradient descent (SNGD) of Amari [1] and its variants often show instability depending on the starting point and learning rate tuning
The tools to define the expected direction set at η ∈ Rk after time T ∈ N are given, so we proceed to its formal definition
We have presented a result that allows us to prove the convergence of stochastic processes
Summary
Along most practical research branches, the solution to a given problem is often entrusted to a function optimization problem, where the effectiveness of a solution is measured by a function to be optimized. Seminal work, Bottou proved the convergence of stochastic gradient descent (SGD). It can be seen that the latter is proving that the function to optimize serves already as a Lyapunov function, similar to latter chapters in [4] Both proofs share some similarity but it is not evident how to raise a connection. The generalized convergence result for stochastic processes in this article is obtained after 2 main concepts. Two corollaries are extracted from this result, which we prove to be equivalent to Bottou’s and Sunehag’s convergence theorems. As random variables are used to describe general random phenomena, stochastic processes indexed by N are usually used to model random sequences
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