Local Optimality and Generalization Guarantees for the Langevin Algorithm via Empirical Metastability
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Proceedings of the 31st Conference On Learning Theory, PMLR 75:857875, 2018.
Abstract
We study the detailed pathwise behavior of the discretetime Langevin algorithm for nonconvex Empirical Risk Minimization (ERM) through the lens of metastability, adopting some techniques from Berglund and Gentz (2003). For a particular local optimum of the empirical risk, with an \textit{arbitrary initialization}, we show that, with high probability, at least one of the following two events will occur: (1) the Langevin trajectory ends up somewhere outside the $\varepsilon$neighborhood of this particular optimum within a short \textit{recurrence time}; (2) it enters this $\varepsilon$neighborhood by the recurrence time and stays there until a potentially exponentially long \textit{escape time}. We call this phenomenon \textit{empirical metastability}. This twotimescale characterization aligns nicely with the existing literature in the following two senses. First, the effective recurrence time (i.e., number of iterations multiplied by stepsize) is dimensionindependent, and resembles the convergence time of continuoustime deterministic Gradient Descent (GD). However unlike GD, the Langevin algorithm does not require strong conditions on local initialization, and has the possibility of eventually visiting all optima. Second, the scaling of the escape time is consistent with the EyringKramers law, which states that the Langevin scheme will eventually visit all local minima, but it will take an exponentially long time to transit among them. We apply this pathwise concentration result in the context of statistical learning to examine local notions of generalization and optimality.
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