Title: Learning in temporally structured environments. Abstract: Natural environments have temporal structure at multiple timescales. This property is reflected in biological learning and memory but typically not in machine learning systems. We advance a multiscale learning method in which each weight in a neural network is decomposed as a sum of subweights with different learning and decay rates. Thus knowledge becomes distributed across different timescales, enabling rapid adaptation to task changes while avoiding catastrophic interference. First, we prove previous models that learn at multiple timescales, but with complex coupling between timescales, are equivalent to multiscale learning via a reparameterization that eliminates this coupling. The same analysis yields a new characterization of momentum learning, as a fast weight with a negative learning rate. Second, we derive a model of Bayesian inference over $1/f$ noise, a common temporal pattern in many online learning domains that involves long-range (power law) autocorrelations. The generative side of the model expresses $1/f$ noise as a sum of diffusion processes at different timescales, and the inferential side tracks these latent processes using a Kalman filter. We then derive a variational approximation to the Bayesian model and show how it is an extension of the multiscale learner. The result is an optimizer that can be used as a drop-in replacement in an arbitrary neural network architecture. Third, we evaluate the ability of these methods to handle nonstationarity by testing them in online prediction tasks characterized by $1/f$ noise in the latent parameters. We find that the Bayesian model significantly outperforms online stochastic gradient descent and two batch heuristics that rely preferentially or exclusively on more recent data. Moreover, the variational approximation performs nearly as well as the full Bayesian model, and with memory requirements that are linear in the size of the network.
