Francesco Triggiano

Marco Romito, Francesco Triggiano

The signature is a fundamental object that describes paths (that is, continuous functions from an interval to a Euclidean space). Likewise, the expected signature provides a statistical description of the law of stochastic processes. We propose a feature extraction model for time series built upon the expected signature. This is computed through a Gaussian processes based data augmentation. One of the main features is that an optimal feature extraction is learnt through the supervised task that uses the model.

Luca Callisti, Marco Romito, Francesco Triggiano

We present a theoretical analysis of some popular adaptive Stochastic Gradient Descent (SGD) methods in the small learning rate regime. Using the stochastic modified equations framework introduced by Li et al., we derive effective continuous stochastic dynamics for these methods. Our key contribution is that sampling-induced noise in SGD manifests in the limit as independent Brownian motions driving the parameter and gradient second momentum evolutions. Furthermore, extending the approach of Malladi et al., we investigate scaling rules between the learning rate and key hyperparameters in adaptive methods, characterising all non-trivial limiting dynamics.