Le Cam meets LeCun: Deficiency and Generic Feature Learning

"Deep Learning" methods attempt to learn generic features in an unsupervised fashion from a large unlabelled data set. These generic features should perform as well as the best hand crafted features for any learning problem that makes use of this data. We provide a definition of generic features, characterize when it is possible to learn them and provide methods closely related to the autoencoder and deep belief network of deep learning. In order to do so we use the notion of deficiency and illustrate its value in studying certain general learning problems.Comment: 25 pages, 2 figure

From Stochastic Mixability to Fast Rates

Empirical risk minimization (ERM) is a fundamental learning rule for statistical learning problems where the data is generated according to some unknown distribution $\mathsf{P}$ and returns a hypothesis $f$ chosen from a fixed class $\mathcal{F}$ with small loss $\ell$. In the parametric setting, depending upon $(\ell, \mathcal{F},\mathsf{P})$ ERM can have slow $(1/\sqrt{n})$ or fast $(1/n)$ rates of convergence of the excess risk as a function of the sample size $n$. There exist several results that give sufficient conditions for fast rates in terms of joint properties of $\ell$, $\mathcal{F}$, and $\mathsf{P}$, such as the margin condition and the Bernstein condition. In the non-statistical prediction with expert advice setting, there is an analogous slow and fast rate phenomenon, and it is entirely characterized in terms of the mixability of the loss $\ell$ (there being no role there for $\mathcal{F}$ or $\mathsf{P}$). The notion of stochastic mixability builds a bridge between these two models of learning, reducing to classical mixability in a special case. The present paper presents a direct proof of fast rates for ERM in terms of stochastic mixability of $(\ell,\mathcal{F}, \mathsf{P})$, and in so doing provides new insight into the fast-rates phenomenon. The proof exploits an old result of Kemperman on the solution to the general moment problem. We also show a partial converse that suggests a characterization of fast rates for ERM in terms of stochastic mixability is possible.Comment: 21 pages, accepted to NIPS 201

Particle Filter Design Using Importance Sampling for Acoustic Source Localisation and Tracking in Reverberant Environments

Sequential Monte Carlo methods have been recently proposed to deal with the problem of acoustic source localisation and tracking using an array of microphones. Previous implementations make use of the basic bootstrap particle filter, whereas a more general approach involves the concept of importance sampling. In this paper, we develop a new particle filter for acoustic source localisation using importance sampling, and compare its tracking ability with that of a bootstrap algorithm proposed previously in the literature. Experimental results obtained with simulated reverberant samples and real audio recordings demonstrate that the new algorithm is more suitable for practical applications due to its reinitialisation capabilities, despite showing a slightly lower average tracking accuracy. A real-time implementation of the algorithm also shows that the proposed particle filter can reliably track a person talking in real reverberant rooms.This paper was performed while Eric A. Lehmann was working with National ICT Australia. National ICT Australia is funded by the Australian Government’s Department of Communications, Information Technology, and the Arts, the Australian Research Council, through Backing Australia’s Ability, and the ICT Centre of Excellence programs