A Tight Excess Risk Bound via a Unified PAC-Bayesian-Rademacher-Shtarkov-MDL Complexity

We present a novel notion of complexity that interpolates between and generalizes some classic existing complexity notions in learning theory: for estimators like empirical risk minimization (ERM) with arbitrary bounded losses, it is upper bounded in terms of data-independent Rademacher complexity; for generalized Bayesian estimators, it is upper bounded by the data-dependent information complexity (also known as stochastic or PAC-Bayesian, $\mathrm{KL}(\text{posterior} \operatorname{\|} \text{prior})$ complexity. For (penalized) ERM, the new complexity reduces to (generalized) normalized maximum likelihood (NML) complexity, i.e. a minimax log-loss individual-sequence regret. Our first main result bounds excess risk in terms of the new complexity. Our second main result links the new complexity via Rademacher complexity to $L_2(P)$ entropy, thereby generalizing earlier results of Opper, Haussler, Lugosi, and Cesa-Bianchi who did the log-loss case with $L_\infty$. Together, these results recover optimal bounds for VC- and large (polynomial entropy) classes, replacing localized Rademacher complexity by a simpler analysis which almost completely separates the two aspects that determine the achievable rates: 'easiness' (Bernstein) conditions and model complexity.Comment: 38 page

Incremental Learning of Nonparametric Bayesian Mixture Models

Clustering is a fundamental task in many vision applications. To date, most clustering algorithms work in a batch setting and training examples must be gathered in a large group before learning can begin. Here we explore incremental clustering, in which data can arrive continuously. We present a novel incremental model-based clustering algorithm based on nonparametric Bayesian methods, which we call Memory Bounded Variational Dirichlet Process (MB-VDP). The number of clusters are determined flexibly by the data and the approach can be used to automatically discover object categories. The computational requirements required to produce model updates are bounded and do not grow with the amount of data processed. The technique is well suited to very large datasets, and we show that our approach outperforms existing online alternatives for learning nonparametric Bayesian mixture models

Learning Bounded Treewidth Bayesian Networks with Thousands of Variables

We present a method for learning treewidth-bounded Bayesian networks from data sets containing thousands of variables. Bounding the treewidth of a Bayesian greatly reduces the complexity of inferences. Yet, being a global property of the graph, it considerably increases the difficulty of the learning process. We propose a novel algorithm for this task, able to scale to large domains and large treewidths. Our novel approach consistently outperforms the state of the art on data sets with up to ten thousand variables

Cutset Sampling for Bayesian Networks

The paper presents a new sampling methodology for Bayesian networks that samples only a subset of variables and applies exact inference to the rest. Cutset sampling is a network structure-exploiting application of the Rao-Blackwellisation principle to sampling in Bayesian networks. It improves convergence by exploiting memory-based inference algorithms. It can also be viewed as an anytime approximation of the exact cutset-conditioning algorithm developed by Pearl. Cutset sampling can be implemented efficiently when the sampled variables constitute a loop-cutset of the Bayesian network and, more generally, when the induced width of the networks graph conditioned on the observed sampled variables is bounded by a constant w. We demonstrate empirically the benefit of this scheme on a range of benchmarks

A new metric for probability distributions

We introduce a metric for probability distributions, which is bounded, information-theoretically motivated, and has a natural Bayesian interpretation. The square root of the well-known chi(2) distance is an asymptotic approximation to it. Moreover, it is a close relative of the capacitory discrimination and Jensen-Shannon divergence.Publisher PDFPeer reviewe