We examine a network of learners which address the same classification task but must learn from different data sets. The learners can share a limited portion of their data sets so as to preserve the network load. We introduce DELCO (standing for Decentralized Ensemble Learning with COpulas), a new approach in which the shared data and the trained models are sent to a central machine that allows to build an ensemble of classifiers. The proposed method aggregates the base classifiers using a probabilistic model relying on Gaussian copulas. Experiments on logistic regressor ensembles demonstrate competing accuracy and increased robustness as compared to gold standard approaches. A companion python implementation can be downloaded at https: github.com john-klein DELCO
In this subsection, we present several approaches for inferring the parameters of the multinomial conditional distributions @math . These approaches are essentially due to Dawid and Skene @cite_8 and were promoted and further developed by Kim and Ghahramani @cite_12 in the context of classifier combination. Inferring parameters of multinomial distributions may not seem challenging at first sight. The problem is that, we need to solve @math such inference problems so the complexity of the problem does not scale well w.r.t. both @math and @math . Applying Bayes formula, we have The estimation of class probabilities is easy but again, the estimation of conditional joint distributions @math has the same complexity as the estimation of the posterior. Linear complexity can be achieved by making conditional independence assumptions that allow each conditional joint distribution to factorize as the product of its marginals, that is
@cite_12 @cite_8
1564629734 2131775048 2127090196 2026924428
We investigate the class of computable probability distributions and explore the fundamental limitations of using this class to describe and compute conditional distributions. In addition to proving the existence of noncomputable conditional distributions, and thus ruling out the possibility of generic probabilistic inference algorithms (even inefficient ones), we highlight some positive results showing that posterior inference is possible in the presence of additional structure like exchangeability and noise, both of which are common in Bayesian hierarchical modeling. This theoretical work bears on the development of probabilistic programming languages (which enable the specification of complex probabilistic models) and their implementations (which can be used to perform Bayesian reasoning). The probabilistic programming approach is particularly well suited for defining infinite-dimensional, recursively-defined stochastic processes of the sort used in nonparametric Bayesian statistics. We present a new construction of the Mondrian process as a partition-valued Markov process in continuous time, which can be viewed as placing a distribution on an infinite kd-tree data structure. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.) We observe a training set Q composed of l labeled samples (X sub 1 , spl theta sub 1 ),...,(X sub l , spl theta sub l ) and u unlabeled samples X sub 1 ',...,X sub u ' . The labels spl theta sub i are independent random variables satisfying Pr spl theta sub i =1 = spl eta , Pr spl theta sub i =2 =1- spl eta . The labeled observations X sub i are independently distributed with conditional density f sub spl theta i ( spl middot ) given spl theta sub i . Let (X sub 0 , spl theta sub 0 ) be a new sample, independently distributed as the samples in the training set. We observe X sub 0 and we wish to infer the classification spl theta sub 0 . In this paper we first assume that the distributions f sub 1 ( spl middot ) and f sub 2 ( spl middot ) are given and that the mixing parameter is unknown. We show that the relative value of labeled and unlabeled samples in reducing the risk of optimal classifiers is the ratio of the Fisher informations they carry about the parameter spl eta . We then assume that two densities g sub 1 ( spl middot ) and g sub 2 ( spl middot ) are given, but we do not know whether g sub 1 ( spl middot )=f sub 1 ( spl middot ) and g sub 2 ( spl middot )=f sub 2 ( spl middot ) or if the opposite holds, nor do we know spl eta . Thus the learning problem consists of both estimating the optimum partition of the observation space and assigning the classifications to the decision regions. Here, we show that labeled samples are necessary to construct a classification rule and that they are exponentially more valuable than unlabeled samples. We introduce a new approach to characterizing the unobserved portion of a distribution, which provides sublinear--sample estimators achieving arbitrarily small additive constant error for a class of properties that includes entropy and distribution support size. Additionally, we show new matching lower bounds. Together, this settles the longstanding question of the sample complexities of these estimation problems, up to constant factors. Our algorithm estimates these properties up to an arbitrarily small additive constant, using O(n log n) samples, where n is a bound on the support size, or in the case of estimating the support size, 1 n is a lower bound on the probability of any element of the domain. Previously, no explicit sublinear--sample algorithms for either of these problems were known. Our algorithm is also computationally extremely efficient, running in time linear in the number of samples used. In the second half of the paper, we provide a matching lower bound of Ω(n log n) samples for estimating entropy or distribution support size to within an additive constant. The previous lower-bounds on these sample complexities were n 2O(√log n). To show our lower bound, we prove two new and natural multivariate central limit theorems (CLTs); the first uses Stein's method to relate the sum of independent distributions to the multivariate Gaussian of corresponding mean and covariance, under the earthmover distance metric (also known as the Wasserstein metric). We leverage this central limit theorem to prove a stronger but more specific central limit theorem for "generalized multinomial" distributions---a large class of discrete distributions, parameterized by matrices, that represents sums of independent binomial or multinomial distributions, and describes many distributions encountered in computer science. Convergence here is in the strong sense of statistical distance, which immediately implies that any algorithm with input drawn from a generalized multinomial distribution behaves essentially as if the input were drawn from a discretized Gaussian with the same mean and covariance. Such tools in the multivariate setting are rare, and we hope this new tool will be of use to the community. where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail.