Transitive consistency is an intrinsic property for collections of linear invertible transformations between Euclidean coordinate frames. In practice, when the transformations are estimated from data, this property is lacking. This work addresses the problem of synchronizing transformations that are not transitively consistent. Once the transformations have been synchronized, they satisfy the transitive consistency condition - a transformation from frame $A$ to frame $C$ is equal to the composite transformation of first transforming A to B and then transforming B to C. The coordinate frames correspond to nodes in a graph and the transformations correspond to edges in the same graph. Two direct or centralized synchronization methods are presented for different graph topologies; the first one for quasi-strongly connected graphs, and the second one for connected graphs. As an extension of the second method, an iterative Gauss-Newton method is presented, which is later adapted to the case of affine and Euclidean transformations. Two distributed synchronization methods are also presented for orthogonal matrices, which can be seen as distributed versions of the two direct or centralized methods; they are similar in nature to standard consensus protocols used for distributed averaging. When the transformations are orthogonal matrices, a bound on the optimality gap can be computed. Simulations show that the gap is almost right, even for noise large in magnitude. This work also contributes on a theoretical level by providing linear algebraic relationships for transitively consistent transformations. One of the benefits of the proposed methods is their simplicity - basic linear algebraic methods are used, e.g., the Singular Value Decomposition (SVD). For a wide range of parameter settings, the methods are numerically validated.

The speed with which a learning algorithm converges as it is presented with more data is a central problem in machine learning --- a fast rate of convergence means less data is needed for the same level of performance. The pursuit of fast rates in online and statistical learning has led to the discovery of many conditions in learning theory under which fast learning is possible. We show that most of these conditions are special cases of a single, unifying condition, that comes in two forms: the central condition for 'proper' learning algorithms that always output a hypothesis in the given model, and stochastic mixability for online algorithms that may make predictions outside of the model. We show that under surprisingly weak assumptions both conditions are, in a certain sense, equivalent. The central condition has a re-interpretation in terms of convexity of a set of pseudoprobabilities, linking it to density estimation under misspecification. For bounded losses, we show how the central condition enables a direct proof of fast rates and we prove its equivalence to the Bernstein condition, itself a generalization of the Tsybakov margin condition, both of which have played a central role in obtaining fast rates in statistical learning. Yet, while the Bernstein condition is two-sided, the central condition is one-sided, making it more suitable to deal with unbounded losses. In its stochastic mixability form, our condition generalizes both a stochastic exp-concavity condition identified by Juditsky, Rigollet and Tsybakov and Vovk's notion of mixability. Our unifying conditions thus provide a substantial step towards a characterization of fast rates in statistical learning, similar to how classical mixability characterizes constant regret in the sequential prediction with expert advice setting.

We provide non-asymptotic bounds for the well-known temporal difference learning algorithm TD(0) with linear function approximators. These include high-probability bounds as well as bounds in expectation. Our analysis suggests that a step-size inversely proportional to the number of iterations cannot guarantee optimal rate of convergence unless we assume (partial) knowledge of the stationary distribution for the Markov chain underlying the policy considered. We also provide bounds for the iterate averaged TD(0) variant, which gets rid of the step-size dependency while exhibiting the optimal rate of convergence. Furthermore, we propose a variant of TD(0) with linear approximators that incorporates a centering sequence, and establish that it exhibits an exponential rate of convergence in expectation. We demonstrate the usefulness of our bounds on two synthetic experimental settings.

We consider the iterated belief change that occurs following an alternating sequence of actions and observations. At each instant, an agent has beliefs about the actions that have occurred as well as beliefs about the resulting state of the world. We represent such problems by a sequence of ranking functions, so an agent assigns a quantitative plausibility value to every action and every state at each point in time. The resulting formalism is able to represent fallible belief, erroneous perception, exogenous actions, and failed actions. We illustrate that our framework is a generalization of several existing approaches to belief change, and it appropriately captures the non-elementary interaction between belief update and belief revision.

Algorithm selection (AS) techniques -- which involve choosing from a set of algorithms the one expected to solve a given problem instance most efficiently -- have substantially improved the state of the art in solving many prominent AI problems, such as SAT, CSP, ASP, MAXSAT and QBF. Although several AS procedures have been introduced, not too surprisingly, none of them dominates all others across all AS scenarios. Furthermore, these procedures have parameters whose optimal values vary across AS scenarios. This holds specifically for the machine learning techniques that form the core of current AS procedures, and for their hyperparameters. Therefore, to successfully apply AS to new problems, algorithms and benchmark sets, two questions need to be answered: (i) how to select an AS approach and (ii) how to set its parameters effectively. We address both of these problems simultaneously by using automated algorithm configuration. Specifically, we demonstrate that we can automatically configure claspfolio 2, which implements a large variety of different AS approaches and their respective parameters in a single, highly-parameterized algorithm framework. Our approach, dubbed AutoFolio, allows researchers and practitioners across a broad range of applications to exploit the combined power of many different AS methods. We demonstrate AutoFolio can significantly improve the performance of claspfolio 2 on 8 out of the 13 scenarios from the Algorithm Selection Library, leads to new state-of-the-art algorithm selectors for 7 of these scenarios, and matches state-of-the-art performance (statistically) on all other scenarios. Compared to the best single algorithm for each AS scenario, AutoFolio achieves average speedup factors between 1.3 and 15.4.

We design and analyze deterministic truthful approximation mechanisms for multi-unit Combinatorial Auctions involving only a constant number of distinct goods, each in arbitrary limited supply. Prospective buyers (bidders) have preferences over multisets of items, i.e., for more than one unit per distinct good. Our objective is to determine allocations of multisets that maximize the Social Welfare. Our main results are for multi-minded and submodular bidders. In the first setting each bidder has a positive value for being allocated one multiset from a prespecified demand set of alternatives. In the second setting each bidder is associated to a submodular valuation function that defines his value for the multiset he is allocated. For multi-minded bidders, we design a truthful FPTAS that fully optimizes the Social Welfare, while violating the supply constraints on goods within factor (1+e), for any fixed e>0 (i.e., the approximation applies to the constraints and not to the Social Welfare). This result is best possible, in that full optimization is impossible without violating the supply constraints. For submodular bidders, we obtain a PTAS that approximates the optimum Social Welfare within factor (1+e), for any fixed e>0, without violating the supply constraints. This result is best possible as well. Our allocation algorithms are Maximal-in-Range and yield truthful mechanisms, when paired with Vickrey-Clarke-Groves payments.

Optimization of very expensive black-box functions requires utilization of maximum information gathered by the process of optimization. Model Guided Sampling Optimization (MGSO) forms a more robust alternative to Jones' Gaussian-process-based EGO algorithm. Instead of EGO's maximizing expected improvement, the MGSO uses sampling the probability of improvement which is shown to be helpful against trapping in local minima. Further, the MGSO can reach close-to-optimum solutions faster than standard optimization algorithms on low dimensional or smooth problems.

A general approach for anomaly detection or novelty detection consists in estimating high density regions or Minimum Volume (MV) sets. The One-Class Support Vector Machine (OCSVM) is a state-of-the-art algorithm for estimating such regions from high dimensional data. Yet it suffers from practical limitations. When applied to a limited number of samples it can lead to poor performance even when picking the best hyperparameters. Moreover the solution of OCSVM is very sensitive to the selection of hyperparameters which makes it hard to optimize in an unsupervised setting. We present a new approach to estimate MV sets using the OCSVM with a different choice of the parameter controlling the proportion of outliers. The solution function of the OCSVM is learnt on a training set and the desired probability mass is obtained by adjusting the offset on a test set to prevent overfitting. Models learnt on different train/test splits are then aggregated to reduce the variance induced by such random splits. Our approach makes it possible to tune the hyperparameters automatically and obtain nested set estimates. Experimental results show that our approach outperforms the standard OCSVM formulation while suffering less from the curse of dimensionality than kernel density estimates. Results on actual data sets are also presented.

Recent advances in cryptography promise to enable secure statistical computation on encrypted data, whereby a limited set of operations can be carried out without the need to first decrypt. We review these homomorphic encryption schemes in a manner accessible to statisticians and machine learners, focusing on pertinent limitations inherent in the current state of the art. These limitations restrict the kind of statistics and machine learning algorithms which can be implemented and we review those which have been successfully applied in the literature. Finally, we document a high performance R package implementing a recent homomorphic scheme in a general framework.

In this paper we consider the training of single hidden layer neural networks by pseudoinversion, which, in spite of its popularity, is sometimes affected by numerical instability issues. Regularization is known to be effective in such cases, so that we introduce, in the framework of Tikhonov regularization, a matricial reformulation of the problem which allows us to use the condition number as a diagnostic tool for identification of instability. By imposing well-conditioning requirements on the relevant matrices, our theoretical analysis allows the identification of an optimal value for the regularization parameter from the standpoint of stability. We compare with the value derived by cross-validation for overfitting control and optimisation of the generalization performance. We test our method for both regression and classification tasks. The proposed method is quite effective in terms of predictivity, often with some improvement on performance with respect to the reference cases considered. This approach, due to analytical determination of the regularization parameter, dramatically reduces the computational load required by many other techniques.