336,674 research outputs found

    The update complexity of selection and related problems

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    We present a framework for computing with input data specified by intervals, representing uncertainty in the values of the input parameters. To compute a solution, the algorithm can query the input parameters that yield more refined estimates in form of sub-intervals and the objective is to minimize the number of queries. The previous approaches address the scenario where every query returns an exact value. Our framework is more general as it can deal with a wider variety of inputs and query responses and we establish interesting relationships between them that have not been investigated previously. Although some of the approaches of the previous restricted models can be adapted to the more general model, we require more sophisticated techniques for the analysis and we also obtain improved algorithms for the previous model. We address selection problems in the generalized model and show that there exist 2-update competitive algorithms that do not depend on the lengths or distribution of the sub-intervals and hold against the worst case adversary. We also obtain similar bounds on the competitive ratio for the MST problem in graphs

    Let's Make Block Coordinate Descent Go Fast: Faster Greedy Rules, Message-Passing, Active-Set Complexity, and Superlinear Convergence

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    Block coordinate descent (BCD) methods are widely-used for large-scale numerical optimization because of their cheap iteration costs, low memory requirements, amenability to parallelization, and ability to exploit problem structure. Three main algorithmic choices influence the performance of BCD methods: the block partitioning strategy, the block selection rule, and the block update rule. In this paper we explore all three of these building blocks and propose variations for each that can lead to significantly faster BCD methods. We (i) propose new greedy block-selection strategies that guarantee more progress per iteration than the Gauss-Southwell rule; (ii) explore practical issues like how to implement the new rules when using "variable" blocks; (iii) explore the use of message-passing to compute matrix or Newton updates efficiently on huge blocks for problems with a sparse dependency between variables; and (iv) consider optimal active manifold identification, which leads to bounds on the "active set complexity" of BCD methods and leads to superlinear convergence for certain problems with sparse solutions (and in some cases finite termination at an optimal solution). We support all of our findings with numerical results for the classic machine learning problems of least squares, logistic regression, multi-class logistic regression, label propagation, and L1-regularization

    Relatedness Measures to Aid the Transfer of Building Blocks among Multiple Tasks

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    Multitask Learning is a learning paradigm that deals with multiple different tasks in parallel and transfers knowledge among them. XOF, a Learning Classifier System using tree-based programs to encode building blocks (meta-features), constructs and collects features with rich discriminative information for classification tasks in an observed list. This paper seeks to facilitate the automation of feature transferring in between tasks by utilising the observed list. We hypothesise that the best discriminative features of a classification task carry its characteristics. Therefore, the relatedness between any two tasks can be estimated by comparing their most appropriate patterns. We propose a multiple-XOF system, called mXOF, that can dynamically adapt feature transfer among XOFs. This system utilises the observed list to estimate the task relatedness. This method enables the automation of transferring features. In terms of knowledge discovery, the resemblance estimation provides insightful relations among multiple data. We experimented mXOF on various scenarios, e.g. representative Hierarchical Boolean problems, classification of distinct classes in the UCI Zoo dataset, and unrelated tasks, to validate its abilities of automatic knowledge-transfer and estimating task relatedness. Results show that mXOF can estimate the relatedness reasonably between multiple tasks to aid the learning performance with the dynamic feature transferring.Comment: accepted by The Genetic and Evolutionary Computation Conference (GECCO 2020

    Parallel Successive Convex Approximation for Nonsmooth Nonconvex Optimization

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    Consider the problem of minimizing the sum of a smooth (possibly non-convex) and a convex (possibly nonsmooth) function involving a large number of variables. A popular approach to solve this problem is the block coordinate descent (BCD) method whereby at each iteration only one variable block is updated while the remaining variables are held fixed. With the recent advances in the developments of the multi-core parallel processing technology, it is desirable to parallelize the BCD method by allowing multiple blocks to be updated simultaneously at each iteration of the algorithm. In this work, we propose an inexact parallel BCD approach where at each iteration, a subset of the variables is updated in parallel by minimizing convex approximations of the original objective function. We investigate the convergence of this parallel BCD method for both randomized and cyclic variable selection rules. We analyze the asymptotic and non-asymptotic convergence behavior of the algorithm for both convex and non-convex objective functions. The numerical experiments suggest that for a special case of Lasso minimization problem, the cyclic block selection rule can outperform the randomized rule
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