Berkeley COMPSCI 294 - Bounds for Linear Multi-Task Learning (16 pages)

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Bounds for Linear Multi-Task Learning



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Bounds for Linear Multi-Task Learning

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Pages:
16
School:
University of California, Berkeley
Course:
Compsci 294 - Special Topics
Special Topics Documents

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Bounds for Linear Multi Task Learning Andreas Maurer Adalbertstr 55 D 80799 M nchen andreasmaurer compuserve com Abstract We give dimension free and data dependent bounds for linear multi task learning where a common linear operator is chosen to preprocess data for a vector of task speci c linear thresholding classi ers The complexity penalty of multi task learning is bounded by a simple expression involving the margins of the task speci c classi ers the Hilbert Schmidt norm of the selected preprocessor and the HilbertSchmidt norm of the covariance operator for the total mixture of all task distributions or alternatively the Frobenius norm of the total Gramian matrix for the data dependent version The results can be compared to state of the art results on linear single task learning 1 Introduction Simultaneous learning of di erent tasks under some common constraint often called multi task learning has been tested in practice with good results under a variety of di erent circumstances see 4 7 14 15 The technique has been analyzed theoretically and in some generality see Baxter 5 and Zhang 15 The purpose of this paper is to improve and clarify some of these theoretical results in the case when input data is represented in a linear potentially in nite dimensional space and the common constraint is a linear preprocessor The simplest conceptual model to understand multi task learning and its potential advantages is perhaps agnostic learning with an input space X and a nite set F of hypotheses f X f0 1g For a hypothesis f 2 F let er f and er f be the expected error and the empirical error on a training sample S of size n drawn iid from the underlying task distribution respectively Combining Hoe ding s inequality with a union bound one shows see e g 1 that with probability greater than 1 we have for every f 2 F the error bound er f 1 p er f p ln jFj ln 1 2n 1 Suppose now that F factors in the following sense There is a set Y and a set G of functions g X Y and a set H of



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