In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
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Question: Common statistical practice has shown that the full power of Bayesian methods is not realized until hierarchical priors are used, as these allow for greater "robustness" and the ability to "share statistical strength." Yet it is an ongoing challenge to provide a learning-theoretically sound formalism of such notions that: offers practical guidance concerning when and how best to utilize hierarchical models; provides insights into what makes for a good hierarchical prior; and, when the form of the prior has been chosen, can guide the choice of hyperparameter settings. We present a set of analytical tools for understanding hierarchical priors in both the online and batch learning settings. We provide regret bounds under log-loss, which show how certain hierarchical models compare, in retrospect, to the best single model in the model class. We also show how to convert a Bayesian log-loss regret bound into a Bayesian risk bound for any bounded loss, a result which may be of independent interest. Risk and regret bounds for Student's t and hierarchical Gaussian priors allow us to formalize the concepts of "robustness" and "sharing statistical strength." Priors for feature selection are investigated as well. Our results suggest that the learning-theoretic benefits of using hierarchical priors can often come at little cost on practical problems.

Answer: Risk and Regret of Hierarchical Bayesian Learners

Question: We consider relative error low rank approximation of tensors with respect to the Frobenius norm. Namely, given an order-q tensor A R q i=1 ni , output a rank-k tensor B for which A BF (1 + ) OPT, where OPT = infrank-k A A AF . Despite much success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is that there may be no rank-k tensor Ak achieving the above infinum. Another, computational issue, is that an efficient relative error low rank approximation algorithm for tensors would allow one to compute the rank of a tensor, which is NP-hard. We bypass these two issues via (1) bicriteria and (2) parameterized complexity solutions: 1. We give an algorithm which outputs a rank k = O((k/ )q1) tensor B for which A BF (1+ ) OPT in nnz(A)+n poly(k/ ) time in the real RAM model, whenever either Ak exists or OPT > 0. Here nnz(A) denotes the number of non-zero entries in A. If both Ak does not exist and OPT = 0, then B instead satisfies A BF < , where is any positive, arbitrarily small function of n. 2. We give an algorithm for any > 0 which outputs a rank k tensor B for which ABF (1+ ) OPT and runs in (nnz(A)+n poly(k/ )+exp(k/ )) n time in the unit cost RAM model, whenever OPT > 2O(n ) and there is a rank-k tensor B = k i=1 ui vi wi for which A BF (1 + /2) OPT and ui2, vi2, wi2 2 ). If OPT 2(n), then B instead satisfies ABF 2(n ). Our first result is polynomial time, and in fact input sparsity time, in n, k, and 1/ , for any k 1 and any 0 < < 1, while our second result is fixed parameter tractable in k and 1/ . For outputting a rank-k tensor, or even a bicriteria solution with rank-Ck for a certain constant C > 1, we show a 2 1o(1)) time lower bound under the Exponential Time Hypothesis. Our results are based on an "iterative existential argument", and give the first relative error low rank approximations for tensors for a large number of error measures for which nothing was known. In particular, we give the first relative error approximation algorithms on tensors for: column row and tube subset selection, entrywise 'p-low rank approximation for 1 p < 2, low rank approximation with respect to sum of Euclidean norms of faces or tubes, weighted low rank approximation, and low rank approximation in distributed and streaming models. We also obtain several new results for matrices, such as nnz(A)-time CUR decompositions, improving the previous nnz(A) log n-time CUR decompositions, which may be of independent interest. Work done while visiting IBM Almaden, and supported in part by UTCS TAship (CS361 Spring 17 Introduction to Computer Security). Supported in part by Simons Foundation, and NSF CCF-1617955. ar X iv :1 70 4. 08 24 6v 1 [ cs .D S] 2 6 A pr 2 01 7

Answer: Relative Error Tensor Low Rank Approximation

Question: In cognitive sciences it is not uncommon to use various games effectively. For example, in artificial intelligence, the RoboCup [14] initiative was to set up to catalyse research on the field of autonomous agent technology. In this paper, we introduce a similar soccer simulation initiative to try to investigate a model of human consciousness and a notion of reality in the form of a cognitive problem. In addition, for example, the home pitch advantage and the objective role of the supporters could be naturally described and discussed in terms of this new soccer simulation model.

Answer:
Quantum Consciousness Soccer Simulator