Becauseofarichvarietyof applications, themaximization ofanonnegativesubmodular function with respect toacardinality constraint (MCC) has a long history of study (Nemhauser et al., 1978). Furthermore, this ratio is optimal under thevalue oracle model (Nemhauser and Wolsey,1978).

A deterministic approximation algorithm is presented for the maximization of non-monotone submodular functions over a ground set of size $n$ subject to cardinality constraint $k$; the algorithm is based upon the idea of interlacing two greedy procedures. The algorithm uses interlaced, thresholded greedy procedures to obtain tight ratio $1/4 - \epsilon$ in $O \left( \frac{n}{\epsilon} \log \left( \frac{k}{\epsilon} \right) \right)$ queries of the objective function, which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem. The algorithm is validated in the context of two applications of non-monotone submodular maximization, on which it outperforms the fastest deterministic and randomized algorithms in prior literature.

We thank the four reviewers for their insightful comments and suggestions. I looked into the paper in ref[12] . . . ": In [12], the greedy algorithm is generic, with no assumptions about models ": Random search leads to a set of For Tab. 1, we ran the Wilcoxon signed-rank test (paired along settings, datasets and model types) and For Tab. 2 (with more costly experiments), we do not have enough runs to apply such We nonetheless report the standard errors in the paper, which seem to indicate significant improvements. ": Those numbers indicate the size of the ensemble; we will clarify this point. ": We thank R1 for the idea and ran our entire benchmark for ResNet-20: ": Hyper ensembles can indeed be viewed as a mixture They typically use Bayes nonparametric priors/posteriors and MCMC; we use mixtures and SGD. ": When used with replacement, the greedy algorithm from Caruana et al. [12, Sec.

SUMMARY The paper introduces a principled approach to generate a diverse set of relevant object proposals. This is formulated as optimizing a function F defined on sets of bounding boxes. This function is defined as a sum of a relevance term (which encourages individual bounding boxes to be likely to contain an object) and a diversity term. For optimization they commit to a greedy approach which iteratively adds the most promising bounding box, until the desired number of proposals was generated. The greedy procedure was motivated by existing theoretical guarantees on the obtained solution when F is submodular and monotone.

A deterministic approximation algorithm is presented for the maximization of non-monotone submodular functions over a ground set of size n subject to cardinality constraint k; the algorithm is based upon the idea of interlacing two greedy procedures. The algorithm uses interlaced, thresholded greedy procedures to obtain tight ratio 1/4 - \epsilon in O \left( \frac{n}{\epsilon} \log \left( \frac{k}{\epsilon} \right) \right) queries of the objective function, which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem. The algorithm is validated in the context of two applications of non-monotone submodular maximization, on which it outperforms the fastest deterministic and randomized algorithms in prior literature.

We introduce a voting model with multi-agent ranked delegations. This model generalises liquid democracy in two aspects: first, an agent's delegation can use the votes of multiple other agents to determine their own -- for instance, an agent's vote may correspond to the majority outcome of the votes of a trusted group of agents; second, agents can submit a ranking over multiple delegations, so that a backup delegation can be used when their preferred delegations are involved in cycles. The main focus of this paper is the study of unravelling procedures that transform the delegation ballots received from the agents into a profile of direct votes, from which a winning alternative can then be determined by using a standard voting rule. We propose and study six such unravelling procedures, two based on optimisation and four using a greedy approach. We study both algorithmic and axiomatic properties, as well as related computational complexity problems of our unravelling procedures for different restrictions on the types of ballots that the agents can submit.

A deterministic approximation algorithm is presented for the maximization of non-monotone submodular functions over a ground set of size $n$ subject to cardinality constraint $k$; the algorithm is based upon the idea of interlacing two greedy procedures. The algorithm uses interlaced, thresholded greedy procedures to obtain tight ratio $1/4 - \epsilon$ in $O \left( \frac{n}{\epsilon} \log \left( \frac{k}{\epsilon} \right) \right)$ queries of the objective function, which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem. The algorithm is validated in the context of two applications of non-monotone submodular maximization, on which it outperforms the fastest deterministic and randomized algorithms in prior literature. Papers published at the Neural Information Processing Systems Conference.

Gaussian mixtures (or so-called radial basis function networks) for density estimation provide a natural counterpart to sigmoidal neural networks for function fitting and approximation. In both cases, it is possible to give simple expressions for the iterative improvement of performance as components of the network are introduced one at a time. In particular, for mixture density estimation we show that a k-component mixture estimated by maximum likelihood (or by an iterative likelihood improvement that we introduce) achieves log-likelihood within order 1/k of the log-likelihood achievable by any convex combination. Consequences for approximation and estimation using Kullback-Leibler risk are also given. A Minimum Description Length principle selects the optimal number of components k that minimizes the risk bound. 1 Introduction In density estimation, Gaussian mixtures provide flexible-basis representations for densities that can be used to model heterogeneous data in high dimensions. Consider a parametric family G { pe(x), x E X C Rd': fJ E The main theme of the paper is to give approximation and estimation bounds of arbitrary densities by finite mixture densities.