We study contests where the designer's objective is an extension of the widely studied objective of maximizing the total output: The designer gets zero marginal utility from a player's output if the output of the player is very low or very high. We model this using two objective functions: binary threshold, where a player's contribution to the designer's utility is 1 if her output is above a certain threshold, and 0 otherwise; and linear threshold, where a player's contribution is linear if her output is between a lower and an upper threshold, and becomes constant below the lower and above the upper threshold. For both of these objectives, we study (1) rank-order allocation contests that use only the ranking of the players to assign prizes and (2) general contests that may use the numerical values of the players' outputs to assign prizes. We characterize the optimal contests that maximize the designer's objective and indicate techniques to efficiently compute them. We also prove that for the linear threshold objective, a contest that distributes the prize equally among a fixed number of top-ranked players offers a factor-2 approximation to the optimal rank-order allocation contest.
相關內容
Convex regression is the problem of fitting a convex function to a data set consisting of input-output pairs. We present a new approach to this problem called spectrahedral regression, in which we fit a spectrahedral function to the data, i.e. a function that is the maximum eigenvalue of an affine matrix expression of the input. This method represents a significant generalization of polyhedral (also called max-affine) regression, in which a polyhedral function (a maximum of a fixed number of affine functions) is fit to the data. We prove bounds on how well spectrahedral functions can approximate arbitrary convex functions via statistical risk analysis. We also analyze an alternating minimization algorithm for the non-convex optimization problem of fitting the best spectrahedral function to a given data set. We show that this algorithm converges geometrically with high probability to a small ball around the optimal parameter given a good initialization. Finally, we demonstrate the utility of our approach with experiments on synthetic data sets as well as real data arising in applications such as economics and engineering design.
In this paper, we investigate the question: Given a small number of datapoints, for example N = 30, how tight can PAC-Bayes and test set bounds be made? For such small datasets, test set bounds adversely affect generalisation performance by withholding data from the training procedure. In this setting, PAC-Bayes bounds are especially attractive, due to their ability to use all the data to simultaneously learn a posterior and bound its generalisation risk. We focus on the case of i.i.d. data with a bounded loss and consider the generic PAC-Bayes theorem of Germain et al. While their theorem is known to recover many existing PAC-Bayes bounds, it is unclear what the tightest bound derivable from their framework is. For a fixed learning algorithm and dataset, we show that the tightest possible bound coincides with a bound considered by Catoni; and, in the more natural case of distributions over datasets, we establish a lower bound on the best bound achievable in expectation. Interestingly, this lower bound recovers the Chernoff test set bound if the posterior is equal to the prior. Moreover, to illustrate how tight these bounds can be, we study synthetic one-dimensional classification tasks in which it is feasible to meta-learn both the prior and the form of the bound to numerically optimise for the tightest bounds possible. We find that in this simple, controlled scenario, PAC-Bayes bounds are competitive with comparable, commonly used Chernoff test set bounds. However, the sharpest test set bounds still lead to better guarantees on the generalisation error than the PAC-Bayes bounds we consider.
The support vector machine (SVM) and minimum Euclidean norm least squares regression are two fundamentally different approaches to fitting linear models, but they have recently been connected in models for very high-dimensional data through a phenomenon of support vector proliferation, where every training example used to fit an SVM becomes a support vector. In this paper, we explore the generality of this phenomenon and make the following contributions. First, we prove a super-linear lower bound on the dimension (in terms of sample size) required for support vector proliferation in independent feature models, matching the upper bounds from previous works. We further identify a sharp phase transition in Gaussian feature models, bound the width of this transition, and give experimental support for its universality. Finally, we hypothesize that this phase transition occurs only in much higher-dimensional settings in the $\ell_1$ variant of the SVM, and we present a new geometric characterization of the problem that may elucidate this phenomenon for the general $\ell_p$ case.
This paper proposes an isogeometric boundary element method (IGBEM) to solve the electromagnetic scattering problems for three-dimensional doubly-periodic multi-layered structures. The main concerns are the constructions of (i) an open surface (between two layers) and (ii) a vector basis function with using the B-spline functions. Regarding (i), we considered an algorithm to generate a doubly-periodic open surface with the tensor product of the B-spline functions of any degree. Regarding (ii), we employed the vector basis function based on the B-spline functions, which was proposed by Buffa et al. (2010), and adapted it to the underlying periodic problems so that it can satisfy the quasi-periodic condition on the boundary of an open surface. The proposed IGBEM worked for solving some numerical examples satisfactorily and proved the applicability to plasmonic simulations.
We study regression discontinuity designs in which many covariates, possibly much more than the number of observations, are available. We provide a two-step algorithm which first selects the set of covariates to be used through a localized Lasso-type procedure, and then, in a second step, estimates the treatment effect by including the selected covariates into the usual local linear estimator. We provide an in-depth analysis of the algorithm's theoretical properties, showing that, under an approximate sparsity condition, the resulting estimator is asymptotically normal, with asymptotic bias and variance that are conceptually similar to those obtained in low-dimensional settings. Bandwidth selection and inference can be carried out using standard methods. We also provide simulations and an empirical application.
The stochastic multi-armed bandit (MAB) problem is a common model for sequential decision problems. In the standard setup, a decision maker has to choose at every instant between several competing arms, each of them provides a scalar random variable, referred to as a "reward." Nearly all research on this topic considers the total cumulative reward as the criterion of interest. This work focuses on other natural objectives that cannot be cast as a sum over rewards, but rather more involved functions of the reward stream. Unlike the case of cumulative criteria, in the problems we study here the oracle policy, that knows the problem parameters a priori and is used to "center" the regret, is not trivial. We provide a systematic approach to such problems, and derive general conditions under which the oracle policy is sufficiently tractable to facilitate the design of optimism-based (upper confidence bound) learning policies. These conditions elucidate an interesting interplay between the arm reward distributions and the performance metric. Our main findings are illustrated for several commonly used objectives such as conditional value-at-risk, mean-variance trade-offs, Sharpe-ratio, and more.
The Glivenko-Cantelli theorem states that the empirical distribution function converges uniformly almost surely to the theoretical distribution for a random variable $X \in \mathbb{R}$. This is an important result because it establishes the fact that sampling does capture the dispersion measure the distribution function $F$ imposes. In essence, sampling permits one to learn and infer the behavior of $F$ by only looking at observations from $X$. The probabilities that are inferred from samples $\mathbf{X}$ will become more precise as the sample size increases and more data becomes available. Therefore, it is valid to study distributions via samples. The proof present here is constructive, meaning that the result is derived directly from the fact that the empirical distribution function converges pointwise almost surely to the theoretical distribution. The work includes a proof of this preliminary statement and attempts to motivate the intuition one gets from sampling techniques when studying the regions in which a model concentrates probability. The sets where dispersion is described with precision by the empirical distribution function will eventually cover the entire sample space.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
We present a new clustering method in the form of a single clustering equation that is able to directly discover groupings in the data. The main proposition is that the first neighbor of each sample is all one needs to discover large chains and finding the groups in the data. In contrast to most existing clustering algorithms our method does not require any hyper-parameters, distance thresholds and/or the need to specify the number of clusters. The proposed algorithm belongs to the family of hierarchical agglomerative methods. The technique has a very low computational overhead, is easily scalable and applicable to large practical problems. Evaluation on well known datasets from different domains ranging between 1077 and 8.1 million samples shows substantial performance gains when compared to the existing clustering techniques.
Both generative adversarial network models and variational autoencoders have been widely used to approximate probability distributions of datasets. Although they both use parametrized distributions to approximate the underlying data distribution, whose exact inference is intractable, their behaviors are very different. In this report, we summarize our experiment results that compare these two categories of models in terms of fidelity and mode collapse. We provide a hypothesis to explain their different behaviors and propose a new model based on this hypothesis. We further tested our proposed model on MNIST dataset and CelebA dataset.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191