We derive lower bounds on the Bayes risk in decentralized estimation, where the estimator does not have direct access to the random samples generated conditionally on the random parameter of interest, but only to the data received from local processors that observe the samples. The received data are subject to communication constraints, due to quantization and the noise in the communication channels from the processors to the estimator. We first derive general lower bounds on the Bayes risk using information-theoretic quantities, such as mutual information, information density, small ball probability, and differential entropy. We then apply these lower bounds to the decentralized case, using strong data processing inequalities to quantify the contraction of information due to communication constraints. We treat the cases of a single processor and of multiple processors, where the samples observed by different processors may be conditionally dependent given the parameter, for noninteractive and interactive communication protocols. Our results recover and improve recent lower bounds on the Bayes risk and the minimax risk for certain decentralized estimation problems, where previously only conditionally independent sample sets and noiseless channels have been considered. Moreover, our results provide a general way to quantify the degradation of estimation performance caused by distributing resources to multiple processors, which is only discussed for specific examples in existing works.
By contrast, we derive general lower bounds on the Bayes risk, which automatically serve as lower bounds on the minimax risk. We use strong data processing inequalities as a unifiying general method for quantifying the contraction of mutual information in decentralized estimation. Our results apply to general priors, sample generating models, and distortion functions. When particularized to the examples in the existing works, our results can lead to sharper lower bounds on both the Bayes and the minimax risk. For example, we improve the lower bound for the mean estimation on the unit cube studied in @cite_19 , as well as the lower bound for the meta-problem of Shamir @cite_17 . Moreover, we consider the situations where the sample sets are conditionally dependent and where the communication channels are noisy. We also provide a general way to quantify the degradation of estimation performance caused by distributing resources to multiple processors, which is only discussed for specific examples in existing works.
@cite_19 @cite_17
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Research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near-ideal estimation performance in various asymptotic frameworks. However, image processing practice has shown that universal thresholding methods are outperformed by simple Bayesian estimators assuming independent wavelet coefficients and heavy-tailed priors such as generalized Gaussian distributions (GGDs). In this paper, we investigate various connections between shrinkage methods and maximum a posteriori (MAP) estimation using such priors. In particular, we state a simple condition under which MAP estimates are sparse. We also introduce a new family of complexity priors based upon Rissanen's universal prior on integers. One particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length (MDL) principle. We develop analytical expressions for the shrinkage rules implied by GGD and complexity priors. This allows us to show the equivalence between universal hard thresholding, MAP estimation using a very heavy-tailed GGD, and MDL estimation using one of the new complexity priors. Theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis-specifications of the prior-a basic concern in image processing applications. We propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions, where the support size @math is unknown and may be comparable with or even much larger than the number of observations @math . We treat the respective regions where the functional is nonsmooth and smooth separately. In the nonsmooth regime, we apply an unbiased estimator for the best polynomial approximation of the functional whereas, in the smooth regime, we apply a bias-corrected version of the maximum likelihood estimator (MLE). We illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures: 1) the entropy @math and 2) @math . We obtain the minimax @math rates for estimating these functionals. In particular, we demonstrate that our estimator achieves the optimal sample complexity @math for entropy estimation. We also demonstrate that the sample complexity for estimating @math n S^ 1 S @math 1 , we show the minimax @math rate for estimating @math is @math for infinite support size, while the maximum @math rate for the MLE is @math . For all the above cases, the behavior of the minimax rate-optimal estimators with @math samples is essentially that of the MLE (plug-in rule) with @math samples, which we term “effective sample size enlargement.” We highlight the practical advantages of our schemes for the estimation of entropy and mutual information. We compare our performance with various existing approaches, and demonstrate that our approach reduces running time and boosts the accuracy. Moreover, we show that the minimax rate-optimal mutual information estimator yielded by our framework leads to significant performance boosts over the Chow–Liu algorithm in learning graphical models. The wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable. PAC-Bayes bounds have been proposed to get risk estimates based on a training sample. In this paper the PAC-Bayes approach is combined with stability of the hypothesis learned by a Hilbert space valued algorithm. The PAC-Bayes setting is used with a Gaussian prior centered at the expected output. Thus a novelty of our paper is using priors defined in terms of the data-generating distribution. Our main result estimates the risk of the randomized algorithm in terms of the hypothesis stability coefficients. We also provide a new bound for the SVM classifier, which is compared to other known bounds experimentally. Ours appears to be the first stability-based bound that evaluates to non-trivial values. In this paper, we propose a non-local structured prior for volumetric multi-view 3D reconstruction. Towards this goal, we present a novel Markov random field model based on ray potentials in which assumptions about large 3D surface patches such as planarity or Manhattan world constraints can be efficiently encoded as probabilistic priors. We further derive an inference algorithm that reasons jointly about voxels, pixels and image segments, and estimates marginal distributions of appearance, occupancy, depth, normals and planarity. Key to tractable inference is a novel hybrid representation that spans both voxel and pixel space and that integrates non-local information from 2D image segmentations in a principled way. We compare our non-local prior to commonly employed local smoothness assumptions and a variety of state-of-the-art volumetric reconstruction baselines on challenging outdoor scenes with textureless and reflective surfaces. Our experiments indicate that regularizing over larger distances has the potential to resolve ambiguities where local regularizers fail.