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Privacy and Statistical Risk: Formalisms and Minimax Bounds  [PDF]
Rina Foygel Barber,John C. Duchi
Statistics , 2014,
Abstract: We explore and compare a variety of definitions for privacy and disclosure limitation in statistical estimation and data analysis, including (approximate) differential privacy, testing-based definitions of privacy, and posterior guarantees on disclosure risk. We give equivalence results between the definitions, shedding light on the relationships between different formalisms for privacy. We also take an inferential perspective, where---building off of these definitions---we provide minimax risk bounds for several estimation problems, including mean estimation, estimation of the support of a distribution, and nonparametric density estimation. These bounds highlight the statistical consequences of different definitions of privacy and provide a second lens for evaluating the advantages and disadvantages of different techniques for disclosure limitation.
Excess risk bounds for multitask learning with trace norm regularization  [PDF]
Andreas Maurer,Massimiliano Pontil
Computer Science , 2012,
Abstract: Trace norm regularization is a popular method of multitask learning. We give excess risk bounds with explicit dependence on the number of tasks, the number of examples per task and properties of the data distribution. The bounds are independent of the dimension of the input space, which may be infinite as in the case of reproducing kernel Hilbert spaces. A byproduct of the proof are bounds on the expected norm of sums of random positive semidefinite matrices with subexponential moments.
Risk Bounds for the Majority Vote: From a PAC-Bayesian Analysis to a Learning Algorithm  [PDF]
Pascal Germain,Alexandre Lacasse,Fran?ois Laviolette,Mario Marchand,Jean-Francis Roy
Computer Science , 2015,
Abstract: We propose an extensive analysis of the behavior of majority votes in binary classification. In particular, we introduce a risk bound for majority votes, called the C-bound, that takes into account the average quality of the voters and their average disagreement. We also propose an extensive PAC-Bayesian analysis that shows how the C-bound can be estimated from various observations contained in the training data. The analysis intends to be self-contained and can be used as introductory material to PAC-Bayesian statistical learning theory. It starts from a general PAC-Bayesian perspective and ends with uncommon PAC-Bayesian bounds. Some of these bounds contain no Kullback-Leibler divergence and others allow kernel functions to be used as voters (via the sample compression setting). Finally, out of the analysis, we propose the MinCq learning algorithm that basically minimizes the C-bound. MinCq reduces to a simple quadratic program. Aside from being theoretically grounded, MinCq achieves state-of-the-art performance, as shown in our extensive empirical comparison with both AdaBoost and the Support Vector Machine.
Fast learning rates in statistical inference through aggregation  [PDF]
Jean-Yves Audibert
Mathematics , 2007,
Abstract: We develop minimax optimal risk bounds for the general learning task consisting in predicting as well as the best function in a reference set G up to the smallest possible additive term, called the convergence rate. When the reference set is finite and when n denotes the size of the training data, we provide minimax convergence rates of the form C ([log |G|]/n)^v with tight evaluation of the positive constant C and with exact v in ]0;1], the latter value depending on the convexity of the loss function and on the level of noise in the output distribution. The risk upper bounds are based on a sequential randomized algorithm, which at each step concentrates on functions having both low risk and low variance with respect to the previous step prediction function. Our analysis puts forward the links between the probabilistic and worst-case viewpoints, and allows to obtain risk bounds unachievable with the standard statistical learning approach. One of the key idea of this work is to use probabilistic inequalities with respect to appropriate (Gibbs) distributions on the prediction function space instead of using them with respect to the distribution generating the data. The risk lower bounds are based on refinements of the Assouad's lemma taking particularly into account the properties of the loss function. Our key example to illustrate the upper and lower bounds is to consider the L_q-regression setting for which an exhaustive analysis of the convergence rates is given while q describes [1;+infinity[.
Fast learning rates in statistical inference through aggregation  [PDF]
Jean-Yves Audibert
Statistics , 2009, DOI: 10.1214/08-AOS623
Abstract: We develop minimax optimal risk bounds for the general learning task consisting in predicting as well as the best function in a reference set $\mathcal{G}$ up to the smallest possible additive term, called the convergence rate. When the reference set is finite and when $n$ denotes the size of the training data, we provide minimax convergence rates of the form $C(\frac{\log|\mathcal{G}|}{n})^v$ with tight evaluation of the positive constant $C$ and with exact $0
Privacy Aware Learning  [PDF]
John C. Duchi,Michael I. Jordan,Martin J. Wainwright
Computer Science , 2012,
Abstract: We study statistical risk minimization problems under a privacy model in which the data is kept confidential even from the learner. In this local privacy framework, we establish sharp upper and lower bounds on the convergence rates of statistical estimation procedures. As a consequence, we exhibit a precise tradeoff between the amount of privacy the data preserves and the utility, as measured by convergence rate, of any statistical estimator or learning procedure.
On Bayes Risk Lower Bounds  [PDF]
Xi Chen,Adityanand Guntuboyina,Yuchen Zhang
Statistics , 2014,
Abstract: This paper provides a general technique to lower bound the Bayes risk for arbitrary loss functions and prior distributions in the standard abstract decision theoretic setting. A lower bound on the Bayes risk not only serves as a lower bound on the minimax risk but also characterizes the fundamental limitations of the statistical difficulty of a decision problem under a given prior. Our bounds are based on the notion of $f$-informativity of the underlying class of probability measures and the prior. Application of our bounds requires upper bounds on the $f$-informativity and we derive new upper bounds on $f$-informativity for a class of $f$ functions which lead to tight Bayes risk lower bounds. Our technique leads to generalizations of a variety of classical minimax bounds (e.g., generalized Fano's inequality). Using our Bayes risk lower bound, we provide a succinct proof to the main result of Chatterjee [2014]: for estimating mean of a Gaussian random vector under convex constraint, least squares estimator is always admissible up to a constant.
Learning with incremental iterative regularization  [PDF]
Lorenzo Rosasco,Silvia Villa
Mathematics , 2014,
Abstract: Within a statistical learning setting, we propose and study an iterative regularization algorithm for least squares defined by an incremental gradient method. In particular, we show that, if all other parameters are fixed a priori, the number of passes over the data (epochs) acts as a regularization parameter, and prove strong universal consistency, i.e. almost sure convergence of the risk, as well as sharp finite sample bounds for the iterates. Our results are a step towards understanding the effect of multiple epochs in stochastic gradient techniques in machine learning and rely on integrating statistical and optimization results.
Passive Learning with Target Risk  [PDF]
Mehrdad Mahdavi,Rong Jin
Computer Science , 2013,
Abstract: In this paper we consider learning in passive setting but with a slight modification. We assume that the target expected loss, also referred to as target risk, is provided in advance for learner as prior knowledge. Unlike most studies in the learning theory that only incorporate the prior knowledge into the generalization bounds, we are able to explicitly utilize the target risk in the learning process. Our analysis reveals a surprising result on the sample complexity of learning: by exploiting the target risk in the learning algorithm, we show that when the loss function is both strongly convex and smooth, the sample complexity reduces to $\O(\log (\frac{1}{\epsilon}))$, an exponential improvement compared to the sample complexity $\O(\frac{1}{\epsilon})$ for learning with strongly convex loss functions. Furthermore, our proof is constructive and is based on a computationally efficient stochastic optimization algorithm for such settings which demonstrate that the proposed algorithm is practically useful.
Statistical Learning of Arbitrary Computable Classifiers  [PDF]
David Soloveichik
Computer Science , 2008,
Abstract: Statistical learning theory chiefly studies restricted hypothesis classes, particularly those with finite Vapnik-Chervonenkis (VC) dimension. The fundamental quantity of interest is the sample complexity: the number of samples required to learn to a specified level of accuracy. Here we consider learning over the set of all computable labeling functions. Since the VC-dimension is infinite and a priori (uniform) bounds on the number of samples are impossible, we let the learning algorithm decide when it has seen sufficient samples to have learned. We first show that learning in this setting is indeed possible, and develop a learning algorithm. We then show, however, that bounding sample complexity independently of the distribution is impossible. Notably, this impossibility is entirely due to the requirement that the learning algorithm be computable, and not due to the statistical nature of the problem.
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