Results 1 
7 of
7
Tight Lower Bound on the Probability of a Binomial Exceeding its Expectation
"... Abstract. We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in learning theory and generalization bounds for unbo ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in learning theory and generalization bounds for unbounded loss functions. 1
Stochastic Convex Optimization with Multiple Objectives
"... In this paper, we are interested in the development of efficient algorithms for convex optimization problems in the simultaneous presence of multiple objectives and stochasticity in the firstorder information. We cast the stochastic multiple objective optimization problem into a constrained optim ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
In this paper, we are interested in the development of efficient algorithms for convex optimization problems in the simultaneous presence of multiple objectives and stochasticity in the firstorder information. We cast the stochastic multiple objective optimization problem into a constrained optimization problem by choosing one function as the objective and try to bound other objectives by appropriate thresholds. We first examine a two stages explorationexploitation based algorithm which first approximates the stochastic objectives by sampling and then solves a constrained stochastic optimization problem by projected gradient method. This method attains a suboptimal convergence rate even under strong assumption on the objectives. Our second approach is an efficient primaldual stochastic algorithm. It leverages on the theory of Lagrangian method in constrained optimization and attains the optimal convergence rate of O(1= p T) in high probability for general Lipschitz continuous objectives.
Comment on “Hypothesis testing by convex optimization”∗
, 2015
"... With the growing size of problems at hand, convexity has become preponderant in modern statistics. Indeed, convex relaxations of NPhard problems have been successfully employed in a variety of statistical problems such as classification [2, 16], linear regression [7, 5], matrix estimation [8, 12], ..."
Abstract
 Add to MetaCart
(Show Context)
With the growing size of problems at hand, convexity has become preponderant in modern statistics. Indeed, convex relaxations of NPhard problems have been successfully employed in a variety of statistical problems such as classification [2, 16], linear regression [7, 5], matrix estimation [8, 12], graphical models [15, 9] or sparse principal component analysis (PCA) [10, 4]. The paper “Hypothesis testing by convex optimization ” by Alexander Goldenshluger, Anatoli Juditsky and Arkadi Nemirovski, hereafter denoted by GJN, brings a new perspective on the role of convexity in a fundamental statistical problem: composite hypothesis testing. The role of this problem is illustrated in the light of several interesting applications in Section 4 of GJN. One of the key insights in GJN is that there exists a pair of distributions, one in each of the composite hypotheses and on which the statistician should focus her efforts. Indeed, Theorem 2.1(ii) guarantees that any test that is optimal for this simple hypothesis problem is also near optimal for the composite hypothesis problem. Moreover, this pair can be found by solving a convex optimization
24th Annual Conference on Learning Theory NeymanPearson classification under a strict constraint
"... Motivated by problems of anomaly detection, this paper implements the NeymanPearson paradigm to deal with asymmetric errors in binary classification with a convex loss. Given a finite collection of classifiers, we combine them and obtain a new classifier that satisfies simultaneously the two follow ..."
Abstract
 Add to MetaCart
(Show Context)
Motivated by problems of anomaly detection, this paper implements the NeymanPearson paradigm to deal with asymmetric errors in binary classification with a convex loss. Given a finite collection of classifiers, we combine them and obtain a new classifier that satisfies simultaneously the two following properties with high probability: (i), its probability of type I error is below a prespecified level and (ii), it has probability of type II error close to the minimum possible. The proposed classifier is obtained by minimizing an empirical objective subject to an empirical constraint. The novelty of the method is that the classifier output by this problem is shown to satisfy the original constraint on type I error. This strict enforcement of the constraint has interesting consequences on the control of the type II error and we develop new techniques to handle this situation. Finally, connections with chance constrained optimization are evident and are investigated.
A Plugin Approach to NeymanPearson Classification
"... The NeymanPearson (NP) paradigm in binary classification treats type I and type II errors with different priorities. It seeks classifiers that minimize type II error, subject to a type I error constraint under a user specified level α. In this paper, plugin classifiers are developed under the NP ..."
Abstract
 Add to MetaCart
The NeymanPearson (NP) paradigm in binary classification treats type I and type II errors with different priorities. It seeks classifiers that minimize type II error, subject to a type I error constraint under a user specified level α. In this paper, plugin classifiers are developed under the NP paradigm. Based on the fundamental NeymanPearson Lemma, we propose two related plugin classifiers which amount to thresholding respectively the class conditional density ratio and the regression function. These two classifiers handle different sampling schemes. This work focuses on theoretical properties of the proposed classifiers; in particular, we derive oracle inequalities that can be viewed as finite sample versions of risk bounds. NP classification can be used to address anomaly detection problems, where asymmetry in errors is an intrinsic property. As opposed to a common practice in anomaly detection that consists of thresholding normal class density, our approach does not assume a specific form for anomaly distributions. Such consideration is particularly necessary when the anomaly class density is far from uniformly distributed.
On Minimum Clinically Important Difference
"... In clinical trials, minimum clinically important difference (MCID) has attracted increasing interest as an important supportive clinical and statistical inference tool. Many estimation methods have been developed based on various intuitions, while little theoretical justification has been establishe ..."
Abstract
 Add to MetaCart
In clinical trials, minimum clinically important difference (MCID) has attracted increasing interest as an important supportive clinical and statistical inference tool. Many estimation methods have been developed based on various intuitions, while little theoretical justification has been established. This paper proposes a new estimation framework of MCID using both diagnostic measurements and patientreported outcomes (PRO’s). The framework first formulates populationbased MCID as a large margin classification problem, and then extends to personalized MCID to allow individualized thresholding value for patients whose clinical profiles may affect their PRO responses. More importantly, the proposed estimation framework is showed to be asymptotically consistent, and a finitesample upper bound is established for its prediction accuracy compared against the ideal MCID. The advantage of our proposed method is also demonstrated in a variety of simulated experiments as well as two phase3 clinical trials.