832,153 research outputs found
Statistical Learning of Arbitrary Computable Classifiers
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.Comment: Expanded the section on prior work and added reference
Statistical learnability of nuclear masses
After more than 80 years from the seminal work of Weizs\"acker and the liquid
drop model of the atomic nucleus, deviations from experiments of mass models
( MeV) are orders of magnitude larger than experimental errors
( keV). Predicting the mass of atomic nuclei with precision is
extremely challenging. This is due to the non--trivial many--body interplay of
protons and neutrons in nuclei, and the complex nature of the nuclear strong
force. Statistical theory of learning will be used to provide bounds to the
prediction errors of model trained with a finite data set. These bounds are
validated with neural network calculations, and compared with state of the art
mass models. Therefore, it will be argued that the nuclear structure models
investigating ground state properties explore a system on the limit of the
knowledgeable, as defined by the statistical theory of learning
Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution
Recent years have seen a flurry of activities in designing provably efficient
nonconvex procedures for solving statistical estimation problems. Due to the
highly nonconvex nature of the empirical loss, state-of-the-art procedures
often require proper regularization (e.g. trimming, regularized cost,
projection) in order to guarantee fast convergence. For vanilla procedures such
as gradient descent, however, prior theory either recommends highly
conservative learning rates to avoid overshooting, or completely lacks
performance guarantees.
This paper uncovers a striking phenomenon in nonconvex optimization: even in
the absence of explicit regularization, gradient descent enforces proper
regularization implicitly under various statistical models. In fact, gradient
descent follows a trajectory staying within a basin that enjoys nice geometry,
consisting of points incoherent with the sampling mechanism. This "implicit
regularization" feature allows gradient descent to proceed in a far more
aggressive fashion without overshooting, which in turn results in substantial
computational savings. Focusing on three fundamental statistical estimation
problems, i.e. phase retrieval, low-rank matrix completion, and blind
deconvolution, we establish that gradient descent achieves near-optimal
statistical and computational guarantees without explicit regularization. In
particular, by marrying statistical modeling with generic optimization theory,
we develop a general recipe for analyzing the trajectories of iterative
algorithms via a leave-one-out perturbation argument. As a byproduct, for noisy
matrix completion, we demonstrate that gradient descent achieves near-optimal
error control --- measured entrywise and by the spectral norm --- which might
be of independent interest.Comment: accepted to Foundations of Computational Mathematics (FOCM
The Impact of Real Options on Willingness to Pay for Investments in Road Safety
Abstract: Public investments are dynamic in nature, and decision making must account for the uncertainty, irreversibility and potential for future learning. In this paper we adapt the theory for investment under uncertainty for a public referendum setting and perform the first empirical test to show that estimates of the value of safety (VSL) from stated preference surveys are highly dependent on the inclusion of the option value. Our results indicate an option value of a major economic magnitude. This implies that previously reported VSL estimates are exaggerated.Value of a Statistical Life; Real Options; Contingent Valuation; Road Safety
Accelerating The Use Of Weblogs As An Alternative Method To Deliver Case-Based Learning
Weblog technology is an alternative medium to deliver the case-based method of learning business concepts. The social nature of this technology can potentially promote active learning and enhance analytical ability of students. The present research investigates the primary factors contributing to the adoption of Weblog technology by students to learn business cases. A theoretical framework is proposed to address this issue based on the Unified Theory of Acceptance and Use of Technology (UTAUT) theory. Statistical evidences show that three major factors can contribute to users' intention to adopt Weblogs: (a) performance expectancy, (b) effort expectancy, and (c)social influence. It is also found that behavioral intention is a significant antecedent to actual use of Weblogs to learn business cases. Implications of the results for educators as well as possible future research paths for researchers are also discusse
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