More data speeds up training time in learning halfspaces over sparse vectors

The increased availability of data in recent years has led several authors to ask whether it is possible to use data as a {\em computational} resource. That is, if more data is available, beyond the sample complexity limit, is it possible to use the extra examples to speed up the computation time required to perform the learning task? We give the first positive answer to this question for a {\em natural supervised learning problem} --- we consider agnostic PAC learning of halfspaces over $3$-sparse vectors in $\{-1,1,0\}^n$. This class is inefficiently learnable using $O\left(n/\epsilon^2\right)$ examples. Our main contribution is a novel, non-cryptographic, methodology for establishing computational-statistical gaps, which allows us to show that, under a widely believed assumption that refuting random $\mathrm{3CNF}$ formulas is hard, it is impossible to efficiently learn this class using only $O\left(n/\epsilon^2\right)$ examples. We further show that under stronger hardness assumptions, even $O\left(n^{1.499}/\epsilon^2\right)$ examples do not suffice. On the other hand, we show a new algorithm that learns this class efficiently using $\tilde{\Omega}\left(n^2/\epsilon^2\right)$ examples. This formally establishes the tradeoff between sample and computational complexity for a natural supervised learning problem.Comment: 13 page

Optimal Testing for Planted Satisfiability Problems

We study the problem of detecting planted solutions in a random satisfiability formula. Adopting the formalism of hypothesis testing in statistical analysis, we describe the minimax optimal rates of detection. Our analysis relies on the study of the number of satisfying assignments, for which we prove new results. We also address algorithmic issues, and give a computationally efficient test with optimal statistical performance. This result is compared to an average-case hypothesis on the hardness of refuting satisfiability of random formulas

A new perspective for the training assessment: Machine learning-based neurometric for augmented user's evaluation

Inappropriate training assessment might have either high social costs and economic impacts, especially in high risks categories, such as Pilots, Air Traffic Controllers, or Surgeons. One of the current limitations of the standard training assessment procedures is the lack of information about the amount of cognitive resources requested by the user for the correct execution of the proposed task. In fact, even if the task is accomplished achieving the maximum performance, by the standard training assessment methods, it would not be possible to gather and evaluate information about cognitive resources available for dealing with unexpected events or emergency conditions. Therefore, a metric based on the brain activity (neurometric) able to provide the Instructor such a kind of information should be very important. As a first step in this direction, the Electroencephalogram (EEG) and the performance of 10 participants were collected along a training period of 3 weeks, while learning the execution of a new task. Specific indexes have been estimated from the behavioral and EEG signal to objectively assess the users' training progress. Furthermore, we proposed a neurometric based on a machine learning algorithm to quantify the user's training level within each session by considering the level of task execution, and both the behavioral and cognitive stabilities between consecutive sessions. The results demonstrated that the proposed methodology and neurometric could quantify and track the users' progresses, and provide the Instructor information for a more objective evaluation and better tailoring of training programs. © 2017 Borghini, Aricò, Di Flumeri, Sciaraffa, Colosimo, Herrero, Bezerianos, Thakor and Babiloni

Order-Revealing Encryption and the Hardness of Private Learning

An order-revealing encryption scheme gives a public procedure by which two ciphertexts can be compared to reveal the ordering of their underlying plaintexts. We show how to use order-revealing encryption to separate computationally efficient PAC learning from efficient $(\epsilon, \delta)$-differentially private PAC learning. That is, we construct a concept class that is efficiently PAC learnable, but for which every efficient learner fails to be differentially private. This answers a question of Kasiviswanathan et al. (FOCS '08, SIAM J. Comput. '11). To prove our result, we give a generic transformation from an order-revealing encryption scheme into one with strongly correct comparison, which enables the consistent comparison of ciphertexts that are not obtained as the valid encryption of any message. We believe this construction may be of independent interest.Comment: 28 page