Still No Free Lunches: The Price to Pay for Tighter PAC-Bayes Bounds

“No free lunch” results state the impossibility of obtaining meaningful bounds on the error of a learning algorithm without prior assumptions and modelling, which is more or less realistic for a given problem. Some models are “expensive” (strong assumptions, such as sub-Gaussian tails), others are “cheap” (simply finite variance). As it is well known, the more you pay, the more you get: in other words, the most expensive models yield the more interesting bounds. Recent advances in robust statistics have investigated procedures to obtain tight bounds while keeping the cost of assumptions minimal. The present paper explores and exhibits what the limits are for obtaining tight probably approximately correct (PAC)-Bayes bounds in a robust setting for cheap models.

Epsilon Best Arm Identification in Spectral Bandits

We propose an analysis of Probably Approximately Correct (PAC) identification of an ϵ-best arm in graph bandit models with Gaussian distributions. We consider finite but potentially very large bandit models where the set of arms is endowed with a graph structure, and we assume that the arms' expectations μ are smooth with respect to this graph. Our goal is to identify an arm whose expectation is at most ϵ below the largest of all means. We focus on the fixed-confidence setting: given a risk parameter δ, we consider sequential strategies that yield an ϵ-optimal arm with probability at least 1-δ. All such strategies use at least T*(μ)log(1/δ) samples, where R is the smoothness parameter. We identify the complexity term T*(μ) as the solution of a min-max problem for which we give a game-theoretic analysis and an approximation procedure. This procedure is the key element required by the asymptotically optimal Track-and-Stop strategy.

Optimal Algorithms for Range Searching over Multi-Armed Bandits

This paper studies a multi-armed bandit (MAB) version of the range-searching problem. In its basic form, range searching considers as input a set of points (on the real line) and a collection of (real) intervals. Here, with each specified point, we have an associated weight, and the problem objective is to find a maximum-weight point within every given interval. The current work addresses range searching with stochastic weights: each point corresponds to an arm (that admits sample access) and the point's weight is the (unknown) mean of the underlying distribution. In this MAB setup, we develop sample-efficient algorithms that find, with high probability, near-optimal arms within the given intervals, i.e., we obtain PAC (probably approximately correct) guarantees. We also provide an algorithm for a generalization wherein the weight of each point is a multi-dimensional vector. The sample complexities of our algorithms depend, in particular, on the size of the {optimal hitting set} of the given intervals. Finally, we establish lower bounds proving that the obtained sample complexities are essentially tight. Our results highlight the significance of geometric constructs (specifically, hitting sets) in our MAB setting.

On the Quantum versus Classical Learnability of Discrete Distributions

Here we study the comparative power of classical and quantum learners for generative modelling within the Probably Approximately Correct (PAC) framework. More specifically we consider the following task: Given samples from some unknown discrete probability distribution, output with high probability an efficient algorithm for generating new samples from a good approximation of the original distribution. Our primary result is the explicit construction of a class of discrete probability distributions which, under the decisional Diffie-Hellman assumption, is provably not efficiently PAC learnable by a classical generative modelling algorithm, but for which we construct an efficient quantum learner. This class of distributions therefore provides a concrete example of a generative modelling problem for which quantum learners exhibit a provable advantage over classical learning algorithms. In addition, we discuss techniques for proving classical generative modelling hardness results, as well as the relationship between the PAC learnability of Boolean functions and the PAC learnability of discrete probability distributions.

PAC-Bayes Bounds on Variational Tempered Posteriors for Markov Models

Datasets displaying temporal dependencies abound in science and engineering applications, with Markov models representing a simplified and popular view of the temporal dependence structure. In this paper, we consider Bayesian settings that place prior distributions over the parameters of the transition kernel of a Markov model, and seek to characterize the resulting, typically intractable, posterior distributions. We present a Probably Approximately Correct (PAC)-Bayesian analysis of variational Bayes (VB) approximations to tempered Bayesian posterior distributions, bounding the model risk of the VB approximations. Tempered posteriors are known to be robust to model misspecification, and their variational approximations do not suffer the usual problems of over confident approximations. Our results tie the risk bounds to the mixing and ergodic properties of the Markov data generating model. We illustrate the PAC-Bayes bounds through a number of example Markov models, and also consider the situation where the Markov model is misspecified.

Property Checking with Interpretable Error Characterization for Recurrent Neural Networks

This paper presents a novel on-the-fly, black-box, property-checking through learning approach as a means for verifying requirements of recurrent neural networks (RNN) in the context of sequence classification. Our technique steps on a tool for learning probably approximately correct (PAC) deterministic finite automata (DFA). The sequence classifier inside the black-box consists of a Boolean combination of several components, including the RNN under analysis together with requirements to be checked, possibly modeled as RNN themselves. On one hand, if the output of the algorithm is an empty DFA, there is a proven upper bound (as a function of the algorithm parameters) on the probability of the language of the black-box to be nonempty. This implies the property probably holds on the RNN with probabilistic guarantees. On the other, if the DFA is nonempty, it is certain that the language of the black-box is nonempty. This entails the RNN does not satisfy the requirement for sure. In this case, the output automaton serves as an explicit and interpretable characterization of the error. Our approach does not rely on a specific property specification formalism and is capable of handling nonregular languages as well. Besides, it neither explicitly builds individual representations of any of the components of the black-box nor resorts to any external decision procedure for verification. This paper also improves previous theoretical results regarding the probabilistic guarantees of the underlying learning algorithm.

The Compound F2-index and the Compound H-index as Extension of the f2 and h-indexes from a Dynamic Perspective

PurposeElaboration of an indicator to include the dynamic aspect of citations in bibliometric indexes.Design/methodology/approachA new bibliometric methodology—the f2-index—is applied at the career level and at the level of the recent 5 years to analyze the dynamic aspect of bibliometrics. The method is applied, as an illustration, to the field of corporate governance.FindingsThe compound F2-index as an extension of the f2-index recognizes past achievements but also values new research work with potential. The method is extended to the h-index and the h2-index. An activity index is defined as the ratio between the recent h’-index to the career h-index.Research limitationsThe compound F2 and H-indexes are PAC, probably approximately correct, and depend on the selection and database.Practical implicationsThe F2- and H compound indexes allow identifying the rising stars of a field from a dynamic perspective. The activity ratio highlights the contribution of younger researchers.Originality/valueThe new methodology demonstrates the underestimated dynamic capacity of bibliometric research.

On the Learnability of Possibilistic Theories

We investigate learnability of possibilistic theories from entailments in light of Angluin’s exact learning model. We consider cases in which only membership, only equivalence, and both kinds of queries can be posed by the learner. We then show that, for a large class of problems, polynomial time learnability results for classical logic can be transferred to the respective possibilistic extension. In particular, it follows from our results that the possibilistic extension of propositional Horn theories is exactly learnable in polynomial time. As polynomial time learnability in the exact model is transferable to the classical probably approximately correct (PAC) model extended with membership queries, our work also establishes such results in this model.