Massive online teaching to bounded learners

Abstract

We consider a model of teaching in which the learners are consistent and have bounded state, but are otherwise arbitrary. The teacher is non-interactive and "massively open": the teacher broadcasts a sequence of examples of an arbitrary target concept, intended for every possible on-line learning algorithm to learn from. We focus on the problem of designing interesting teachers: efficient sequences of examples that lead all capable and consistent learners to learn concepts, regardless of the underlying algorithm used by the learner. We use two measures of teaching efficiency: the number of mistakes made by the worst-case learner, and the maximum length of the example sequence needed for the worst-case learner. Our results are summarized as follows:• Given a uniform random sequence of examples of an nbit concept function, learners (capable of consistently learning the concept) with s(n) bits of state are guaranteed to make only O(n · s(n)) mistakes and exactly learn the concept, with high probability. This theorem has interesting corollaries; for instance, every concept c has a sequence of examples can teach c to all capable consistent on-line learners implementable with s(n)-size circuits, such that every learner makes onlyÕ(s(n) 2 ) mistakes. That is, all resourcebounded algorithms capable of consistently learning a concept can be simultaneously taught that concept with few mistakes, on a single example sequence.We also show how to efficiently generate such a sequence of examples on-line: using Nisan's pseudorandom generator, each example in the sequence can be generated with polynomial-time overhead per example, with an O(n · s(n))-bit initial seed.• To justify our use of randomness, we prove that any non-trivial derandomization of our sequences would imply new circuit lower bounds. For instance, if there is a deterministic 2 n O(1) time algorithm that generates a sequence of examples, such that all consistent and capable polynomialsize circuit learners learn the all-zeroes concept with less than 2 n mistakes, then EXP ⊂ P/poly.• We present examples illustrating that the key differences in our model -our focus on mistakes rather than the total number of examples, and our use of a state bound -must be considered together to obtain our results.• We show that for every consistent s(n)-state bounded learner A, and every n-bit concept that A is capable of learning, there is a custom "tutoring" sequence of only O(n · s(n)) examples that teaches A the concept. That is, in principle, there are no slow learners, only bad teachers: if a statebounded learner is capable of learning a concept at all, then it can always be taught that concept quickly via some short sequence of examples.

Communication with Contextual Uncertainty

“…Remarkably, especially in the natural setting, context can seemingly be used to compress communication, even when it is not shared perfectly. This ability to communicate despite a major source of uncertainty has led to a series of works attempting to model various forms of communication amid uncertainty, starting with Goldreich, Juba and Sudan [JS08,GJS12] followed by [JKKS11,JS11,JW13,HS14,CGMS15]. This current work introduces a new theme to this series of works by introducing a functional notion of uncertainty and studying this model.…”

Communication with Contextual Uncertainty

“…Remarkably, especially in the natural setting, context can seemingly be used to compress communication, even when it is not shared perfectly. This ability to communicate despite a major source of uncertainty has led to a series of works attempting to model various forms of communication amid uncertainty, starting with Goldreich, Juba and Sudan [8,5] followed by [7,9,10,6,4]. This current work introduces a new theme to this series of works by introducing a functional notion of uncertainty and studying this model.…”