Valiant's 1984 paper is widely credited with introducing the PAC learning model, but it, in fact, introduced a different model: unlike PAC learning, the learner receives only positives, may issue membership queries, and must output a hypothesis with no false positives. Prior work characterized variants, including the case without queries. We revisit Valiant's original model and ask: *Which classes are learnable in it?* For every finite domain, including Valiant's Boolean-hypercube setting, we show that a class is learnable if and only if every realizable positive sample can be certified by a poly-size adaptive query-compression scheme. This is a new variant of sample compression where the learner certifies samples via a short interaction with the membership oracle. Our characterization shows that learnability in Valiant's model is strictly sandwiched between learnability in the PAC model and the variant of Valiant's model without membership queries. This is one of the rare cases where introducing membership queries changes the set of learnable classes, and not just the sample or computational complexity. Next, we study the natural extension of the model to arbitrary domains. While we do not obtain an exact characterization, our techniques readily generalize and show that the same strict sandwiching persists. Finally, we show that $d$-dimensional halfspaces, which are not learnable without queries, are learnable with queries: we give a $\mathrm{poly}(d) \tilde{O}(1/ε)$ sample and $\mathrm{poly}(d) \mathrm{polylog}(1/ε)$ query algorithm, and prove that at least $Ω(d)$ samples or queries are necessary. To our knowledge, this is the first algorithm for halfspaces in Valiant's model. Together, these results uncover a surprisingly rich theory behind Valiant's original notion of learnability and introduce ideas that may be of independent interest in learning theory.

The fuzzy or soft k-means objective is a popular generalization of the well-known kmeans problem, extending the clustering capability of the k-means to datasets that are uncertain, vague and otherwise hard to cluster. In this paper, we propose a semisupervised active clustering framework, where the learner is allowed to interact with an oracle (domain expert), asking for the similarity between a certain set of chosen items. We study the query and computational complexities of clustering in this framework. We prove that having a few of such similarity queries enables one to get a polynomial-time approximation algorithm to an otherwise conjecturally NP-hard problem. In particular, we provide algorithms for fuzzy clustering in this setting that ask O(poly(k)logn)similarity queries and run with polynomialtime-complexity, where nis the number of items. The fuzzy k-means objective is nonconvex, with k-means as a special case, and is equivalent to some other generic nonconvex problem such as non-negative matrix factorization. The ubiquitous Lloyd-type algorithms (or alternating-minimization algorithms) can get stuck at a local minima. Our results show that by making few similarity queries, the problem becomes easier to solve. Finally, we test our algorithms over real-world datasets, showing their effectiveness in real-world applications.

Neural Information Processing Systems http://nips.cc/

Membership queries (MQ) often yield speedups for learning tasks, particularly in the distribution-specific setting. We show that in the \emph{testable learning} model of Rubinfeld and Vasilyan [RV23], membership queries cannot decrease the time complexity of testable learning algorithms beyond the complexity of sample-only distribution-specific learning. In the testable learning model, the learner must output a hypothesis whenever the data distribution satisfies a desired property, and if it outputs a hypothesis, the hypothesis must be near-optimal. We give a general reduction from sample-based \emph{refutation} of boolean concept classes, as presented in [Vadhan17, KL18], to testable learning with queries (TL-Q). This yields lower bounds for TL-Q via the reduction from learning to refutation given in [KL18]. The result is that, relative to a concept class and a distribution family, no $m$-sample TL-Q algorithm can be super-polynomially more time-efficient than the best $m$-sample PAC learner. Finally, we define a class of ``statistical'' MQ algorithms that encompasses many known distribution-specific MQ learners, such as those based on influence estimation or subcube-conditional statistical queries. We show that TL-Q algorithms in this class imply efficient statistical-query refutation and learning algorithms. Thus, combined with known SQ dimension lower bounds, our results imply that these efficient membership query learners cannot be made testable.

Automata learning has many applications in artificial intelligence and software engineering. Central to these applications is the $L^*$ algorithm, introduced by Angluin. The $L^*$ algorithm learns deterministic finite-state automata (DFAs) in polynomial time when provided with a minimally adequate teacher. Unfortunately, the $L^*$ algorithm can only learn DFAs over finite alphabets, which limits its applicability. In this paper, we extend $L^*$ to learn symbolic automata whose transitions use predicates over rational numbers, i.e., over infinite and dense alphabets. Our result makes the $L^*$ algorithm applicable to new settings like (real) RGX, and time series. Furthermore, our proposed algorithm is optimal in the sense that it asks a number of queries to the teacher that is at most linear with respect to the number of transitions, and to the representation size of the predicates.

Leslie Valiant's 1984 paper "A Theory of the Learnable" [Val84], reproduced in this volume, has the unusual distinction of having changed the course of several scientific disciplines. Within theoretical computer science it was one of the key works giving rise to the field now known as computational learning theory, which may loosely be defined as the rigorous study of learning processes and phenomena from the computer science perspective of efficient algorithms and computational complexity. In the decades since the publication of [Val84], computational learning theory has grown into a rich field with strong connections to many other theoretical disciplines such as mathematical probability and statistics, information theory, decision theory and more. Beyond the realm of theory, Valiant's paper and the Probably Approximately Correct (PAC) model which he introduced in it have also had a great impact on the subsequent development of machine learning, a field which has already transformed many aspects of science and human society and seems certain to have an even greater influence in the future. This chapter gives an overview of the Probably Approximately Correct learning model that Valiant introduced in [Val84], explaining some of the major results and directions that the field has taken in the years since that work.

Despite the theoretical significance and wide practical use of regular expressions, the computational complexity of learning them has been largely unexplored. We study the computational hardness of improperly learning regular expressions in the PAC model and with membership queries. We show that PAC learning is hard even under the uniform distribution on the hypercube, and also prove hardness of distribution-free learning with membership queries. Furthermore, if regular expressions are extended with complement or intersection, we establish hardness of learning with membership queries even under the uniform distribution. We emphasize that these results do not follow from existing hardness results for learning DFAs or NFAs, since the descriptive complexity of regular languages can differ exponentially between DFAs, NFAs, and regular expressions.

We start with two negative results. We show that no non-trivial concept class can be robustly learned in the distribution-free setting against an adversary who can perturb just a single input bit.