We introduce definitions of computable PAC learning for binary classification over computable metric spaces. We provide sufficient conditions for learners that are empirical risk minimizers (ERM) to be computable, and bound the strong Weihrauch degree of an ERM learner under more general conditions. We also give a presentation of a hypothesis class that does not admit any proper computable PAC learner with computable sample function, despite the underlying class being PAC learnable.

This work addressed the problem of learning a network with communication between vertices. The communication between vertices is presented in the form of perturbation on the measure. We studied the scenario where samples are drawn from a uniform ergodic Random Graph Process (RGPs for short), which provides a natural mathematical context for the problem of interest. For the binary classification problem, the result we obtained gives uniform learn-ability as the worst-case theoretical limits. We introduced the structural Rademacher complexity, which naturally fused into the VC theory to upperbound the first moment. With the martingale method and Marton's coupling, we establish the tail bound for uniform convergence and give consistency guarantee for empirical risk minimizer. The technique used in this work to obtain high probability bounds is of independent interest to other mixing processes with and without network structure.

In this work, we investigate the expressiveness of the "conditional mutual information" (CMI) framework of Steinke and Zakynthinou (2020) and the prospect of using it to provide a unified framework for proving generalization bounds in the realizable setting. We first demonstrate that one can use this framework to express non-trivial (but sub-optimal) bounds for any learning algorithm that outputs hypotheses from a class of bounded VC dimension. We prove that the CMI framework yields the optimal bound on the expected risk of Support Vector Machines (SVMs) for learning halfspaces. This result is an application of our general result showing that stable compression schemes Bousquet al. (2020) of size $k$ have uniformly bounded CMI of order $O(k)$. We further show that an inherent limitation of proper learning of VC classes contradicts the existence of a proper learner with constant CMI, and it implies a negative resolution to an open problem of Steinke and Zakynthinou (2020). We further study the CMI of empirical risk minimizers (ERMs) of class $H$ and show that it is possible to output all consistent classifiers (version space) with bounded CMI if and only if $H$ has a bounded star number (Hanneke and Yang (2015)). Moreover, we prove a general reduction showing that "leave-one-out" analysis is expressible via the CMI framework. As a corollary we investigate the CMI of the one-inclusion-graph algorithm proposed by Haussler et al. (1994). More generally, we show that the CMI framework is universal in the sense that for every consistent algorithm and data distribution, the expected risk vanishes as the number of samples diverges if and only if its evaluated CMI has sublinear growth with the number of samples.

We study fast rates of convergence in the setting of nonparametric online regression, namely where regret is defined with respect to an arbitrary function class which has bounded complexity. Our contributions are two-fold: - In the realizable setting of nonparametric online regression with the absolute loss, we propose a randomized proper learning algorithm which gets a near-optimal mistake bound in terms of the sequential fat-shattering dimension of the hypothesis class. In the setting of online classification with a class of Littlestone dimension $d$, our bound reduces to $d \cdot {\rm poly} \log T$. This result answers a question as to whether proper learners could achieve near-optimal mistake bounds; previously, even for online classification, the best known mistake bound was $\tilde O( \sqrt{dT})$. Further, for the real-valued (regression) setting, the optimal mistake bound was not even known for improper learners, prior to this work. - Using the above result, we exhibit an independent learning algorithm for general-sum binary games of Littlestone dimension $d$, for which each player achieves regret $\tilde O(d^{3/4} \cdot T^{1/4})$. This result generalizes analogous results of Syrgkanis et al. (2015) who showed that in finite games the optimal regret can be accelerated from $O(\sqrt{T})$ in the adversarial setting to $O(T^{1/4})$ in the game setting. To establish the above results, we introduce several new techniques, including: a hierarchical aggregation rule to achieve the optimal mistake bound for real-valued classes, a multi-scale extension of the proper online realizable learner of Hanneke et al. (2021), an approach to show that the output of such nonparametric learning algorithms is stable, and a proof that the minimax theorem holds in all online learnable games.

Electrification was, without a doubt, the greatest engineering marvel of the 20th century. The electric motor was invented way back in 1821, and the electrical circuit was mathematically analyzed in 1827. But factory electrification, household electrification, and railway electrification all started slowly several decades later. The field of AI was formally founded in 1956. But it's only now--more than six decades later--that AI is expected to revolutionize the way humanity will live and work in the coming decades.

Machine learning models that are developed to be invariant under certain types of data transformations have shown improved generalization in practice. However, a principled understanding of why invariance benefits generalization is limited. Given a dataset, there is often no principled way to select "suitable" data transformations under which model invariance guarantees better generalization. This paper studies the generalization benefit of model invariance by introducing the sample cover induced by transformations, i.e., a representative subset of a dataset that can approximately recover the whole dataset using transformations. For any data transformations, we provide refined generalization bounds for invariant models based on the sample cover. We also characterize the "suitability" of a set of data transformations by the sample covering number induced by transformations, i.e., the smallest size of its induced sample covers. We show that we may tighten the generalization bounds for "suitable" transformations that have a small sample covering number. In addition, our proposed sample covering number can be empirically evaluated and thus provides a guidance for selecting transformations to develop model invariance for better generalization. In experiments on multiple datasets, we evaluate sample covering numbers for some commonly used transformations and show that the smaller sample covering number for a set of transformations (e.g., the 3D-view transformation) indicates a smaller gap between the test and training error for invariant models, which verifies our propositions.

The equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory. With variants ranging from classical settings like PAC learning and regression to recent trends such as adversarially robust and private learning, it's surprising that we still lack a unified theory; traditional proofs of the equivalence tend to be disparate, and rely on strong model-specific assumptions like uniform convergence and sample compression. In this work, we give the first model-independent framework explaining the equivalence of realizable and agnostic learnability: a three-line blackbox reduction that simplifies, unifies, and extends our understanding across a wide variety of settings. This includes models with no known characterization of learnability such as learning with arbitrary distributional assumptions or general loss, as well as a host of other popular settings such as robust learning, partial learning, fair learning, and the statistical query model. More generally, we argue that the equivalence of realizable and agnostic learning is actually a special case of a broader phenomenon we call property generalization: any desirable property of a learning algorithm (e.g.\ noise tolerance, privacy, stability) that can be satisfied over finite hypothesis classes extends (possibly in some variation) to any learnable hypothesis class.

If you want to cultivate a mathematical understanding of machine learning theory while covering basic probability, matrices and calculus, then this is …

Electrification was, without a doubt, the greatest engineering marvel of the 20th century. The electric motor was invented way back in 1821, and the electrical circuit was mathematically analyzed in 1827. But factory electrification, household electrification, and railway electrification all started slowly several decades later. The field of AI was formally founded in 1956. But it's only now--more than six decades later--that AI is expected to revolutionize the way humanity will live and work in the coming decades.

Modern CDCL SAT solvers learn clauses rapidly, and an important heuristic is the clause deletion scheme. Most current solvers have two (or more) stores of clauses. One has ``valuable'' clauses which are never deleted. Most learned clauses are added to the other, with an aggressive deletion strategy to restrict its size. Recent solvers in the MapleSAT family, have comparatively complex deletion scheme, and perform well. Many solvers store only binary clauses permanently, but MapleLCMDistChronoBT stores clauses with small LBD permanently. We report an experimental study of the permanent clause store in MapleLCMDistChronoBT. We observe that this store can get quite large, but several methods for limiting its size reduced performance. We also show that alternate size and LBD based criteria improve performance, while still having large permanent stores. In particular, saving clauses up to size 8, and adding small numbers of high-centrality clauses, both improved performance, with the best improvement using both methods.