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Computational Learning Theory: Instructional Materials


Data Science & Machine Learning(Theory+Projects)A-Z 90 HOURS

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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.


The Lifecycle of a Statistical Model: Model Failure Detection, Identification, and Refitting

arXiv.org Machine Learning

The statistical machine learning community has demonstrated considerable resourcefulness over the years in developing highly expressive tools for estimation, prediction, and inference. The bedrock assumptions underlying these developments are that the data comes from a fixed population and displays little heterogeneity. But reality is significantly more complex: statistical models now routinely fail when released into real-world systems and scientific applications, where such assumptions rarely hold. Consequently, we pursue a different path in this paper vis-a-vis the well-worn trail of developing new methodology for estimation and prediction. In this paper, we develop tools and theory for detecting and identifying regions of the covariate space (subpopulations) where model performance has begun to degrade, and study intervening to fix these failures through refitting. We present empirical results with three real-world data sets -- including a time series involving forecasting the incidence of COVID-19 -- showing that our methodology generates interpretable results, is useful for tracking model performance, and can boost model performance through refitting. We complement these empirical results with theory proving that our methodology is minimax optimal for recovering anomalous subpopulations as well as refitting to improve accuracy in a structured normal means setting.


A Survey of Methods for Automated Algorithm Configuration

arXiv.org Artificial Intelligence

Algorithm configuration (AC) is concerned with the automated search of the most suitable parameter configuration of a parametrized algorithm. There is currently a wide variety of AC problem variants and methods proposed in the literature. Existing reviews do not take into account all derivatives of the AC problem, nor do they offer a complete classification scheme. To this end, we introduce taxonomies to describe the AC problem and features of configuration methods, respectively. We review existing AC literature within the lens of our taxonomies, outline relevant design choices of configuration approaches, contrast methods and problem variants against each other, and describe the state of AC in industry. Finally, our review provides researchers and practitioners with a look at future research directions in the field of AC.


Structure-preserving GANs

arXiv.org Machine Learning

Generative adversarial networks (GANs), a class of distribution-learning methods based on a two-player game between a generator and a discriminator, can generally be formulated as a minmax problem based on the variational representation of a divergence between the unknown and the generated distributions. We introduce structure-preserving GANs as a data-efficient framework for learning distributions with additional structure such as group symmetry, by developing new variational representations for divergences. Our theory shows that we can reduce the discriminator space to its projection on the invariant discriminator space, using the conditional expectation with respect to the $\sigma$-algebra associated to the underlying structure. In addition, we prove that the discriminator space reduction must be accompanied by a careful design of structured generators, as flawed designs may easily lead to a catastrophic "mode collapse" of the learned distribution. We contextualize our framework by building symmetry-preserving GANs for distributions with intrinsic group symmetry, and demonstrate that both players, namely the equivariant generator and invariant discriminator, play important but distinct roles in the learning process. Empirical experiments and ablation studies across a broad range of data sets, including real-world medical imaging, validate our theory, and show our proposed methods achieve significantly improved sample fidelity and diversity -- almost an order of magnitude measured in Fr\'echet Inception Distance -- especially in the small data regime.


Exact learning for infinite families of concepts

arXiv.org Artificial Intelligence

In this paper, based on results of exact learning, test theory, and rough set theory, we study arbitrary infinite families of concepts each of which consists of an infinite set of elements and an infinite set of subsets of this set called concepts. We consider the notion of a problem over a family of concepts that is described by a finite number of elements: for a given concept, we should recognize which of the elements under consideration belong to this concept. As algorithms for problem solving, we consider decision trees of five types: (i) using membership queries, (ii) using equivalence queries, (iii) using both membership and equivalence queries, (iv) using proper equivalence queries, and (v) using both membership and proper equivalence queries. As time complexity, we study the depth of decision trees. In the worst case, with the growth of the number of elements in the problem description, the minimum depth of decision trees of the first type either grows as a logarithm or linearly, and the minimum depth of decision trees of each of the other types either is bounded from above by a constant or grows as a logarithm, or linearly. The obtained results allow us to distinguish seven complexity classes of infinite families of concepts.


Learning Logic Programs From Noisy Failures

arXiv.org Artificial Intelligence

Inductive Logic Programming (ILP) is a form of machine learning (ML) which in contrast to many other state of the art ML methods typically produces highly interpretable and reusable models. However, many ILP systems lack the ability to naturally learn from any noisy or partially misclassified training data. We introduce the relaxed learning from failures approach to ILP, a noise handling modification of the previously introduced learning from failures (LFF) approach which is incapable of handling noise. We additionally introduce the novel Noisy Popper ILP system which implements this relaxed approach and is a modification of the existing Popper system. Like Popper, Noisy Popper takes a generate-test-constrain loop to search its hypothesis space wherein failed hypotheses are used to construct hypothesis constraints. These constraints are used to prune the hypothesis space, making the hypothesis search more efficient. However, in the relaxed setting, constraints are generated in a more lax fashion as to avoid allowing noisy training data to lead to hypothesis constraints which prune optimal hypotheses. Constraints unique to the relaxed setting are generated via hypothesis comparison. Additional constraints are generated by weighing the accuracy of hypotheses against their sizes to avoid overfitting through an application of the minimum description length. We support this new setting through theoretical proofs as well as experimental results which suggest that Noisy Popper improves the noise handling capabilities of Popper but at the cost of overall runtime efficiency.


Teaching an Active Learner with Contrastive Examples

arXiv.org Machine Learning

We study the problem of active learning with the added twist that the learner is assisted by a helpful teacher. We consider the following natural interaction protocol: At each round, the learner proposes a query asking for the label of an instance $x^q$, the teacher provides the requested label $\{x^q, y^q\}$ along with explanatory information to guide the learning process. In this paper, we view this information in the form of an additional contrastive example ($\{x^c, y^c\}$) where $x^c$ is picked from a set constrained by $x^q$ (e.g., dissimilar instances with the same label). Our focus is to design a teaching algorithm that can provide an informative sequence of contrastive examples to the learner to speed up the learning process. We show that this leads to a challenging sequence optimization problem where the algorithm's choices at a given round depend on the history of interactions. We investigate an efficient teaching algorithm that adaptively picks these contrastive examples. We derive strong performance guarantees for our algorithm based on two problem-dependent parameters and further show that for specific types of active learners (e.g., a generalized binary search learner), the proposed teaching algorithm exhibits strong approximation guarantees. Finally, we illustrate our bounds and demonstrate the effectiveness of our teaching framework via two numerical case studies.


PAC-Learning Uniform Ergodic Communicative Networks

arXiv.org Machine Learning

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.


Towards a Unified Information-Theoretic Framework for Generalization

arXiv.org Machine Learning

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.


Fast Rates for Nonparametric Online Learning: From Realizability to Learning in Games

arXiv.org Machine Learning

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.