In this paper, we address the critical challenges of double-spending and selfish mining attacks in blockchain-based digital currencies. Double-spending is a problem where the same tender is spent multiple times during a digital currency transaction, while selfish mining is an intentional alteration of a blockchain to increase rewards to one miner or a group of miners. We introduce a new attack that combines both these attacks and propose a machine learning-based solution to mitigate the risks associated with them. Specifically, we use the learning automaton, a powerful online learning method, to develop two models, namely the SDTLA and WVBM, which can effectively defend against selfish mining attacks. Our experimental results show that the SDTLA method increases the profitability threshold of selfish mining up to 47$\%$, while the WVBM method performs even better and is very close to the ideal situation where each miner's revenue is proportional to their shared hash processing power. Additionally, we demonstrate that both methods can effectively reduce the risks of double-spending by tuning the $Z$ Parameter. Our findings highlight the potential of SDTLA and WVBM as promising solutions for enhancing the security and efficiency of blockchain networks.

We advance a recently flourishing line of work at the intersection of learning theory and computational economics by studying the learnability of two classes of mechanisms prominent in economics, namely menus of lotteries and two-part tariffs. The former is a family of randomized mechanisms designed for selling multiple items, known to achieve revenue beyond deterministic mechanisms, while the latter is designed for selling multiple units (copies) of a single item with applications in real-world scenarios such as car or bike-sharing services. We focus on learning high-revenue mechanisms of this form from buyer valuation data in both distributional settings, where we have access to buyers' valuation samples up-front, and the more challenging and less-studied online settings, where buyers arrive one-at-a-time and no distributional assumption is made about their values. Our main contribution is proposing the first online learning algorithms for menus of lotteries and two-part tariffs with strong regret-bound guarantees. In the general case, we provide a reduction to a finite number of experts, and in the limited buyer type case, we show a reduction to online linear optimization, which allows us to obtain no-regret guarantees by presenting buyers with menus that correspond to a barycentric spanner. In addition, we provide algorithms with improved running times over prior work for the distributional settings. The key difficulty when deriving learning algorithms for these settings is that the relevant revenue functions have sharp transition boundaries. In stark contrast with the recent literature on learning such unstructured functions, we show that simple discretization-based techniques are sufficient for learning in these settings.

We study the problem of PAC learning $\gamma$-margin halfspaces with Random Classification Noise. We establish an information-computation tradeoff suggesting an inherent gap between the sample complexity of the problem and the sample complexity of computationally efficient algorithms. Concretely, the sample complexity of the problem is $\widetilde{\Theta}(1/(\gamma^2 \epsilon))$. We start by giving a simple efficient algorithm with sample complexity $\widetilde{O}(1/(\gamma^2 \epsilon^2))$. Our main result is a lower bound for Statistical Query (SQ) algorithms and low-degree polynomial tests suggesting that the quadratic dependence on $1/\epsilon$ in the sample complexity is inherent for computationally efficient algorithms. Specifically, our results imply a lower bound of $\widetilde{\Omega}(1/(\gamma^{1/2} \epsilon^2))$ on the sample complexity of any efficient SQ learner or low-degree test.

Despite significant effort, the quantum machine learning community has only demonstrated quantum learning advantages for artificial cryptography-inspired datasets when dealing with classical data. In this paper we address the challenge of finding learning problems where quantum learning algorithms can achieve a provable exponential speedup over classical learning algorithms. We reflect on computational learning theory concepts related to this question and discuss how subtle differences in definitions can result in significantly different requirements and tasks for the learner to meet and solve. We examine existing learning problems with provable quantum speedups and find that they largely rely on the classical hardness of evaluating the function that generates the data, rather than identifying it. To address this, we present two new learning separations where the classical difficulty primarily lies in identifying the function generating the data. Furthermore, we explore computational hardness assumptions that can be leveraged to prove quantum speedups in scenarios where data is quantum-generated, which implies likely quantum advantages in a plethora of more natural settings (e.g., in condensed matter and high energy physics). We also discuss the limitations of the classical shadow paradigm in the context of learning separations, and how physically-motivated settings such as characterizing phases of matter and Hamiltonian learning fit in the computational learning framework.

Suppose we are given access to $n$ independent samples from distribution $\mu$ and we wish to output one of them with the goal of making the output distributed as close as possible to a target distribution $\nu$. In this work we show that the optimal total variation distance as a function of $n$ is given by $\tilde\Theta(\frac{D}{f'(n)})$ over the class of all pairs $\nu,\mu$ with a bounded $f$-divergence $D_f(\nu\|\mu)\leq D$. Previously, this question was studied only for the case when the Radon-Nikodym derivative of $\nu$ with respect to $\mu$ is uniformly bounded. We then consider an application in the seemingly very different field of smoothed online learning, where we show that recent results on the minimax regret and the regret of oracle-efficient algorithms still hold even under relaxed constraints on the adversary (to have bounded $f$-divergence, as opposed to bounded Radon-Nikodym derivative). Finally, we also study efficacy of importance sampling for mean estimates uniform over a function class and compare importance sampling with rejection sampling.

Inspired by Solomonoffs theory of inductive inference, we propose a prior based on circuit complexity. There are several advantages to this approach. First, it relies on a complexity measure that does not depend on the choice of UTM. There is one universal definition for Boolean circuits involving an universal operation such as nand with simple conversions to alternative definitions such as and, or, and not. Second, there is no analogue of the halting problem. The output value of a circuit can be calculated recursively by computer in time proportional to the number of gates, while a short program may run for a very long time. Our prior assumes that a Boolean function, or equivalently, Boolean string of fixed length, is generated by some Bayesian mixture of circuits. This model is appropriate for learning Boolean functions from partial information, a problem often encountered within machine learning as "binary classification." We argue that an inductive bias towards simple explanations as measured by circuit complexity is appropriate for this problem.

We establish generic uniform convergence guarantees for Gaussian data in terms of the Rademacher complexity of the hypothesis class and the Lipschitz constant of the square root of the scalar loss function. We show how these guarantees substantially generalize previous results based on smoothness (Lipschitz constant of the derivative), and allow us to handle the broader class of square-root-Lipschitz losses, which includes also non-smooth loss functions appropriate for studying phase retrieval and ReLU regression, as well as rederive and better understand "optimistic rate" and interpolation learning guarantees.

We study the problem of sequential prediction in the stochastic setting with an adversary that is allowed to inject clean-label adversarial (or out-of-distribution) examples. Algorithms designed to handle purely stochastic data tend to fail in the presence of such adversarial examples, often leading to erroneous predictions. This is undesirable in many high-stakes applications such as medical recommendations, where abstaining from predictions on adversarial examples is preferable to misclassification. On the other hand, assuming fully adversarial data leads to very pessimistic bounds that are often vacuous in practice. To capture this motivation, we propose a new model of sequential prediction that sits between the purely stochastic and fully adversarial settings by allowing the learner to abstain from making a prediction at no cost on adversarial examples. Assuming access to the marginal distribution on the non-adversarial examples, we design a learner whose error scales with the VC dimension (mirroring the stochastic setting) of the hypothesis class, as opposed to the Littlestone dimension which characterizes the fully adversarial setting. Furthermore, we design a learner for VC dimension~1 classes, which works even in the absence of access to the marginal distribution. Our key technical contribution is a novel measure for quantifying uncertainty for learning VC classes, which may be of independent interest.

Determining the optimal sample complexity of PAC learning in the realizable setting was a central open problem in learning theory for decades. Finally, the seminal work by Hanneke (2016) gave an algorithm with a provably optimal sample complexity. His algorithm is based on a careful and structured sub-sampling of the training data and then returning a majority vote among hypotheses trained on each of the sub-samples. While being a very exciting theoretical result, it has not had much impact in practice, in part due to inefficiency, since it constructs a polynomial number of sub-samples of the training data, each of linear size. In this work, we prove the surprising result that the practical and classic heuristic bagging (a.k.a. bootstrap aggregation), due to Breiman (1996), is in fact also an optimal PAC learner. Bagging pre-dates Hanneke's algorithm by twenty years and is taught in most undergraduate machine learning courses. Moreover, we show that it only requires a logarithmic number of sub-samples to reach optimality.

Despite extraordinary progress, current machine learning systems have been shown to be brittle against adversarial examples: seemingly innocuous but carefully crafted perturbations of test examples that cause machine learning predictors to misclassify. Can we learn predictors robust to adversarial examples? and how? There has been much empirical interest in this contemporary challenge in machine learning, and in this thesis, we address it from a theoretical perspective. In this thesis, we explore what robustness properties can we hope to guarantee against adversarial examples and develop an understanding of how to algorithmically guarantee them. We illustrate the need to go beyond traditional approaches and principles such as empirical risk minimization and uniform convergence, and make contributions that can be categorized as follows: (1) introducing problem formulations capturing aspects of emerging practical challenges in robust learning, (2) designing new learning algorithms with provable robustness guarantees, and (3) characterizing the complexity of robust learning and fundamental limitations on the performance of any algorithm.