Database indexes facilitate data retrieval and benefit broad applications in real-world systems. Recently, a new family of index, named learned index, is proposed to learn hidden yet useful data distribution and incorporate such information into the learning of indexes, which leads to promising performance improvements. However, the "learning" process of learned indexes is still under-explored. In this paper, we propose a formal machine learning based framework to quantify the index learning objective, and study two general and pluggable techniques to enhance the learning efficiency and learning effectiveness for learned indexes. With the guidance of the formal learning objective, we can efficiently learn index by incorporating the proposed sampling technique, and learn precise index with enhanced generalization ability brought by the proposed result-driven gap insertion technique. We conduct extensive experiments on real-world datasets and compare several indexing methods from the perspective of the index learning objective. The results show the ability of the proposed framework to help to design suitable indexes for different scenarios. Further, we demonstrate the effectiveness of the proposed sampling technique, which achieves up to 78x construction speedup while maintaining non-degraded indexing performance. Finally, we show the gap insertion technique can enhance both the static and dynamic indexing performances of existing learned index methods with up to 1.59x query speedup. We will release our codes and processed data for further study, which can enable more exploration of learned indexes from both the perspectives of machine learning and database.

We study the problem of learning the causal relationships between a set of observed variables in the presence of latents, while minimizing the cost of interventions on the observed variables. We assume access to an undirected graph $G$ on the observed variables whose edges represent either all direct causal relationships or, less restrictively, a superset of causal relationships (identified, e.g., via conditional independence tests or a domain expert). Our goal is to recover the directions of all causal or ancestral relations in $G$, via a minimum cost set of interventions. It is known that constructing an exact minimum cost intervention set for an arbitrary graph $G$ is NP-hard. We further argue that, conditioned on the hardness of approximate graph coloring, no polynomial time algorithm can achieve an approximation factor better than $\Theta(\log n)$, where $n$ is the number of observed variables in $G$. To overcome this limitation, we introduce a bi-criteria approximation goal that lets us recover the directions of all but $\epsilon n^2$ edges in $G$, for some specified error parameter $\epsilon > 0$. Under this relaxed goal, we give polynomial time algorithms that achieve intervention cost within a small constant factor of the optimal. Our algorithms combine work on efficient intervention design and the design of low-cost separating set systems, with ideas from the literature on graph property testing.

We study {\em online} active learning of homogeneous $s$-sparse halfspaces in $\mathbb{R}^d$ with adversarial noise \cite{kearns1992toward}, where the overall probability of a noisy label is constrained to be at most $\nu$ and the marginal distribution over unlabeled data is unchanged. Our main contribution is a state-of-the-art online active learning algorithm that achieves near-optimal attribute efficiency, label and sample complexity under mild distributional assumptions. In particular, under the conditions that the marginal distribution is isotropic log-concave and $\nu = \Omega(\epsilon)$, where $\epsilon \in (0, 1)$ is the target error rate, we show that our algorithm PAC learns the underlying halfspace in polynomial time with near-optimal label complexity bound of $\tilde{O}\big(s \cdot polylog(d, \frac{1}{\epsilon})\big)$ and sample complexity bound of $\tilde{O}\big(\frac{s}{\epsilon} \cdot polylog(d)\big)$. Prior to this work, existing online algorithms designed for tolerating the adversarial noise are either subject to label complexity polynomial in $d$ or $\frac{1}{\epsilon}$, or work under the restrictive uniform marginal distribution. As an immediate corollary of our main result, we show that under the more challenging agnostic model \cite{kearns1992toward} where no assumption is made on the noise rate, our active learner achieves an error rate of $O(OPT) + \epsilon$ with the same running time and label and sample complexity, where $OPT$ is the best possible error rate achievable by any homogeneous $s$-sparse halfspace. Our algorithm builds upon the celebrated Perceptron while leveraging novel localized sampling and semi-random gradient update to tolerate the adversarial noise. We believe that our algorithmic design and analysis are of independent interest, and may shed light on learning halfspaces with broader noise models.

Algorithms for noiseless collaborative PAC learning have been analyzed and optimized in recent years with respect to sample complexity. In this paper, we study collaborative PAC learning with the goal of reducing communication cost at essentially no penalty to the sample complexity. We develop communication efficient collaborative PAC learning algorithms using distributed boosting. We then consider the communication cost of collaborative learning in the presence of classification noise. As an intermediate step, we show how collaborative PAC learning algorithms can be adapted to handle classification noise. With this insight, we develop communication efficient algorithms for collaborative PAC learning robust to classification noise.

In the problem of learning a class ratio from unlabeled data, which we call CR learning, the training data is unlabeled, and only the ratios, or proportions, of examples receiving each label are given. The goal is to learn a hypothesis that predicts the proportions of labels on the distribution underlying the sample. This model of learning is applicable to a wide variety of settings, including predicting the number of votes for candidates in political elections from polls. In this paper, we formally define this class and resolve foundational questions regarding the computational complexity of CR learning and characterize its relationship to PAC learning. Among our results, we show, perhaps surprisingly, that for finite VC classes what can be efficiently CR learned is a strict subset of what can be learned efficiently in PAC, under standard complexity assumptions. We also show that there exist classes of functions whose CR learnability is independent of ZFC, the standard set theoretic axioms. This implies that CR learning cannot be easily characterized (like PAC by VC dimension).

We study the complexity of PAC learning halfspaces in the presence of Massart (bounded) noise. Specifically, given labeled examples $(x, y)$ from a distribution $D$ on $\mathbb{R}^{n} \times \{ \pm 1\}$ such that the marginal distribution on $x$ is arbitrary and the labels are generated by an unknown halfspace corrupted with Massart noise at rate $\eta<1/2$, we want to compute a hypothesis with small misclassification error. Characterizing the efficient learnability of halfspaces in the Massart model has remained a longstanding open problem in learning theory. Recent work gave a polynomial-time learning algorithm for this problem with error $\eta+\epsilon$. This error upper bound can be far from the information-theoretically optimal bound of $\mathrm{OPT}+\epsilon$. More recent work showed that {\em exact learning}, i.e., achieving error $\mathrm{OPT}+\epsilon$, is hard in the Statistical Query (SQ) model. In this work, we show that there is an exponential gap between the information-theoretically optimal error and the best error that can be achieved by a polynomial-time SQ algorithm. In particular, our lower bound implies that no efficient SQ algorithm can approximate the optimal error within any polynomial factor.

These are the notes for the lectures that I was giving during Fall 2020 at the Moscow Institute of Physics and Technology (MIPT) and at the Yandex School of Data Analysis (YSDA). The notes cover some aspects of initialization, loss landscape, generalization, and a neural tangent kernel theory. While many other topics (e.g. expressivity, a mean-field theory, a double descent phenomenon) are missing in the current version, we plan to add them in future revisions.

Learning pseudo-Boolean (PB) constraints in PB solvers exploiting cutting planes based inference is not as well understood as clause learning in conflict-driven clause learning solvers. In this paper, we show that PB constraints derived using cutting planes may contain \emph{irrelevant literals}, i.e., literals whose assigned values (whatever they are) never change the truth value of the constraint. Such literals may lead to infer constraints that are weaker than they should be, impacting the size of the proof built by the solver, and thus also affecting its performance. This suggests that current implementations of PB solvers based on cutting planes should be reconsidered to prevent the generation of irrelevant literals. Indeed, detecting and removing irrelevant literals is too expensive in practice to be considered as an option (the associated problem is NP-hard.

Can we infer all the failed components of an infrastructure network, given a sample of reachable nodes from supply nodes? One of the most critical post-disruption processes after a natural disaster is to quickly determine the damage or failure states of critical infrastructure components. However, this is non-trivial, considering that often only a fraction of components may be accessible or observable after a disruptive event. Past work has looked into inferring failed components given point probes, i.e. with a direct sample of failed components. In contrast, we study the harder problem of inferring failed components given partial information of some `serviceable' reachable nodes and a small sample of point probes, being the first often more practical to obtain. We formulate this novel problem using the Minimum Description Length (MDL) principle, and then present a greedy algorithm that minimizes MDL cost effectively. We evaluate our algorithm on domain-expert simulations of real networks in the aftermath of an earthquake. Our algorithm successfully identify failed components, especially the critical ones affecting the overall system performance.

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.