Constrained counting and sampling are two fundamental problems in Computer Science with numerous applications, including network reliability, privacy, probabilistic reasoning, and constrained-random verification. In constrained counting, the task is to compute the total weight, subject to a given weighting function, of the set of solutions of the given constraints. In constrained sampling, the task is to sample randomly, subject to a given weighting function, from the set of solutions to a set of given constraints. Consequently, constrained counting and sampling have been subject to intense theoretical and empirical investigations over the years. Prior work, however, offered either heuristic techniques with poor guarantees of accuracy or approaches with proven guarantees but poor performance in practice. In this thesis, we introduce a novel hashing-based algorithmic framework for constrained sampling and counting that combines the classical algorithmic technique of universal hashing with the dramatic progress made in combinatorial reasoning tools, in particular, SAT and SMT, over the past two decades. The resulting frameworks for counting (ApproxMC2) and sampling (UniGen) can handle formulas with up to million variables representing a significant boost up from the prior state of the art tools' capability to handle few hundreds of variables. If the initial set of constraints is expressed as Disjunctive Normal Form (DNF), ApproxMC is the only known Fully Polynomial Randomized Approximation Scheme (FPRAS) that does not involve Monte Carlo steps. By exploiting the connection between definability of formulas and variance of the distribution of solutions in a cell defined by 3-universal hash functions, we introduced an algorithmic technique, MIS, that reduced the size of XOR constraints employed in the underlying universal hash functions by as much as two orders of magnitude.

Searching, browsing, and recommendations are common ways in which the "choice overload" faced by users in the online marketplace can be mitigated. In this paper we propose the use of hierarchical item categories, obtained from implicit feedback data, to enable efficient browsing and recommendations. We present a method of creating hierarchical item categories from implicit feedback data only i.e., without any other information on the items like name, genre etc. Categories created in this fashion are based on users' co-consumption of items. Thus, they can be more useful for users in finding interesting and relevant items while they are browsing through the hierarchy. We also show that this item hierarchy can be useful in making category based recommendations, which makes the recommendations more explainable and increases the diversity of the recommendations without compromising much on the accuracy. Item hierarchy can also be useful in the creation of an automatic item taxonomy skeleton by bypassing manual labeling and annotation. This can especially be useful for small vendors. Our data-driven hierarchical categories are based on hierarchical latent tree analysis (HLTA) which has been previously used for text analysis. We present a scaled up learning algorithm \emph{HLTA-Forest} so that HLTA can be applied to implicit feedback data.

As artificial agents proliferate, it is becoming increasingly important to ensure that their interactions with one another are well-behaved. In this paper, we formalize a common-sense notion of when algorithms are well-behaved: an algorithm is safe if it does no harm. Motivated by recent progress in deep learning, we focus on the specific case where agents update their actions according to gradient descent. The paper shows that that gradient descent converges to a Nash equilibrium in safe games. The main contribution is to define strongly-typed agents and show they are guaranteed to interact safely, thereby providing sufficient conditions to guarantee safe interactions. A series of examples show that strong-typing generalizes certain key features of convexity, is closely related to blind source separation, and introduces a new perspective on classical multilinear games based on tensor decomposition.

Trains have been a prominent mode of long-distance travel for decades, especially in the countries with a significant land area and large population. India, with a population of 1.324 billion people in 2016, has a railway system of network route length of 66, 687 kilometers, with 11, 122 locomotives, 7, 216 stations, that served 8.107 billion ridership in 2016 [7]. The Indian railway system is fourth largest in the world in terms of network size. However its trains are plagued with endemic delays that can be credited to (a) obsolete technology, e.g., dated rail engines, (b) size, e.g., large network structure and high railway traffic, (c) weather, e.g., fog in winter months in north India and rains during summer monsoons countrywide. In this paper, we take the initial steps in understanding and predicting train delays.

Generating novel graph structures that optimize given objectives while obeying some given underlying rules is fundamental for chemistry, biology and social science research. This is especially important in the task of molecular graph generation, whose goal is to discover novel molecules with desired properties such as drug-likeness and synthetic accessibility, while obeying physical laws such as chemical valency. However, designing models to find molecules that optimize desired properties while incorporating highly complex and non-differentiable rules remains to be a challenging task. Here we propose Graph Convolutional Policy Network (GCPN), a general graph convolutional network based model for goal-directed graph generation through reinforcement learning. The model is trained to optimize domain-specific rewards and adversarial loss through policy gradient, and acts in an environment that incorporates domain-specific rules. Experimental results show that GCPN can achieve 61% improvement on chemical property optimization over state-of-the-art baselines while resembling known molecules, and achieve 184% improvement on the constrained property optimization task.

Temporal difference learning (TD) is a simple iterative algorithm used to estimate the value function corresponding to a given policy in a Markov decision process. Although TD is one of the most widely used algorithms in reinforcement learning, its theoretical analysis has proved challenging and few guarantees on its statistical efficiency are available. In this work, we provide a simple and explicit finite time analysis of temporal difference learning with linear function approximation. Except for a few key insights, our analysis mirrors standard techniques for analyzing stochastic gradient descent algorithms, and therefore inherits the simplicity and elegance of that literature. A final section of the paper shows that all of our main results extend to the study of Q-learning applied to high-dimensional optimal stopping problems.

This paper introduces the variational implicit processes (VIPs), a Bayesian nonparametric method based on a class of highly flexible priors over functions. Similar to Gaussian processes (GPs), in implicit processes (IPs), an implicit multivariate prior (data simulators, Bayesian neural networks, etc.) is placed over any finite collections of random variables. A novel and efficient variational inference algorithm for IPs is derived using wake-sleep updates, which gives analytic solutions and allows scalable hyper-parameter learning with stochastic optimization. Experiments on real-world regression datasets demonstrate that VIPs return better uncertainty estimates and superior performance over existing inference methods for GPs and Bayesian neural networks. With a Bayesian LSTM as the implicit prior, the proposed approach achieves state-of-the-art results on predicting power conversion efficiency of molecules based on raw chemical formulas.

We propose a single neural probabilistic model based on variational autoencoder that can be conditioned on an arbitrary subset of observed features and then sample the remaining features in "one shot". The features may be both real-valued and categorical. Training of the model is performed by stochastic variational Bayes. The experimental evaluation on synthetic data, as well as feature imputation and image inpainting problems, shows the effectiveness of the proposed approach and diversity of the generated samples.

Linear regression is the task of modeling the relationship between a result variable and some explanatory variables by a linear rule. Linear regression is a standard tool of statistical analysis, with widespread applications spanning essentially all of the sciences. While the standard linear regression task seeks to model the majority of the data, we consider problems where a regression fit could exist for some subset or portion of the data, that does not necessarily model the majority of the data. We will consider cases in which the subset with a linear model is described by some simple condition; in other words, we desire to perform linear regression on this conditional distribution. Note that neither the condition nor the model is known in advance.

We present the very first robust Bayesian Online Changepoint Detection algorithm through General Bayesian Inference (GBI) with $\beta$-divergences. The resulting inference procedure is doubly robust for both the predictive and the changepoint (CP) posterior, with linear time and constant space complexity. We provide a construction for exponential models and demonstrate it on the Bayesian Linear Regression model. In so doing, we make two additional contributions: Firstly, we make GBI scalable using Structural Variational approximations that are exact as $\beta \to 0$. Secondly, we give a principled way of choosing the divergence parameter $\beta$ by minimizing expected predictive loss on-line. We offer the state of the art and improve the False Discovery Rate of CPs by more than 80% on real world data.