Efficient Strongly Polynomial Algorithms for Quantile Regression

Linear Regression is a seminal technique in statistics and machine learning, where the objective is to build linear predictive models between a response (i.e., dependent) variable and one or more predictor (i.e., independent) variables. In this paper, we revisit the classical technique of Quantile Regression (QR), which is statistically a more robust alternative to the other classical technique of Ordinary Least Square Regression (OLS). However, while there exist efficient algorithms for OLS, almost all of the known results for QR are only weakly polynomial. Towards filling this gap, this paper proposes several efficient strongly polynomial algorithms for QR for various settings. For two dimensional QR, making a connection to the geometric concept of $k$-set, we propose an algorithm with a deterministic worst-case time complexity of $\mathcal{O}(n^{4/3} polylog(n))$ and an expected time complexity of $\mathcal{O}(n^{4/3})$ for the randomized version. We also propose a randomized divide-and-conquer algorithm -- RandomizedQR with an expected time complexity of $\mathcal{O}(n\log^2{(n)})$ for two dimensional QR problem. For the general case with more than two dimensions, our RandomizedQR algorithm has an expected time complexity of $\mathcal{O}(n^{d-1}\log^2{(n)})$.

Approximate Query Processing using Deep Generative Models

Data is generated at an unprecedented rate surpassing our ability to analyze them. One viable solution that was pioneered by the database community is Approximate Query Processing (AQP). AQP seeks to provide approximate answers to queries in a fraction of time needed for computing exact answers. This is often achieved by running the query on a pre-computed or on-demand derived sample and generating estimates for the entire dataset based on the result. In this work, we explore a novel approach for AQP utilizing deep learning (DL). We use deep generative models, an unsupervised learning based approach, to learn the data distribution faithfully in a compact manner (typically few hundred KBs). Queries could be answered approximately by generating samples from the learned model. This approach eliminates the dependency of AQP to a sample of fixed size and allows us to satisfy arbitrary accuracy requirements by generating as many samples as needed very fast. While we specifically focus on variational autoencoders (VAE), we demonstrate how our approach could also be used for other popular DL models such as generative adversarial networks (GAN) and deep Bayesian networks (DBN). Our other contributions include (a) identifying model bias and minimizing it through a rejection sampling based approach (b) An algorithm to build model ensembles for AQP for improved accuracy and (c) an analysis of VAE latent space to understand its suitability to AQP. Our extensive experiments show that deep learning is a very promising approach for AQP.

Multi-Attribute Selectivity Estimation Using Deep Learning

Selectivity estimation - the problem of estimating the result size of queries - is a fundamental yet challenging problem in databases. Accurate estimation of query selectivity involving multiple correlated attributes is especially challenging. Poor cardinality estimates could result in the selection of bad plans by the query optimizer. In this paper, we investigate the feasibility of using deep learning based approaches for challenging scenarios such as queries involving multiple predicates and/or low selectivity. Specifically, we propose two complementary approaches. Our first approach considers selectivity as an unsupervised deep density estimation problem. We successfully introduce techniques from neural density estimation for this purpose. The key idea is to decompose the joint distribution into a set of tractable conditional probability distributions such that they satisfy the autoregressive property. Our second approach formulates selectivity estimation as a supervised deep learning problem that predicts the selectivity of a given query. We also introduce and address a number of practical challenges arising when adapting deep learning for relational data. These include query/data featurization, incorporating query workload information in a deep learning framework and the dynamic scenario where both data and workload queries could be updated. Our extensive experiments with a special emphasis on queries with a large number of predicates and/or small result sizes demonstrates that deep learning based techniques are a promising research avenue for selectivity estimation worthy of further investigation.