We present a meta-learning approach based on learning an adaptive, high-dimensional loss function that can generalize across multiple tasks and different model architectures. We develop a fully differentiable pipeline for learning a loss function targeted at maximizing the performance of an optimizee trained using this loss function. We observe that the loss landscape produced by our learned loss significantly improves upon the original task-specific loss. We evaluate our method on supervised and reinforcement learning tasks. Furthermore, we show that our pipeline is able to operate in sparse reward and self-supervised reinforcement learning scenarios.

This thesis presents new methods for unsupervised learning of distributed representations of words and entities from text and knowledge bases. The first algorithm presented in the thesis is a multi-view algorithm for learning representations of words called Multiview Latent Semantic Analysis (MVLSA). By incorporating up to 46 different types of co-occurrence statistics for the same vocabulary of english words, I show that MVLSA outperforms other state-of-the-art word embedding models. Next, I focus on learning entity representations for search and recommendation and present the second method of this thesis, Neural Variational Set Expansion (NVSE). NVSE is also an unsupervised learning method, but it is based on the Variational Autoencoder framework. Evaluations with human annotators show that NVSE can facilitate better search and recommendation of information gathered from noisy, automatic annotation of unstructured natural language corpora. Finally, I move from unstructured data and focus on structured knowledge graphs. I present novel approaches for learning embeddings of vertices and edges in a knowledge graph that obey logical constraints.

Understanding temporal and causal relations between events is a fundamental natural language understanding task. Because a cause must be before its effect in time, temporal and causal relations are closely related and one relation even dictates the other one in many cases. However, limited attention has been paid to studying these two relations jointly. This paper presents a joint inference framework for them using constrained conditional models (CCMs). Specifically, we formulate the joint problem as an integer linear programming (ILP) problem, enforcing constraints inherently in the nature of time and causality. We show that the joint inference framework results in statistically significant improvement in the extraction of both temporal and causal relations from text.

Transportation cost is an attractive similarity measure between probability distributions due to its many useful theoretical properties. However, solving optimal transport exactly can be prohibitively expensive. Therefore, there has been significant effort towards the design of scalable approximation algorithms. Previous combinatorial results [Sharathkumar, Agarwal STOC '12, Agarwal, Sharathkumar STOC '14] have focused primarily on the design of strongly polynomial multiplicative approximation algorithms. There has also been an effort to design approximate solutions with additive errors [Cuturi NIPS '13, Altschuler et. al NIPS '17, Dvurechensky et al., ICML '18, Quanrud, SOSA '19] within a time bound that is linear in the size of the cost matrix and polynomial in $C/\delta$; here $C$ is the largest value in the cost matrix and $\delta$ is the additive error. We present an adaptation of the classical graph algorithm of Gabow and Tarjan and provide a novel analysis of this algorithm that bounds its execution time by $O(\frac{n^2 C}{\delta}+ \frac{nC^2}{\delta^2})$. Our algorithm is extremely simple and executes, for an arbitrarily small constant $\varepsilon$, only $\lfloor \frac{2C}{(1-\varepsilon)\delta}\rfloor + 1$ iterations, where each iteration consists only of a Dijkstra search followed by a depth-first search. We also provide empirical results that suggest our algorithm significantly outperforms existing approaches in execution time.

We study how language on social media is linked to diseases such as atherosclerotic heart disease (AHD), diabetes and various types of cancer. Our proposed model leverages state-of-the-art sentence embeddings, followed by a regression model and clustering, without the need of additional labelled data. It allows to predict community-level medical outcomes from language, and thereby potentially translate these to the individual level. The method is applicable to a wide range of target variables and allows us to discover known and potentially novel correlations of medical outcomes with life-style aspects and other socioeconomic risk factors.

Space partitioning methods such as random forests and the Mondrian process are powerful machine learning methods for multi-dimensional and relational data, and are based on recursively cutting a domain. The flexibility of these methods is often limited by the requirement that the cuts be axis aligned. The Ostomachion process and the self-consistent binary space partitioning-tree process were recently introduced as generalizations of the Mondrian process for space partitioning with non-axis aligned cuts in the two dimensional plane. Motivated by the need for a multi-dimensional partitioning tree with non-axis aligned cuts, we propose the Random Tessellation Process (RTP), a framework that includes the Mondrian process and the binary space partitioning-tree process as special cases. We derive a sequential Monte Carlo algorithm for inference, and provide random forest methods. Our process is self-consistent and can relax axis-aligned constraints, allowing complex inter-dimensional dependence to be captured. We present a simulation study, and analyse gene expression data of brain tissue, showing improved accuracies over other methods.

In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the "topological signal" and the short intervals represent "noise". We give evidence to dispute this thesis, showing that the short intervals encode geometric information. Specifically, we prove that persistent homology detects the curvature of disks from which points have been sampled. We describe a general computational framework for solving inverse problems using the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space. In the present application, the average persistence landscapes of points sampled from disks of constant curvature results in a path in this Hilbert space which may be learned using standard tools from statistical and machine learning.

A series of recent works suggest that deep neural networks (DNNs), of fixed depth, are equivalent to certain Gaussian Processes (NNGP/NTK) in the highly over-parameterized regime (width or number-of-channels going to infinity). Other works suggest that this limit is relevant for real-world DNNs. These results invite further study into the generalization properties of Gaussian Processes of the NNGP and NTK type. Here we make several contributions along this line. First, we develop a formalism, based on field theory tools, for calculating learning curves perturbatively in one over the dataset size. For the case of NNGPs, this formalism naturally extends to finite width corrections. Second, in cases where one can diagonalize the covariance-function of the NNGP/NTK, we provide analytic expressions for the asymptotic learning curves of any given target function. These go beyond the standard equivalence kernel results. Last, we provide closed analytic expressions for the eigenvalues of NNGP/NTK kernels of depth 2 fully-connected ReLU networks. For datasets on the hypersphere, the eigenfunctions of such kernels, at any depth, are hyperspherical harmonics. A simple coherent picture emerges wherein fully-connected DNNs have a strong entropic bias towards functions which are low order polynomials of the input.

Recently, natural policy gradient algorithms gained widespread recognition due to their strong performance in reinforcement learning tasks [12, 13]. However, their major drawback is the need to secure the policy being in a "trust region" and meanwhile allowing for sufficient exploration. The main objective of this study was to present an approach which models dynamical isometry of agents policies by estimating conditioning of its Jacobian at individual points in the environment space. We present a Jacobian Policy Optimization algorithm for policy optimization, which dynamically adapts the trust interval with respect to policy conditioning. The suggested approach was tested across a range of Atari environments. This paper offers some important insights into an improvement of policy optimization in reinforcement learning tasks.

We study the problem of fair binary classification using the notion of Equal Opportunity. It requires the true positive rate to distribute equally across the sensitive groups. Within this setting we show that the fair optimal classifier is obtained by recalibrating the Bayes classifier by a group-dependent threshold. We provide a constructive expression for the threshold. This result motivates us to devise a plug-in classification procedure based on both unlabeled and labeled datasets. While the latter is used to learn the output conditional probability, the former is used for calibration. The overall procedure can be computed in polynomial time and it is shown to be statistically consistent both in terms of classification error and fairness measure. Finally, we present numerical experiments which indicate that our method is often superior or competitive with the state-of-the-art methods on benchmark datasets.