This thesis reviews numerical optimization methods with machine learning problems in mind. Since machine learning models are highly parametrized, we focus on methods suited for high dimensional optimization. We build intuition on quadratic models to figure out which methods are suited for non-convex optimization, and develop convergence proofs on convex functions for this selection of methods. With this theoretical foundation for stochastic gradient descent and momentum methods, we try to explain why the methods used commonly in the machine learning field are so successful. Besides explaining successful heuristics, the last chapter also provides a less extensive review of more theoretical methods, which are not quite as popular in practice. So in some sense this work attempts to answer the question: Why are the default Tensorflow optimizers included in the defaults?

In this paper, we present a novel algorithm to address the Network Alignment problem. It is inspired from a previous message passing framework of Bayati et al. [2] and includes several modifications designed to significantly speed up the message updates as well as to enforce their convergence. Experiments show that our proposed model outperforms other state-of-the-art solvers. Finally, we propose an application of our method in order to address the Binary Diffing problem. We show that our solution provides better assignment than the reference differs in almost all submitted instances and outline the importance of leveraging the graphical structure of binary programs.

We consider Bayesian optimization of the output of a network of functions, where each function takes as input the output of its parent nodes, and where the network takes significant time to evaluate. Such problems arise, for example, in reinforcement learning, engineering design, and manufacturing. While the standard Bayesian optimization approach observes only the final output, our approach delivers greater query efficiency by leveraging information that the former ignores: intermediate output within the network. This is achieved by modeling the nodes of the network using Gaussian processes and choosing the points to evaluate using, as our acquisition function, the expected improvement computed with respect to the implied posterior on the objective. Although the non-Gaussian nature of this posterior prevents computing our acquisition function in closed form, we show that it can be efficiently maximized via sample average approximation. In addition, we prove that our method is asymptotically consistent, meaning that it finds a globally optimal solution as the number of evaluations grows to infinity, thus generalizing previously known convergence results for the expected improvement. Notably, this holds even though our method might not evaluate the domain densely, instead leveraging problem structure to leave regions unexplored. Finally, we show that our approach dramatically outperforms standard Bayesian optimization methods in several synthetic and real-world problems.

An improvement in technology is linearly related to time and time-relevant problems. It has been seen that as time progresses, the number of problems humans face also increases. However, technology to resolve these problems tends to improve as well. One of the earliest existing problems which started with the invention of vehicles was parking. The ease of resolving this problem using technology has evolved over the years but the problem of parking still remains unsolved. The main reason behind this is that parking does not only involve one problem but it consists of a set of problems within itself. One of these problems is the occupancy detection of the parking slots in a distributed parking ecosystem. In a distributed system, users would find preferable parking spaces as opposed to random parking spaces. In this paper, we propose a web-based application as a solution for parking space detection in different parking spaces. The solution is based on Computer Vision (CV) and is built using the Django framework written in Python 3.0. The solution works to resolve the occupancy detection problem along with providing the user the option to determine the block based on availability and his preference. The evaluation results for our proposed system are promising and efficient. The proposed system can also be integrated with different systems and be used for solving other relevant parking problems.

In this paper, we address the problem of finding a correspondence, or matching, between the functions of two programs in binary form, which is one of the most common task in binary diffing. We introduce a new formulation of this problem as a particular instance of a graph edit problem over the call graphs of the programs. In this formulation, the quality of a mapping is evaluated simultaneously with respect to both function content and call graph similarities. We show that this formulation is equivalent to a network alignment problem. We propose a solving strategy for this problem based on max-product belief propagation. Finally, we implement a prototype of our method, called QBinDiff, and propose an extensive evaluation which shows that our approach outperforms state of the art diffing tools.

In this paper, we focus on the decentralized optimization problem over the Stiefel manifold, which is defined on a connected network of $d$ agents. The objective is an average of $d$ local functions, and each function is privately held by an agent and encodes its data. The agents can only communicate with their neighbors in a collaborative effort to solve this problem. In existing methods, multiple rounds of communications are required to guarantee the convergence, giving rise to high communication costs. In contrast, this paper proposes a decentralized algorithm, called DESTINY, which only invokes a single round of communications per iteration. DESTINY combines gradient tracking techniques with a novel approximate augmented Lagrangian function. The global convergence to stationary points is rigorously established. Comprehensive numerical experiments demonstrate that DESTINY has a strong potential to deliver a cutting-edge performance in solving a variety of testing problems.

We study a multivariate version of trend filtering, called Kronecker trend filtering or KTF, for the case in which the design points form a lattice in $d$ dimensions. KTF is a natural extension of univariate trend filtering (Steidl et al., 2006; Kim et al., 2009; Tibshirani, 2014), and is defined by minimizing a penalized least squares problem whose penalty term sums the absolute (higher-order) differences of the parameter to be estimated along each of the coordinate directions. The corresponding penalty operator can be written in terms of Kronecker products of univariate trend filtering penalty operators, hence the name Kronecker trend filtering. Equivalently, one can view KTF in terms of an $\ell_1$-penalized basis regression problem where the basis functions are tensor products of falling factorial functions, a piecewise polynomial (discrete spline) basis that underlies univariate trend filtering. This paper is a unification and extension of the results in Sadhanala et al. (2016, 2017). We develop a complete set of theoretical results that describe the behavior of $k^{\mathrm{th}}$ order Kronecker trend filtering in $d$ dimensions, for every $k \geq 0$ and $d \geq 1$. This reveals a number of interesting phenomena, including the dominance of KTF over linear smoothers in estimating heterogeneously smooth functions, and a phase transition at $d=2(k+1)$, a boundary past which (on the high dimension-to-smoothness side) linear smoothers fail to be consistent entirely. We also leverage recent results on discrete splines from Tibshirani (2020), in particular, discrete spline interpolation results that enable us to extend the KTF estimate to any off-lattice location in constant-time (independent of the size of the lattice $n$).

In dynamic programming (DP) and reinforcement learning (RL), an agent learns to act optimally in terms of expected long-term return by sequentially interacting with its environment modeled by a Markov decision process (MDP). More generally in distributional reinforcement learning (DRL), the focus is on the whole distribution of the return, not just its expectation. Although DRL-based methods produced state-of-the-art performance in RL with function approximation, they involve additional quantities (compared to the non-distributional setting) that are still not well understood. As a first contribution, we introduce a new class of distributional operators, together with a practical DP algorithm for policy evaluation, that come with a robust MDP interpretation. Indeed, our approach reformulates through an augmented state space where each state is split into a worst-case substate and a best-case substate, whose values are maximized by safe and risky policies respectively. Finally, we derive distributional operators and DP algorithms solving a new control task: How to distinguish safe from risky optimal actions in order to break ties in the space of optimal policies?

This paper proposes a Generalized Power Method (GPM) to tackle the problem of community detection and group synchronization simultaneously in a direct non-convex manner. Under the stochastic group block model (SGBM), theoretical analysis indicates that the algorithm is able to exactly recover the ground truth in $O(n\log^2n)$ time, sharply outperforming the benchmark method of semidefinite programming (SDP) in $O(n^{3.5})$ time. Moreover, a lower bound of parameters is given as a necessary condition for exact recovery of GPM. The new bound breaches the information-theoretic threshold for pure community detection under the stochastic block model (SBM), thus demonstrating the superiority of our simultaneous optimization algorithm over the trivial two-stage method which performs the two tasks in succession. We also conduct numerical experiments on GPM and SDP to evidence and complement our theoretical analysis.

Several widely-used first-order saddle point optimization methods yield an identical continuous-time ordinary differential equation (ODE) to that of the Gradient Descent Ascent (GDA) method when derived naively. However, their convergence properties are very different even on simple bilinear games. We use a technique from fluid dynamics called High-Resolution Differential Equations (HRDEs) to design ODEs of several saddle point optimization methods. On bilinear games, the convergence properties of the derived HRDEs correspond to that of the starting discrete methods. Using these techniques, we show that the HRDE of Optimistic Gradient Descent Ascent (OGDA) has last-iterate convergence for general monotone variational inequalities. To our knowledge, this is the first continuous-time dynamics shown to converge for such a general setting. Moreover, we provide the rates for the best-iterate convergence of the OGDA method, relying solely on the first-order smoothness of the monotone operator.