We analyze a new spectral graph matching algorithm, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), for recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. Extending the exact recovery guarantees established in the companion paper for Gaussian weights, in this work, we prove the universality of these guarantees for a general correlated Wigner model. In particular, for two Erd\H{o}s-R\'enyi graphs with edge correlation coefficient $1-\sigma^2$ and average degree at least $\operatorname{polylog}(n)$, we show that GRAMPA exactly recovers the latent vertex correspondence with high probability when $\sigma \lesssim 1/\operatorname{polylog}(n)$. Moreover, we establish a similar guarantee for a variant of GRAMPA, corresponding to a tighter quadratic programming relaxation of the quadratic assignment problem. Our analysis exploits a resolvent representation of the GRAMPA similarity matrix and local laws for the resolvents of sparse Wigner matrices.

Graph matching aims at finding the vertex correspondence between two unlabeled graphs that maximizes the total edge weight correlation. This amounts to solving a computationally intractable quadratic assignment problem. In this paper we propose a new spectral method, GRAph Matching by Pairwise eigen-Alignments (GRAMPA). Departing from prior spectral approaches that only compare top eigenvectors, or eigenvectors of the same order, GRAMPA first constructs a similarity matrix as a weighted sum of outer products between all pairs of eigenvectors of the two graphs, with weights given by a Cauchy kernel applied to the separation of the corresponding eigenvalues, then outputs a matching by a simple rounding procedure. The similarity matrix can also be interpreted as the solution to a regularized quadratic programming relaxation of the quadratic assignment problem. For the Gaussian Wigner model in which two complete graphs on $n$ vertices have Gaussian edge weights with correlation coefficient $1-\sigma^2$, we show that GRAMPA exactly recovers the correct vertex correspondence with high probability when $\sigma = O(\frac{1}{\log n})$. This matches the state of the art of polynomial-time algorithms, and significantly improves over existing spectral methods which require $\sigma$ to be polynomially small in $n$. The superiority of GRAMPA is also demonstrated on a variety of synthetic and real datasets, in terms of both statistical accuracy and computational efficiency. Universality results, including similar guarantees for dense and sparse Erd\H{o}s-R\'{e}nyi graphs, are deferred to the companion paper.

Fundamental mathematical constants like $e$ and $\pi$ are ubiquitous in diverse fields of science, from abstract mathematics and geometry to physics, biology and chemistry. Nevertheless, for centuries new mathematical formulas relating fundamental constants have been scarce and usually discovered sporadically. In this paper we propose a novel and systematic approach that leverages algorithms for deriving mathematical formulas for fundamental constants and help reveal their underlying structure. Our algorithms find dozens of well-known as well as previously unknown continued fraction representations of $\pi$, $e$, and the Riemann zeta function values. Two conjectures produced by our algorithm, along with many others, are: \begin{equation*} \frac{e}{e-2} = 4 - \frac{1}{5-\frac{2}{6-\frac{3}{7-\frac{4}{8-\ldots}}}} \quad\quad,\quad\quad \frac{4}{3\pi-8} = 3-\frac{1\cdot1}{6-\frac{2\cdot3}{9-\frac{3\cdot5}{12-\frac{4\cdot 7}{15-\ldots}}}} \end{equation*} We present two algorithms that proved useful in finding conjectures: a variant of the Meet-In-The-Middle (MITM) algorithm and a Gradient Descent (GD) tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values and thus they conjecture formulas without providing proofs and without requiring any prior knowledge on any underlaying mathematical structure. This approach is especially attractive for fundamental constants for which no mathematical structure is known, as it reverses the conventional approach of sequential logic in formal proofs. Instead, our work presents a new conceptual approach for research: computer algorithms utilizing numerical data to unveil mathematical structures, thus trying to play the role of intuition of great mathematicians of the past, providing leads to new mathematical research.

A main driver behind the digitization of industry and society is the belief that data-driven model building and decision making can contribute to higher degrees of automation and more informed decisions. Building such models from data often involves the application of some form of machine learning. Thus, there is an ever growing demand in work force with the necessary skill set to do so. This demand has given rise to a new research topic concerned with fitting machine learning models fully automatically - AutoML. This paper gives an overview of the state of the art in AutoML with a focus on practical applicability in a business context, and provides recent benchmark results on the most important AutoML algorithms.

We develop tools for utilizing correspondence experiments to detect illegal discrimination by individual employers. Employers violate US employment law if their propensity to contact applicants depends on protected characteristics such as race or sex. We establish identification of higher moments of the causal effects of protected characteristics on callback rates as a function of the number of fictitious applications sent to each job ad. These moments are used to bound the fraction of jobs that illegally discriminate. Applying our results to three experimental datasets, we find evidence of significant employer heterogeneity in discriminatory behavior, with the standard deviation of gaps in job-specific callback probabilities across protected groups averaging roughly twice the mean gap. In a recent experiment manipulating racially distinctive names, we estimate that at least 85% of jobs that contact both of two white applications and neither of two black applications are engaged in illegal discrimination. To assess the tradeoff between type I and II errors presented by these patterns, we consider the performance of a series of decision rules for investigating suspicious callback behavior under a simple two-type model that rationalizes the experimental data. Though, in our preferred specification, only 17% of employers are estimated to discriminate on the basis of race, we find that an experiment sending 10 applications to each job would enable accurate detection of 7-10% of discriminators while falsely accusing fewer than 0.2% of non-discriminators. A minimax decision rule acknowledging partial identification of the joint distribution of callback rates yields higher error rates but more investigations than our baseline two-type model. Our results suggest illegal labor market discrimination can be reliably monitored with relatively small modifications to existing audit designs.

An optimal feedback controller for a given Markov decision process (MDP) can in principle be synthesized by value or policy iteration. However, if the system dynamics and the reward function are unknown, a learning agent must discover an optimal controller via direct interaction with the environment. Such interactive data gathering commonly leads to divergence towards dangerous or uninformative regions of the state space unless additional regularization measures are taken. Prior works proposed bounding the information loss measured by the Kullback-Leibler (KL) divergence at every policy improvement step to eliminate instability in the learning dynamics. In this paper, we consider a broader family of $f$-divergences, and more concretely $\alpha$-divergences, which inherit the beneficial property of providing the policy improvement step in closed form at the same time yielding a corresponding dual objective for policy evaluation. Such entropic proximal policy optimization view gives a unified perspective on compatible actor-critic architectures. In particular, common least-squares value function estimation coupled with advantage-weighted maximum likelihood policy improvement is shown to correspond to the Pearson $\chi^2$-divergence penalty. Other actor-critic pairs arise for various choices of the penalty-generating function $f$. On a concrete instantiation of our framework with the $\alpha$-divergence, we carry out asymptotic analysis of the solutions for different values of $\alpha$ and demonstrate the effects of the divergence function choice on common standard reinforcement learning problems.

Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures accuracy between predicted values and ground truth values. Decision-focused learning explicitly integrates the downstream decision problem when training the predictive model, in order to optimize the quality of decisions induced by the predictions. It has been successfully applied to several limited combinatorial problem classes, such as those that can be expressed as linear programs (LP), and submodular optimization. However, these previous applications have uniformly focused on problems from specific classes with simple constraints. Here, we enable decision-focused learning for the broad class of problems that can be encoded as a Mixed Integer Linear Program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables. We show how to differentiate through a MIP by employing a cutting planes solution approach, which is an exact algorithm that iteratively adds constraints to a continuous relaxation of the problem until an integral solution is found. We evaluate our new end-to-end approach on several real world domains and show that it outperforms the standard two phase approaches that treat prediction and prescription separately, as well as a baseline approach of simply applying decision-focused learning to the LP relaxation of the MIP.

Inspired from recent insights into the common ground of machine learning, optimization and decision-making, this paper proposes an easy-to-implement, but effective procedure to enhance both the quality of renewable energy forecasts and the competitive edge of renewable energy producers in electricity markets with a dual-price settlement of imbalances. The quality and economic gains brought by the proposed procedure essentially stem from the utilization of valuable predictors (also known as features) in a data-driven newsvendor model that renders a computationally inexpensive linear program. We illustrate the proposed procedure and numerically assess its benefits on a realistic case study that considers the aggregate wind power production in the Danish DK1 bidding zone as the variable to be predicted and traded. Within this context, our procedure leverages, among others, spatial information in the form of wind power forecasts issued by transmission system operators (TSO) in surrounding bidding zones and publicly available in online platforms. We show that our method is able to improve the quality of the wind power forecast issued by the Danish TSO by several percentage points (when measured in terms of the mean absolute or the root mean square error) and to significantly reduce the balancing costs incurred by the wind power producer.

We study the problem of repeated play in a zero-sum game in which the payoff matrix may change, in a possibly adversarial fashion, on each round; we call these Online Matrix Games. Finding the Nash Equilibrium (NE) of a two player zero-sum game is core to many problems in statistics, optimization, and economics, and for a fixed game matrix this can be easily reduced to solving a linear program. But when the payoff matrix evolves over time our goal is to find a sequential algorithm that can compete with, in a certain sense, the NE of the long-term-averaged payoff matrix. We design an algorithm with small NE regret--that is, we ensure that the long-term payoff of both players is close to minimax optimum in hindsight. Our algorithm achieves near-optimal dependence with respect to the number of rounds and depends poly-logarithmically on the number of available actions of the players. Additionally, we show that the naive reduction, where each player simply minimizes its own regret, fails to achieve the stated objective regardless of which algorithm is used. We also consider the so-called bandit setting, where the feedback is significantly limited, and we provide an algorithm with small NE regret using one-point estimates of each payoff matrix.

Optimization of conflicting functions is of paramount importance in decision making, and real world applications frequently involve data that is uncertain or unknown, resulting in multi-objective optimization (MOO) problems of stochastic type. We study the stochastic multi-gradient (SMG) method, seen as an extension of the classical stochastic gradient method for single-objective optimization. At each iteration of the SMG method, a stochastic multi-gradient direction is calculated by solving a quadratic subproblem, and it is shown that this direction is biased even when all individual gradient estimators are unbiased. We establish rates to compute a point in the Pareto front, of order similar to what is known for stochastic gradient in both convex and strongly convex cases. The analysis handles the bias in the multi-gradient and the unknown a priori weights of the limiting Pareto point. The SMG method is framed into a Pareto-front type algorithm for the computation of the entire Pareto front. The Pareto-front SMG algorithm is capable of robustly determining Pareto fronts for a number of synthetic test problems. One can apply it to any stochastic MOO problem arising from supervised machine learning, and we report results for logistic binary classification where multiple objectives correspond to distinct-sources data groups.