Edward Rothberg is CEO and Co-Founder of Gurobi Optimization, which produces the world's fastest mathematical optimization solver. This is a question that -- as the CEO of a mathematical optimization software company -- I get asked all the time. Although it seems like a simple question, it's actually quite difficult to come up with a concise, coherent answer. Indeed, mathematical optimization and machine learning are two tools that at first glance -- like scissors and pliers -- may seem to have a lot in common. When you look closely at their fundamental features and actual applications, however, you'll see some important differences.

A significant body of research in the data sciences considers unfair discrimination against social categories such as race or gender that could occur or be amplified as a result of algorithmic decisions. Simultaneously, real-world disparities continue to exist, even before algorithmic decisions are made. In this work, we draw on insights from the social sciences and humanistic studies brought into the realm of causal modeling and constrained optimization, and develop a novel algorithmic framework for tackling pre-existing real-world disparities. The purpose of our framework, which we call the "impact remediation framework," is to measure real-world disparities and discover the optimal intervention policies that could help improve equity or access to opportunity for those who are underserved with respect to an outcome of interest. We develop a disaggregated approach to tackling pre-existing disparities that relaxes the typical set of assumptions required for the use of social categories in structural causal models. Our approach flexibly incorporates counterfactuals and is compatible with various ontological assumptions about the nature of social categories. We demonstrate impact remediation with a real-world case study and compare our disaggregated approach to an existing state-of-the-art approach, comparing its structure and resulting policy recommendations. In contrast to most work on optimal policy learning, we explore disparity reduction itself as an objective, explicitly focusing the power of algorithms on reducing inequality.

In this thesis, we explore and apply methods inspired by the free energy principle to two important areas in machine learning and neuroscience. The free energy principle is a general mathematical theory of the necessary information-theoretic behaviours of systems which maintain a separation from their environment. A core postulate of the theory is that complex systems can be seen as performing variational Bayesian inference and minimizing an information-theoretic quantity called the variational free energy. The free energy principle originated in, and has been extremely influential in theoretical neuroscience, having spawned a number of neurophysiologically realistic process theories, and maintaining close links with Bayesian Brain viewpoints. The thesis is split into three main parts where we apply methods and insights from the free energy principle to understand questions first in perception, then action, and finally learning. Specifically, in the first section, we focus on the theory of predictive coding, a neurobiologically plausible process theory derived from the free energy principle under certain assumptions, which argues that the primary function of the brain is to minimize prediction errors. We focus on scaling up predictive coding architectures and simulate large-scale predictive coding networks for perception on machine learning benchmarks; we investigate predictive coding's relationship to other classical filtering algorithms, and we demonstrate that many biologically implausible aspects of current models of predictive coding can be relaxed without unduly harming the performance of predictive coding models which allows for a potentially more literal translation of predictive coding theory into cortical microcircuits. In the second part of the thesis, we focus on the application of methods deriving from the free energy principle to action. We study the extension of methods of'active inference', a neurobiologically grounded account of action through variational message passing, to utilize deep artificial neural networks, allowing these methods to'scale up' to be competitive with state of the art deep reinforcement learning methods.

Probabilistic model checking aims to prove whether a Markov decision process (MDP) satisfies a temporal logic specification. The underlying methods rely on an often unrealistic assumption that the MDP is precisely known. Consequently, parametric MDPs (pMDPs) extend MDPs with transition probabilities that are functions over unspecified parameters. The parameter synthesis problem is to compute an instantiation of these unspecified parameters such that the resulting MDP satisfies the temporal logic specification. We formulate the parameter synthesis problem as a quadratically constrained quadratic program (QCQP), which is nonconvex and is NP-hard to solve in general. We develop two approaches that iteratively obtain locally optimal solutions. The first approach exploits the so-called convex-concave procedure (CCP), and the second approach utilizes a sequential convex programming (SCP) method. The techniques improve the runtime and scalability by multiple orders of magnitude compared to black-box CCP and SCP by merging ideas from convex optimization and probabilistic model checking. We demonstrate the approaches on a satellite collision avoidance problem with hundreds of thousands of states and tens of thousands of parameters and their scalability on a wide range of commonly used benchmarks.

The search for binary sequences with low paper, we present a computational approach that uses peak sidelobe level value represents a formidable computational a stochastic algorithm. To locate better sequences for this our approach cannot provide optimal solutions but in a problem, we designed a stochastic algorithm that uses reasonable time we can locate optimal or near-optimal two fitness functions. Therefore, our approach is also suitable for of the autocorrelation function has a different impact on solving larger instances of the problem. It is defined with the value of the of length L in our problem is defined as follows: exponent over the autocorrelation function values. The main goal of a binary sequences problem with low peak sidelobe level is to find an optimal sequence that has the minimal PSL value, as shown in Eq. (4). 1 Introduction The binary sequences with low peak sidelobe level value S In this sequences with length L. From Eq. (1) it is evident that the number of sequences with length L is 2 The exhaustive search was also applied under the restriction The remainder of the paper is organized as follows. of m-sequence [7].

We study the stochastic bilinear minimax optimization problem, presenting an analysis of the Stochastic ExtraGradient (SEG) method with constant step size, and presenting variations of the method that yield favorable convergence. We first note that the last iterate of the basic SEG method only contracts to a fixed neighborhood of the Nash equilibrium, independent of the step size. This contrasts sharply with the standard setting of minimization where standard stochastic algorithms converge to a neighborhood that vanishes in proportion to the square-root (constant) step size. Under the same setting, however, we prove that when augmented with iteration averaging, SEG provably converges to the Nash equilibrium, and such a rate is provably accelerated by incorporating a scheduled restarting procedure. In the interpolation setting, we achieve an optimal convergence rate up to tight constants. We present numerical experiments that validate our theoretical findings and demonstrate the effectiveness of the SEG method when equipped with iteration averaging and restarting.

Optimization in the latent space of variational autoencoders is a promising approach to generate high-dimensional discrete objects that maximize an expensive black-box property (e.g., drug-likeness in molecular generation, function approximation with arithmetic expressions). However, existing methods lack robustness as they may decide to explore areas of the latent space for which no data was available during training and where the decoder can be unreliable, leading to the generation of unrealistic or invalid objects. We propose to leverage the epistemic uncertainty of the decoder to guide the optimization process. This is not trivial though, as a naive estimation of uncertainty in the high-dimensional and structured settings we consider would result in high estimator variance. To solve this problem, we introduce an importance sampling-based estimator that provides more robust estimates of epistemic uncertainty. Our uncertainty-guided optimization approach does not require modifications of the model architecture nor the training process. It produces samples with a better trade-off between black-box objective and validity of the generated samples, sometimes improving both simultaneously. We illustrate these advantages across several experimental settings in digit generation, arithmetic expression approximation and molecule generation for drug design.

We provide an explicit formula for the Levi-Civita connection and Riemannian Hessian for a Riemannian manifold that is a quotient of a manifold embedded in an inner product space with a non-constant metric function. Together with a classical formula for projection, this allows us to evaluate Riemannian gradient and Hessian for several families of metrics on classical manifolds, including a family of metrics on Stiefel manifolds connecting both the constant and canonical ambient metrics with closed-form geodesics. Using these formulas, we derive Riemannian optimization frameworks on quotients of Stiefel manifolds, including flag manifolds, and a new family of complete quotient metrics on the manifold of positive-semidefinite matrices of fixed rank, considered as a quotient of a product of Stiefel and positive-definite matrix manifold with affine-invariant metrics. The method is procedural, and in many instances, the Riemannian gradient and Hessian formulas could be derived by symbolic calculus. The method extends the list of potential metrics that could be used in manifold optimization and machine learning.

Compressive phase retrieval is a popular variant of the standard compressive sensing problem, in which the measurements only contain magnitude information. In this paper, motivated by recent advances in deep generative models, we provide recovery guarantees with order-optimal sample complexity bounds for phase retrieval with generative priors. We first show that when using i.i.d. Gaussian measurements and an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs, roughly $O(k \log L)$ samples suffice to guarantee that the signal is close to any vector that minimizes an amplitude-based empirical loss function. Attaining this sample complexity with a practical algorithm remains a difficult challenge, and a popular spectral initialization method has been observed to pose a major bottleneck. To partially address this, we further show that roughly $O(k \log L)$ samples ensure sufficient closeness between the signal and any {\em globally optimal} solution to an optimization problem designed for spectral initialization (though finding such a solution may still be challenging). We adapt this result to sparse phase retrieval, and show that $O(s \log n)$ samples are sufficient for a similar guarantee when the underlying signal is $s$-sparse and $n$-dimensional, matching an information-theoretic lower bound. While our guarantees do not directly correspond to a practical algorithm, we propose a practical spectral initialization method motivated by our findings, and experimentally observe significant performance gains over various existing spectral initialization methods of sparse phase retrieval.

Gradual pattern mining allows for extraction of attribute correlations through gradual rules such as: "the more X, the more Y". Such correlations are useful in identifying and isolating relationships among the attributes that may not be obvious through quick scans on a data set. For instance, a researcher may apply gradual pattern mining to determine which attributes of a data set exhibit unfamiliar correlations in order to isolate them for deeper exploration or analysis. In this work, we propose an ant colony optimization technique which uses a popular probabilistic approach that mimics the behavior biological ants as they search for the shortest path to find food in order to solve combinatorial problems. In our second contribution, we extend an existing gradual pattern mining technique to allow for extraction of gradual patterns together with an approximated temporal lag between the affected gradual item sets. Such a pattern is referred to as a fuzzy-temporal gradual pattern and it may take the form: "the more X, the more Y, almost 3 months later". In our third contribution, we propose a data crossing model that allows for integration of mostly gradual pattern mining algorithm implementations into a Cloud platform. This contribution is motivated by the proliferation of IoT applications in almost every area of our society and this comes with provision of large-scale time-series data from different sources.