We propose a continuous optimization framework for discovering a latent directed acyclic graph (DAG) from observational data. Our approach optimizes over the polytope of permutation vectors, the so-called Permutahedron, to learn a topological ordering. Edges can be optimized jointly, or learned conditional on the ordering via a non-differentiable subroutine. Compared to existing continuous optimization approaches our formulation has a number of advantages including: 1. validity: optimizes over exact DAGs as opposed to other relaxations optimizing approximate DAGs; 2. modularity: accommodates any edge-optimization procedure, edge structural parameterization, and optimization loss; 3. end-to-end: either alternately iterates between node-ordering and edge-optimization, or optimizes them jointly. We demonstrate, on real-world data problems in protein-signaling and transcriptional network discovery, that our approach lies on the Pareto frontier of two key metrics, the SID and SHD. In many domains, including cell biology (Sachs et al., 2005), finance (Sanford & Moosa, 2012), and genetics (Zhang et al., 2013), the data generating process is thought to be represented by an underlying directed acylic graph (DAG). Many models rely on DAG assumptions, e.g., causal modeling uses DAGs to model distribution shifts, ensure predictor fairness among subpopulations, or learn agents more sample-efficiently (Kaddour et al., 2022). A key question, with implications ranging from better modeling to causal discovery, is how to recover this unknown DAG from observed data alone. Learning DAGs from observational data alone is fundamentally difficult for two reasons. This riddles the search space with local minima; (ii) Computation: DAG discovery is a costly combinatorial optimization problem over an exponentially large solution space and subject to global acyclicity constraints. To address issue (ii), recent work has proposed continuous relaxations of the DAG learning problem.

Entropic regularization is a method for large-scale linear programming. Geometrically, one traces intersections of the feasible polytope with scaled toric varieties, starting at the Birch point. We compare this to log-barrier methods, with reciprocal linear spaces, starting at the analytic center. We revisit entropic regularization for unbalanced optimal transport, and we develop the use of optimal conic couplings. We compute the degree of the associated toric variety, and we explore algorithms like iterative scaling.

The vanishing ideal of a set of points $X = \{\mathbf{x}_1, \ldots, \mathbf{x}_m\}\subseteq \mathbb{R}^n$ is the set of polynomials that evaluate to $0$ over all points $\mathbf{x} \in X$ and admits an efficient representation by a finite subset of generators. In practice, to accommodate noise in the data, algorithms that construct generators of the approximate vanishing ideal are widely studied but their computational complexities remain expensive. In this paper, we scale up the oracle approximate vanishing ideal algorithm (OAVI), the only generator-constructing algorithm with known learning guarantees. We prove that the computational complexity of OAVI is not superlinear, as previously claimed, but linear in the number of samples $m$. In addition, we propose two modifications that accelerate OAVI's training time: Our analysis reveals that replacing the pairwise conditional gradients algorithm, one of the solvers used in OAVI, with the faster blended pairwise conditional gradients algorithm leads to an exponential speed-up in the number of features $n$. Finally, using a new inverse Hessian boosting approach, intermediate convex optimization problems can be solved almost instantly, improving OAVI's training time by multiple orders of magnitude in a variety of numerical experiments.

A perception-aware Nonlinear Model Predictive Control (NMPC) strategy aimed at performing vision-based target tracking and collision avoidance with a multi-rotor aerial vehicle is presented in this paper. The proposed control strategy considers both realistic actuation limits at the torque level and visual perception constraints to enforce the visibility coverage of a target while complying with the mission objectives. Furthermore, the approach allows to safely navigate in a workspace area populated by dynamic obstacles with a ballistic motion. The formulation is meant to be generic and set upon a large class of multi-rotor vehicles that covers both coplanar designs like quadrotors as well as fully-actuated platforms with tilted propellers. The feasibility and effectiveness of the control strategy are demonstrated via closed-loop simulations achieved in MATLAB.

A Nonlinear Model Predictive Control (NMPC) strategy aimed at controlling a small-scale car model for autonomous racing competitions is presented in this paper. The proposed control strategy is concerned with minimizing the lap time while keeping the vehicle within track boundaries. The optimization problem considers both the vehicle's actuation limits and the lateral and longitudinal forces acting on the car modeled through the Pacejka's magic formula and a simple drivetrain model. Furthermore, the approach allows to safely race on a track populated by static obstacles generating collision-free trajectories and tracking them while enhancing the lap timing performance. Gazebo simulations using the F1/10 simulator showcase the feasibility and validity of the proposed control strategy. The code is released as open-source making it possible to replicate the obtained results.

We develop new tools in the theory of nonlinear random matrices and apply them to study the performance of the Sum of Squares (SoS) hierarchy on average-case problems. The SoS hierarchy is a powerful optimization technique that has achieved tremendous success for various problems in combinatorial optimization, robust statistics and machine learning. It's a family of convex relaxations that lets us smoothly trade off running time for approximation guarantees. In recent works, it's been shown to be extremely useful for recovering structure in high dimensional noisy data. It also remains our best approach towards refuting the notorious Unique Games Conjecture. In this work, we analyze the performance of the SoS hierarchy on fundamental problems stemming from statistics, theoretical computer science and statistical physics. In particular, we show subexponential-time SoS lower bounds for the problems of the Sherrington-Kirkpatrick Hamiltonian, Planted Slightly Denser Subgraph, Tensor Principal Components Analysis and Sparse Principal Components Analysis. These SoS lower bounds involve analyzing large random matrices, wherein lie our main contributions. These results offer strong evidence for the truth of and insight into the low-degree likelihood ratio hypothesis, an important conjecture that predicts the power of bounded-time algorithms for hypothesis testing. We also develop general-purpose tools for analyzing the behavior of random matrices which are functions of independent random variables. Towards this, we build on and generalize the matrix variant of the Efron-Stein inequalities. In particular, our general theorem on matrix concentration recovers various results that have appeared in the literature. We expect these random matrix theory ideas to have other significant applications.

The Right to Explanation and the Right to be Forgotten are two important principles outlined to regulate algorithmic decision making and data usage in real-world applications. While the right to explanation allows individuals to request an actionable explanation for an algorithmic decision, the right to be forgotten grants them the right to ask for their data to be deleted from all the databases and models of an organization. Intuitively, enforcing the right to be forgotten may trigger model updates which in turn invalidate previously provided explanations, thus violating the right to explanation. In this work, we investigate the technical implications arising due to the interference between the two aforementioned regulatory principles, and propose the first algorithmic framework to resolve the tension between them. To this end, we formulate a novel optimization problem to generate explanations that are robust to model updates due to the removal of training data instances by data deletion requests. We then derive an efficient approximation algorithm to handle the combinatorial complexity of this optimization problem. We theoretically demonstrate that our method generates explanations that are provably robust to worst-case data deletion requests with bounded costs in case of linear models and certain classes of non-linear models. Extensive experimentation with real-world datasets demonstrates the efficacy of the proposed framework.

A framework for computing feasible and constrained trajectories for a fleet of quad-rotors leveraging on Signal Temporal Logic (STL) specifications for power line inspection tasks is proposed in this paper. The planner allows the formulation of complex missions that avoid obstacles and maintain a safe distance between drones while performing the planned mission. An optimization problem is set to generate optimal strategies that satisfy these specifications and also take vehicle constraints into account. Further, an event-triggered replanner is proposed to reply to unforeseen events and external disturbances. An energy minimization term is also considered to implicitly save quad-rotors battery life while carrying out the mission. Numerical simulations in MATLAB and experimental results show the validity and the effectiveness of the proposed approach, and demonstrate its applicability in real-world scenarios.

Classical epidemiological models assume homogeneous populations. There have been important extensions to model heterogeneous populations, when the identity of the sub-populations is known, such as age group or geographical location. Here, we propose two new methods to model the number of people infected with COVID-19 over time, each as a linear combination of latent sub-populations -- i.e., when we do not know which person is in which sub-population, and the only available observations are the aggregates across all sub-populations. Method #1 is a dictionary-based approach, which begins with a large number of pre-defined sub-population models (each with its own starting time, shape, etc), then determines the (positive) weight of small (learned) number of sub-populations. Method #2 is a mixture-of-$M$ fittable curves, where $M$, the number of sub-populations to use, is given by the user. Both methods are compatible with any parametric model; here we demonstrate their use with first (a)~Gaussian curves and then (b)~SIR trajectories. We empirically show the performance of the proposed methods, first in (i) modeling the observed data and then in (ii) forecasting the number of infected people 1 to 4 weeks in advance. Across 187 countries, we show that the dictionary approach had the lowest mean absolute percentage error and also the lowest variance when compared with classical SIR models and moreover, it was a strong baseline that outperforms many of the models developed for COVID-19 forecasting.

Machine learning systems have been extensively used as auxiliary tools in domains that require critical decision-making, such as healthcare and criminal justice. The explainability of decisions is crucial for users to develop trust on these systems. In recent years, the globally-consistent rule-based summary-explanation and its max-support (MS) problem have been proposed, which can provide explanations for particular decisions along with useful statistics of the dataset. However, globally-consistent summary-explanations with limited complexity typically have small supports, if there are any. In this paper, we propose a relaxed version of summary-explanation, i.e., the $q$-consistent summary-explanation, which aims to achieve greater support at the cost of slightly lower consistency. The challenge is that the max-support problem of $q$-consistent summary-explanation (MSqC) is much more complex than the original MS problem, resulting in over-extended solution time using standard branch-and-bound solvers. To improve the solution time efficiency, this paper proposes the weighted column sampling~(WCS) method based on solving smaller problems by sampling variables according to their simplified increase support (SIS) values. Experiments verify that solving MSqC with the proposed SIS-based WCS method is not only more scalable in efficiency, but also yields solutions with greater support and better global extrapolation effectiveness.