This paper develops methods for proving Lyapunov stability of dynamical systems subject to disturbances with an unknown distribution. We assume only a finite set of disturbance samples is available and that the true online disturbance realization may be drawn from a different distribution than the given samples. We formulate an optimization problem to search for a sum-of-squares (SOS) Lyapunov function and introduce a distributionally robust version of the Lyapunov function derivative constraint. We show that this constraint may be reformulated as several SOS constraints, ensuring that the search for a Lyapunov function remains in the class of SOS polynomial optimization problems. For general systems, we provide a distributionally robust chance-constrained formulation for neural network Lyapunov function search. Simulations demonstrate the validity and efficiency of either formulation on non-linear uncertain dynamical systems.

We give query complexity lower bounds for convex optimization and the related feasibility problem. We show that quadratic memory is necessary to achieve the optimal oracle complexity for first-order convex optimization. In particular, this shows that center-of-mass cutting-planes algorithms in dimension $d$ which use $\tilde O(d^2)$ memory and $\tilde O(d)$ queries are Pareto-optimal for both convex optimization and the feasibility problem, up to logarithmic factors. Precisely, we prove that to minimize $1$-Lipschitz convex functions over the unit ball to $1/d^4$ accuracy, any deterministic first-order algorithms using at most $d^{2-\delta}$ bits of memory must make $\tilde\Omega(d^{1+\delta/3})$ queries, for any $\delta\in[0,1]$. For the feasibility problem, in which an algorithm only has access to a separation oracle, we show a stronger trade-off: for at most $d^{2-\delta}$ memory, the number of queries required is $\tilde\Omega(d^{1+\delta})$. This resolves a COLT 2019 open problem of Woodworth and Srebro.

Collision detection between objects is critical for simulation, control, and learning for robotic systems. However, existing collision detection routines are inherently non-differentiable, limiting their applications in gradient-based optimization tools. In this work, we propose DCOL: a fast and fully differentiable collision-detection framework that reasons about collisions between a set of composable and highly expressive convex primitive shapes. This is achieved by formulating the collision detection problem as a convex optimization problem that solves for the minimum uniform scaling applied to each primitive before they intersect. The optimization problem is fully differentiable with respect to the configurations of each primitive and is able to return a collision detection metric and contact points on each object, agnostic of interpenetration. We demonstrate the capabilities of DCOL on a range of robotics problems from trajectory optimization and contact physics, and have made an open-source implementation available.

Causal effect estimation is important for many tasks in the natural and social sciences. We design algorithms for the continuous partial identification problem: bounding the effects of multivariate, continuous treatments when unmeasured confounding makes identification impossible. Specifically, we cast causal effects as objective functions within a constrained optimization problem, and minimize/maximize these functions to obtain bounds. We combine flexible learning algorithms with Monte Carlo methods to implement a family of solutions under the name of stochastic causal programming. In particular, we show how the generic framework can be efficiently formulated in settings where auxiliary variables are clustered into pre-treatment and post-treatment sets, where no fine-grained causal graph can be easily specified. In these settings, we can avoid the need for fully specifying the distribution family of hidden common causes. Monte Carlo computation is also much simplified, leading to algorithms which are more computationally stable against alternatives.

We prove Wasserstein inverse reinforcement learning enables the learner's reward values to imitate the expert's reward values in a finite iteration for multi-objective optimizations. Moreover, we prove Wasserstein inverse reinforcement learning enables the learner's optimal solutions to imitate the expert's optimal solutions for multi-objective optimizations with lexicographic order.

Ensuring solution feasibility is a key challenge in developing Deep Neural Network (DNN) schemes for solving constrained optimization problems, due to inherent DNN prediction errors. In this paper, we propose a ``preventive learning'' framework to guarantee DNN solution feasibility for problems with convex constraints and general objective functions without post-processing, upon satisfying a mild condition on constraint calibration. Without loss of generality, we focus on problems with only inequality constraints. We systematically calibrate inequality constraints used in DNN training, thereby anticipating prediction errors and ensuring the resulting solutions remain feasible. We characterize the calibration magnitudes and the DNN size sufficient for ensuring universal feasibility. We propose a new Adversarial-Sample Aware training algorithm to improve DNN's optimality performance without sacrificing feasibility guarantee. Overall, the framework provides two DNNs. The first one from characterizing the sufficient DNN size can guarantee universal feasibility while the other from the proposed training algorithm further improves optimality and maintains DNN's universal feasibility simultaneously. We apply the framework to develop DeepOPF+ for solving essential DC optimal power flow problems in grid operation. Simulation results over IEEE test cases show that it outperforms existing strong DNN baselines in ensuring 100% feasibility and attaining consistent optimality loss ($<$0.19%) and speedup (up to $\times$228) in both light-load and heavy-load regimes, as compared to a state-of-the-art solver. We also apply our framework to a non-convex problem and show its performance advantage over existing schemes.

Inertial-based navigation refers to the navigation methods or systems that have inertial information or sensors as the core part and integrate a spectrum of other kinds of sensors for enhanced performance. Through a series of papers, the authors attempt to explore information blending of inertial-based navigation by a polynomial optimization method. The basic idea is to model rigid motions as finite-order polynomials and then attacks the involved navigation problems by optimally solving their coefficients, taking into considerations the constraints posed by inertial sensors and others. In the current paper, a continuous-time attitude estimation approach is proposed, which transforms the attitude estimation into a constant parameter determination problem by the polynomial optimization. Specifically, the continuous attitude is first approximated by a Chebyshev polynomial, of which the unknown Chebyshev coefficients are determined by minimizing the weighted residuals of initial conditions, dynamics and measurements. We apply the derived estimator to the attitude estimation with the magnetic and inertial sensors. Simulation and field tests show that the estimator has much better stability and faster convergence than the traditional extended Kalman filter does, especially in the challenging large initial state error scenarios.

Deep learning models form one of the most powerful machine learning models for the extraction of important features. Most of the designs of deep neural models, i.e., the initialization of parameters, are still manually tuned. Hence, obtaining a model with high performance is exceedingly time-consuming and occasionally impossible. Optimizing the parameters of the deep networks, therefore, requires improved optimization algorithms with high convergence rates. The single objective-based optimization methods generally used are mostly time-consuming and do not guarantee optimum performance in all cases. Mathematical optimization problems containing multiple objective functions that must be optimized simultaneously fall under the category of multi-objective optimization sometimes referred to as Pareto optimization. Multi-objective optimization problems form one of the alternatives yet useful options for parameter optimization. However, this domain is a bit less explored. In this survey, we focus on exploring the effectiveness of multi-objective optimization strategies for parameter optimization in conjunction with deep neural networks. The case studies used in this study focus on how the two methods are combined to provide valuable insights into the generation of predictions and analysis in multiple applications.

This paper reconsiders end-to-end learning approaches to the Optimal Power Flow (OPF). Existing methods, which learn the input/output mapping of the OPF, suffer from scalability issues due to the high dimensionality of the output space. This paper first shows that the space of optimal solutions can be significantly compressed using principal component analysis (PCA). It then proposes Compact Learning, a new method that learns in a subspace of the principal components before translating the vectors into the original output space. This compression reduces the number of trainable parameters substantially, improving scalability and effectiveness. Compact Learning is evaluated on a variety of test cases from the PGLib with up to 30,000 buses. The paper also shows that the output of Compact Learning can be used to warm-start an exact AC solver to restore feasibility, while bringing significant speed-ups.

We show the convergence of Wasserstein inverse reinforcement learning for multi-objective optimizations with the projective subgradient method by formulating an inverse problem of the multi-objective optimization problem. In addition, we prove convergence of inverse reinforcement learning (maximum entropy inverse reinforcement learning, guided cost learning) with gradient descent and the projective subgradient method.