We present Theseus, an efficient application-agnostic open source library for differentiable nonlinear least squares (DNLS) optimization built on PyTorch, providing a common framework for end-to-end structured learning in robotics and vision. Existing DNLS implementations are application specific and do not always incorporate many ingredients important for efficiency. Theseus is application-agnostic, as we illustrate with several example applications that are built using the same underlying differentiable components, such as second-order optimizers, standard costs functions, and Lie groups. For efficiency, Theseus incorporates support for sparse solvers, automatic vectorization, batching, GPU acceleration, and gradient computation with implicit differentiation and direct loss minimization. We do extensive performance evaluation in a set of applications, demonstrating significant efficiency gains and better scalability when these features are incorporated.

Many types of data from fields including natural language processing, computer vision, and bioinformatics, are well represented by discrete, compositional structures such as trees, sequences, or matchings. Latent structure models are a powerful tool for learning to extract such representations, offering a way to incorporate structural bias, discover insight about the data, and interpret decisions. However, effective training is challenging, as neural networks are typically designed for continuous computation. This text explores three broad strategies for learning with discrete latent structure: continuous relaxation, surrogate gradients, and probabilistic estimation. Our presentation relies on consistent notations for a wide range of models. As such, we reveal many new connections between latent structure learning strategies, showing how most consist of the same small set of fundamental building blocks, but use them differently, leading to substantially different applicability and properties.

Uncertainty is a pervasive challenge in decision and risk management and it is usually studied by quantification and modeling. Interestingly, engineers and other decision makers usually manage uncertainty with strategies such as incorporating robustness, or by employing decision heuristics. The focus of this paper is then to develop a systematic understanding of such strategies, determine their range of application, and develop a framework to better employ them. Based on a review of a dataset of 100 decision problems, this paper found that many decision problems have pivotal properties, i.e. properties that enable solution strategies, and finds 14 such properties. Therefore, an analyst can first find these properties in a given problem, and then utilize the strategies they enable. Multi-objective optimization methods could be used to make investment decisions quantitatively. The analytical complexity of decision problems can also be scored by evaluating how many of the pivotal properties are available. Overall, we find that in the light of pivotal properties, complex problems under uncertainty frequently appear surprisingly tractable.

Restless multi-arm bandits (RMABs) is a popular decision-theoretic framework that has been used to model real-world sequential decision making problems in public health, wildlife conservation, communication systems, and beyond. Deployed RMAB systems typically operate in two stages: the first predicts the unknown parameters defining the RMAB instance, and the second employs an optimization algorithm to solve the constructed RMAB instance. In this work we provide and analyze the results from a first-of-its-kind deployment of an RMAB system in public health domain, aimed at improving maternal and child health. Our analysis is focused towards understanding the relationship between prediction accuracy and overall performance of deployed RMAB systems. This is crucial for determining the value of investing in improving predictive accuracy towards improving the final system performance, and is useful for diagnosing, monitoring deployed RMAB systems. Using real-world data from our deployed RMAB system, we demonstrate that an improvement in overall prediction accuracy may even be accompanied by a degradation in the performance of RMAB system -- a broad investment of resources to improve overall prediction accuracy may not yield expected results. Following this, we develop decision-focused evaluation metrics to evaluate the predictive component and show that it is better at explaining (both empirically and theoretically) the overall performance of a deployed RMAB system.

Motivated by recent applications requiring differential privacy over adaptive streams, we investigate the question of optimal instantiations of the matrix mechanism in this setting. We prove fundamental theoretical results on the applicability of matrix factorizations to adaptive streams, and provide a parameter-free fixed-point algorithm for computing optimal factorizations. We instantiate this framework with respect to concrete matrices which arise naturally in machine learning, and train user-level differentially private models with the resulting optimal mechanisms, yielding significant improvements in a notable problem in federated learning with user-level differential privacy.

We study a class of rearrangement problems under a novel pick-n-swap prehensile manipulation model, in which a robotic manipulator, capable of carrying an item and making item swaps, is tasked to sort items stored in lattices of variable dimensions in a time-optimal manner. We systematically analyze the intrinsic optimality structure, which is fairly rich and intriguing, under different levels of item distinguishability (fully labeled, where each item has a unique label, or partially labeled, where multiple items may be of the same type) and different lattice dimensions. Focusing on the most practical setting of one and two dimensions, we develop low polynomial time cycle-following-based algorithms that optimally perform rearrangements on 1D lattices under both fully- and partially-labeled settings. On the other hand, we show that rearrangement on 2D and higher-dimensional lattices become computationally intractable to optimally solve. Despite their NP-hardness, we prove that efficient cycle-following-based algorithms remain optimal in the asymptotic sense for 2D fully- and partially-labeled settings, in expectation, using the interesting fact that random permutations induce only a small number of cycles. We further improve these algorithms to provide $1.x$-optimality when the number of items is small. Simulation studies corroborate the effectiveness of our algorithms. The implementation of the algorithms from the paper can be found at github.com/arc-l/lattice-rearrangement.

Annealing machines specialized for combinatorial optimization problems have been developed, and some companies offer services to use those machines. Such specialized machines can only handle binary variables, and their input format is the quadratic unconstrained binary optimization (QUBO) formulation. Therefore, discretization is necessary to solve problems with continuous variables. However, there is a severe constraint on the number of binary variables with such machines. Although the simple binary expansion in the previous research requires many binary variables, we need to reduce the number of such variables in the QUBO formulation due to the constraint. We propose a discretization method that involves using correlations of continuous variables. We numerically show that the proposed method reduces the number of necessary binary variables in the QUBO formulation without a significant loss in prediction accuracy.

Handling an infinite number of inequality constraints in infinite-dimensional spaces occurs in many fields, from global optimization to optimal transport. These problems have been tackled individually in several previous articles through kernel Sum-Of-Squares (kSoS) approximations. We propose here a unified theorem to prove convergence guarantees for these schemes. Inequalities are turned into equalities to a class of nonnegative kSoS functions. This enables the use of scattering inequalities to mitigate the curse of dimensionality in sampling the constraints, leveraging the assumed smoothness of the functions appearing in the problem. This approach is illustrated in learning vector fields with side information, here the invariance of a set.

Polyak and Juditsky [30] famously showed that the stochastic gradient method for minimizing smooth and strongly convex functions enjoys a central limit theorem: the error between the running average of the iterates and the minimizer, normalized by the square root of the iteration counter, converges to a normal random vector. Moreover, the covariance matrix of the limiting distribution is in a precise sense "optimal" among any estimation procedure. A long standing open question is whether similar guarantees - asymptotic normality and optimality - exist for nonsmooth optimization and, more generally, for equilibrium problems. In this work, we obtain such guarantees under mild conditions that hold both in concrete circumstances (e.g.

Abstract--Electrical smart grids are units that supply electricity from power plants to the users to yield reduced costs, power failures/loss, and maximized energy management. Smart grids (SGs) are well-known devices due to their exceptional benefits such as bi-directional communication, stability, detection of power failures, and inter-connectivity with appliances for monitoring purposes. Hence, the importance of SGs as a research field is increasing with every passing year. This paper focuses on novel features and applications of smart grids in domestic and industrial sectors. Specifically, we focused on Genetic algorithm, Particle Swarm Optimization, and Grey Wolf Optimization to study the efforts made up till date for maximized energy management and cost minimization in SGs. Many counter Smart grids refers to an electric grid that delivers the attack solutions such as secure data collectors, broadcast authentication, electricity from utility (power generator sources/company) to and secure DoS-resistant broadcast authentication the users (residential/industrial). A simple smart grid connection protocols have been studied to secure the data collection and is shown in Figure 1, with bi-directional communication coping the demands of users in efficient ways [9], [10]. The process of electricity other challenges are faced by both utility and users (energy delivery is capable of monitoring, modeling, controlling, data supply and energy demand) such as energy management, filtering, and data processing with help of number of intelligent cost efficiency, reducing power losses, and reducing pollutant features such as Artificial Intelligence (AI) or Computational emissions [11], [12]. The aforementioned challenges can be Intelligence (CI) as shown in Figure 2. SGs allow users to addressed using optimization techniques in SGs to maximize schedule the appliances depending upon pricing hours and the profit (for both users and utility) by managing electricity its demand that helps in saving energy, increasing reliability, distribution and reducing emissions. Furthermore, SGs support Optimization in SGs is employed to find the conditions with bidirectional power line communications such as Home Area maximum benefits while (at the same time) minimizing the Network (HAN) or Wide Area Network (WAN), and wireless electricity wastage and cost [13]. Hence, optimization problem communications such as ZigBee, 6LowPAN, Z-wave, IoT in SGs is defined as a scenario (i.e., an objective function) that networks, etc. [3]-[6]. For future work, we aim to expand our research for other optimization algorithms (i.e., ABC, ACO). Our contributions in this paper are: fluenced by a set of variables and/or constraints.