The paper introduces DiSProD, an online planner developed for environments with probabilistic transitions in continuous state and action spaces. DiSProD builds a symbolic graph that captures the distribution of future trajectories, conditioned on a given policy, using independence assumptions and approximate propagation of distributions. The symbolic graph provides a differentiable representation of the policy's value, enabling efficient gradient-based optimization for long-horizon search. The propagation of approximate distributions can be seen as an aggregation of many trajectories, making it well-suited for dealing with sparse rewards and stochastic environments. An extensive experimental evaluation compares DiSProD to state-of-the-art planners in discrete-time planning and real-time control of robotic systems. The proposed method improves over existing planners in handling stochastic environments, sensitivity to search depth, sparsity of rewards, and large action spaces. Additional real-world experiments demonstrate that DiSProD can control ground vehicles and surface vessels to successfully navigate around obstacles.

We consider statistical inference of equality-constrained stochastic nonlinear optimization problems. We develop a fully online stochastic sequential quadratic programming (StoSQP) method to solve the problems, which can be regarded as applying Newton's method to the first-order optimality conditions (i.e., the KKT conditions). Motivated by recent designs of numerical second-order methods, we allow StoSQP to adaptively select any random stepsize $\bar{\alpha}_t$, as long as $\beta_t\leq \bar{\alpha}_t \leq \beta_t+\chi_t$, for some control sequences $\beta_t$ and $\chi_t=o(\beta_t)$. To reduce the dominant computational cost of second-order methods, we additionally allow StoSQP to inexactly solve quadratic programs via efficient randomized iterative solvers that utilize sketching techniques. Notably, we do not require the approximation error to diminish as iteration proceeds. For the developed method, we show that under mild assumptions (i) computationally, it can take at most $O(1/\epsilon^4)$ iterations (same as samples) to attain $\epsilon$-stationarity; (ii) statistically, its primal-dual sequence $1/\sqrt{\beta_t}\cdot (x_t - x^\star, \lambda_t - \lambda^\star)$ converges to a mean-zero Gaussian distribution with a nontrivial covariance matrix depending on the underlying sketching distribution. Additionally, we establish the almost-sure convergence rate of the iterate $(x_t, \lambda_t)$ along with the Berry-Esseen bound; the latter quantitatively measures the convergence rate of the distribution function. We analyze a plug-in limiting covariance matrix estimator, and demonstrate the performance of the method both on benchmark nonlinear problems in CUTEst test set and on linearly/nonlinearly constrained regression problems.

This is part II of a two-part paper. Part I presented a universal Birkhoff theory for fast and accurate trajectory optimization. The theory rested on two main hypotheses. In this paper, it is shown that if the computational grid is selected from any one of the Legendre and Chebyshev family of node points, be it Lobatto, Radau or Gauss, then, the resulting collection of trajectory optimization methods satisfy the hypotheses required for the universal Birkhoff theory to hold. All of these grid points can be generated at an $\mathcal{O}(1)$ computational speed. Furthermore, all Birkhoff-generated solutions can be tested for optimality by a joint application of Pontryagin's- and Covector-Mapping Principles, where the latter was developed in Part~I. More importantly, the optimality checks can be performed without resorting to an indirect method or even explicitly producing the full differential-algebraic boundary value problem that results from an application of Pontryagin's Principle. Numerical problems are solved to illustrate all these ideas. The examples are chosen to particularly highlight three practically useful features of Birkhoff methods: (1) bang-bang optimal controls can be produced without suffering any Gibbs phenomenon, (2) discontinuous and even Dirac delta covector trajectories can be well approximated, and (3) extremal solutions over dense grids can be computed in a stable and efficient manner.

Second-order optimization algorithms exhibit excellent convergence properties for training deep learning models, but often incur significant computation and memory overheads. This can result in lower training efficiency than the first-order counterparts such as stochastic gradient descent (SGD). In this work, we present a memory- and time-efficient second-order algorithm named Eva with two novel techniques: 1) we construct the second-order information with the Kronecker factorization of small stochastic vectors over a mini-batch of training data to reduce memory consumption, and 2) we derive an efficient update formula without explicitly computing the inverse of matrices using the Sherman-Morrison formula. We further extend Eva to a general vectorized approximation framework to improve the compute and memory efficiency of two existing second-order algorithms (FOOF and Shampoo) without affecting their convergence performance. Extensive experimental results on different models and datasets show that Eva reduces the end-to-end training time up to 2.05x and 2.42x compared to first-order SGD and second-order algorithms (K-FAC and Shampoo), respectively.

In recent years, Variational Quantum Algorithms (VQAs) have emerged as a promising approach for solving optimization problems on quantum computers in the NISQ era. However, one limitation of VQAs is their reliance on fixed-structure circuits, which may not be taylored for specific problems or hardware configurations. A leading strategy to address this issue are Adaptative VQAs, which dynamically modify the circuit structure by adding and removing gates, and optimize their parameters during the training. Several Adaptative VQAs, based on heuristics such as circuit shallowness, entanglement capability and hardware compatibility, have already been proposed in the literature, but there is still lack of a systematic comparison between the different methods. In this paper, we aim to fill this gap by analyzing three Adaptative VQAs: Evolutionary Variational Quantum Eigensolver (EVQE), Variable Ansatz (VAns), already proposed in the literature, and Random Adapt-VQE (RA-VQE), a random approach we introduce as a baseline. In order to compare these algorithms to traditional VQAs, we also include the Quantum Approximate Optimization Algorithm (QAOA) in our analysis. We apply these algorithms to QUBO problems and study their performance by examining the quality of the solutions found and the computational times required. Additionally, we investigate how the choice of the hyperparameters can impact the overall performance of the algorithms, highlighting the importance of selecting an appropriate methodology for hyperparameter tuning. Our analysis sets benchmarks for Adaptative VQAs designed for near-term quantum devices and provides valuable insights to guide future research in this area.

The turbulent jet ignition concept using prechambers is a promising solution to achieve stable combustion at lean conditions in large gas engines, leading to high efficiency at low emission levels. Due to the wide range of design and operating parameters for large gas engine prechambers, the preferred method for evaluating different designs is computational fluid dynamics (CFD), as testing in test bed measurement campaigns is time-consuming and expensive. However, the significant computational time required for detailed CFD simulations due to the complexity of solving the underlying physics also limits its applicability. In optimization settings similar to the present case, i.e., where the evaluation of the objective function(s) is computationally costly, Bayesian optimization has largely replaced classical design-of-experiment. Thus, the present study deals with the computationally efficient Bayesian optimization of large gas engine prechambers design using CFD simulation. Reynolds-averaged-Navier-Stokes simulations are used to determine the target values as a function of the selected prechamber design parameters. The results indicate that the chosen strategy is effective to find a prechamber design that achieves the desired target values.

We consider convex relaxations for recovering low-rank tensors based on constrained minimization over a ball induced by the tensor nuclear norm, recently introduced in \cite{tensor_tSVD}. We build on a recent line of results that considered convex relaxations for the recovery of low-rank matrices and established that under a strict complementarity condition (SC), both the convergence rate and per-iteration runtime of standard gradient methods may improve dramatically. We develop the appropriate strict complementarity condition for the tensor nuclear norm ball and obtain the following main results under this condition: 1. When the objective to minimize is of the form $f(\mX)=g(\mA\mX)+\langle{\mC,\mX}\rangle$ , where $g$ is strongly convex and $\mA$ is a linear map (e.g., least squares), a quadratic growth bound holds, which implies linear convergence rates for standard projected gradient methods, despite the fact that $f$ need not be strongly convex. 2. For a smooth objective function, when initialized in certain proximity of an optimal solution which satisfies SC, standard projected gradient methods only require SVD computations (for projecting onto the tensor nuclear norm ball) of rank that matches the tubal rank of the optimal solution. In particular, when the tubal rank is constant, this implies nearly linear (in the size of the tensor) runtime per iteration, as opposed to super linear without further assumptions. 3. For a nonsmooth objective function which admits a popular smooth saddle-point formulation, we derive similar results to the latter for the well known extragradient method. An additional contribution which may be of independent interest, is the rigorous extension of many basic results regarding tensors of arbitrary order, which were previously obtained only for third-order tensors.

Temporal data, representing chronological observations of complex systems, has always been a typical data structure that can be widely generated by many domains, such as industry, medicine and finance. Analyzing this type of data is extremely valuable for various applications. Thus, different temporal data analysis tasks, eg, classification, clustering and prediction, have been proposed in the past decades. Among them, causal discovery, learning the causal relations from temporal data, is considered an interesting yet critical task and has attracted much research attention. Existing causal discovery works can be divided into two highly correlated categories according to whether the temporal data is calibrated, ie, multivariate time series causal discovery, and event sequence causal discovery. However, most previous surveys are only focused on the time series causal discovery and ignore the second category. In this paper, we specify the correlation between the two categories and provide a systematical overview of existing solutions. Furthermore, we provide public datasets, evaluation metrics and new perspectives for temporal data causal discovery.

We consider stochastic approximation algorithms where the expected progress toward the optimum is proportional to the algorithm's step size. For instance, a stochastic gradient descent algorithm applied to a convex function will satisfy this property when bounded away from the optimum. This property can continue to hold when the optimum is on the corner of a convex constraint set or when the function is not smooth at the optimum or when the objective function lies within a cone. In such settings, we will show that a projected stochastic gradient descent algorithm can have a different rate of convergence than would be anticipated by standard results for stochastic gradient descent with a smooth objective and smooth constraints. We develop new results whose methods are typically used in probability to analyze random walks or in applied probability to analyze queueing networks. For stochastic approximation, our results establish new exponential concentration bounds. We now summarize the background and problems where our results apply.

We deal with a general distributed constrained online learning problem with privacy over time-varying networks, where a class of nondecomposable objectives are considered. Under this setting, each node only controls a part of the global decision, and the goal of all nodes is to collaboratively minimize the global cost over a time horizon $T$ while guarantees the security of the transmitted information. For such problems, we first design a novel generic algorithm framework, named as DPSDA, of differentially private distributed online learning using the Laplace mechanism and the stochastic variants of dual averaging method. Note that in the dual updates, all nodes of DPSDA employ the noise-corrupted gradients for more generality. Then, we propose two algorithms, named as DPSDA-C and DPSDA-PS, under this framework. In DPSDA-C, the nodes implement a circulation-based communication in the primal updates so as to alleviate the disagreements over time-varying undirected networks. In addition, for the extension to time-varying directed ones, the nodes implement the broadcast-based push-sum dynamics in DPSDA-PS, which can achieve average consensus over arbitrary directed networks. Theoretical results show that both algorithms attain an expected regret upper bound in $\mathcal{O}( \sqrt{T} )$ when the objective function is convex, which matches the best utility achievable by cutting-edge algorithms. Finally, numerical experiment results on both synthetic and real-world datasets verify the effectiveness of our algorithms.