We evaluate the state estimation performance of the RL-ROE for systems governed by the Burgers equation and Navier-Stokes equations. For each system, we first compute various solution trajectories corresponding to different physical parameter values, which we use to construct the ROM and train the RL-ROE following the procedure outlined in Section 2.4. The trained RL-ROE is finally deployed online and compared against a time-dependent Kalman filter constructed from the same ROM, which we refer to as KF-ROE. The KF-ROE is given by equations (3a) and (4), with the calculation of the time-varying Kalman gain detailed in Appendix C of the supplementary materials. Before proceeding to the results, we discuss our choice of baseline. The ensemble Kalman filter and 4D-Var are two estimation techniques for high-dimensional systems such as those governed by PDEs (Lorenc, 2003). Although they are commonly employed for data assimilation in numerical weather prediction, they require large computational resources since they involve repeated solutions of the high-dimensional dynamics (1). Thus, they are not applicable in the context of embedded control systems, whose limited resources call for an inexpensive model such as the ROM (2). Since the ROM that we consider has linear dynamics, extensions of the Kalman filter for nonlinear dynamics such as the extended or unscented Kalman filters (Wan & Van Der Merwe, 2000; Julier & Uhlmann, 2004) are not relevant, and the vanilla Kalman filter remains the best choice of baseline.

In this paper, we propose a novel probabilistic variant of iterative closest point (ICP) dubbed as CoBigICP. The method leverages both local geometrical information and global noise characteristics. Locally, the 3D structure of both target and source clouds are incorporated into the objective function through bidirectional correspondence. Globally, error metric of correntropy is introduced as noise model to resist outliers. Importantly, the close resemblance between normal-distributions transform (NDT) and correntropy is revealed. To ease the minimization step, an on-manifold parameterization of the special Euclidean group is proposed. Extensive experiments validate that CoBigICP outperforms several well-known and state-of-the-art methods.

Network function virtualization (NFV) and software-defined network (SDN) have become emerging network paradigms, allowing virtualized network function (VNF) deployment at a low cost. Even though VNF deployment can be flexible, it is still challenging to optimize VNF deployment due to its high complexity. Several studies have approached the task as dynamic programming, e.g., integer linear programming (ILP). However, optimizing VNF deployment for highly complex networks remains a challenge. Alternatively, reinforcement learning (RL) based approaches have been proposed to optimize this task, especially to employ a scaling action-based method which can deploy VNFs within less computational time. However, the model architecture can be improved further to generalize to the different networking settings. In this paper, we propose an enhanced model which can be adapted to more general network settings. We adopt the improved GNN architecture and a few techniques to obtain a better node representation for the VNF deployment task. Furthermore, we apply a recently proposed RL method, phasic policy gradient (PPG), to leverage the shared representation of the service function chain (SFC) generation model from the value function. We evaluate the proposed method in various scenarios, achieving a better QoS with minimum resource utilization compared to the previous methods. Finally, as a qualitative evaluation, we analyze our proposed encoder's representation for the nodes, which shows a more disentangled representation.

We present in this paper a way to transform a constrained statistical inference problem into an unconstrained one in order to be able to use modern computational methods, such as those based on automatic differentiation, GPU computing, stochastic gradients with mini-batch. Unlike the parametrizations classically used in Machine Learning, the parametrizations introduced here are all bijective and are even diffeomorphisms, thus allowing to keep the important properties from a statistical inference point of view, first of all identifiability. This cookbook presents a set of recipes to use to transform a constrained problem into a unconstrained one. For an easy use of parametrizations, this paper is at the same time a cookbook, and a Python package allowing the use of parametrizations with numpy, but also JAX and PyTorch, as well as a high level and expressive interface allowing to easily describe a parametrization to transform a difficult problem of statistical inference into an easier problem addressable with modern optimization tools.

Wide machine learning tasks can be formulated as non-convex multi-player games, where Nash equilibrium (NE) is an acceptable solution to all players, since no one can benefit from changing its strategy unilaterally. Attributed to the non-convexity, obtaining the existence condition of global NE is challenging, let alone designing theoretically guaranteed realization algorithms. This paper takes conjugate transformation to the formulation of non-convex multi-player games, and casts the complementary problem into a variational inequality (VI) problem with a continuous pseudo-gradient mapping. We then prove the existence condition of global NE: the solution to the VI problem satisfies a duality relation. Based on this VI formulation, we design a conjugate-based ordinary differential equation (ODE) to approach global NE, which is proved to have an exponential convergence rate. To make the dynamics more implementable, we further derive a discretized algorithm. We apply our algorithm to two typical scenarios: multi-player generalized monotone game and multi-player potential game. In the two settings, we prove that the step-size setting is required to be $\mathcal{O}(1/k)$ and $\mathcal{O}(1/\sqrt k)$ to yield the convergence rates of $\mathcal{O}(1/ k)$ and $\mathcal{O}(1/\sqrt k)$, respectively. Extensive experiments in robust neural network training and sensor localization are in full agreement with our theory.

Decentralized bilevel optimization has received increasing attention recently due to its foundational role in many emerging multi-agent learning paradigms (e.g., multi-agent meta-learning and multi-agent reinforcement learning) over peer-to-peer edge networks. However, to work with the limited computation and communication capabilities of edge networks, a major challenge in developing decentralized bilevel optimization techniques is to lower sample and communication complexities. This motivates us to develop a new decentralized bilevel optimization called DIAMOND (decentralized single-timescale stochastic approximation with momentum and gradient-tracking). The contributions of this paper are as follows: i) our DIAMOND algorithm adopts a single-loop structure rather than following the natural double-loop structure of bilevel optimization, which offers low computation and implementation complexity; ii) compared to existing approaches, the DIAMOND algorithm does not require any full gradient evaluations, which further reduces both sample and computational complexities; iii) through a careful integration of momentum information and gradient tracking techniques, we show that the DIAMOND algorithm enjoys $\mathcal{O}(\epsilon^{-3/2})$ in sample and communication complexities for achieving an $\epsilon$-stationary solution, both of which are independent of the dataset sizes and significantly outperform existing works. Extensive experiments also verify our theoretical findings.

This paper investigates the problem of computing the equilibrium of competitive games, which is often modeled as a constrained saddle-point optimization problem with probability simplex constraints. Despite recent efforts in understanding the last-iterate convergence of extragradient methods in the unconstrained setting, the theoretical underpinnings of these methods in the constrained settings, especially those using multiplicative updates, remain highly inadequate, even when the objective function is bilinear. Motivated by the algorithmic role of entropy regularization in single-agent reinforcement learning and game theory, we develop provably efficient extragradient methods to find the quantal response equilibrium (QRE) -- which are solutions to zero-sum two-player matrix games with entropy regularization -- at a linear rate. The proposed algorithms can be implemented in a decentralized manner, where each player executes symmetric and multiplicative updates iteratively using its own payoff without observing the opponent's actions directly. In addition, by controlling the knob of entropy regularization, the proposed algorithms can locate an approximate Nash equilibrium of the unregularized matrix game at a sublinear rate without assuming the Nash equilibrium to be unique. Our methods also lead to efficient policy extragradient algorithms for solving (entropy-regularized) zero-sum Markov games at similar rates. All of our convergence rates are nearly dimension-free, which are independent of the size of the state and action spaces up to logarithm factors, highlighting the positive role of entropy regularization for accelerating convergence.

Recent neural methods for vehicle routing problems always train and test the deep models on the same instance distribution (i.e., uniform). To tackle the consequent cross-distribution generalization concerns, we bring the knowledge distillation to this field and propose an Adaptive Multi-Distribution Knowledge Distillation (AMDKD) scheme for learning more generalizable deep models. Particularly, our AMDKD leverages various knowledge from multiple teachers trained on exemplar distributions to yield a light-weight yet generalist student model. Meanwhile, we equip AMDKD with an adaptive strategy that allows the student to concentrate on difficult distributions, so as to absorb hard-to-master knowledge more effectively. Extensive experimental results show that, compared with the baseline neural methods, our AMDKD is able to achieve competitive results on both unseen in-distribution and out-of-distribution instances, which are either randomly synthesized or adopted from benchmark datasets (i.e., TSPLIB and CVRPLIB). Notably, our AMDKD is generic, and consumes less computational resources for inference.

Bayesian inference for high-dimensional inverse problems is computationally costly and requires selecting a suitable prior distribution. Amortized variational inference addresses these challenges via a neural network that approximates the posterior distribution not only for one instance of data, but a distribution of data pertaining to a specific inverse problem. During inference, the neural network -- in our case a conditional normalizing flow -- provides posterior samples at virtually no cost. However, the accuracy of amortized variational inference relies on the availability of high-fidelity training data, which seldom exists in geophysical inverse problems due to the Earth's heterogeneity. In addition, the network is prone to errors if evaluated over out-of-distribution data. As such, we propose to increase the resilience of amortized variational inference in the presence of moderate data distribution shifts. We achieve this via a correction to the latent distribution that improves the posterior distribution approximation for the data at hand. The correction involves relaxing the standard Gaussian assumption on the latent distribution and parameterizing it via a Gaussian distribution with an unknown mean and (diagonal) covariance. These unknowns are then estimated by minimizing the Kullback-Leibler divergence between the corrected and the (physics-based) true posterior distributions. While generic and applicable to other inverse problems, by means of a linearized seismic imaging example, we show that our correction step improves the robustness of amortized variational inference with respect to changes in the number of seismic sources, noise variance, and shifts in the prior distribution. This approach provides a seismic image with limited artifacts and an assessment of its uncertainty at approximately the same cost as five reverse-time migrations.

Differential dynamic programming (DDP) is a widely used and powerful trajectory optimization technique, however, due to its internal structure, it is not exempt from local minima. In this paper, we present Differential Dynamic Programming with Escape Network (DDPEN) - a novel approach to avoid DDP local minima by utilising an additional term used in the optimization criteria pointing towards the direction where robot should move in order to escape local minima. In order to produce the aforementioned directions, we propose to utilize a deep model that takes as an input the map of the environment in the form of a costmap together with the desired goal position. The Model produces possible future directions that will lead to the goal, avoiding local minima which is possible to run in real time conditions. The model is trained on a synthetic dataset and overall the system is evaluated at the Gazebo simulator. In this work we show that our proposed method allows avoiding local minima of trajectory optimization algorithm and successfully execute a trajectory 278 m long with various convex and nonconvex obstacles.