Learning Based NMPC Adaptation for Autonomous Driving using Parallelized Digital Twin

In this work, we address the problem of transferring an autonomous driving (AD) module from one domain to another, in particular from simulation to the real world (Sim2Real). We propose a data-efficient method for online and on-the-fly learning based adaptation for parametrizable control architectures such that the target closed-loop performance is optimized under several uncertainty sources such as model mismatches, environment changes and task choice. The novelty of the work resides in leveraging black-box optimization enabled by executable digital twins, with data-driven hyper-parameter tuning through derivative-free methods to directly adapt in real-time the AD module. Our proposed method requires a minimal amount of interaction with the real-world in the randomization and online training phase. Specifically, we validate our approach in real-world experiments and show the ability to transfer and safely tune a nonlinear model predictive controller in less than 10 minutes, eliminating the need of day-long manual tuning and hours-long machine learning training phases. Our results show that the online adapted NMPC directly compensates for disturbances, avoids overtuning in simulation and for one specific task, and it generalizes for less than 15cm of tracking accuracy over a multitude of trajectories, and leads to 83% tracking improvement.

Adaptive proximal gradient methods are universal without approximation

We show that adaptive proximal gradient methods for convex problems are not restricted to traditional Lipschitzian assumptions. Our analysis reveals that a class of linesearch-free methods is still convergent under mere local H\"older gradient continuity, covering in particular continuously differentiable semi-algebraic functions. To mitigate the lack of local Lipschitz continuity, popular approaches revolve around $\varepsilon$-oracles and/or linesearch procedures. In contrast, we exploit plain H\"older inequalities not entailing any approximation, all while retaining the linesearch-free nature of adaptive schemes. Furthermore, we prove full sequence convergence without prior knowledge of local H\"older constants nor of the order of H\"older continuity. In numerical experiments we present comparisons to baseline methods on diverse tasks from machine learning covering both the locally and the globally H\"older setting.

Convergence of the Chambolle-Pock Algorithm in the Absence of Monotonicity

The Chambolle-Pock algorithm (CPA), also known as the primal-dual hybrid gradient method (PDHG), has surged in popularity in the last decade due to its success in solving convex/monotone structured problems. This work provides convergence results for problems with varying degrees of (non)monotonicity, quantified through a so-called oblique weak Minty condition on the associated primal-dual operator. Our results reveal novel stepsize and relaxation parameter ranges which do not only depend on the norm of the linear mapping, but also on its other singular values. In particular, in nonmonotone settings, in addition to the classical stepsize conditions for CPA, extra bounds on the stepsizes and relaxation parameters are required. On the other hand, in the strongly monotone setting, the relaxation parameter is allowed to exceed the classical upper bound of two. Moreover, sufficient convergence conditions are obtained when the individual operators belong to the recently introduced class of semimonotone operators. Since this class of operators encompasses many traditional operator classes including (hypo)- and co(hypo)monotone operators, this analysis recovers and extends existing results for CPA. Several examples are provided for the aforementioned problem classes to demonstrate and establish tightness of the proposed stepsize ranges.

On the convergence of adaptive first order methods: proximal gradient and alternating minimization algorithms

Building upon recent works on linesearch-free adaptive proximal gradient methods, this paper proposes AdaPG$^{\pi,r}$, a framework that unifies and extends existing results by providing larger stepsize policies and improved lower bounds. Different choices of the parameters $\pi$ and $r$ are discussed and the efficacy of the resulting methods is demonstrated through numerical simulations. In an attempt to better understand the underlying theory, its convergence is established in a more general setting that allows for time-varying parameters. Finally, an adaptive alternating minimization algorithm is presented by exploring the dual setting. This algorithm not only incorporates additional adaptivity, but also expands its applicability beyond standard strongly convex settings.

Zeroth-order Asynchronous Learning with Bounded Delays with a Use-case in Resource Allocation in Communication Networks

Distributed optimization has experienced a significant surge in interest due to its wide-ranging applications in distributed learning and adaptation. While various scenarios, such as shared-memory, local-memory, and consensus-based approaches, have been extensively studied in isolation, there remains a need for further exploration of their interconnections. This paper specifically concentrates on a scenario where agents collaborate toward a unified mission while potentially having distinct tasks. Each agent's actions can potentially impact other agents through interactions. Within this context, the objective for the agents is to optimize their local parameters based on the aggregate of local reward functions, where only local zeroth-order oracles are available. Notably, the learning process is asynchronous, meaning that agents update and query their zeroth-order oracles asynchronously while communicating with other agents subject to bounded but possibly random communication delays. This paper presents theoretical convergence analyses and establishes a convergence rate for the proposed approach. Furthermore, it addresses the relevant issue of deep learning-based resource allocation in communication networks and conducts numerical experiments in which agents, acting as transmitters, collaboratively train their individual (possibly unique) policies to maximize a common performance metric.

A Deep Learning Based Resource Allocator for Communication Systems with Dynamic User Utility Demands

Deep learning (DL) based resource allocation (RA) has recently gained a lot of attention due to its performance efficiency. However, most of the related studies assume an ideal case where the number of users and their utility demands, e.g., data rate constraints, are fixed and the designed DL based RA scheme exploits a policy trained only for these fixed parameters. A computationally complex policy retraining is required whenever these parameters change. Therefore, in this paper, a DL based resource allocator (ALCOR) is introduced, which allows users to freely adjust their utility demands based on, e.g., their application layer. ALCOR employs deep neural networks (DNNs), as the policy, in an iterative optimization algorithm. The optimization algorithm aims to optimize the on-off status of users in a time-sharing problem to satisfy their utility demands in expectation. The policy performs unconstrained RA (URA) -- RA without taking into account user utility demands -- among active users to maximize the sum utility (SU) at each time instant. Based on the chosen URA scheme, ALCOR can perform RA in a model-based or model-free manner and in a centralized or distributed scenario. Derived convergence analyses provide guarantees for the convergence of ALCOR, and numerical experiments corroborate its effectiveness.

Nonlinear SVD with Asymmetric Kernels: feature learning and asymmetric Nyström method

Asymmetric data naturally exist in real life, such as directed graphs. Different from the common kernel methods requiring Mercer kernels, this paper tackles the asymmetric kernel-based learning problem. We describe a nonlinear extension of the matrix Singular Value Decomposition through asymmetric kernels, namely KSVD. First, we construct two nonlinear feature mappings w.r.t. rows and columns of the given data matrix. The proposed optimization problem maximizes the variance of each mapping projected onto the subspace spanned by the other, subject to a mutual orthogonality constraint. Through Lagrangian duality, we show that it can be solved by the left and right singular vectors in the feature space induced by the asymmetric kernel. Moreover, we start from the integral equations with a pair of adjoint eigenfunctions corresponding to the singular vectors on an asymmetrical kernel, and extend the Nystr\"om method to asymmetric cases through the finite sample approximation, which can be applied to speedup the training in KSVD. Experiments show that asymmetric KSVD learns features outperforming Mercer-kernel based methods that resort to symmetrization, and also verify the effectiveness of the asymmetric Nystr\"om method.

Combining Primal and Dual Representations in Deep Restricted Kernel Machines Classifiers

In contrast to deep networks, kernel methods cannot directly take advantage of depth. In this regard, the deep Restricted Kernel Machine (DRKM) framework allows multiple levels of kernel PCA (KPCA) and Least-Squares Support Vector Machines (LSSVM) to be combined into a deep architecture using visible and hidden units. We propose a new method for DRKM classification coupling the objectives of KPCA and classification levels, with the hidden feature matrix lying on the Stiefel manifold. The classification level can be formulated as an LSSVM or as an MLP feature map, combining depth in terms of levels and layers. The classification level is expressed in its primal formulation, as the deep KPCA levels can embed the most informative components of the data in a much lower dimensional space. In the experiments on benchmark datasets with few available training points, we show that our deep method improves over the LSSVM/MLP and that models with multiple KPCA levels can outperform models with a single level.

Extending Kernel PCA through Dualization: Sparsity, Robustness and Fast Algorithms

The goal of this paper is to revisit Kernel Principal Component Analysis (KPCA) through dualization of a difference of convex functions. This allows to naturally extend KPCA to multiple objective functions and leads to efficient gradient-based algorithms avoiding the expensive SVD of the Gram matrix. Particularly, we consider objective functions that can be written as Moreau envelopes, demonstrating how to promote robustness and sparsity within the same framework. The proposed method is evaluated on synthetic and real-world benchmarks, showing significant speedup in KPCA training time as well as highlighting the benefits in terms of robustness and sparsity.

Distributionally Robust Optimization using Cost-Aware Ambiguity Sets

We present a novel class of ambiguity sets for distributionally robust optimization (DRO). These ambiguity sets, called cost-aware ambiguity sets, are defined as halfspaces which depend on the cost function evaluated at an independent estimate of the optimal solution, thus excluding only those distributions that are expected to have significant impact on the obtained worst-case cost. We show that the resulting DRO method provides both a high-confidence upper bound and a consistent estimator of the out-of-sample expected cost, and demonstrate empirically that it results in less conservative solutions compared to divergence-based ambiguity sets.