Contractivity and linear convergence in bilinear saddle-point problems: An operator-theoretic approach

We study the convex-concave bilinear saddle-point problem $\min_x \max_y f(x) + y^\top Ax - g(y)$, where both, only one, or none of the functions $f$ and $g$ are strongly convex, and suitable rank conditions on the matrix $A$ hold. The solution of this problem is at the core of many machine learning tasks. By employing tools from operator theory, we systematically prove the contractivity (in turn, the linear convergence) of several first-order primal-dual algorithms, including the Chambolle-Pock method. Our approach results in concise and elegant proofs, and it yields new convergence guarantees and tighter bounds compared to known results.

Maximum likelihood inference for high-dimensional problems with multiaffine variable relations

Maximum Likelihood Estimation of continuous variable models can be very challenging in high dimensions, due to potentially complex probability distributions. The existence of multiple interdependencies among variables can make it very difficult to establish convergence guarantees. This leads to a wide use of brute-force methods, such as grid searching and Monte-Carlo sampling and, when applicable, complex and problem-specific algorithms. In this paper, we consider inference problems where the variables are related by multiaffine expressions. We propose a novel Alternating and Iteratively-Reweighted Least Squares (AIRLS) algorithm, and prove its convergence for problems with Generalized Normal Distributions. We also provide an efficient method to compute the variance of the estimates obtained using AIRLS. Finally, we show how the method can be applied to graphical statistical models. We perform numerical experiments on several inference problems, showing significantly better performance than state-of-the-art approaches in terms of scalability, robustness to noise, and convergence speed due to an empirically observed super-linear convergence rate.

Learning Diffusion at Lightspeed

Diffusion regulates a phenomenal number of natural processes and the dynamics of many successful generative models. Existing models to learn the diffusion terms from observational data rely on complex bilevel optimization problems and properly model only the drift of the system. We propose a new simple model, JKOnet*, which bypasses altogether the complexity of existing architectures while presenting significantly enhanced representational capacity: JKOnet* recovers the potential, interaction, and internal energy components of the underlying diffusion process. JKOnet* minimizes a simple quadratic loss, runs at lightspeed, and drastically outperforms other baselines in practice. Additionally, JKOnet* provides a closed-form optimal solution for linearly parametrized functionals. Our methodology is based on the interpretation of diffusion processes as energy-minimizing trajectories in the probability space via the so-called JKO scheme, which we study via its first-order optimality conditions, in light of few-weeks-old advancements in optimization in the probability space.

When to Sense and Control? A Time-adaptive Approach for Continuous-Time RL

Reinforcement learning (RL) excels in optimizing policies for discrete-time Markov decision processes (MDP). However, various systems are inherently continuous in time, making discrete-time MDPs an inexact modeling choice. In many applications, such as greenhouse control or medical treatments, each interaction (measurement or switching of action) involves manual intervention and thus is inherently costly. Therefore, we generally prefer a time-adaptive approach with fewer interactions with the system. In this work, we formalize an RL framework, Time-adaptive Control & Sensing (TaCoS), that tackles this challenge by optimizing over policies that besides control predict the duration of its application. Our formulation results in an extended MDP that any standard RL algorithm can solve. We demonstrate that state-of-the-art RL algorithms trained on TaCoS drastically reduce the interaction amount over their discrete-time counterpart while retaining the same or improved performance, and exhibiting robustness over discretization frequency. Finally, we propose OTaCoS, an efficient model-based algorithm for our setting. We show that OTaCoS enjoys sublinear regret for systems with sufficiently smooth dynamics and empirically results in further sample-efficiency gains.

NeoRL: Efficient Exploration for Nonepisodic RL

We study the problem of nonepisodic reinforcement learning (RL) for nonlinear dynamical systems, where the system dynamics are unknown and the RL agent has to learn from a single trajectory, i.e., without resets. We propose Nonepisodic Optimistic RL (NeoRL), an approach based on the principle of optimism in the face of uncertainty. NeoRL uses well-calibrated probabilistic models and plans optimistically w.r.t. the epistemic uncertainty about the unknown dynamics. Under continuity and bounded energy assumptions on the system, we provide a first-of-its-kind regret bound of $\setO(\beta_T \sqrt{T \Gamma_T})$ for general nonlinear systems with Gaussian process dynamics. We compare NeoRL to other baselines on several deep RL environments and empirically demonstrate that NeoRL achieves the optimal average cost while incurring the least regret.

Bridging the Sim-to-Real Gap with Bayesian Inference

We present SIM-FSVGD for learning robot dynamics from data. As opposed to traditional methods, SIM-FSVGD leverages low-fidelity physical priors, e.g., in the form of simulators, to regularize the training of neural network models. While learning accurate dynamics already in the low data regime, SIM-FSVGD scales and excels also when more data is available. We empirically show that learning with implicit physical priors results in accurate mean model estimation as well as precise uncertainty quantification. We demonstrate the effectiveness of SIM-FSVGD in bridging the sim-to-real gap on a high-performance RC racecar system. Using model-based RL, we demonstrate a highly dynamic parking maneuver with drifting, using less than half the data compared to the state of the art.

Towards a Systems Theory of Algorithms

Traditionally, numerical algorithms are seen as isolated pieces of code confined to an {\em in silico} existence. However, this perspective is not appropriate for many modern computational approaches in control, learning, or optimization, wherein {\em in vivo} algorithms interact with their environment. Examples of such {\em open} include various real-time optimization-based control strategies, reinforcement learning, decision-making architectures, online optimization, and many more. Further, even {\em closed} algorithms in learning or optimization are increasingly abstracted in block diagrams with interacting dynamic modules and pipelines. In this opinion paper, we state our vision on a to-be-cultivated {\em systems theory of algorithms} and argue in favour of viewing algorithms as open dynamical systems interacting with other algorithms, physical systems, humans, or databases. Remarkably, the manifold tools developed under the umbrella of systems theory also provide valuable insights into this burgeoning paradigm shift and its accompanying challenges in the algorithmic world. We survey various instances where the principles of algorithmic systems theory are being developed and outline pertinent modeling, analysis, and design challenges.

Urban traffic congestion control: a DeePC change

Urban traffic congestion remains a pressing challenge in our rapidly expanding cities, despite the abundance of available data and the efforts of policymakers. By leveraging behavioral system theory and data-driven control, this paper exploits the DeePC algorithm in the context of urban traffic control performed via dynamic traffic lights. To validate our approach, we consider a high-fidelity case study using the state-of-the-art simulation software package Simulation of Urban MObility (SUMO). Preliminary results indicate that DeePC outperforms existing approaches across various key metrics, including travel time and CO$_2$ emissions, demonstrating its potential for effective traffic management

Efficient Exploration in Continuous-time Model-based Reinforcement Learning

Reinforcement learning algorithms typically consider discrete-time dynamics, even though the underlying systems are often continuous in time. In this paper, we introduce a model-based reinforcement learning algorithm that represents continuous-time dynamics using nonlinear ordinary differential equations (ODEs). We capture epistemic uncertainty using well-calibrated probabilistic models, and use the optimistic principle for exploration. Our regret bounds surface the importance of the measurement selection strategy(MSS), since in continuous time we not only must decide how to explore, but also when to observe the underlying system. Our analysis demonstrates that the regret is sublinear when modeling ODEs with Gaussian Processes (GP) for common choices of MSS, such as equidistant sampling. Additionally, we propose an adaptive, data-dependent, practical MSS that, when combined with GP dynamics, also achieves sublinear regret with significantly fewer samples. We showcase the benefits of continuous-time modeling over its discrete-time counterpart, as well as our proposed adaptive MSS over standard baselines, on several applications.

Designing Optimal Personalized Incentive for Traffic Routing using BIG Hype algorithm

We study the problem of optimally routing plug-in electric and conventional fuel vehicles on a city level. In our model, commuters selfishly aim to minimize a local cost that combines travel time, from a fixed origin to a desired destination, and the monetary cost of using city facilities, parking or service stations. The traffic authority can influence the commuters' preferred routing choice by means of personalized discounts on parking tickets and on the energy price at service stations. We formalize the problem of designing these monetary incentives optimally as a large-scale bilevel game, where constraints arise at both levels due to the finite capacities of city facilities and incentives budget. Then, we develop an efficient decentralized solution scheme with convergence guarantees based on BIG Hype, a recently-proposed hypergradient-based algorithm for hierarchical games. Finally, we validate our model via numerical simulations over the Anaheim's network, and show that the proposed approach produces sensible results in terms of traffic decongestion and it is able to solve in minutes problems with more than 48000 variables and 110000 constraints.