In two-phase image segmentation, convex relaxation has allowed global minimisers to be computed for a variety of data fitting terms. Many efficient approaches exist to compute a solution quickly. However, we consider whether the nature of the data fitting in this formulation allows for reasonable assumptions to be made about the solution that can improve the computational performance further. In particular, we employ a well known dual formulation of this problem and solve the corresponding equations in a restricted domain. We present experimental results that explore the dependence of the solution on this restriction and quantify imrovements in the computational performance. This approach can be extended to analogous methods simply and could provide an efficient alternative for problems of this type.

Our premise is that autonomous vehicles must optimize communications and motion planning jointly. Specifically, a vehicle must adapt its motion plan staying cognizant of communications rate related constraints and adapt the use of communications while being cognizant of motion planning related restrictions that may be imposed by the on-road environment. To this end, we formulate a reinforcement learning problem wherein an autonomous vehicle jointly chooses (a) a motion planning action that executes on-road and (b) a communications action of querying sensed information from the infrastructure. The goal is to optimize the driving utility of the autonomous vehicle. We apply the Q-learning algorithm to make the vehicle learn the optimal policy, which makes the optimal choice of planning and communications actions at any given time. We demonstrate the ability of the optimal policy to smartly adapt communications and planning actions, while achieving large driving utilities, using simulations.

In this paper, we design nonlinear state feedback controllers for discrete-time polynomial dynamical systems via the occupation measure approach. We propose the discrete-time controlled Liouville equation, and use it to formulate the controller synthesis problem as an infinite-dimensional linear programming problem on measures, which is then relaxed as finite-dimensional semidefinite programming problems on moments of measures and their duals on sums-of-squares polynomials. Nonlinear controllers can be extracted from the solutions to the relaxed problems. The advantage of the occupation measure approach is that we solve convex problems instead of generally non-convex problems, and the computational complexity is polynomial in the state and input dimensions, and hence the approach is more scalable. In addition, we show that the approach can be applied to over-approximating the backward reachable set of discrete-time autonomous polynomial systems and the controllable set of discrete-time polynomial systems under known state feedback control laws. We illustrate our approach on several dynamical systems.

Kalman filters are routinely used for many data fusion applications including navigation, tracking, and simultaneous localization and mapping problems. However, significant time and effort is frequently required to tune various Kalman filter model parameters, e.g. process noise covariance, pre-whitening filter models for non-white noise, etc. Conventional optimization techniques for tuning can get stuck in poor local minima and can be expensive to implement with real sensor data. To address these issues, a new "black box" Bayesian optimization strategy is developed for automatically tuning Kalman filters. In this approach, performance is characterized by one of two stochastic objective functions: normalized estimation error squared (NEES) when ground truth state models are available, or the normalized innovation error squared (NIS) when only sensor data is available. By intelligently sampling the parameter space to both learn and exploit a nonparametric Gaussian process surrogate function for the NEES/NIS costs, Bayesian optimization can efficiently identify multiple local minima and provide uncertainty quantification on its results.

Fourier-transform infra-red (FTIR) spectra of samples from 7 plant species were used to explore the influence of preprocessing and feature extraction on efficiency of machine learning algorithms. Wavelet Tensor Train (WTT) and Discrete Wavelet Transforms (DWT) were compared as feature extraction techniques for FTIR data of medicinal plants. Various combinations of signal processing steps showed different behavior when applied to classification and clustering tasks. Best results for WTT and DWT found through grid search were similar, significantly improving quality of clustering as well as classification accuracy for tuned logistic regression in comparison to original spectra. Unlike DWT, WTT has only one parameter to be tuned (rank), making it a more versatile and easier to use as a data processing tool in various signal processing applications.

Swarm systems constitute a challenging problem for reinforcement learning (RL) as the algorithm needs to learn decentralized control policies that can cope with limited local sensing and communication abilities of the agents. While it is often difficult to directly define the behavior of the agents, simple communication protocols can be defined more easily using prior knowledge about the given task. In this paper, we propose a number of simple communication protocols that can be exploited by deep reinforcement learning to find decentralized control policies in a multi-robot swarm environment. The protocols are based on histograms that encode the local neighborhood relations of the agents and can also transmit task-specific information, such as the shortest distance and direction to a desired target. In our framework, we use an adaptation of Trust Region Policy Optimization to learn complex collaborative tasks, such as formation building and building a communication link. We evaluate our findings in a simulated 2D-physics environment, and compare the implications of different communication protocols.

Distribution grids are currently challenged by frequent voltage excursions induced by intermittent solar generation. Smart inverters have been advocated as a fast-responding means to regulate voltage and minimize ohmic losses. Since optimal inverter coordination may be computationally challenging and preset local control rules are subpar, the approach of customized control rules designed in a quasi-static fashion features as a golden middle. Departing from affine control rules, this work puts forth non-linear inverter control policies. Drawing analogies to multi-task learning, reactive control is posed as a kernel-based regression task. Leveraging a linearized grid model and given anticipated data scenarios, inverter rules are jointly designed at the feeder level to minimize a convex combination of voltage deviations and ohmic losses via a linearly-constrained quadratic program. Numerical tests using real-world data on a benchmark feeder demonstrate that nonlinear control rules driven also by a few non-local readings can attain near-optimal performance.

Nowadays, with the availability of massive amount of trade data collected, the dynamics of the financial markets pose both a challenge and an opportunity for high frequency traders. In order to take advantage of the rapid, subtle movement of assets in High Frequency Trading (HFT), an automatic algorithm to analyze and detect patterns of price change based on transaction records must be available. The multichannel, time-series representation of financial data naturally suggests tensor-based learning algorithms. In this work, we investigate the effectiveness of two multilinear methods for the mid-price prediction problem against other existing methods. The experiments in a large scale dataset which contains more than 4 millions limit orders show that by utilizing tensor representation, multilinear models outperform vector-based approaches and other competing ones.

A novel solution is obtained to solve the rigid 3D registration problem, motivated by previous eigen-decomposition approaches. Different from existing solvers, the proposed algorithm does not require sophisticated matrix operations e.g. singular value decomposition or eigenvalue decomposition. Instead, the optimal eigenvector of the point cross-covariance matrix can be computed within several iterations. It is also proven that the optimal rotation matrix can be directly computed for cases without need of quaternion. The simple framework provides very easy approach of integer-implementation on embedded platforms. Simulations on noise-corrupted point clouds have verified the robustness and computation speed of the proposed method. The final results indicate that the proposed algorithm is accurate, robust and owns over $60\% \sim 80\%$ less computation time than representatives. It has also been applied to real-world applications for faster relative robotic navigation.

This article is a study about the existence and the uniqueness of solutions of a specific quadratic first-order ODE that frequently appears in multiple reconstruction problems. It is called the \emph{planar-perspective equation} due to the duality with the geometric problem of reconstruction of planar-perspective curves from their modulus. Solutions of the \emph{planar-perspective equation} are related with planar curves parametrized with perspective parametrization due to this geometric interpretation. The article proves the existence of only two local solutions to the \emph{initial value problem} with \emph{regular initial conditions} and a maximum of two analytic solutions with \emph{critical initial conditions}. The article also gives theorems to extend the local definition domain where the existence of both solutions are guaranteed. It introduces the \emph{maximal depth function} as a function that upper-bound all possible solutions of the \emph{planar-perspective equation} and contains all its possible \emph{critical points}. Finally, the article describes the \emph{maximal-depth solution problem} that consists of finding the solution of the referred equation that has maximum the depth and proves its uniqueness. It is an important problem as it does not need initial conditions to obtain the unique solution and its the frequent solution that practical algorithms of the state-of-the-art give.

While many studies and tools target the basic stabilizability problem of networked control systems (NCS), nowadays modern systems require more sophisticated objectives such as those expressed as formulae in linear temporal logic or as automata on infinite strings. One general technique to achieve this is based on so-called symbolic models, where complex systems are approximated by finite abstractions, and then, correct-by-construction controllers are automatically synthesized for them. We present tool SENSE for the construction of finite abstractions for NCS and the automated synthesis of controllers. Constructed controllers enforce complex specifications over plants in NCS by taking into account several non-idealities of the communication channels. Given a symbolic model of the plant and network parameters, SENSE can efficiently construct a symbolic model of the NCS, by employing operations on binary decision diagrams (BDDs). Then, it synthesizes symbolic controllers satisfying a class of specifications. It has interfaces for the simulation and the visualization of the resulting closed-loop systems using OMNETPP and MATLAB. Additionally, SENSE can generate ready-to-implement VHDL/Verilog or C/C++ codes from the synthesized controllers.

In this paper, we present two variants of DCA (Different of Convex functions Algorithm) to solve the constrained sum of differentiable function and composite functions minimization problem, with the aim of increasing the convergence speed of DCA. In the first variant, DCA-Like, we introduce a new technique to iteratively modify the decomposition of the objective function. This successive decomposition could lead to a better majorization and consequently a better convergence speed than the basic DCA. We then incorporate the Nesterov's acceleration technique into DCA-Like to give rise to the second variant, named Accelerated DCA-Like. The convergence properties and the convergence rate under Kudyka-Lojasiewicz assumption of both variants are rigorously studied. As an application, we investigate our algorithms for the t-distributed stochastic neighbor embedding. Numerical experiments on several benchmark datasets illustrate the efficiency of our algorithms.

Industrial cyber-physical systems are hybrid systems with strict safety requirements. Despite not having a formal semantics, most of these systems are modeled using Stateflow/Simulink for mainly two reasons: (1) it is easier to model, test, and simulate using these tools, and (2) dynamics of these systems are not supported by most other tools. Furthermore, with the ever growing complexity of cyber-physical systems, grows the gap between what can be modeled using an automatic formal verification tool and models of industrial cyber-physical systems. In this paper, we present a simple formal model for self-deriving cars. While after some simplification, safety of this system has already been proven manually, to the best of our knowledge, no automatic formal verification tool supports its dynamics. We hope this serves as a challenge problem for formal verification tools targeting industrial applications.

The recent popularity of post-denitrification processes in the greater Paris area wastewater treatment plants has caused a resurgence of the presence of nitrite in the Seine river. Controlling the production of nitrite during the post-denitrification has thus become a major technical issue. Research studies have been led in the MOCOPEE program (www.mocopee.com) to better understand the underlying mechanisms behind the production of nitrite during wastewater denitrification and to develop technical tools (measurement and control solutions) to assist on-site reductions of nitrite productions. Prior studies have shown that typical methanol dosage strategies produce a varying carbon-to-nitrogen ratio in the reactor, which in turn leads to unstable nitrite concentrations in the effluent. The possibility of adding a model-free control to the actual classical dosage strategy has thus been tested on the SimBio model, which simulates the behavior of wastewater biofilters. The corresponding "intelligent" feedback loop, which is using effluent nitrite concentrations, compensates the classical strategy only when needed. Simulation results show a clear improvement in average nitrite concentration level and level stability in the effluent, without a notable overcost in methanol.

This paper proposes a new method for solving the well-known rank aggregation problem from pairwise comparisons using the method of low-rank matrix completion. The partial and noisy data of pairwise comparisons is transformed into a matrix form. We then use tools from matrix completion, which has served as a major component in the low-rank completion solution of the Netflix challenge, to construct the preference of the different objects. In our approach, the data of multiple comparisons is used to create an estimate of the probability of object i to win (or be chosen) over object j, where only a partial set of comparisons between N objects is known. The data is then transformed into a matrix form for which the noiseless solution has a known rank of one. An alternating minimization algorithm, in which the target matrix takes a bilinear form, is then used in combination with maximum likelihood estimation for both factors. The reconstructed matrix is used to obtain the true underlying preference intensity. This work demonstrates the improvement of our proposed algorithm over the current state-of-the-art in both simulated scenarios and real data.

The objective of this paper is to design novel multi-layer neural network architectures for multiscale simulations of flows taking into account the observed data and physical modeling concepts. Our approaches use deep learning concepts combined with local multiscale model reduction methodologies to predict flow dynamics. Using reduced-order model concepts is important for constructing robust deep learning architectures since the reduced-order models provide fewer degrees of freedom. Flow dynamics can be thought of as multi-layer networks. More precisely, the solution (e.g., pressures and saturations) at the time instant $n+1$ depends on the solution at the time instant $n$ and input parameters, such as permeability fields, forcing terms, and initial conditions. One can regard the solution as a multi-layer network, where each layer, in general, is a nonlinear forward map and the number of layers relates to the internal time steps. We will rely on rigorous model reduction concepts to define unknowns and connections for each layer. In each layer, our reduced-order models will provide a forward map, which will be modified ("trained") using available data. It is critical to use reduced-order models for this purpose, which will identify the regions of influence and the appropriate number of variables. Because of the lack of available data, the training will be supplemented with computational data as needed and the interpolation between data-rich and data-deficient models. We will also use deep learning algorithms to train the elements of the reduced model discrete system. We will present main ingredients of our approach and numerical results. Numerical results show that using deep learning and multiscale models, we can improve the forward models, which are conditioned to the available data.

We consider networked control systems consisting of multiple independent controlled subsystems, operating over a shared communication network. Such systems are ubiquitous in cyber-physical systems, Internet of Things, and large-scale industrial systems. In many large-scale settings, the size of the communication network is smaller than the size of the system. In consequence, scheduling issues arise. The main contribution of this paper is to develop a deep reinforcement learning-based \emph{control-aware} scheduling (\textsc{DeepCAS}) algorithm to tackle these issues. We use the following (optimal) design strategy: First, we synthesize an optimal controller for each subsystem; next, we design a learning algorithm that adapts to the chosen subsystems (plants) and controllers. As a consequence of this adaptation, our algorithm finds a schedule that minimizes the \emph{control loss}. We present empirical results to show that \textsc{DeepCAS} finds schedules with better performance than periodic ones.

A supervised learning framework is proposed to approximate a model predictive controller (MPC) with reduced computational complexity and guarantees on stability and constraint satisfaction. The framework can be used for a wide class of nonlinear systems. Any standard supervised learning technique (e.g. neural networks) can be employed to approximate the MPC from samples. In order to obtain closed-loop guarantees for the learned MPC, a robust MPC design is combined with statistical learning bounds. The MPC design ensures robustness to inaccurate inputs within given bounds, and Hoeffding's Inequality is used to validate that the learned MPC satisfies these bounds with high confidence. The result is a closed-loop statistical guarantee on stability and constraint satisfaction for the learned MPC. The proposed learning-based MPC framework is illustrated on a nonlinear benchmark problem, for which we learn a neural network controller with guarantees.

This paper presents a novel method for reformulating non-differentiable collision avoidance constraints into smooth nonlinear constraints using strong duality of convex optimization. We focus on a controlled object whose goal is to avoid obstacles while moving in an n-dimensional space. The proposed reformulation does not introduce approximations, and applies to general obstacles and controlled objects that can be represented in an n-dimensional space as the finite union of convex sets. Furthermore, we connect our results with the notion of signed distance, which is widely used in traditional trajectory generation algorithms. Our method can be used in generic navigation and trajectory planning tasks, and the smoothness property allows the use of general-purpose gradient- and Hessian-based optimization algorithms. Finally, in case a collision cannot be avoided, our framework allows us to find "least-intrusive" trajectories, measured in terms of penetration. We demonstrate the efficacy of our framework on a quadcopter navigation and automated parking problem, and our numerical experiments suggest that the proposed methods enable real-time optimization-based trajectory planning problems in tight environments. Source code of our implementation is provided at https://github.com/XiaojingGeorgeZhang/OBCA.

There are many advantages to use probability method for nonlinear system identification, such as the noises and outliers in the data set do not affect the probability models significantly; the input features can be extracted in probability forms. The biggest obstacle of the probability model is the probability distributions are not easy to be obtained. In this paper, we form the nonlinear system identification into solving the conditional probability. Then we modify the restricted Boltzmann machine (RBM), such that the joint probability, input distribution, and the conditional probability can be calculated by the RBM training. Binary encoding and continue valued methods are discussed. The universal approximation analysis for the conditional probability based modelling is proposed. We use two benchmark nonlinear systems to compare our probability modelling method with the other black-box modeling methods. The results show that this novel method is much better when there are big noises and the system dynamics are complex.

An excellent self-driving car is expected to take its passengers safely and efficiently from one place to another. However, different ways of defining safety and efficiency may significantly affect the conclusion we make. In this paper, we give formal definitions to the safe state of a road and safe state of a vehicle using the syntax of linear temporal logic (LTL). We then propose the concept of safe driving throughput (SDT) and safe driving capacity (SDC) which measure the amount of vehicles in the safe state on a road. We analyze how SDT is affected by different factors. We show the analytic difference of SDC between the road with perception-based vehicles (PBV) and the road with cooperative-based vehicles (CBV). We claim that through proper design, the SDC of the road filled with PBVs will be upper-bounded by the SDC of the road filled with CBVs.

We outline a new approach for solving optimization problems which enforce triangle inequalities on output variables. We refer to this as metric-constrained optimization, and give several examples where problems of this form arise in machine learning applications and theoretical approximation algorithms for graph clustering. Although these problem are interesting from a theoretical perspective, they are challenging to solve in practice due to the high memory requirement of black-box solvers. In order to address this challenge we first prove that the metric-constrained linear program relaxation of correlation clustering is equivalent to a special case of the metric nearness problem. We then developed a general solver for metric-constrained linear and quadratic programs by generalizing and improving a simple projection algorithm originally developed for metric nearness. We give several novel approximation guarantees for using our framework to find lower bounds for optimal solutions to several challenging graph clustering problems. We also demonstrate the power of our framework by solving optimizing problems involving up to 10^{8} variables and 10^{11} constraints.

We consider a network scenario in which agents can evaluate each other according to a score graph that models some interactions. The goal is to design a distributed protocol, run by the agents, that allows them to learn their unknown state among a finite set of possible values. We propose a Bayesian framework in which scores and states are associated to probabilistic events with unknown parameters and hyperparameters, respectively. We show that each agent can learn its state by means of a local Bayesian classifier and a (centralized) Maximum-Likelihood (ML) estimator of parameter-hyperparameter that combines plain ML and Empirical Bayes approaches. By using tools from graphical models, which allow us to gain insight on conditional dependencies of scores and states, we provide a relaxed probabilistic model that ultimately leads to a parameter-hyperparameter estimator amenable to distributed computation. To highlight the appropriateness of the proposed relaxation, we demonstrate the distributed estimators on a social interaction set-up for user profiling.

In this paper a new strategy based on disturbance and uncertainty (DU) estimation and attenuation technique is proposed and tested on the nonlinear kinematic model of the differential drive mobile robot (DDMR). The proposed technique is an improved version of the Active Disturbance Rejection Control (ADRC) strategy suggested by J. Han. The ADRC is used to actively reject disturbances caused by the unknown exogenous signals and the matched uncertainties of the system model, which are lumped all together and attributed as a total disturbance. In this work, the considered system is assumed to be affine and the total disturbance and the input are considered to be on different channels. To deal with the mismatched disturbances and uncertainties, the total disturbance has been converted into a matched one. Then, based on the improved ADRC (IADRC), the dynamic performance of the DDMR has been enhanced by estimating the total disturbance and canceling it from the system. Through digital simulations, different performance measures are applied, and they all indicate the effectiveness of the proposed IADRC by almost removing the chattering phenomenon and providing a high immunity in the closed-loop system against torque disturbance.

A core novelty of Alpha Zero is the interleaving of tree search and deep learning, which has proven very successful in board games like Chess, Shogi and Go. These games have a discrete action space. However, many real-world reinforcement learning domains have continuous action spaces, for example in robotic control, navigation and self-driving cars. This paper presents the necessary theoretical extensions of Alpha Zero to deal with continuous action space. We also provide some preliminary experiments on the Pendulum swing-up task, empirically showing the feasibility of our approach. Thereby, this work provides a first step towards the application of iterated search and learning in domains with a continuous action space.

We consider the problem of optimization of cost functionals on the infinite-dimensional manifold of diffeomorphisms. We present a new class of optimization methods, valid for any optimization problem setup on the space of diffeomorphisms by generalizing Nesterov accelerated optimization to the manifold of diffeomorphisms. While our framework is general for infinite dimensional manifolds, we specifically treat the case of diffeomorphisms, motivated by optical flow problems in computer vision. This is accomplished by building on a recent variational approach to a general class of accelerated optimization methods by Wibisono, Wilson and Jordan, which applies in finite dimensions. We generalize that approach to infinite dimensional manifolds. We derive the surprisingly simple continuum evolution equations, which are partial differential equations, for accelerated gradient descent, and relate it to simple mechanical principles from fluid mechanics. Our approach has natural connections to the optimal mass transport problem. This is because one can think of our approach as an evolution of an infinite number of particles endowed with mass (represented with a mass density) that moves in an energy landscape. The mass evolves with the optimization variable, and endows the particles with dynamics. This is different than the finite dimensional case where only a single particle moves and hence the dynamics does not depend on the mass. We derive the theory, compute the PDEs for accelerated optimization, and illustrate the behavior of these new accelerated optimization schemes.

In low-rank tensor completion tasks, due to the underlying multiple large-scale singular value decomposition (SVD) operations and rank selection problem of the traditional methods, they suffer from high computational cost and high sensitivity of model complexity. In this paper, taking advantages of high compressibility of the recently proposed tensor ring (TR) decomposition, we propose a new model for tensor completion problem. This is achieved through introducing convex surrogates of tensor low-rank assumption on latent tensor ring factors, which makes it possible for the Schatten norm regularization based models to be solved at much smaller scale. We propose two algorithms which apply different structured Schatten norms on tensor ring factors respectively. By the alternating direction method of multipliers (ADMM) scheme, the tensor ring factors and the predicted tensor can be optimized simultaneously. The experiments on synthetic data and real-world data show the high performance and efficiency of the proposed approach.

The supervisory control of probabilistic discrete event systems (PDESs) is investigated under the assumptions that the supervisory controller (supervisor) is probabilistic and has a partial observation. The probabilistic P-supervisor is defined, which specifies a probability distribution on the control patterns for each observation. The notions of the probabilistic controllability and observability are proposed and demonstrated to be a necessary and sufficient conditions for the existence of the probabilistic P-supervisors. Moreover, the polynomial verification algorithms for the probabilistic controllability and observability are put forward. In addition, the infimal probabilistic controllable and observable superlanguage is introduced and computed as the solution of the optimal control problem of PDESs. Several examples are presented to illustrate the results obtained.

In this paper, we consider an adversarial scenario where one agent seeks to achieve an objective and its adversary seeks to learn the agent's intentions and prevent the agent from achieving its objective. The agent has an incentive to try to deceive the adversary about its intentions, while at the same time working to achieve its objective. The primary contribution of this paper is to introduce a mathematically rigorous framework for the notion of deception within the context of optimal control. The central notion introduced in the paper is that of a belief-induced reward: a reward dependent not only on the agent's state and action, but also adversary's beliefs. Design of an optimal deceptive strategy then becomes a question of optimal control design on the product of the agent's state space and the adversary's belief space. The proposed framework allows for deception to be defined in an arbitrary control system endowed with a reward function, as well as with additional specifications limiting the agent's control policy. In addition to defining deception, we discuss design of optimally deceptive strategies under uncertainties in agent's knowledge about the adversary's learning process. In the latter part of the paper, we focus on a setting where the agent's behavior is governed by a Markov decision process, and show that the design of optimally deceptive strategies under lack of knowledge about the adversary naturally reduces to previously discussed problems in control design on partially observable or uncertain Markov decision processes. Finally, we present two examples of deceptive strategies: a "cops and robbers" scenario and an example where an agent may use camouflage while moving. We show that optimally deceptive strategies in such examples follow the intuitive idea of how to deceive an adversary in the above settings.

To operate effectively in tomorrow's smart cities, autonomous vehicles (AVs) must rely on intra-vehicle sensors such as camera and radar as well as inter-vehicle communication. Such dependence on sensors and communication links exposes AVs to cyber-physical (CP) attacks by adversaries that seek to take control of the AVs by manipulating their data. Thus, to ensure safe and optimal AV dynamics control, the data processing functions at AVs must be robust to such CP attacks. To this end, in this paper, the state estimation process for monitoring AV dynamics, in presence of CP attacks, is analyzed and a novel adversarial deep reinforcement learning (RL) algorithm is proposed to maximize the robustness of AV dynamics control to CP attacks. The attacker's action and the AV's reaction to CP attacks are studied in a game-theoretic framework. In the formulated game, the attacker seeks to inject faulty data to AV sensor readings so as to manipulate the inter-vehicle optimal safe spacing and potentially increase the risk of AV accidents or reduce the vehicle flow on the roads. Meanwhile, the AV, acting as a defender, seeks to minimize the deviations of spacing so as to ensure robustness to the attacker's actions. Since the AV has no information about the attacker's action and due to the infinite possibilities for data value manipulations, the outcome of the players' past interactions are fed to long-short term memory (LSTM) blocks. Each player's LSTM block learns the expected spacing deviation resulting from its own action and feeds it to its RL algorithm. Then, the the attacker's RL algorithm chooses the action which maximizes the spacing deviation, while the AV's RL algorithm tries to find the optimal action that minimizes such deviation.