## 2018 |

Tal, Ezra; Gorodetsky, Alex; Karaman, Sertac Continuous Tensor Train-Based Dynamic Programming for High-Dimensional Zero-Sum Differential Games Inproceedings Forthcoming American Control Conference (ACC), Milwaukee, WI, Forthcoming. Abstract | Links | BibTeX | Tags: Differential Games, Stochastic Optimal Control, Tensor decompositions @inproceedings{Tal2018, title = {Continuous Tensor Train-Based Dynamic Programming for High-Dimensional Zero-Sum Differential Games}, author = {Ezra Tal and Alex Gorodetsky and Sertac Karaman}, url = {http://www.alexgorodetsky.com/wp-content/uploads/2018/03/tal-ea-ACC2018.pdf}, year = {2018}, date = {2018-06-28}, booktitle = {American Control Conference (ACC)}, address = {Milwaukee, WI}, abstract = {Zero-sum differential games are a prominent research topic in a wide variety of fields ranging from motion planning under adversarial conditions to economics modeling. In the field of robust control, synthesis using game theory can also be used to obtain performance guarantees. However, most existing computational methods for differential games suffer from the curse of dimensionality, i.e., the computational requirements grow exponentially with the dimensionality of the state space. In the present work, we aim to alleviate the curse of dimensionality with a novel dynamic-programming-based algorithm that uses a continuous tensor-train approximation to represent the value function. This approximation method can represent and compute with high-dimensional tensors using computational resources that grow polynomially with dimensionality and polynomially with the rank of the value function. The proposed algorithm is shown to converge to optimal solutions with arbitrary bounds for a general class of differential games. The new algorithm is demonstrated using several game scenarios, achieving more than five orders of magnitude savings in memory and three order of magnitude in computational cost in a six-dimensional problem, when compared to the standard value iteration.}, keywords = {Differential Games, Stochastic Optimal Control, Tensor decompositions}, pubstate = {forthcoming}, tppubtype = {inproceedings} } Zero-sum differential games are a prominent research topic in a wide variety of fields ranging from motion planning under adversarial conditions to economics modeling. In the field of robust control, synthesis using game theory can also be used to obtain performance guarantees. However, most existing computational methods for differential games suffer from the curse of dimensionality, i.e., the computational requirements grow exponentially with the dimensionality of the state space. In the present work, we aim to alleviate the curse of dimensionality with a novel dynamic-programming-based algorithm that uses a continuous tensor-train approximation to represent the value function. This approximation method can represent and compute with high-dimensional tensors using computational resources that grow polynomially with dimensionality and polynomially with the rank of the value function. The proposed algorithm is shown to converge to optimal solutions with arbitrary bounds for a general class of differential games. The new algorithm is demonstrated using several game scenarios, achieving more than five orders of magnitude savings in memory and three order of magnitude in computational cost in a six-dimensional problem, when compared to the standard value iteration. |

Gorodetsky, Alex; Karaman, Sertac; Marzouk, Youssef High-Dimensional Stochastic Optimal Control using Continuous Tensor Decompositions Journal Article International Journal of Robotics Research, 37 (2-3), pp. 340-377, 2018. Abstract | Links | BibTeX | Tags: Dynamic Programming, Markov Decision Processes, Stochastic Optimal Control, Tensor decompositions @article{Gorodetsky2018, title = {High-Dimensional Stochastic Optimal Control using Continuous Tensor Decompositions}, author = {Alex Gorodetsky and Sertac Karaman and Youssef Marzouk}, url = {https://alexgorodetsky.com/wp-content/uploads/2018/02/1611.04706.pdf}, doi = {10.1177/0278364917753994}, year = {2018}, date = {2018-03-19}, journal = {International Journal of Robotics Research}, volume = {37}, number = {2-3}, pages = {340-377}, abstract = {Motion planning and control problems are embedded and essential in almost all robotics applications. These problems are often formulated as stochastic optimal control problems and solved using dynamic programming algorithms. Unfortunately, most existing algorithms that guarantee convergence to optimal solutions suffer from the curse of dimensionality: the run time of the algorithm grows exponentially with the dimension of the state space of the system. We propose novel dynamic programming algorithms that alleviate the curse of dimensionality in problems that exhibit certain low-rank structure. The proposed algorithms are based on continuous tensor decompositions recently developed by the authors. Essentially, the algorithms represent high-dimensional functions (e.g., the value function) in a compressed format, and directly perform dynamic programming computations (e.g., value iteration, policy iteration) in this format. Under certain technical assumptions, the new algorithms guarantee convergence towards optimal solutions with arbitrary precision. Furthermore, the run times of the new algorithms scale polynomially with the state dimension and polynomially with the ranks of the value function. This approach realizes substantial computational savings in "compressible" problem instances, where value functions admit low-rank approximations. We demonstrate the new algorithms in a wide range of problems, including a simulated six-dimensional agile quadcopter maneuvering example and a seven-dimensional aircraft perching example. In some of these examples, we estimate computational savings of up to ten orders of magnitude over standard value iteration algorithms. We further demonstrate the algorithms running in real time on board a quadcopter during a flight experiment under motion capture.}, keywords = {Dynamic Programming, Markov Decision Processes, Stochastic Optimal Control, Tensor decompositions}, pubstate = {published}, tppubtype = {article} } Motion planning and control problems are embedded and essential in almost all robotics applications. These problems are often formulated as stochastic optimal control problems and solved using dynamic programming algorithms. Unfortunately, most existing algorithms that guarantee convergence to optimal solutions suffer from the curse of dimensionality: the run time of the algorithm grows exponentially with the dimension of the state space of the system. We propose novel dynamic programming algorithms that alleviate the curse of dimensionality in problems that exhibit certain low-rank structure. The proposed algorithms are based on continuous tensor decompositions recently developed by the authors. Essentially, the algorithms represent high-dimensional functions (e.g., the value function) in a compressed format, and directly perform dynamic programming computations (e.g., value iteration, policy iteration) in this format. Under certain technical assumptions, the new algorithms guarantee convergence towards optimal solutions with arbitrary precision. Furthermore, the run times of the new algorithms scale polynomially with the state dimension and polynomially with the ranks of the value function. This approach realizes substantial computational savings in "compressible" problem instances, where value functions admit low-rank approximations. We demonstrate the new algorithms in a wide range of problems, including a simulated six-dimensional agile quadcopter maneuvering example and a seven-dimensional aircraft perching example. In some of these examples, we estimate computational savings of up to ten orders of magnitude over standard value iteration algorithms. We further demonstrate the algorithms running in real time on board a quadcopter during a flight experiment under motion capture. |

## 2017 |

Gorodetsky, Alex Massachusetts Institute of Technology, 2017. Abstract | Links | BibTeX | Tags: Dynamic Programming, Kalman Filtering, Stochastic Optimal Control, Surrogate models, Tensor decompositions @phdthesis{Gorodetsky2017b, title = {Continuous low-rank tensor decompositions, with applications to stochastic optimal control and data assimilation}, author = {Alex Gorodetsky}, url = {http://hdl.handle.net/1721.1/108918}, year = {2017}, date = {2017-02-02}, address = {Cambridge, MA}, school = {Massachusetts Institute of Technology}, abstract = {Optimal decision making under uncertainty is critical for control and optimization of complex systems. However, many techniques for solving problems such as stochastic optimal control and data assimilation encounter the curse of dimensionality when too many state variables are involved. In this thesis, we propose a framework for computing with high-dimensional functions that mitigates this exponential growth in complexity for problems with separable structure. Our framework tightly integrates two emerging areas: tensor decompositions and continuous computation. Tensor decompositions are able to effectively compress and operate with low-rank multidimensional arrays. Continuous computation is a paradigm for computing with functions instead of arrays, and it is best realized by Chebfun, a MATLAB package for computing with functions of up to three dimensions. Continuous computation provides a natural framework for building numerical algorithms that effectively, naturally, and automatically adapt to problem structure. The first part of this thesis describes a compressed continuous computation framework centered around a continuous analogue to the (discrete) tensor-train decomposition called the function-train decomposition. Computation with the function-train requires continuous matrix factorizations and continuous numerical linear algebra. Continuous analogues are presented for performing cross approximation; rounding; multilinear algebra operations such as addition, multiplication, integration, and differentiation; and continuous, rank-revealing, alternating least squares. Advantages of the function-train over the tensor-train include the ability to adaptively approximate functions and the ability to compute with functions that are parameterized differently. For example, while elementwise multiplication between tensors of different sizes is undefined, functions in FT format can be readily multiplied together. Next, we develop compressed versions of value iteration, policy iteration, and multilevel algorithms for solving dynamic programming problems arising in stochastic optimal control. These techniques enable computing global solutions to a broader set of problems, for example those with non-affine control inputs, than previously possible. Examples are presented for motion planning with robotic systems that have up to seven states. Finally, we use the FT to extend integration-based Gaussian filtering to larger state spaces than previously considered. Examples are presented for dynamical systems with up to twenty states.}, keywords = {Dynamic Programming, Kalman Filtering, Stochastic Optimal Control, Surrogate models, Tensor decompositions}, pubstate = {published}, tppubtype = {phdthesis} } Optimal decision making under uncertainty is critical for control and optimization of complex systems. However, many techniques for solving problems such as stochastic optimal control and data assimilation encounter the curse of dimensionality when too many state variables are involved. In this thesis, we propose a framework for computing with high-dimensional functions that mitigates this exponential growth in complexity for problems with separable structure. Our framework tightly integrates two emerging areas: tensor decompositions and continuous computation. Tensor decompositions are able to effectively compress and operate with low-rank multidimensional arrays. Continuous computation is a paradigm for computing with functions instead of arrays, and it is best realized by Chebfun, a MATLAB package for computing with functions of up to three dimensions. Continuous computation provides a natural framework for building numerical algorithms that effectively, naturally, and automatically adapt to problem structure. The first part of this thesis describes a compressed continuous computation framework centered around a continuous analogue to the (discrete) tensor-train decomposition called the function-train decomposition. Computation with the function-train requires continuous matrix factorizations and continuous numerical linear algebra. Continuous analogues are presented for performing cross approximation; rounding; multilinear algebra operations such as addition, multiplication, integration, and differentiation; and continuous, rank-revealing, alternating least squares. Advantages of the function-train over the tensor-train include the ability to adaptively approximate functions and the ability to compute with functions that are parameterized differently. For example, while elementwise multiplication between tensors of different sizes is undefined, functions in FT format can be readily multiplied together. Next, we develop compressed versions of value iteration, policy iteration, and multilevel algorithms for solving dynamic programming problems arising in stochastic optimal control. These techniques enable computing global solutions to a broader set of problems, for example those with non-affine control inputs, than previously possible. Examples are presented for motion planning with robotic systems that have up to seven states. Finally, we use the FT to extend integration-based Gaussian filtering to larger state spaces than previously considered. Examples are presented for dynamical systems with up to twenty states. |

## 2016 |

Alora, John Irvin; Gorodetsky, Alex; Karaman, Sertac; Marzouk, Youssef; Lowry, Nathan Automated synthesis of low-rank control systems from sc-LTL specifications using tensor-train decompositions Inproceedings 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 1131-1138, Las Vegas, NV, USA, 2016, ISBN: 978-1-5090-1837-6. Abstract | Links | BibTeX | Tags: Linear Temporal Logic, Stochastic Optimal Control, Tensor decompositions @inproceedings{Alora2016, title = {Automated synthesis of low-rank control systems from sc-LTL specifications using tensor-train decompositions}, author = {John Irvin Alora and Alex Gorodetsky and Sertac Karaman and Youssef Marzouk and Nathan Lowry}, url = {https://alexgorodetsky.com/wp-content/uploads/2018/02/Alora.Gorodetsky.ea_.CDC16.pdf}, doi = {10.1109/CDC.2016.7798419}, isbn = {978-1-5090-1837-6}, year = {2016}, date = {2016-12-12}, booktitle = {2016 IEEE 55th Conference on Decision and Control (CDC)}, pages = {1131-1138}, address = {Las Vegas, NV, USA}, abstract = {Correct-by-design automated construction of control systems has attracted a tremendous amount of attention. However, most existing algorithms for automated construction suffer from the curse of dimensionality, i.e., their run time scales exponentially with increasing dimensionality of the state space. As a result, typically, systems with only a few degrees of freedom are considered. In this paper, we propose a novel algorithm based on the tensor-train decomposition that solves stochastic optimal control problems with syntactically co-safe linear temporal logic specifications. We show that, under certain conditions, the run time of the proposed algorithm scales polynomially with the dimensionality of the state space and the rank of the optimal cost-to-go function. We demonstrate the algorithm in a six-dimensional problem instance involving a simple airplane model. In this example, the proposed algorithm provides up to four orders of computational savings when compared to the standard value iteration algorithm.}, keywords = {Linear Temporal Logic, Stochastic Optimal Control, Tensor decompositions}, pubstate = {published}, tppubtype = {inproceedings} } Correct-by-design automated construction of control systems has attracted a tremendous amount of attention. However, most existing algorithms for automated construction suffer from the curse of dimensionality, i.e., their run time scales exponentially with increasing dimensionality of the state space. As a result, typically, systems with only a few degrees of freedom are considered. In this paper, we propose a novel algorithm based on the tensor-train decomposition that solves stochastic optimal control problems with syntactically co-safe linear temporal logic specifications. We show that, under certain conditions, the run time of the proposed algorithm scales polynomially with the dimensionality of the state space and the rank of the optimal cost-to-go function. We demonstrate the algorithm in a six-dimensional problem instance involving a simple airplane model. In this example, the proposed algorithm provides up to four orders of computational savings when compared to the standard value iteration algorithm. |

## 2015 |

Gorodetsky, Alex; Karaman, Sertac; Marzouk, Youssef Efficient High-Dimensional Stochastic Optimal Motion Control using Tensor-Train Decomposition Inproceedings Proceedings of Robotics: Science and Systems, Rome, Italy, 2015. Abstract | Links | BibTeX | Tags: Dynamic Programming, Motion Planning, Stochastic Optimal Control, Tensor-train decomposition @inproceedings{Gorodetsky2015, title = {Efficient High-Dimensional Stochastic Optimal Motion Control using Tensor-Train Decomposition}, author = {Alex Gorodetsky and Sertac Karaman and Youssef Marzouk}, url = {https://alexgorodetsky.com/wp-content/uploads/2018/02/p15.pdf}, doi = {10.15607/RSS.2015.XI.015}, year = {2015}, date = {2015-07-01}, booktitle = {Proceedings of Robotics: Science and Systems}, address = {Rome, Italy}, abstract = {Stochastic optimal control problems frequently arise as motion control problems in the context of robotics. Unfortunately, all existing approaches that guarantee arbitrary precision suffer from the curse of dimensionality: the computational effort invested by the algorithm grows exponentially fast with increasing dimensionality of the state space of the underlying dynamic system governing the robot. In this paper, we propose a novel algorithm that utilizes compressed representations to efficiently solve stochastic optimal control problems with arbitrary precision. The running time of the new algorithms scale linearly with increasing dimensionality of the state space! The running time also depends polynomially on the rank of the value function, a measure that quantifies the intrinsic geometric complexity of the value function, due to the geometry and physics embedded in the problem instance at hand. The new algorithms are based on the recent analysis and algorithms for tensor decomposition, generalizing matrix decomposition algorithms, e.g., the singular value decomposition, to three or more dimensions. In computational experiments, we show the computational effort of the new algorithm also scales linearly with the discretization resolution of the state space. We also demonstrate the new algorithm on a problem involving the perching of an aircraft, represented by a nonlinear non-holonomic longitudinal model with a seven-dimensional state space, the full numerical solution to which was not obtained before. In this example, we estimate that the proposed algorithm runs more than seven orders of magnitude faster, when compared to the naive value iteration}, keywords = {Dynamic Programming, Motion Planning, Stochastic Optimal Control, Tensor-train decomposition}, pubstate = {published}, tppubtype = {inproceedings} } Stochastic optimal control problems frequently arise as motion control problems in the context of robotics. Unfortunately, all existing approaches that guarantee arbitrary precision suffer from the curse of dimensionality: the computational effort invested by the algorithm grows exponentially fast with increasing dimensionality of the state space of the underlying dynamic system governing the robot. In this paper, we propose a novel algorithm that utilizes compressed representations to efficiently solve stochastic optimal control problems with arbitrary precision. The running time of the new algorithms scale linearly with increasing dimensionality of the state space! The running time also depends polynomially on the rank of the value function, a measure that quantifies the intrinsic geometric complexity of the value function, due to the geometry and physics embedded in the problem instance at hand. The new algorithms are based on the recent analysis and algorithms for tensor decomposition, generalizing matrix decomposition algorithms, e.g., the singular value decomposition, to three or more dimensions. In computational experiments, we show the computational effort of the new algorithm also scales linearly with the discretization resolution of the state space. We also demonstrate the new algorithm on a problem involving the perching of an aircraft, represented by a nonlinear non-holonomic longitudinal model with a seven-dimensional state space, the full numerical solution to which was not obtained before. In this example, we estimate that the proposed algorithm runs more than seven orders of magnitude faster, when compared to the naive value iteration |