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Symbolic Regression for Reinforcement Learning and Dynamic System Modeling Robert Babuka 1 Research interests Clustering for building locally linear models Reinforcement learning for continuous dynamic systems Neural

  1. Symbolic Regression for Reinforcement Learning and Dynamic System Modeling Robert Babuška 1

  2. Research interests • Clustering for building locally linear models • Reinforcement learning for continuous dynamic systems • Neural networks, deep learning • Genetic programming, symbolic regression • Applications in robotics and motion control 2

  3. Deep reinforcement learning + Excellent for state representation using high-dimensional input - Many hyper-parameters to tune - Unpredictable and difficult to reproduce - High computational costs Useful to investigate other representations! Genetic programming and symbolic regression are tools that definitely deserve more attention. 3

  4. Genetic Programming, Symbolic Regression 4

  5. Symbolic Regression -3.141592654 -30 -23.34719731 f = -15.42978401 + 2.42980826 * ((x1 – (x1 * -2.932153143 -30 -22.67195916 -1.49416733 + x2 * 0.51196778 + 0.00000756)) + -2.722713633 -30 -22.07798667 (sqrt(power((x1 – (x1 * -1.49416733 + x2 * -2.513274123 -30 -21.63117778 0.51196778 + 0.00000756)), 2) + 1) – 1) / 2) ... -2.303834613 -30 -21.2992009 ... ... ... 5

  6. Symbolic Regression Algorithms – � � + –  / * 𝑧 � � 𝛽 � 𝐺 � �𝑦 � , … , 𝑦 � � x / x x x + x + ��� x x x x Multiple Regression Genetic Programming [1] • Evolutionary Feature Synthesis [2] • Multi-Gene Genetic Programming [3] • Single Node Genetic Programming [4, 5] • [1] I. Arnaldo et al.: Multiple regression genetic programming (2014) • [2] I. Arnaldo et al.: Building predictive models via feature synthesis (2015) • [3] M. Hinchliffe et al.: Modelling chemical process systems using a multi-gene genetic programming • algorithm (1996) [4] D. Jackson: Single node genetic programming on problems with side effects (2012) • [5] J. Kubalík et al.: An improved Single Node Genetic Programming for symbolic regression (2015) • 6

  7. Symbolic Regression Algorithms – � � + –  / * 𝑧 � � 𝛽 � 𝐺 � �𝑦 � , … , 𝑦 � � x / x x x + x + ��� x x x x Multiple Regression Genetic Programming [1] • Evolutionary Feature Synthesis [2] • Multi-Gene Genetic Programming (MGGP) [3] • Single Node Genetic Programming (SNGP) [4, 5] • [1] I. Arnaldo et al.: Multiple regression genetic programming (2014) • [2] I. Arnaldo et al.: Building predictive models via feature synthesis (2015) • [3] M. Hinchliffe et al.: Modelling chemical process systems using a multi-gene genetic programming • algorithm (1996) [4] D. Jackson: Single node genetic programming on problems with side effects (2012) • [5] J. Kubalík et al.: An improved Single Node Genetic Programming for symbolic regression (2015) • 7

  8. Basic SNGP Σ 𝛽 � 𝛽 � � � – +  / – * 𝑁 � � 𝛽 � 𝐺 � �𝑦 � , … , 𝑦 � � x / x x x + ��� x + x x F 2 F 1 x x J. Kubalík et al.: Hybrid single node genetic programming for symbolic regression (2016) 8

  9. Modifications and extensions SNGP and MGGP with affine transformation of input variables [1,2] • MGGP: Backpropagation for model tuning and tracking dynamic data [2] • SNGP with partitioned population [3] • Multi-objective SNGP [4] • [1] J. Kubalík et al.: Enhanced Symbolic Regression Through Local Variable Transformations (2017) • [2] J. Žegklitz, P. Pošík: Symbolic Regression in Dynamic Scenarios with Gradually Changing Targets • (2019) [3] Alibekov et al.: Symbolic Method for Deriving Policy in Reinforcement Learning (2016). • [4] J. Kubalík et al.: Learning Accurate Robot Models via Combination of Prior Knowledge and Data • (submitted, 2019) 9

  10. Affine transformation of inputs: motivation 10

  11. Extended SNGP population Standard SNGP : Partitioned population and transformed inputs: 11

  12. Benefits of transformed inputs 2 𝑔 𝑦 � , 𝑦 � � 0.1�0.5𝑦 � � 0.5𝑦 � � � 1 � 𝑓 ���.�� � ��.�� � � Transformed input variables: Original SNGP: f = 1.27297628 * sigmoid(x1 + x2 – 0.0625 * f = -2.6 + 0.1 * (36.0 + v1) – 2.0 * (0.5 – x1) – 0.38266172 * (power((0.0625 * x1), 3) – sigmoid(v1)) – 9.0E-8 * (sigmoid(v2 – 81.0) (0.22340393 * ((x1 + x2) – (0.0625 * x1)))) – * 0.00195313) 2.7355E-4 * ((power(x1, 2) * x2 – x1 – (30.25 v1 = 0.5 * x1 + 0.5 * x2 * (x1 + sigmoid(x2))))) + 0.35937439 v2 = 0.07105142 * x1 + 0.07105142 * x2 + 4.24664016 RMSE = 5.78E-2 RMSE = 6.31E-10 12

  13. Solving Bellman equation via genetic programming 13

  14. Solve Bellman equation by using GP Generate data: Bellman equation in terms of the data: 14

  15. Direct solution of Bellman equation Fitness function: Use GP to find a symbolic representation of V 15

  16. Symbolic value iteration (SVI) Symbolic V-function Target data from previous iteration Symbolic – regression – cos / x 1 x 2 + x 3 x 2 x 1 16

  17. Pendulum swing-up: symbolic value iteration

  18. V function for 1-DOF pendulum swing-up 89 parameters 18

  19. V-function for 1-DOF pendulum swing-up 89 parameters 961 parameters 19

  20. V-function for 1-DOF pendulum swing-up Baseline V-function Symbolic V-function Smooth swing-up trajectory Less smooth trajectory 20

  21. Comparison with a neural network Neural network V-function Symbolic V-function 89 parameters 201 parameters 21

  22. Swing-up experiment on the real system Control action Pendulum angle Performance very close to theoretically optimal bang-bang control 22

  23. Conclusions on symbolic value functions Compact and typically very smooth V-functions. Analytic, can be plugged • in other algorithms. Near optimal control performance, outperforms other approximators • (basis functions, DNN). High computational costs, comparable to NN. • So far tested on systems with a small number of state variables. • Challenges: Direct solution, high-dimensional state spaces, convergence guarantees, model-free variant. 23

  24. Genetic programming for building dynamic models 24

  25. Symbolic regression for modeling dynamic systems Predicted output Past outputs Past inputs Nonlinear autoregressive with exogenous input model (NARX) 25

  26. Challenges of model building for dynamic systems Use short data sequences • Consistent models of multi-variable systems • Include prior knowledge • Automatically select data for updating models • Model accuracy – complexity tradeoff • 26

  27. Challenges of model building for dynamic systems Use short data sequences • Consistent models of multi-variable systems • Include prior knowledge • Automatically select data for updating models • Model accuracy – complexity tradeoff • 27

  28. Mobile robot experiments Mechanistic model: Mechanistic model correctly represents the physics, but is inaccurate as • a prediction model (actuator nonlinearities). Data-driven model constructed via symbolic regression is accurate, but • does not necessarily respect the physical constraints. 28

  29. Motion planning with Motion planning with mechanistic model data-driven model 29

  30. Solution: include prior knowledge Generate synthetic data representing physical constraints, use MO GP Examples: Equilibrium under zero input • Non-holonomic constraint (robot cannot move sideways) • 30

  31. Conclusions on symbolic model construction Accurate and compact models from small data sets • Model structure can be constrained to a specific model class • Challenges: Effective incorporation of prior knowledge, computational costs, multi-dimensional models. 31

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