Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
97 tokens/sec
GPT-4o
53 tokens/sec
Gemini 2.5 Pro Pro
44 tokens/sec
o3 Pro
5 tokens/sec
GPT-4.1 Pro
47 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Deep Model Predictive Optimization (2310.04590v1)

Published 6 Oct 2023 in cs.RO, cs.LG, cs.AI, cs.SY, and eess.SY

Abstract: A major challenge in robotics is to design robust policies which enable complex and agile behaviors in the real world. On one end of the spectrum, we have model-free reinforcement learning (MFRL), which is incredibly flexible and general but often results in brittle policies. In contrast, model predictive control (MPC) continually re-plans at each time step to remain robust to perturbations and model inaccuracies. However, despite its real-world successes, MPC often under-performs the optimal strategy. This is due to model quality, myopic behavior from short planning horizons, and approximations due to computational constraints. And even with a perfect model and enough compute, MPC can get stuck in bad local optima, depending heavily on the quality of the optimization algorithm. To this end, we propose Deep Model Predictive Optimization (DMPO), which learns the inner-loop of an MPC optimization algorithm directly via experience, specifically tailored to the needs of the control problem. We evaluate DMPO on a real quadrotor agile trajectory tracking task, on which it improves performance over a baseline MPC algorithm for a given computational budget. It can outperform the best MPC algorithm by up to 27% with fewer samples and an end-to-end policy trained with MFRL by 19%. Moreover, because DMPO requires fewer samples, it can also achieve these benefits with 4.3X less memory. When we subject the quadrotor to turbulent wind fields with an attached drag plate, DMPO can adapt zero-shot while still outperforming all baselines. Additional results can be found at https://tinyurl.com/mr2ywmnw.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (95)
  1. M. A. OpenAI, B. Baker, M. Chociej, R. Józefowicz, B. McGrew, J. W. Pachocki, J. Pachocki, A. Petron, M. Plappert, G. Powell, et al., “Learning Dexterous In-Hand Manipulation,” arXiv preprint arXiv:1808.00177, 2018.
  2. K. Huang, R. Rana, G. Shi, A. Spitzer, and B. Boots, “DATT: Deep Adaptive Trajectory Tracking for Quadrotor Control,” in Conference on Robot Learning (CoRL), 2023.
  3. E. Kaufmann, L. Bauersfeld, A. Loquercio, M. Müller, V. Koltun, and D. Scaramuzza, “Champion-Level Drone Racing Using Deep Reinforcement Learning,” Nature, vol. 620, no. 7976, pp. 982–987, 2023.
  4. M. O’Connell, G. Shi, X. Shi, K. Azizzadenesheli, A. Anandkumar, Y. Yue, and S.-J. Chung, “Neural-fly enables rapid learning for agile flight in strong winds,” Science Robotics, vol. 7, no. 66, p. eabm6597, 2022.
  5. G. Shi, X. Shi, M. O’Connell, R. Yu, K. Azizzadenesheli, A. Anandkumar, Y. Yue, and S.-J. Chung, “Neural lander: Stable drone landing control using learned dynamics,” in 2019 international conference on robotics and automation (icra).   IEEE, 2019, pp. 9784–9790.
  6. J. Tobin, R. Fong, A. Ray, J. Schneider, W. Zaremba, and P. Abbeel, “Domain Randomization for Transferring Deep Neural Networks from Simulation to the Real World,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).   IEEE, 2017, pp. 23–30.
  7. X. B. Peng, M. Andrychowicz, W. Zaremba, and P. Abbeel, “Sim-to-Real Transfer of Robotic Control with Dynamics Randomization,” in IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2018, pp. 3803–3810.
  8. X. Chen, J. Hu, C. Jin, L. Li, and L. Wang, “Understanding Domain Randomization for Sim-to-Real Transfer,” arXiv preprint arXiv:2110.03239, 2021.
  9. M. Morari and J. H. Lee, “Model predictive control: past, present and future,” Computers & chemical engineering, vol. 23, no. 4-5, pp. 667–682, 1999.
  10. G. Williams, N. Wagener, B. Goldfain, P. Drews, J. M. Rehg, B. Boots, and E. A. Theodorou, “Information theoretic MPC for model-based reinforcement learning,” in IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2017, pp. 1714–1721.
  11. C. Yu, G. Shi, S.-J. Chung, Y. Yue, and A. Wierman, “The power of predictions in online control,” Advances in Neural Information Processing Systems, vol. 33, pp. 1994–2004, 2020.
  12. N. Wagener, C.-A. Cheng, J. Sacks, and B. Boots, “An Online Learning Approach to Model Predictive Control,” arXiv preprint arXiv:1902.08967, 2019.
  13. T. Erez, Y. Tassa, and E. Todorov, “Infinite-Horizon Model Predictive Control for Periodic Tasks with Contacts,” Robotics: Science and systems VII, p. 73, 2012.
  14. N. Jiang, A. Kulesza, S. Singh, and R. Lewis, “The Dependence of Effective Planning Horizon on Model Accuracy,” in Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems, 2015, pp. 1181–1189.
  15. A. Tamar, G. Thomas, T. Zhang, S. Levine, and P. Abbeel, “Learning from the Hindsight Plan — Episodic MPC Improvement,” in IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2017, pp. 336–343.
  16. T. Li, R. Yang, G. Qu, G. Shi, C. Yu, A. Wierman, and S. Low, “Robustness and consistency in linear quadratic control with untrusted predictions,” Proceedings of the ACM on Measurement and Analysis of Computing Systems, vol. 6, no. 1, pp. 1–35, 2022.
  17. A. Jain, L. Chan, D. S. Brown, and A. D. Dragan, “Optimal Cost Design for Model Predictive Control,” in Learning for Dynamics and Control.   PMLR, 2021, pp. 1205–1217.
  18. J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov, “Proximal Policy Optimization Algorithms,” arXiv preprint arXiv:1707.06347, 2017.
  19. J. Schulman, S. Levine, P. Abbeel, M. Jordan, and P. Moritz, “Trust Region Policy Optimization,” in International Conference on Machine Learning (ICML).   PMLR, 2015, pp. 1889–1897.
  20. G. Williams, P. Drews, B. Goldfain, J. M. Rehg, and E. A. Theodorou, “Aggressive driving with model predictive path integral control,” in IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2016, pp. 1433–1440.
  21. J. Sacks and B. Boots, “Learning to Optimize in Model Predictive Control,” in IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2022, pp. 10 549–10 556.
  22. M. Andrychowicz, M. Denil, S. Gomez, M. W. Hoffman, D. Pfau, T. Schaul, B. Shillingford, and N. De Freitas, “Learning to learn by gradient descent by gradient descent,” in Advances in Neural Information Processing Systems (NeurIPS), 2016, pp. 3981–3989.
  23. S. Ravi and H. Larochelle, “Optimization as a Model for Few-Shot Learning,” International Conference on Learning Representations (ICLR), 2016.
  24. K. Lv, S. Jiang, and J. Li, “Learning Gradient Descent: Better Generalization and Longer Horizons,” in International Conference on Machine Learning (ICML).   PMLR, 2017, pp. 2247–2255.
  25. T. Chen, W. Zhang, Z. Jingyang, S. Chang, S. Liu, L. Amini, and Z. Wang, “Training Stronger Baselines for Learning to Optimize,” Advances in Neural Information Processing Systems (NeurIPS), vol. 33, 2020.
  26. S. Flennerhag, Y. Schroecker, T. Zahavy, H. van Hasselt, D. Silver, and S. Singh, “Bootstrapped Meta-Learning,” arXiv preprint arXiv:2109.04504, 2021.
  27. J. Yang, T. Chen, M. Zhu, F. He, D. Tao, Y. Liang, and Z. Wang, “Learning to Generalize Provably in Learning to Optimize,” in International Conference on Artificial Intelligence and Statistics.   PMLR, 2023, pp. 9807–9825.
  28. X. Chen, T. Chen, Y. Cheng, W. Chen, A. Awadallah, and Z. Wang, “Scalable Learning to Optimize: A Learned Optimizer Can Train Big Models,” in European Conference on Computer Vision.   Springer, 2022, pp. 389–405.
  29. L. Metz, C. D. Freeman, N. Maheswaranathan, and J. Sohl-Dickstein, “Training Learned Optimizers with Randomly Initialized Learned Optimizers,” arXiv preprint arXiv:2101.07367, 2021.
  30. L. Metz, N. Maheswaranathan, C. D. Freeman, B. Poole, and J. Sohl-Dickstein, “Tasks, Stability, Architecture, and Compute: Training More Effective Learned Optimizers, and Using Them to Train Themselves,” arXiv preprint arXiv:2009.11243, 2020.
  31. O. Wichrowska, N. Maheswaranathan, M. W. Hoffman, S. G. Colmenarejo, M. Denil, N. Freitas, and J. Sohl-Dickstein, “Learned Optimizers that Scale and Generalize,” in International Conference on Machine Learning (ICML).   PMLR, 2017, pp. 3751–3760.
  32. L. Metz, N. Maheswaranathan, J. Nixon, D. Freeman, and J. Sohl-Dickstein, “Understanding and Correcting Pathologies in the Training of Learned Optimizers,” in International Conference on Machine Learning (ICML).   PMLR, 2019, pp. 4556–4565.
  33. K. Li and J. Malik, “Learning to Optimize,” arXiv preprint arXiv:1606.01885, 2016.
  34. ——, “Learning to Optimize Neural Nets,” arXiv preprint arXiv:1703.00441, 2017.
  35. C. Lu, J. Kuba, A. Letcher, L. Metz, C. Schroeder de Witt, and J. Foerster, “Discovered Policy Optimisation,” Advances in Neural Information Processing Systems (NeurIPS), vol. 35, pp. 16 455–16 468, 2022.
  36. J. Oh, M. Hessel, W. M. Czarnecki, Z. Xu, H. P. van Hasselt, S. Singh, and D. Silver, “Discovering Reinforcement Learning Algorithms,” Advances in Neural Information Processing Systems, vol. 33, pp. 1060–1070, 2020.
  37. L. Kirsch, S. van Steenkiste, and J. Schmidhuber, “Improving Generalization in Meta Reinforcement Learning Using Learned Objectives,” arXiv preprint arXiv:1910.04098, 2019.
  38. R. Houthooft, Y. Chen, P. Isola, B. Stadie, F. Wolski, O. Jonathan Ho, and P. Abbeel, “Evolved Policy Gradients,” Advances in Neural Information Processing Systems, vol. 31, 2018.
  39. J. D. Co-Reyes, Y. Miao, D. Peng, E. Real, S. Levine, Q. V. Le, H. Lee, and A. Faust, “Evolving Reinforcement Learning Algorithms,” arXiv preprint arXiv:2101.03958, 2021.
  40. J. X. Wang, Z. Kurth-Nelson, D. Tirumala, H. Soyer, J. Z. Leibo, R. Munos, C. Blundell, D. Kumaran, and M. Botvinick, “Learning to reinforcement learn,” arXiv preprint arXiv:1611.05763, 2016.
  41. Z. Erickson, H. M. Clever, G. Turk, C. K. Liu, and C. C. Kemp, “Deep Haptic Model Predictive Control for Robot-Assisted Dressing,” in IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2018, pp. 4437–4444.
  42. I. Lenz, R. A. Knepper, and A. Saxena, “DeepMPC: Learning Deep Latent Features for Model Predictive Control,” in Robotics: Science and Systems (R:SS).   Rome, Italy, 2015.
  43. J. Kocijan, R. Murray-Smith, C. E. Rasmussen, and A. Girard, “Gaussian process model based predictive control,” in American Control Conference (ACC), vol. 3.   IEEE, 2004, pp. 2214–2219.
  44. J. Fu, S. Levine, and P. Abbeel, “One-Shot Learning of Manipulation Skills with Online Dynamics Adaptation and Neural Network Priors,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).   IEEE, 2016, pp. 4019–4026.
  45. K. Chua, R. Calandra, R. McAllister, and S. Levine, “Deep Reinforcement Learning in a Handful of Trials using Probabilistic Dynamics Models,” arXiv preprint arXiv:1805.12114, 2018.
  46. A. Nagabandi, G. Kahn, R. S. Fearing, and S. Levine, “Neural Network Dynamics for Model-Based Deep Reinforcement Learning with Model-Free Fine-Tuning,” in IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2018, pp. 7559–7566.
  47. C. Finn and S. Levine, “Deep Visual Foresight for Planning Robot Motion,” in IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2017, pp. 2786–2793.
  48. N. Wahlström, T. B. Schön, and M. P. Deisenroth, “From Pixels to Torques: Policy Learning with Deep Dynamical Models,” arXiv preprint arXiv:1502.02251, 2015.
  49. M. Watter, J. T. Springenberg, J. Boedecker, and M. Riedmiller, “Embed to Control: A Locally Linear Latent Dynamics Model for Control from Raw Images,” arXiv preprint arXiv:1506.07365, 2015.
  50. E. Banijamali, R. Shu, H. Bui, and A. Ghodsi, “Robust Locally-Linear Controllable Embedding,” in International Conference on Artificial Intelligence and Statistics (AISTATS).   PMLR, 2018, pp. 1751–1759.
  51. F. Ebert, C. Finn, S. Dasari, A. Xie, A. Lee, and S. Levine, “Visual Foresight: Model-Based Deep Reinforcement Learning for Vision-Based Robotic Control,” arXiv preprint arXiv:1812.00568, 2018.
  52. J.-S. Ha, Y.-J. Park, H.-J. Chae, S.-S. Park, and H.-L. Choi, “Adaptive Path-Integral Autoencoder: Representation Learning and Planning for Dynamical Systems,” Advances in Neural Information Processing Systems (NeurIPS), 2018.
  53. D. Hafner, T. Lillicrap, I. Fischer, R. Villegas, D. Ha, H. Lee, and J. Davidson, “Learning Latent Dynamics for Planning from Pixels,” in International Conference on Machine Learning (ICML).   PMLR, 2019, pp. 2555–2565.
  54. M. Zhong, M. Johnson, Y. Tassa, T. Erez, and E. Todorov, “Value Function Approximation and Model Predictive Control,” in IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning (ADPRL).   IEEE, 2013, pp. 100–107.
  55. U. Rosolia and F. Borrelli, “Learning Model Predictive Control for Iterative Tasks. A Data-Driven Control Framework,” IEEE Transactions on Automatic Control, vol. 63, no. 7, pp. 1883–1896, 2017.
  56. K. Lowrey, A. Rajeswaran, S. Kakade, E. Todorov, and I. Mordatch, “Plan Online, Learn Offline: Efficient Learning and Exploration via Model-Based Control,” arXiv preprint arXiv:1811.01848, 2018.
  57. M. Bhardwaj, S. Choudhury, and B. Boots, “Blending MPC & Value Function Approximation for Efficient Reinforcement Learning,” arXiv preprint arXiv:2012.05909, 2020.
  58. M. Bhardwaj, A. Handa, D. Fox, and B. Boots, “Information Theoretic Model Predictive Q-Learning,” in Learning for Dynamics & Control (L4DC).   PMLR, 2020, pp. 840–850.
  59. B. Amos and D. Yarats, “The Differentiable Cross-Entropy Method,” in International Conference on Machine Learning (ICML).   PMLR, 2020, pp. 291–302.
  60. P. Karkus, D. Hsu, and W. S. Lee, “QMDP-Net: Deep Learning for Planning under Partial Observability,” arXiv preprint arXiv:1703.06692, 2017.
  61. M. Okada, L. Rigazio, and T. Aoshima, “Path Integral Networks: End-to-End Differentiable Optimal Control,” arXiv preprint arXiv:1706.09597, 2017.
  62. B. Amos, I. D. J. Rodriguez, J. Sacks, B. Boots, and J. Z. Kolter, “Differentiable MPC for End-to-end Planning and Control,” arXiv preprint arXiv:1810.13400, 2018.
  63. M. Okada and T. Taniguchi, “Acceleration of Gradient-based Path Integral Method for Efficient Optimal and Inverse Optimal Control,” in IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2018, pp. 3013–3020.
  64. M. Pereira, D. D. Fan, G. N. An, and E. Theodorou, “MPC-Inspired Neural Network Policies for Sequential Decision Making,” arXiv preprint arXiv:1802.05803, 2018.
  65. T. Power and D. Berenson, “Variational Inference MPC for Robot Motion with Normalizing Flows,” Advances in Neural Information Processing Systems (NeurIPS) Workshop on Robot Learning: Self-Supervised and Lifelong Learning, 2021.
  66. ——, “Variational Inference MPC using Normalizing Flows and Out-of-Distribution Projection,” arXiv preprint arXiv:2205.04667, 2022.
  67. J. Sacks and B. Boots, “Learning Sampling Distributions for Model Predictive Control,” in Conference on Robot Learning (CoRL).   PMLR, 2023, pp. 1733–1742.
  68. M. Okada and T. Taniguchi, “Variational Inference MPC for Bayesian Model-based Reinforcement Learning,” in Conference on Robot Learning (CoRL).   PMLR, 2020, pp. 258–272.
  69. A. Lambert, A. Fishman, D. Fox, B. Boots, and F. Ramos, “Stein Variational Model Predictive Control,” arXiv preprint arXiv:2011.07641, 2020.
  70. T. Power and D. Berenson, “Constrained Stein Variational Trajectory Optimization,” arXiv preprint arXiv:2308.12110, 2023.
  71. D. M. Asmar, R. Senanayake, S. Manuel, and M. J. Kochenderfer, “Model Predictive Optimized Path Integral Strategies,” in IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2023, pp. 3182–3188.
  72. J. Yin, Z. Zhang, E. Theodorou, and P. Tsiotras, “Trajectory Distribution Control for Model Predictive Path Integral Control Using Covariance Steering,” in 2022 International Conference on Robotics and Automation (ICRA).   IEEE, 2022, pp. 1478–1484.
  73. I. S. Mohamed, J. Xu, G. Sukhatme, and L. Liu, “Towards Efficient MPPI Trajectory Generation with Unscented Guidance: U-MPPI Control Strategy,” arXiv preprint arXiv:2306.12369, 2023.
  74. K. Honda, N. Akai, K. Suzuki, M. Aoki, H. Hosogaya, H. Okuda, and T. Suzuki, “Stein Variational Guided Model Predictive Path Integral Control: Proposal and Experiments with Fast Maneuvering Vehicles,” arXiv preprint arXiv:2309.11040, 2023.
  75. G. Lee, B. Hou, S. Choudhury, and S. S. Srinivasa, “Bayesian Residual Policy Optimization:: Scalable Bayesian Reinforcement Learning with Clairvoyant Experts,” in 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).   IEEE, 2021, pp. 5611–5618.
  76. T. Johannink, S. Bahl, A. Nair, J. Luo, A. Kumar, M. Loskyll, J. A. Ojea, E. Solowjow, and S. Levine, “Residual Reinforcement Learning for Robot Ccontrol,” in 2019 International Conference on Robotics and Automation (ICRA).   IEEE, 2019, pp. 6023–6029.
  77. T. Silver, K. Allen, J. Tenenbaum, and L. Kaelbling, “Residual Policy Learning,” arXiv preprint arXiv:1812.06298, 2018.
  78. Y. Yang, X. Meng, W. Yu, T. Zhang, J. Tan, and B. Boots, “Continuous Versatile Jumping Using Learned Action Residuals,” in Learning for Dynamics & Control (L4DC).   PMLR, 2023, pp. 770–782.
  79. Y. Yang, G. Shi, X. Meng, W. Yu, T. Zhang, J. Tan, and B. Boots, “Cajun: Continuous adaptive jumping using a learned centroidal controller,” arXiv preprint arXiv:2306.09557, 2023.
  80. J. Pravitra, K. A. Ackerman, C. Cao, N. Hovakimyan, and E. A. Theodorou, “L1-Adaptive MPPI Architecture for Robust and Agile Control of Multirotors,” in 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).   IEEE, 2020, pp. 7661–7666.
  81. K. Lee, J. Gibson, and E. A. Theodorou, “Aggressive Perception-Aware Navigation Using Deep Optical Flow Dynamics and PixelMPC,” IEEE Robotics and Automation Letters, vol. 5, no. 2, pp. 1207–1214, 2020.
  82. D. Hanover, P. Foehn, S. Sun, E. Kaufmann, and D. Scaramuzza, “Performance, Precision, and Payloads: Adaptive Nonlinear MPC for Quadrotors,” IEEE Robotics and Automation Letters, vol. 7, no. 2, pp. 690–697, 2021.
  83. S. Sun, A. Romero, P. Foehn, E. Kaufmann, and D. Scaramuzza, “A Comparative Study of Nonlinear MPC and Differential-Flatness-Based Control for Quadrotor Agile Flight,” IEEE Transactions on Robotics, vol. 38, no. 6, pp. 3357–3373, 2022.
  84. E. Kaufmann, A. Loquercio, R. Ranftl, M. Müller, V. Koltun, and D. Scaramuzza, “Deep Drone Acrobatics,” arXiv preprint arXiv:2006.05768, 2020.
  85. Y. Zhang, W. Wang, P. Huang, and Z. Jiang, “Monocular Vision-Based Sense and Avoid of UAV Using Nonlinear Model Predictive Control,” Robotica, vol. 37, no. 9, pp. 1582–1594, 2019.
  86. A. Molchanov, T. Chen, W. Hönig, J. A. Preiss, N. Ayanian, and G. S. Sukhatme, “Sim-to-(Multi)-Real: Transfer of Low-Level Robust Control Policies to Multiple Quadrotors,” in 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).   IEEE, 2019, pp. 59–66.
  87. D. Zhang, A. Loquercio, X. Wu, A. Kumar, J. Malik, and M. W. Mueller, “Learning a Single Near-hover Position Controller for Vastly Different Quadcopters,” in 2023 IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2023, pp. 1263–1269.
  88. J. Hwangbo, I. Sa, R. Siegwart, and M. Hutter, “Control of a Quadrotor with Reinforcement Learning,” IEEE Robotics and Automation Letters, vol. 2, no. 4, pp. 2096–2103, 2017.
  89. E. Kaufmann, L. Bauersfeld, and D. Scaramuzza, “A Benchmark Comparison of Learned Control Policies for Agile Quadrotor Flight,” in 2022 International Conference on Robotics and Automation (ICRA).   IEEE, 2022, pp. 10 504–10 510.
  90. A. Romero, Y. Song, and D. Scaramuzza, “Actor-Critic Model Predictive Control,” arXiv preprint arXiv:2306.09852, 2023.
  91. Y. Song and D. Scaramuzza, “Policy Search for Model Predictive Control with Application to Agile Drone Flight,” IEEE Transactions on Robotics, vol. 38, no. 4, pp. 2114–2130, 2022.
  92. A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al., “PyTorch: An Imperative Style, High-Performance Deep Learning Library,” Advances in Neural Information Processing Systems (NeurIPS), vol. 32, pp. 8026–8037, 2019.
  93. J. Schulman, P. Moritz, S. Levine, M. Jordan, and P. Abbeel, “High-Dimensional Continuous Control Using Generalized Advantage Estimation,” arXiv preprint arXiv:1506.02438, 2015.
  94. D. P. Kingma and J. Ba, “Adam: A Method for Stochastic Optimization,” arXiv preprint arXiv:1412.6980, 2014.
  95. J. H. Halton, “Algorithm 247: Radical-inverse quasi-random point sequence,” Communications of the ACM, vol. 7, no. 12, pp. 701–702, 1964.
User Edit Pencil Streamline Icon: https://streamlinehq.com
Authors (6)
  1. Jacob Sacks (7 papers)
  2. Rwik Rana (5 papers)
  3. Kevin Huang (45 papers)
  4. Alex Spitzer (3 papers)
  5. Guanya Shi (54 papers)
  6. Byron Boots (120 papers)
Citations (4)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now