Limits...
A Theory of Cheap Control in Embodied Systems.

Montúfar G, Ghazi-Zahedi K, Ay N - PLoS Comput. Biol. (2015)

Bottom Line: This embodied universal approximation is compared with the classical non-embodied universal approximation.To exemplify our approach, we present a detailed quantitative case study for policy models defined in terms of conditional restricted Boltzmann machines.The experiments indicate that the controller complexity predicted by our theory is close to the minimal sufficient value, which means that the theory has direct practical implications.

View Article: PubMed Central - PubMed

Affiliation: Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany.

ABSTRACT
We present a framework for designing cheap control architectures of embodied agents. Our derivation is guided by the classical problem of universal approximation, whereby we explore the possibility of exploiting the agent's embodiment for a new and more efficient universal approximation of behaviors generated by sensorimotor control. This embodied universal approximation is compared with the classical non-embodied universal approximation. To exemplify our approach, we present a detailed quantitative case study for policy models defined in terms of conditional restricted Boltzmann machines. In contrast to non-embodied universal approximation, which requires an exponential number of parameters, in the embodied setting we are able to generate all possible behaviors with a drastically smaller model, thus obtaining cheap universal approximation. We test and corroborate the theory experimentally with a six-legged walking machine. The experiments indicate that the controller complexity predicted by our theory is close to the minimal sufficient value, which means that the theory has direct practical implications.

No MeSH data available.


Related in: MedlinePlus

Experimental results.Performance of the best CRBM for different complexity parameters m in comparison to the performance of the target behavior (horizontal orange line). The vertical blue line indicates the m estimated from the data (see supporting information S2 Text).
© Copyright Policy
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4556690&req=5

pcbi.1004427.g009: Experimental results.Performance of the best CRBM for different complexity parameters m in comparison to the performance of the target behavior (horizontal orange line). The vertical blue line indicates the m estimated from the data (see supporting information S2 Text).

Mentions: Before the experiments can be described, there is an important note to make. This work is concerned with the minimal complexity that is sufficient for controlling a set of desirable behaviors (this set may consist of all possible behaviors or of just one specific behavior). Here, we are not concerned with the question how these CRBMs should be trained optimally. This is why we used a standard training algorithm for RBMs [30, 31] and conducted a large scan over different complexity parameters m. For each m = 1, 2, 3, …, 100 we trained 100 CRBMs with the following learning parameters: epochs = 20000, batch size = 50, learning rate α = 1.0, momentum = 0.1, Gaussian distributed noise on sensor data = 0.01, weight cost = 0.001, CRBM Gibbs updates for sampling = 10, on a data set of 104 pairs of sensor and actuator values. Each trained CRBM was evaluated ten times, by applying it to the hexapod and recording the distance covered in 30 seconds. The performance of the CRBMs is measured against the target tripod walking gait, which achieves 20.6 meters during the same time. As we are concerned with the performance that is in principle possible for a given m, we choose the policy that covered the most distance at one of the 10 trials (out of the 100 trained policies, for each m). The plot in Fig 9 (left-hand side) shows the best performance of the best policy for all scanned values of m. The plot in Fig 9 (right-hand side) shows the average performance of the best policy and the standard deviation, over 10 different evaluations, for all values of m. The results show that our estimation is fairly tight, which means the performance of the CRBMs converges to the optimal behavior close to the estimated value of m = 65. The supporting information S1–S4 Videos show the performance of the best CRBM for m = 5, 15, 65, 75.


A Theory of Cheap Control in Embodied Systems.

Montúfar G, Ghazi-Zahedi K, Ay N - PLoS Comput. Biol. (2015)

Experimental results.Performance of the best CRBM for different complexity parameters m in comparison to the performance of the target behavior (horizontal orange line). The vertical blue line indicates the m estimated from the data (see supporting information S2 Text).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4556690&req=5

pcbi.1004427.g009: Experimental results.Performance of the best CRBM for different complexity parameters m in comparison to the performance of the target behavior (horizontal orange line). The vertical blue line indicates the m estimated from the data (see supporting information S2 Text).
Mentions: Before the experiments can be described, there is an important note to make. This work is concerned with the minimal complexity that is sufficient for controlling a set of desirable behaviors (this set may consist of all possible behaviors or of just one specific behavior). Here, we are not concerned with the question how these CRBMs should be trained optimally. This is why we used a standard training algorithm for RBMs [30, 31] and conducted a large scan over different complexity parameters m. For each m = 1, 2, 3, …, 100 we trained 100 CRBMs with the following learning parameters: epochs = 20000, batch size = 50, learning rate α = 1.0, momentum = 0.1, Gaussian distributed noise on sensor data = 0.01, weight cost = 0.001, CRBM Gibbs updates for sampling = 10, on a data set of 104 pairs of sensor and actuator values. Each trained CRBM was evaluated ten times, by applying it to the hexapod and recording the distance covered in 30 seconds. The performance of the CRBMs is measured against the target tripod walking gait, which achieves 20.6 meters during the same time. As we are concerned with the performance that is in principle possible for a given m, we choose the policy that covered the most distance at one of the 10 trials (out of the 100 trained policies, for each m). The plot in Fig 9 (left-hand side) shows the best performance of the best policy for all scanned values of m. The plot in Fig 9 (right-hand side) shows the average performance of the best policy and the standard deviation, over 10 different evaluations, for all values of m. The results show that our estimation is fairly tight, which means the performance of the CRBMs converges to the optimal behavior close to the estimated value of m = 65. The supporting information S1–S4 Videos show the performance of the best CRBM for m = 5, 15, 65, 75.

Bottom Line: This embodied universal approximation is compared with the classical non-embodied universal approximation.To exemplify our approach, we present a detailed quantitative case study for policy models defined in terms of conditional restricted Boltzmann machines.The experiments indicate that the controller complexity predicted by our theory is close to the minimal sufficient value, which means that the theory has direct practical implications.

View Article: PubMed Central - PubMed

Affiliation: Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany.

ABSTRACT
We present a framework for designing cheap control architectures of embodied agents. Our derivation is guided by the classical problem of universal approximation, whereby we explore the possibility of exploiting the agent's embodiment for a new and more efficient universal approximation of behaviors generated by sensorimotor control. This embodied universal approximation is compared with the classical non-embodied universal approximation. To exemplify our approach, we present a detailed quantitative case study for policy models defined in terms of conditional restricted Boltzmann machines. In contrast to non-embodied universal approximation, which requires an exponential number of parameters, in the embodied setting we are able to generate all possible behaviors with a drastically smaller model, thus obtaining cheap universal approximation. We test and corroborate the theory experimentally with a six-legged walking machine. The experiments indicate that the controller complexity predicted by our theory is close to the minimal sufficient value, which means that the theory has direct practical implications.

No MeSH data available.


Related in: MedlinePlus