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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

Ambiguity and redundancy of the sensor measurement.In this example, an agent navigates the 20 × 20 maze shown in the left panel. The agent is endowed with two sensors (eyes), Sleft and Sright. Each sensor measures a weighted average of the walls in the immediate vicinity, illustrated in the central panel, and outputs one of 8 possible numerical values, as shown in the right panel. There are 400 possible locations in the maze but only 8 × 8 = 64 joint sensor states. This implies that the sensor measurement is highly ambiguous about the world state. Furthermore, the outputs of both sensors are not independent; they always have the same value at the decimal place. Due to this redundancy, the factual number of joint sensor states is 15, instead of 64.
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pcbi.1004427.g003: Ambiguity and redundancy of the sensor measurement.In this example, an agent navigates the 20 × 20 maze shown in the left panel. The agent is endowed with two sensors (eyes), Sleft and Sright. Each sensor measures a weighted average of the walls in the immediate vicinity, illustrated in the central panel, and outputs one of 8 possible numerical values, as shown in the right panel. There are 400 possible locations in the maze but only 8 × 8 = 64 joint sensor states. This implies that the sensor measurement is highly ambiguous about the world state. Furthermore, the outputs of both sensors are not independent; they always have the same value at the decimal place. Due to this redundancy, the factual number of joint sensor states is 15, instead of 64.

Mentions: The sensors are usually insensitive to a large number of variations of the world state w. This means that β outputs the same distribution for several different w. Furthermore, the sensors implement a certain degree of redundancy, meaning that, for each 𝒲 the probability distribution β(w; ⋅) ∈ Δ𝒮 has certain types of symmetries. Consider, for example, the 20 × 20 maze shown in Fig 3 (left-hand side). The world state includes the location of the agent in the maze, (i, j) ∈ {1, …, 20}2. The agent is endowed with two sensors: a left eye and a right eye. Each eye measures a weighted sum of the light intensity arriving from the walls in the immediate vicinity of the agent. The left eye outputs the value Sleft = 0.8 xw + 0.2 xn + 0.1 xs + 0 xe (with probability one), where xw,n,s,e = 1 if there is a wall to the immediate west, north, south, east, respectively, and 0 otherwise. Similarly, Sright = 0 xw + 0.2 xn + 0.1 xs + 0.8 xe. Each eye can produce a total of 8 states: 0,0.1,0.2,0.3,1,1.1,1.2,1.3. The naive number of joint sensor states (Sleft, Sright) is 8 × 8 = 64. However, both eyes are partially redundant, and the actual total number of possible joint states is 15 (the case of four walls surrounding a location is excluded). In this example, 400 world states are mapped onto 15 sensor states, which implies that the rank of β is 15. This example illustrates the two typical properties of the sensor measurement mentioned above: the ambiguity of the measurement, mapping several world states to the same sensor state, and the redundancy, by which several sensors measure partially overlapping information about the world state.


A Theory of Cheap Control in Embodied Systems.

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

Ambiguity and redundancy of the sensor measurement.In this example, an agent navigates the 20 × 20 maze shown in the left panel. The agent is endowed with two sensors (eyes), Sleft and Sright. Each sensor measures a weighted average of the walls in the immediate vicinity, illustrated in the central panel, and outputs one of 8 possible numerical values, as shown in the right panel. There are 400 possible locations in the maze but only 8 × 8 = 64 joint sensor states. This implies that the sensor measurement is highly ambiguous about the world state. Furthermore, the outputs of both sensors are not independent; they always have the same value at the decimal place. Due to this redundancy, the factual number of joint sensor states is 15, instead of 64.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi.1004427.g003: Ambiguity and redundancy of the sensor measurement.In this example, an agent navigates the 20 × 20 maze shown in the left panel. The agent is endowed with two sensors (eyes), Sleft and Sright. Each sensor measures a weighted average of the walls in the immediate vicinity, illustrated in the central panel, and outputs one of 8 possible numerical values, as shown in the right panel. There are 400 possible locations in the maze but only 8 × 8 = 64 joint sensor states. This implies that the sensor measurement is highly ambiguous about the world state. Furthermore, the outputs of both sensors are not independent; they always have the same value at the decimal place. Due to this redundancy, the factual number of joint sensor states is 15, instead of 64.
Mentions: The sensors are usually insensitive to a large number of variations of the world state w. This means that β outputs the same distribution for several different w. Furthermore, the sensors implement a certain degree of redundancy, meaning that, for each 𝒲 the probability distribution β(w; ⋅) ∈ Δ𝒮 has certain types of symmetries. Consider, for example, the 20 × 20 maze shown in Fig 3 (left-hand side). The world state includes the location of the agent in the maze, (i, j) ∈ {1, …, 20}2. The agent is endowed with two sensors: a left eye and a right eye. Each eye measures a weighted sum of the light intensity arriving from the walls in the immediate vicinity of the agent. The left eye outputs the value Sleft = 0.8 xw + 0.2 xn + 0.1 xs + 0 xe (with probability one), where xw,n,s,e = 1 if there is a wall to the immediate west, north, south, east, respectively, and 0 otherwise. Similarly, Sright = 0 xw + 0.2 xn + 0.1 xs + 0.8 xe. Each eye can produce a total of 8 states: 0,0.1,0.2,0.3,1,1.1,1.2,1.3. The naive number of joint sensor states (Sleft, Sright) is 8 × 8 = 64. However, both eyes are partially redundant, and the actual total number of possible joint states is 15 (the case of four walls surrounding a location is excluded). In this example, 400 world states are mapped onto 15 sensor states, which implies that the rank of β is 15. This example illustrates the two typical properties of the sensor measurement mentioned above: the ambiguity of the measurement, mapping several world states to the same sensor state, and the redundancy, by which several sensors measure partially overlapping information about the world state.

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