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Equation 5 · Part 7 · Comparing the Main Approaches to Robotics and Embodied AI

Symbol r

a1:H∗=arg⁡max⁡a1:H  Ep^θ[∑t=1Hγt r(st,at)]a^{*}_{1:H} = \arg\max_{a_{1:H}} \; \mathbb{E}_{\hat{p}_\theta} \left[ \sum_{t=1}^{H} \gamma^{t} \, r(s_t, a_t) \right]
rr

What this part means

r appears inside an expected value, so its contribution is averaged under the distribution or condition shown by that operator.

Its job in the formula

r appears inside an expected value, so its contribution is averaged under the distribution or condition shown by that operator.

The passage around this formula

Nothing in this update requires knowing or estimating p(s' ∣\mid s, a) ; it only requires having experienced (s, a, r, s') . A model-based approach instead fits an explicit dynamics model p^θ(s′∣s,a)\hat{p}_\theta(s' \mid s, a) — a “world model” — and plans against it directly: a1:H∗=arg⁡max⁡a1:H  Ep^θ[∑t=1Hγt r(st,at)]a^{*}_{1:H} = \arg\max_{a_{1:H}} \; \mathbb{E}_{\hat{p}_\theta} \left[ \sum_{t=1}^{H} \gamma^{t} \, r(s_t, a_t) \right]. A comprehensive survey of the model-based literature frames the trade this way: fitting and planning against p^θ\hat{p}_\theta…

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A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the surrounding passage

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