# 強化学習におけるベルマン演算子は何ですか？

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このトピックに関連するいくつかの論文は、大規模動的プログラミングの機能ベースの方法（John N. TsitsiklisおよびBenjamin Van Roy、1996）、関数近似による時間差学習の分析（John N. TsitsiklisおよびBenjamin Vanによる）です。 Roy、1997）および最小二乗ポリシー反復（Michail G. LagoudakisおよびRonald Parr、2003年）。
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$\begin{array}{}\text{(1)}& {v}_{\pi }\left(s\right)=\sum _{a\in \mathcal{A}}\pi \left(a|s\right)\left({\mathcal{R}}_{\mathcal{s}}^{\mathcal{a}}\mathcal{+}\gamma \sum _{{\mathcal{s}}^{\prime }\in \mathcal{S}}{\mathcal{P}}_{\mathcal{s}{\mathcal{s}}^{\prime }}^{\mathcal{a}}{\mathcal{v}}_{\pi }\mathcal{\left(}{\mathcal{s}}^{\prime }\mathcal{\right)}\right)\end{array}$

させたら 

$\begin{array}{}\mathcal{\text{(2)}}& {\mathcal{P}}_{\mathcal{s}{\mathcal{s}}^{\prime }}^{\pi }\mathcal{=}\sum _{\mathcal{a}\in \mathcal{A}}\pi \mathcal{\left(}\mathcal{a}\mathcal{|}\mathcal{s}\mathcal{\right)}{\mathcal{P}}_{\mathcal{s}{\mathcal{s}}^{\prime }}^{\mathcal{a}}\end{array}$
そして $$Rπs=∑a∈Aπ(a|s)Ras(3)(3)Rsπ=∑a∈Aπ(a|s)Rsa\cal{R}_{s}^\pi = \sum\limits_{a \in \cal{A}} \pi(a|s)\cal{R}_{s}^a \tag 3$$ その後、書き換えることができます $$(1)(1)(1)$$ なので



$\begin{array}{}\mathcal{\text{(4)}}& {v}_{\pi }\left(s\right)={\mathcal{R}}_{\mathcal{s}}^{\pi }\mathcal{+}\gamma \sum _{{\mathcal{s}}^{\prime }\in \mathcal{S}}{\mathcal{P}}_{\mathcal{s}{\mathcal{s}}^{\prime }}^{\pi }{\mathcal{v}}_{\pi }\mathcal{\left(}{\mathcal{s}}^{\prime }\mathcal{\right)}\end{array}$

これは行列形式で書くことができます



$\begin{array}{}\text{(5)}& \left[\begin{array}{c}{v}_{\pi }\left(1\right)\\ ⋮\\ {v}_{\pi }\left(n\right)\end{array}\right]=\left[\begin{array}{c}{\mathcal{R}}_{\mathcal{1}}^{\pi }\\ ⋮\\ {\mathcal{R}}_{\mathcal{n}}^{\pi }\end{array}\right]+\gamma \left[\begin{array}{ccc}{\mathcal{P}}_{\mathcal{11}}^{\pi }& \dots & {\mathcal{P}}_{\mathcal{1}\mathcal{n}}^{\pi }\\ ⋮& \ddots & ⋮\\ {\mathcal{P}}_{\mathcal{n}\mathcal{1}}^{\pi }& \dots & {\mathcal{P}}_{\mathcal{n}\mathcal{n}}^{\pi }\end{array}\right]\left[\begin{array}{c}{v}_{\pi }\left(1\right)\\ ⋮\\ {v}_{\pi }\left(n\right)\end{array}\right]\end{array}$

Or, more compactly,



$\begin{array}{}\mathcal{\text{(6)}}& {v}_{\pi }={\mathcal{R}}^{\pi }\mathcal{+}\gamma {\mathcal{P}}^{\pi }{\mathcal{v}}_{\pi }\end{array}$

Notice that both sides of $$(6)(6)(6)$$ are $$nnn$$-dimensional vectors. Here $$n=|S|n=|S|n=|\cal{S}|$$ is the size of the state space. We can then define an operator $$Tπ:Rn→RnTπ:Rn→Rn\cal{T}^\pi:\mathbb{R}^n\to\mathbb{R}^n$$ as



$\begin{array}{}\mathcal{\text{(7)}}& {\mathcal{T}}^{\pi }\mathcal{\left(}\mathcal{v}\mathcal{\right)}\mathcal{=}{\mathcal{R}}^{\pi }\mathcal{+}\gamma {\mathcal{P}}^{\pi }\mathcal{v}\end{array}$

for any $$v∈Rnv∈Rnv\in \mathbb{R}^n$$. This is the expected Bellman operator.

Similarly, you can rewrite the Bellman optimality equation



$\begin{array}{}\text{(8)}& {v}_{\ast }\left(s\right)=\underset{a\in \mathcal{A}}{max}\left({\mathcal{R}}_{\mathcal{s}}^{\mathcal{a}}\mathcal{+}\gamma \sum _{{\mathcal{s}}^{\prime }\in \mathcal{S}}{\mathcal{P}}_{\mathcal{s}{\mathcal{s}}^{\prime }}^{\mathcal{a}}{\mathcal{v}}_{\mathcal{\ast }}\mathcal{\left(}{\mathcal{s}}^{\prime }\mathcal{\right)}\right)\end{array}$

as the Bellman optimality operator



$\begin{array}{}\mathcal{\text{(9)}}& {\mathcal{T}}^{\mathcal{\ast }}\mathcal{\left(}\mathcal{v}\mathcal{\right)}\mathcal{=}\underset{\mathcal{a}\in \mathcal{A}}{max}\left({\mathcal{R}}^{\mathcal{a}}\mathcal{+}\gamma {\mathcal{P}}^{\mathcal{a}}\mathcal{v}\right)\end{array}$

The Bellman operators are "operators" in that they are mappings from one point to another within the vector space of state values, $$RnRn\mathbb{R}^n$$.

Rewriting the Bellman equations as operators is useful for proving that certain dynamic programming algorithms (e.g. policy iteration, value iteration) converge to a unique fixed point. This usefulness comes in the form of a body of existing work in operator theory, which allows us to make use of special properties of the Bellman operators.

Specifically, the fact that the Bellman operators are contractions gives the useful results that, for any policy $$ππ\pi$$ and any initial vector $$vvv$$,



$\begin{array}{}\mathcal{\text{(10)}}& \underset{k\to \mathrm{\infty }}{lim}\left({\mathcal{T}}^{\pi }{\mathcal{\right)}}^{\mathcal{k}}\mathcal{v}\mathcal{=}{\mathcal{v}}_{\pi }\end{array}$



$\begin{array}{}\mathcal{\text{(11)}}& \underset{k\to \mathrm{\infty }}{lim}\left({\mathcal{T}}^{\mathcal{\ast }}{\mathcal{\right)}}^{\mathcal{k}}\mathcal{v}\mathcal{=}{\mathcal{v}}_{\mathcal{\ast }}\end{array}$

where $$vπvπv_\pi$$ is the value of policy $$ππ\pi$$ and $$v∗v∗v_*$$ is the value of an optimal policy $$π∗π∗\pi^*$$. The proof is due to the contraction mapping theorem.

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