21 Introduction
POMDPs, Dynamic programming, Information state
So far, we have considered a setup where the decision maker perfectly observes the state of the system. In many applications, the decision maker may not directly observe the state of the system but only observe a noisy version of it. Such systems are modeled as partially observable Markov decision processes (POMDPs). We will describe the simplest model of POMDPs, which builds upon the model of MDPs descibed earlier.
We assume that the system has a state \(S_t \in \ALPHABET S\), control input \(A_t \in \ALPHABET A\), and process noise \(W_t \in \ALPHABET W\). The state evolves as \[\begin{equation} \label{eq:state} S_{t+1} = f_t(S_t, A_t, W_t) \end{equation}\] However, unlike the MDP setup, the assumption is that the decision maker does not observe \(S_t\); rather, the observation of the decision maker at time \(t\) is given by \[\begin{equation} \label{eq:obs} Y_t = \ell_t(S_t, N_t) \end{equation}\] where \(Y_t \in \ALPHABET Y\) is the observation and \(N_t \in \ALPHABET N\) is called the observation noise. As in the case of MDPs, we assume that the primitive random varaibles \((S_1, W_1, \dots, W_T\), \(N_1, \dots, N_T)\) are defined on a common probability space and are mutually independent. This assumption is critical for the results to go through.
As in the case of MDPs, we assume that the controller can be as sophisticated as we want. It can analyze the entire history of observations and control actions to choose the current control action. Thus, the control action can be written as \[ A_t = π_t(Y_{1:t}, A_{1:t-1}). \]
At each time, the system incurs a cost \(c_t(S_t, A_t)\) which depends on the current state and the current action. The system operates for a finite horizon \(T\) and incurs a total cost \[ \sum_{t=1}^T c_t(S_t, A_t). \]
Given the above system model, we want to choose a control strategy \(π = (π_1, \dots, π_T)\) to minimize the expected total cost \[ J(π) := \EXP\Bigl[ \sum_{t=1}^T c_t(S_t, A_t) \Bigr]. \] How should we proceed?
Note that the only difference from the MDP model is decision maker observes \(Y_t\) instead of \(S_t\). Apart from this, the other modeling assumptions are the same. So, the conceptual difficulties of the model are the same as that of MDPs:
Recall that for MDPs, we first showed that there is no loss of optimality in restricting attention to Markov strategies. That structural result was instrumental in developing an efficient search algorithm (dynamic programming). So, what is the equivalent result for POMDPs?
21.1 History dependent dynamic program
Our first step to develop an efficient dynamic programming decomposition is to simply ignore efficiency and develop a dynamic programming decomposition. We start by deriving a recursive formula to compute the performance of a generic history dependent strategy \(π = (π_1, \dots, π_T)\).
Performance of history-dependent strategies
Let \(H_t = (Y_{1:t}, A_{1:t-1})\) denote all the information available to the decision maker at time \(t\). Thus, given any history dependent strategy \(π\), we can write \(A_t = π_t(H_t)\). Define the cost-to-go functions as follows: \[ J_t(h_t; π) = \EXP^π\biggl[ \sum_{s=t}^T c_s(S_s, A_s) \biggm| H_t = h_t \biggr]. \] Note that \(J_t(h_t; π)\) only depends on the future strategy \((π_t, \dots, π_T)\). These functions can be computed recursively as follows: \[\begin{align*} J_t(h_t; π) &= \EXP^π\biggl[ \sum_{s=t}^T c_s(H_s, π_s(H_s)) \biggm| H_t = h_t \biggr] \\ &\stackrel{(a)}= \EXP^π \biggl[ c_t(h_t, π_t(h_t)) + \EXP^π\biggl[ \sum_{s=t+1}^T c_s(S_s, π_s(S_s)) \biggm| H_{t+1} \biggr] \biggm| H_t = h_t \biggr] \\ &= \EXP^π[ c_t(h_t, π_t(h_t)) + J_{t+1}(H_{t+1}; π) \mid H_t = h_t ], \end{align*}\] where \((a)\) follows from the towering property of conditional expectation and the fact that \(H_t \subseteq H_{t+1}\).
Thus, we can use the following dynamic program to recursively compute the performance of a history-dependent strategy: \(J_{T+1}(h_{T+1}) = 0\) and for \(t \in \{T, \dots, 1\}\), \[ J_t(h_t; π) = \EXP^π [ c_t(h_t, π_t(h_t)) + J_{t+1}(H_{t+1}; π) \mid H_t = h_t ]. \]
History-dependent dynamic programming decomposition
We can use the above recursive formulation for performance evaluation to derive a history-dependent dynamic program.
Theorem 21.1 Recursively define value functions \(\{V_t\}_{t = 1}^{T+1}\), where \(V_t \colon \ALPHABET H_t \to \reals\) as follows: \[\begin{equation} V_{T+1}(h_{T+1}) = 0 \end{equation}\] and for \(t \in \{T, \dots, 1\}\): \[\begin{align} Q_t(h_t, a_t) &= \EXP[ c_t(S_t, a_t) + V_{t+1}(H_{t+1}) \mid H_t = h_t, A_t = a_t ] \\ V_t(h_t) &= \min_{a_t \in \ALPHABET A} Q_t(h_t, a_t) \end{align}\] Then, a history-dependent policy \(π\) is optimal if and only if it satisfies \[\begin{equation} \label{eq:history-verification} π_t(h_t) \in \arg \min_{a_t \in \ALPHABET A} Q_t(h_t, a_t). \end{equation}\]
The proof idea is similar to the proof for MDPs. Instead of proving the above result, we prove a related result.
Theorem 21.2 (The comparison principle) For any history-dependent strategy \(π\) \[ J_t(h_t; π) \ge V_t(h_t) \] with equality at \(t\) if and only if the future straegy \(π_{t:T}\) satisfies the verification step \eqref{eq:history-verification}.
Note that the comparison principle immediately implies that the strategy obtained using dynamic program of Theorem 21.1 is optimal. The proof of the comparison principle is almost identical to the proof for MDPs.
The proof proceeds by backward induction. Consider any history dependent policy \(π = (π_1, \dots, π_T)\). For \(t = T+1\), the comparison principle is satisfied by definition and this forms the basis of induction. We assume that the result holds for time \(t+1\), which is the induction hypothesis. Then for time \(t\), we have \[\begin{align*} V_t(h_t) &= \min_{a_t \in \ALPHABET A} Q_t(h_t, a_t) \\ &\stackrel{(a)}= \min_{a_t \in \ALPHABET A} \EXP^π[ c_t(S_t, π_t(h_t)) + V_{t+1}(H_{t+1}) \mid H_t = h_t, A_t = π_t(h_t) ] \\ &\stackrel{(b)}\le \EXP^π[ c_t(S_t, π_t(h_t)) + V_{t+1}(H_{t+1}) \mid H_t = h_t, A_t = π_t(h_t)] \\ &\stackrel{(c)}\le \EXP^π[ c_t(S_t, π_t(h_t)) + J_{t+1}(H_{t+1}; π) \mid H_t = h_t, A_t = π_t(h_t)] \\ &= J_t(h_t, π). \end{align*}\] where \((a)\) follows from the definition of the \(Q\)-function; \((b)\) follows from the definition of minimization; and \((c)\) follows from the induction hyothesis. We have the equality at step \((b)\) iff \(π_t\) satisfies the verification step \eqref{eq:history-verification} and have the equality in step \((c)\) iff \(π_{t+1:T}\) is optimal (this is part of the induction hypothesis). Thus, the result is true for time \(t\) and, by the principle of induction, is true for all time.
21.2 The notion of an information state
Now that we have obtained a dynamic programming decomposition, let’s try to simplify it. To do so, we define the notion of an information state.
A stochastic process \(\{Z_t\}_{t = 1}^T\), \(Z_t \in \ALPHABET Z\), is called an information state if \(Z_t\) be a function of \(H_t\) (which we denote by \(Z_t = φ_t(H_t)\)) and satisfies the following two properties:
P1. Sufficient for performance evaluation, i.e., \[ \EXP^π[ c_t(S_t, A_t) \mid H_t = h_t, A_t = a_t] = \EXP[ c_t(S_t, A_t) \mid Z_t = φ_t(h_t), A_t = a_t ] \]
P2. Sufficient to predict itself, i.e., for any Borel measurable subset \(B\) of \(\ALPHABET Z\), we have \[ \PR^π(Z_{t+1} \in B \mid H_t = h_t, A_t = a_t) = \PR(Z_{t+1} \in B \mid Z_t = φ_t(h_t), A_t = a_t). \]
Instead of (P2), the following sufficient conditions are easier to verify in some models:
P2a. Evolves in a state-like manner, i.e., there exist measurable functions \(\{ψ_t\}_{t=1}^T\) such that \[ Z_{t+1} = ψ_t(Z_t, Y_{t+1}, A_t). \]
P2b. Is sufficient for predicting future observations, i.e., for any Borel subset \(B\) of \(\ALPHABET Y\), \[ \PR^π(Y_{t+1} \in B | H_t = h_t, A_t = a_t) = \PR(Y_{t+1} \in B | Z_t = φ_t(h_t), A_t = a_t). \]
The right hand sides of (P1) and (P2) as well as (P2a) and (P2b) do not depend on the choice of the policy \(π\).
Proposition 21.1 : (P2a) and (P2b) imply (P2).
For any Borel measurable subset \(B\) of \(\ALPHABET Z\), we have \[\begin{align*} \hskip 1em & \hskip -1em \PR(Z_{t+1} \in B \mid H_t = h_t, A_t = a_t) \stackrel{(a)}= \sum_{y_{t+1} \in \ALPHABET Y} \PR(Y_{t+1} = y_{t+1}, Z_{t+1} \in B \mid H_t = h_t, A_t = a_t ] \\ &\stackrel{(b)}= \sum_{y_{t+1} \in \ALPHABET Y} \IND\{ ψ_t(φ_t(h_t), y_{t+1}, a_t) \} \PR(Y_{t+1} = y_{t+1} \mid H_t = h_t, A_t = a_t) \\ &\stackrel{(c)}= \sum_{y_{t+1} \in \ALPHABET Y} \IND\{ ψ_t(φ_t(h_t), y_{t+1}, a_t) \} \PR(Y_{t+1} = y_{t+1} \mid Z_t = φ_t(h_t), A_t = a_t) \\ &\stackrel{(d)}= \PR(Z_{t+1} \in B \mid Z_t = φ_t(h_t), A_t = a_t) \end{align*}\] where \((a)\) follows from the law of total probability, \((b)\) follows from (P2a), \((c)\) follows from (P2b), and \((d)\) from the law of total probability.
21.3 Examples of an information state
We start by define the belief state \(b_t \in Δ(\ALPHABET S)\) as follows: for any \(s \in \ALPHABET S\) \[ b_t(s) = \PR^π(S_t = s \mid H_t = h_t). \] The belief state is a function of the history \(h_t\). When we want to explicitly show the dependence of \(b_t\) on \(h_t\), we write it as \(b_t[h_t]\).
Lemma 21.1 The belief state \(b_t\) does not depend on the policy \(π\).
This is an extremely important result which has wide-ranging implications in stochastic control. For a general discussion of this point, see Witsenhausen (1975).
From the law of total probability and Bayes rule, we have \[\begin{equation} \label{eq:belief} \PR(s_t | y_{1:t}, a_{1:t-1}) = \sum_{s_{1:t-1}} \PR(s_{1:t} | y_{1:t}, a_{1:t-1}) = \sum_{s_{1:t-1}} \frac{\PR(s_{1:t}, y_{1:t}, a_{1:t-1})} {\sum_{s'_{1:t}} \PR(s'_{1:t}, y_{1:t}, a_{1:t-1})} \end{equation}\]
Now consider \[\begin{align*} \PR(s_{1:t}, y_{1:t}, a_{1:t-1}) &= \PR(s_1) \PR(y_1 | s_1) \IND\{ a_1 = π_1(y_1) \} \\ & \times \PR(s_2 | s_1, a_1) \PR(y_2 | s_2) \IND \{ a_2 = π_2(y_{1:2}, a_1)\} \\ & \times \cdots \\ & \times \PR(s_{t-1} | s_{t-2}, a_{t-2}) \PR(y_{t-1} | s_{t-1}) \IND \{ a_{t-1} = π_{t-1}(y_{1:t-1}, a_{1:t-2}) \} \\ & \times \PR(s_{t} | s_{t-1}, a_{t-1}) \PR(y_{t} | s_{t}). \end{align*}\] Substitute the above expression in both the numerator and the denominator of \eqref{eq:belief}. Observe that the terms of the form \(\IND\{ a_s = π_s(y_{1:s}, a_{1:s-1})\) are common to both the numerator and the denominator and cancel each other. Thus, \[\begin{equation} \label{eq:belief-fn} \PR(s_t | y_{1:t}, a_{1:t-1}) = \sum_{s_{1:t-1}} \frac{ \prod_{s=1}^t \PR(s_s \mid s_{s-1}, a_{s-1}) \PR(y_s \mid s_s) } { \sum_{s'_{1:t}} \prod_{s=1}^t \PR(s'_s \mid s'_{s-1}, a_{s-1}) \PR(y_s \mid s'_s) }. \end{equation}\] None of the terms here depend on the policy \(π\). Hence, the belief state does not depend on the policy \(π\).
Lemma 21.2 The belief state \(b_t\) updates in a state like manner. In particular, for any \(s_{t+1} \in \ALPHABET S\), we have \[ b_{t+1}(s_{t+1}) = \sum_{s_t \in \ALPHABET S} \frac{ \PR(y_{t+1} | s_{t+1}) \PR(s_{t+1} | s_t, a_t) b_t(s_t) } { \sum_{s'_{t:t+1}} \PR(y_{t+1} | s'_{t+1}) \PR(s'_{t+1} | s'_t, a_t) b_t(s'_t) }. \]
For any \(s_{t+1} \in \ALPHABET S\), consider
\[\begin{align} b_{t+1}(s_{t+1}) &= \PR(s_{t+1} | y_{1:t+1}, a_{1:t}) \notag \\ &= \sum_{s_t} \PR(s_{t:t+1} | y_{1:t+1}, a_{1:t}) \notag \\ &= \sum_{s_t} \frac{ \PR(s_{t:t+1}, y_{t+1}, a_t | y_{1:t}, a_{1:t-1}) } {\sum_{s'_{t:t+1}}\PR(s'_{t:t+1}, y_{t+1}, a_t | y_{1:t}, a_{1:t-1}) }. \label{eq:update-1} \end{align}\]
Now consider \[\begin{align} \hskip 1em & \hskip -1em \PR(s_{t:t+1}, y_{t+1}, a_t | y_{1:t}, a_{1:t-1}) \notag \\ &= \PR(y_{t+1} | s_{t+1}) \PR(s_{t+1} | s_t, a_t) \IND\{ a_t = π_t(y_{1:t}, a_{1:t-1}) \} \PR(s_t | y_{1:t}, a_{1_t-1}) \notag \\ &= \PR(y_{t+1} | s_{t+1}) \PR(s_{t+1} | s_t, a_t) \IND\{ a_t = π_t(y_{1:t}, a_{1:t-1}) \} b_t(s_t). \label{eq:belief-2} \end{align}\] Substitute the above expression in both the numerator and the denominator of \eqref{eq:update-1}. Observe that \(\IND\{ a_t = π_t(y_{1:t}, a_{1:t-1}) \}\) is common to both the numerator and the denominator and cancels out. Thus, we get the result of the lemma.
Now, we present three examples of information state here. See the Exercises for more examples.
Example 21.1 The complete history \(H_t\) is an information state.
We will prove that \(Z_t = H_t\) satisfies properties (P1), (P2a), and (P2b).
P1. \(\displaystyle \EXP^π[ c_t(S_t, A_t) | H_t = h_t, A_t = a_t ] = \sum_{s_t \in \ALPHABET S} c_t(s_t, a_t) b_t[h_t](s_t)\).
P2a. \(H_{t+1} = (H_t, Y_{t+1}, A_t)\)
P2b. \(\displaystyle \PR^π(y_{t+1} | y_{1:t}, a_{1:t}) = \sum_{s_{t:t+1}} \PR(y_{t+1} | s_{t+1}) \PR( s_{t+1} | s_t, a_t) \PR(s_t | y_{1:t}, a_{1:t})\). Note that in the last term \(\PR^π(s_t | y_{1:t}, a_{1:t})\) we can drop \(a_t\) from the conditioning because it is a function of \((y_{1:t}, a_{1:t-1})\). Thus, \[ \PR^π(s_t | y_{1:t}, a_{1:t}) = \PR^π(s_t | y_{1:t}, a_{1:t-1}) = b_t[h_t](s_t).\] Note that in the last step, we have used Lemma 21.1. Thus, \(\displaystyle \PR^π(y_{t+1} | y_{1:t}, a_{1:t}) = \sum_{s_{t:t+1}} \PR(y_{t+1} | s_{t+1}) \PR( s_{t+1} | s_t, a_t) b_t[h_t](s_t)\).
Example 21.2 The belief state \(b_t\) is an information state.
The belief state \(b_t\) is a function of the history \(h_t\). (The exact form of this function is given by \eqref{eq:belief-fn}). In the proof of Example 21.1, we have already shown that \(b_t\) satisfies (P1) and (P2b). Moreover Lemma 21.2 implies that the belief update satisfies (P2a).
Both the above information states are generic information states which work for all models. For specific models, it is possible to identify other information states as well. We present some examples of such an information state below.
Example 21.3 An MDP is a special case of a POMDP where \(Y_t = S_t\). For an MDP \(Z_t = S_t\) is an information state.
We will show that \(Z_t = S_t\) satisfies (P1) and (P2).
(P1) is satisfied because the per-step cost is a function of the \((S_t, A_t)\). (P2) is equivalent to the control Markov property.
21.4 Information state based dynamic program
The main feature of an information state is that one can always write a dynamic program based on an information state.
Theorem 21.3 Let \(\{Z_t\}_{t=1}^T\) be any information state, where \(Z_t = φ_t(H_t)\). Recursively define value functions \(\{ \hat V_t \}_{t=1}^T\), where \(\hat V_t \colon \ALPHABET Z \to \reals\), as follows: \[ \hat V_{T+1}(z_{T+1}) = 0 \] and for \(t \in \{T, \dots, 1\}\): \[\begin{align} \hat Q_t(z_t, a_t) &= \EXP[ c_t(S_t, A_t) + \hat V_{t+1}(Z_{t+1}) \mid Z_t = z_t, A_t = a_t] \\ \hat V_t(z_t) &= \min_{a_t \in \ALPHABET A} \hat Q_t(z_t, a_t). \end{align}\] Then, we have the following: for any \(h_t\) and \(a_t\), \[\begin{equation} \label{eq:history-info} Q_t(h_t, a_t) = \hat Q_t(φ_t(h_t), a_t) \quad\text{and}\quad V_t(h_t) = \hat V_t(φ_t(h_t)). \end{equation}\] Any strategy \(\hat π = (\hat π_1, \dots, \hat π_T)\), where \(\hat π_t \colon \ALPHABET Z \to \ALPHABET A\), is optimal if and only if \[\begin{equation}\label{eq:info-verification} \hat π_t(z_t) \in \arg\min_{a_t \in \ALPHABET A} \hat Q_t(z_t, a_t). \end{equation}\]
As usual, we prove the result by backward induction. By construction, Eq. \eqref{eq:history-info} is true at time \(T+1\). This forms the basis of induction. Now assume that \eqref{eq:history-info} is true at time \(t+1\) and consider the system at time \(t\). Then, \[\begin{align*} Q_t(h_t, a_t) &= \EXP[ c_t(S_t, A_t) + V_{t+1}(H_{t+1}) | H_t = h_t, A_t = a_t ] \\ &\stackrel{(a)}= \EXP[ c_t(S_t, A_t) + \hat V_{t+1}( φ_t(H_{t+1}) ) | H_t = h_t, A_t = a_t ] \\ &\stackrel{(b)}= \EXP[ c_t(S_t, A_t) + \hat V_{t+1}( φ_t(H_{t+1}) ) | Z_t = φ_t(h_t), A_t = a_t ] \\ &\stackrel{(c)}= \hat Q_t(φ_t(h_t), a_t), \end{align*}\] where \((a)\) follows from the induction hypothesis, \((b)\) follows from the properties (P1) and (P2) of the information state, and \((c)\) follows from the definition of \(\hat Q_t\). This shows that the action value functions are equal. By minimizing over the actions, we get that the value functions are also equal.
21.5 Belief state based dynamic program
As shown in Example 21.2, the belief state \(b_t\) is an information state. Therefore, Theorem 21.3 implies that we can write a dynamic program based on \(b_t\). This is an important and commonly used formulation, so we study it separately and present some properties of the value functions. The belief state based dynamic program is given by: \(V_{T+1}(b_{T+1}) = 0\) and for \(t \in \{T, \dots, 1\}\), \[ Q_t(b_t, a_t) = \EXP [ c_t(S_t, A_t) + V_{t+1}(B_{t+1}) \mid B_t = b_t, A_t = a_t ]. \] and \[ V_t(b_t) = \min_{a_t \in \ALPHABET A} Q_t(b_t, a_t). \]
Define \[ \PR(y_{t+1} | b_t, a_t) =
\sum_{s_{t:t+1}} \PR(y_{t+1} | s_{t+1}) \PR(s_{t+1} | s_t, a_t) b_t(s_t).
\] Then, the belief update expression in Lemma 21.2 can be written as: \[
b_{t+1}(s_{t+1}) =
\frac{ \PR(y_{t+1} | s_{t+1}) \sum_{s_t} \PR(s_{t+1} | s_t, a_t) b_t(s_t) }
{ \PR(y_{t+1} | b_t, a_t) }.
\] For the ease of notation, we write this expression as \(b_{t+1} = ψ(b_t,
y_{t+1}, a_t)\).
\[\begin{align*}
Q_t(b_t, a_t) &= \sum_{s_t \in \ALPHABET S} c_t(s_t, a_t) b_t(s_t) \\
& \quad + \sum_{y_{t+1} \in \ALPHABET Y} \PR(y_{t+1} | b_t, a_t)
V_{t+1}( φ(b_t, y_{t+1}, a_t) ).
\end{align*}\]
A key property of the belief-state based value functions is the following.
Theorem 21.4 The belief-state based value functions are piecewise linear and concave.
As usual, we prove the result using backward induction. For any \(a_T\), \[ Q_T(b_T, a_T) = \sum_{s_T \in \ALPHABET S} c_T(s_T, a_T) b_T(s_T) \] is linear in \(b_T\). Therefore, \[ V_T(b_T) = \min_{a_T \in \ALPHABET A} Q_T(b_T, a_T) \] is the minimum of a finite number of linear functions. Hence \(V_T(b_T)\) is piecewise linear and concave.
Now assume that \(V_{t+1}(b_{t+1})\) is piecewise linear and concave (PWLC). Any PWLC function can be represented as a minimum of a finite number of hyperplanes. Therefore, we can find a finite set of vectors \(\{ A_i \}_{i \in I}\) indexed by finite set \(I\) such that \[ V_{t+1}(b) = \min_{i \in I} \langle A_i, b \rangle. \]
We need to show that \(V_t(b_t)\) is piecewise linear and concave (PWLC). We first show that \(Q_t(b_t, a_t)\) is PWLC. For any fixed \(a_t\), the first term \(\sum_{s_t} c_t(s_t, a_t) b_t(s_t)\) is linear in \(b_t\). Now consider the second term: \[\begin{align*} \hskip 1em & \hskip -1em \sum_{y_{t+1} \in \ALPHABET Y} \PR(y_{t+1} | b_t, a_t) V_{t+1}( φ(b_t, y_{t+1}, a_t) ) \\ &= \sum_{y_{t+1} \in \ALPHABET Y} \PR(y_{t+1} | b_t, a_t) \min_{i \in I} \left\langle A_i, \frac{ \PR(y_{t+1} | s_{t+1}) \sum_{s_t} \PR(s_{t+1} | s_t, a_t) b_t(s_t) } { \PR(y_{t+1} | b_t, a_t) } \right\rangle \\ &= \sum_{y_{t+1} \in \ALPHABET Y} \min_{i \in I} \Big\langle A_i, \PR(y_{t+1} | s_{t+1}) \sum_{s_t} \PR(s_{t+1} | s_t, a_t) b_t(s_t) \Big\rangle \end{align*}\] which is the sum of PWLC functions of \(b_t\) and therefore PWLC in \(b_t\).
Thus, \(Q_t(b_t, a_t)\) is PWLC. Hence, \(V_t(b_t)\) which is the pointwise minimum of PWLC functions is PWLC. Hence, the result holds due to principle of induction.
Exercises
Exercise 21.1 (MDP with irrelevant component) Consider an MDP with state space \((\ALPHABET S^1 × \ALPHABET S^2)\) and action space \(\ALPHABET A\). Suppose the transition matrix satisfies \[\begin{multline*} \PR(S^1_{t+1} = s^1_{t+1}, S^2_{t+1} = s^2_{t+1} \mid S^1_t = s^1_t, S^2_t = s^2_t, A_t = a_t) \\ = \PR(S^1_{t+1} = s^1_{t+1} \mid S^1_t = s^1_t, A_t = a_t) \PR(S^2_{t+1} = s^2_{t+1} \mid S^1_t = s^1_t, S^2_t = s^2_t, A_t = a_t) \end{multline*}\] and the per-step costs does not depend on \(s^2\), i.e., \[ c(s^1,s^2,a) = c(s^1,a). \] Then, \(\{S^1_t\}_{t \ge 1}\) is an information state.
Exercise 21.2 (Folded MDP) Consider an MDP where the state space \(\ALPHABET S\) is a symmetric subset of integers of the form \(\{-L, -L + 1, \dots, L -1 , L\}\) and the action space \(\ALPHABET A\) is discrete. Suppose the transition matrix \(P(a)\) and the cost function \(c_t(s,a)\) satisfy properties (A1) and (A2) of Exercise 8.8. Show that \(Z_t = |S_t|\) is an information state.
Exercise 21.3 (MDP with delayed state observation) Consider an MDP where the agent observes the state \(S_t\) with a delay \(δ\), i.e., the observation \(Y_t\) at time \(t\) equals \(S_{t-δ}\). Show that \((S_{t-δ}, A_{t-δ:t-1})\) is an information state.
Exercise 21.4 (POMDP with delayed observations) Consider an POMDP where the agent observes the observation \(Y_t\) with a delay \(δ\), i.e., the observation \(Y_t\) at time \(t\) equals \(\ell_t(S_{t-δ}, N_{t-δ})\). Show that \((B_{t-δ}, A_{t-δ:t-1})\) is an information state.
Exercise 21.5 (LQG control) Consider a linear system with state \(x_t \in \reals^n\), observations \(y_t \in \reals^p\), and action \(u_t \in \reals^m\). Note that we will follow the standard notation of linear systems and denote the system variables by lower case letters \((x,u)\) rather than upper case letter \((S,A)\). The dynamics of the system are given by \[\begin{align*} x_{t+1} &= A x_t + B u_t + w_t \\ y_t &= C x_t + n_t \end{align*}\] where \(A\), \(B\), and \(C\) are matrices of appropriate dimensions. The per-step cost is given by \[ c(x_t, u_t) = x_t^\TRANS Q x_t + u_t^\TRANS R u_t, \] where \(Q\) is a positive semi-definite matrix and \(R\) is a positive definite matrix. We make the standard assumption that the primitive random variables \(\{s_1, w_1, \dots, w_T, n_1, \dots, n_T \}\) are independent.
Show that if the primitive variables are Guassian, then the conditional estimate of the state \[ \hat x_t = \EXP[ x_t | y_{1:t}, u_{1:t-1} ] \] is an information state.
Exercise 21.6 (Machine repair) Consider a machine which can be in one of \(n\) ordered state where the first state is the best and the last state is the worst. The production cost increases with the state of the machine. The state evolves in a Markovian manner. At each time, an agent has the option to either run the machine or stop and inspect it for a cost. After inspection, the agent may either repair the machine (at a cost that depends on the state) or replace it (at a fixed cost). The objective is to identify a maintenance policy to minimize the cost of production, inspection, repair, and replacement.
Let \(τ\) denote the time of last inspection and \(S_τ\) denote the state of the machine after inspection, repair, or replacement. Show that \((S_τ, t-τ)\) is an information state.
Notes
The discussion in this section is taken from Subramanian et al. (2022). Information state may be viewed as a generalization of the traditional notion of state Nerode (1958), which is defined as a statistic (i.e., a function of the observations) sufficient for input-output mapping. In contrast, we define an information state as a statistic sufficient for performance evaluation (and, therefore, for dynamic programming). Such a definition is hinted in Witsenhausen (1976). The notion of information state is also related to sufficient statistics for optimal control defined in Striebel (1965) for systems with state space models.
As far as we are aware, the informal definition of information state was first proposed by Kwakernaak (1965) for adaptive control systems. Formal definitions for linear control systems were given by Bohlin (1970) for discrete time systems and by Davis and Varaiya (1972) for continuous time systems. Kumar and Varaiya (1986) define an information state as a compression of past history which satisfies property (P2a) but do not formally show that such an information state always leads to a dynamic programming decomposition.
Exercise 21.1 is due to Feinberg (2005). As explained in Feinberg (2005), such models arise in control of queues and transformation of continuous time MDPs to discrete time MDPs using uniformization. It may also be considered an instance of what is called the noisy TV problem in the RL literature (Burda et al. 2018).
Exercise 21.2 is due to Chakravorty and Mahajan (2018). Exercise 21.3 is due to Altman and Nain (1992). Exercise 21.4 is due to Bander and White (1999). Exercise 21.6 is an instance of incrementally expanding representations for POMDPs presented in Arabneydi and Mahajan (2015).