• Policy \(\pi\) chooses an action based on the current state so \(a_t=a\) with probability \(\pi(a|s_t)\)
  • Shorthand for deterministic policy: \(a_t=\pi(s_t)\)

• Probability of trajectory \(\tau =(s_0, a_0, s_1, ... s_t, a_t)\) $$ \mathbb{P}_{\mu_0}^\pi (\tau) = \mu_0(s_0)\pi(a_0 \mid s_0) \cdot \displaystyle\prod_{i=1}^t {P}(s_i \mid s_{i-1}, a_{i-1}) \pi(a_i \mid s_i) $$
• Probability of \((s, a)\) at \(t\) $$ \mathbb{P}^\pi_t(s, a ; \mu_0) = \displaystyle\sum_{\substack{s_{0:t-1}\\ a_{0:t-1}}} \mathbb{P}^\pi_{\mu_0} (s_{0:t-1}, a_{0:t-1}, s_t, a_t \mid s_t = s, a_t = a) $$
• Discounted "steady-state" distribution $$ d^\pi_{\mu_0}(s, a) = (1 - \gamma) \displaystyle\sum_{t=0}^\infty \gamma^t \mathbb{P}^\pi_t(s, a; \mu_0) $$

Food for thought:

• How do these distributions change under two different policies \(\pi\) and \(\pi'\)? (HW2)
• How to write the distributions \(\mathbb{P}^\pi_t\) and \(d^\pi_{\mu_0}\) over the state only?

• Evaluate policy by cumulative reward
  • \(V^\pi(s) = \mathbb E[\sum_{t=0}^\infty \gamma^t r_t | s_0=s]\)
  • \(Q^\pi(s, a) = \mathbb E[\sum_{t=0}^\infty \gamma^t r_t | s_0=s, a_0=a]\)
• For finite horizon, for \(t=0,...H-1\),
  • \(V_t^\pi(s) = \mathbb E[\sum_{k=t}^{H-1} r_k | s_t=s]\)
  • \(Q_t^\pi(s, a) = \mathbb E[\sum_{k=t}^{H-1} r_t | s_t=s, a_t=a]\)

Recursive Bellman Expectation Equation:

• Discounted Infinite Horizon
  •  \(V^{\pi}(s) = \mathbb{E}_{a \sim \pi(s)} \left[ r(s, a) + \gamma \mathbb{E}_{s' \sim P( s, a)} [V^\pi(s')] \right]\)
  • \(Q^{\pi}(s, a) = r(s, a) + \gamma \mathbb{E}_{s' \sim P( s, a)} \left[ V^\pi(s') \right]\)
• Finite Horizon,  for \(t=0,\dots H-1\),
  • \(V^{\pi}_t(s) = \mathbb{E}_{a \sim\pi_t(s) } \left[ r(s, a) + \mathbb{E}_{s' \sim P(s, a)} [V^\pi_{t+1}(s')] \right]\)
  • \(Q^{\pi}_t(s) =  r(s, a) + \mathbb{E}_{s' \sim P(s, a)} [V^\pi_{t+1}(s')] \)

• Recursive computation: \(V^{\pi} = R^{\pi} + \gamma P^{\pi} V^\pi\)
  • Exact Policy Evaluation: \(V^{\pi} = (I- \gamma P^{\pi} )^{-1}R^{\pi}\)
  • Iterative Policy Evaluation: \(V^{\pi}_{t+1} = R^{\pi} + \gamma P^{\pi} V^\pi_t\)
• Backwards-Iterative computation in finite horizon:
  • Initialize \(V^{\pi}_H = 0\)
  • For \(t=H-1, H-2, ... 0\)
    • \(V^{\pi}_t = R^{\pi} +P^{\pi} V^\pi_{t+1}\)

• An optimal policy \(\pi^*\) is one where \(V^{\pi^*}(s) \geq V^{\pi}(s)\) for all \(s\) and policies \(\pi\)
• Equivalent condition: Bellman Optimality
  • \(V^*(s) = \max_{a\in\mathcal A} \left[r(s, a) + \gamma \mathbb{E}_{s' \sim P(s, a)} \left[V^*(s') \right]\right]\)
  • \( Q^*(s, a) = r(s, a) + \gamma \mathbb{E}_{s' \sim P(s, a)} \left[ \max_{a'\in\mathcal A} Q^*(s', a') \right]\)
• Optimal policy \(\pi^*(s) = \argmax_{a\in \mathcal A} Q^*(s, a)\)

• Finite horizon, for \(t=0,\dots H-1\),
  • \(V_t^*(s) = \max_{a\in\mathcal A} \left[r(s, a) + \mathbb{E}_{s' \sim P(s, a)} \left[V_{t+1}^*(s') \right]\right]\)
  • \(Q_t^*(s, a) = r(s, a) + \mathbb{E}_{s' \sim P(s, a)} \left[ \max_{a'\in\mathcal A} Q_{t+1}^*(s', a') \right]\)
• Optimal policy \(\pi_t^*(s) = \argmax_{a\in \mathcal A} Q_t^*(s, a)\)
• Can directly solve with Dynamic Programming
  • Iterate backwards in time from \(V^*_{H}=0\)

• Infinite horizon: algorithms for recursion in the Bellman Optimality equation
• Value Iteration
  • Initialize \(Q_0\). For \(t=0,1,\dots\),
    • \(Q^{t+1}(s,a) =r(s, a) + \gamma\mathbb{E}_{s' \sim P(s, a)} \left[ \max_{a'\in\mathcal A} Q^{t}(s', a') \right]\)
• Policy Iteration
  • Initialize \(\pi_0\). For \(t=0,1,\dots\),
    • \(Q^{t}= \) PolicyEval(\(\pi^t\))
    • \(\pi^{t+1}(s) = \argmax_{a\in\mathcal A} Q^{t}(s,a)\)

• Value Iteration
  • Fixed point iteration (like Iterative Policy Iteration) from Bellman Q Optimality
  • Contraction in Q: \(\|Q^{t+1} - Q^*\|_\infty \leq \gamma \|Q^t - Q^*\|_\infty\)
• Policy Iteration
  • Monotone Improvement: \(Q^{t+1}(s,a) \geq Q^{t}(s,a)\)
  • Contraction in V: \(\|V^{t+1} - V^*\|_\infty \leq \gamma \|V^t - V^*\|_\infty\)

• Linear Dynamics: $$s_{t+1} = A s_t + Ba_t + w_t,\quad w_t\sim \mathcal N(0,\sigma^2 I)$$
• Unrolled dynamics $$ s_{t} = A^ts_0 + \sum_{k=0}^{t-1} A^k (Ba_{t-k-1} + w_{t-k-1})$$
• Stability of uncontrolled \(s_{t+1}=As_t\):
  • stable if \(\rho(A)< 1\)
  • unstable if \(\rho(A) > 1\)
  • marginally unstable if \(\rho(A) = 1\)

Finite Horizon LQR: Application of Dynamic Programming

• Initialize \(V^{\pi}_H(s) = 0\)
• For \(t=H-1, H-2, ... 0\)
  • \(Q^{*}_t(s,a) = c(s,a) +\mathbb E_{s'\sim P(s,a)}[ V^*_{t+1}(s')]\)
  • \(\pi^*(s) = \argmin_{a\in\mathcal A} Q^{*}_t(s,a)\)
  • \(V^*_t =  Q^{*}_t(s,\pi^*(s)) \)

1. Add and subtract: $$ \|f(x) - g(y)\| \leq  \|f(x)-f(y)\| +\|f(y)-g(y)\| $$
2. Contractions (induction) $$ \|x_{t+1}\|\leq \gamma \|x_t\| \implies \|x_t\|\leq \gamma^t\|x_0\|$$
3. Additive induction $$ \|x_{t+1}\| \leq \delta_t + \|x_t\| \implies \|x_t\|\leq \sum_{k=0}^{t-1} \delta_k + \|x_0\|  $$
4. Basic Inequalities (HW0) $$|\mathbb E[f(x)] - \mathbb E[g(x)]| \leq \mathbb E[|f(x)-g(x)|] $$ $$|\max f(x) - \max g(x)| \leq \max |f(x)-g(x)| $$ $$ \mathbb E[f(x)] \leq \max f(x) $$

1. Move on if stuck!
2. Write explanations and show steps for partial credit
3. Multipart questions: can be done mostly independently
   • ex: 1) show \(\|x_{t+1}\|\leq \gamma \|x_t\|\)
           2) give a bound on \(\|x_t\|\) in terms of \(\|x_0\|\)