• 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 $$ d^\pi_{\mu_0}(s, a) = (1 - \gamma) \displaystyle\sum_{t=0}^\infty \gamma^t \mathbb{P}^\pi_t(s, a; \mu_0) $$
  • Finite horizon: \(d^\pi_{\mu_0}(s, a) =\frac{1}{H}\sum_{t=0}^{H-1} \mathbb{P}^\pi_t(s, a; \mu_0) \)

Food for thought:

• How do these distributions change under two different transition models \(P\) and \(\widehat P\) (Simulation Lemma) or two different policies (PDL, Prelim, HW2)?
• How to write the distribution \(\mathbb{P}^\pi_t\) in terms of \(\mathbb{P}^\pi_{t-1}\)?

• 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 \(t=0,...H-1\),
  • \(V_t^\pi(s) = \mathbb E[\sum_{k=t}^{H-1} r_k | s_t=s]\)
  • \(Q_t^\pi(s, a) = \mathbb E[\sum_{k=t}^{H-1} r_k | s_t=s, a_t=a]\)

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 \(V^*_{H}=0\)

• 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 \(s_{t+1}=As_t\): determined by whether \(\rho(A)<1\)
• Finite Horizon LQR: Application of Dynamic Programming

  • Basis for approximation-based algorithms (local linearization and iLQR)

• What do we want to learn? \(\mathcal M = \{\mathcal S,\mathcal A,P,r,\gamma\}\)
  • Unknown transitions \(P(s'|s,a)\) or reward function \(r(s,a)\)
  • Value/Q function
    • of policy \(V^\pi(s)\) or \(Q^\pi(s,a)\)
    • optimal \(V^*(s)\) or \(Q^*(s,a)\)
  • Optimal Policy \(\pi^*(s)\)
• Given a dataset with features \(x_i\) and labels \(y_i\)
• Fitting a model:
  • Via counting: \(\widehat f(x) = \sum_{i=1}^N y_i \mathbf 1\{x=x_i\} / \sum_{i=1}^N\mathbf 1\{x=x_i\}  \)
  • Function approx: \(\widehat f(x) = \min_{f\in\mathcal F} \frac{1}{N} \sum_{i=1}^N (f(x_i)-y_i)^2   \)

Model-Based RL

• Features are \((s,a)\) and label is \(s'\)
• Tabular setting: \(\widehat P\) via counting
• Simulation Lemma
  • Translate error in \(\widehat P\) vs \(P\) into difference in performance \(\widehat V\) vs \(V\)

• Features are \((s_i,a_i)\)
  • \((s_i,a_i)  = (s_{h_1}, a_{h_1}) \sim d^\pi_{\mu_0}\)
• Labels constructed as:
  • Rollout based (MC): \(y_i = \sum_{t=h_1}^{h_1+h_2} r_t\)
  • Bellman Exp based (TD): \(y_t =r_t + \gamma \widehat Q(s_{t+1},a_{t+1}) \)
  • Bellman Opt based (TD): \(y_t =r_t + \gamma \max_a \widehat Q(s_{t+1},a) \)
• On vs. off policy (Recall HW)
• \(\widehat Q =\arg\min \frac{1}{N}\sum_{i=1}^N (Q(s_i,a_i)-y_i)^2\)

• Approximate Dynamic Programming
  For \(t=0,1...\):
  1. \(\widehat Q^t = \mathsf{SampleEval}(\pi^t)\)
  2. \(\pi^t = \mathsf{Improvement}(\widehat Q^t)\)
• Approximate Policy Iteration
  • Greedy improvement, could oscillate
• Conservative Policy Iteration
  • Incremental improvement
• Performance Difference Lemma

• Policy Gradient Meta-Algorithm
  for \(t=0,1,...\)
  1. collect rollouts using \(\theta_t\)
  2. estimate gradient with \(g_t\)
  3. \(\theta_{t+1} = \theta_t + \alpha g_t\)
• Trust regions and Natural PG

Imitation Learning with BC

• Inverse RL: Principle of maximum entropy
  
  
   
• Soft-VI (entropy weighted) replaces \(\max\) with softmax
• Max-Ent IRL: For \(k=0,\dots,K-1\):
  1. \(\pi^k = \mathsf{SoftVI}(w_k^\top \varphi)\)
  2. \(w_{k+1} = w_k + \eta (\mathbb E_{d^{\pi^*}_\mu}[\varphi (s,a)] - \mathbb E_{d^{\pi^k}_\mu}[\varphi (s,a)])\)
• Lagrange Formulation to constrained optimization

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) $$|\mathbb E[f(x)] - \mathbb E[g(x)]| \leq \mathbb E[|f(x)-g(x)|] $$ $$|\max f(x) - \max g(x)| \leq \max |f(x)-g(x)| $$ $$ \mathbb E[f(x)] \leq \max f(x) $$