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In RL, *agents* act within an *environment*.
*Rewards* are received for how good the environment state is. The
goal of RL is to maximize the total sum of rewards, called the
return.

**States/observations:**- A
*state*is the full information of the world the agent is in - The agent gets to see an
*observation*of the state - This observation might exclude some information
- Environments where information is excluded in the observation are
*partially-observed*, other states are*fully-observed*

- A
**Action Spaces:***Action spaces*are the set of all valid actions within an environment- There are discrete and continuous action spaces
- E.g. chess vs driving

**Policies:**- A
*policy*is the algorithm an agent uses for choosing an action in a state - There are
*deterministic*and*stochastic*policies: - In deep RL, policies are parameterized (\(\theta\)) using a neural network
- Deterministic policies:
- ArgMax of the output of a neural network
- Usually denoted using \(\mu\), as: \(a_t = \mu_\theta (s_t)\)

- Stochastic policies:
- Sample from the outputs of the network
- Usually denoted using \(\pi\), as: \(a_t \sim \pi_\theta (\cdot | s_t)\)
- Categorical stochastic policies:
- For discrete action spaces
- Neural network with softmaxed outputs which are sampled from as a probability distribution
- Log-likelihood is given by taking the log of a logit

- Diagonal gaussian stochastic policies:
- For continuous action spaces
- Probability distributions are gaussians with covariance 0 between them
- Therefore, the gaussians are parameterized by a mean vector \(\mu\) and diagonal of the covariance matrix, \(\sigma\)
- Log standard deviations either are parameters or outputs of the
neural network dependent upon state
- Log (\(\log \sigma\)) is used as it can be \(-\infty, \infty\) instead of \(\sigma\) which must be non-negative

- The mean action vector is always an output of the neural network
- All together, these make up gaussians which can be sampled from
- Log-likelihood: \(\log \pi_\theta (a | s) = -\frac{1}{2} (\sum^k_{i=1}(\frac{(a_i - \mu_i)^2}{\sigma^2_i} + 2 \log \sigma_i) + k \log 2 \pi)\)

- A
**Trajectories:**- A trajectory (\(\tau\)) is a history of states and actions taken in order
- The first state is sampled from the start-state distribution (\(s_0 \sim \rho_0(\cdot)\)
- After the start state, the next state is governed by the
*state-transition function*based on only the current state and action taken- Only these are needed because of the Markov property

- The state-transition function can be deterministic (\(s_{t+1} = f(s_t, a_t)\)) or stochastic (\(s_{t+1} \sim P(\cdot | s_t, a_t)\))
- Trajectories are also known as “episodes” or “rollouts”

**Reward/return:***Reward*is dependent on*current state*,*current action*, and*next state*: \(r_t = R(s_t, a_t, s_{t+1})\)- It can often be simplified to drop the dependency on next state (\(r_t = R(s_t, a_t)\)) or action (\(r_t = R(s_t)\))

- Rewards over a trajectory can be
*finite*or*infinite*horizon and*discounted*or*un-discounted* - Finite-horizon undiscounted return: \(R(\tau) = \sum^T_{t=0}r_t\)
- Infinite-horizon discounted return: \(R(\tau) = \sum^\infty_{t=0} \gamma^t r_t, \quad \gamma \in (0, 1)\)

**Expected return:**- Probability of a trajectory: \(P(\tau | \pi) = \rho_0(s_0) \prod^{T-1}_{T=0} P(s_{t+1} | s_t, a_t) \pi (a_t | s_t)\)
- Expected return: \(J(\pi) = \int_{\tau} P(\tau | \pi) R(\tau) = \mathbb{E}_{\tau \sim \pi} [R(\tau)]\)

**Value functions:**- Value functions measure how “good” a state or state action pair is
- The inputs are either states or state-action pairs, they either follow the optimal policy or a given policy
- On-policy value function: \(V^\pi(s) = \mathbb{E}_{\tau \sim \pi}[R(\tau)|s_0 = s]\)
- On-policy action-value function: \(Q^\pi(s, a) = \mathbb{E}_{\tau \sim \pi} [R(\tau) | s_0 = s, a_0 = a)]\)
- Optimal value function: \(V^*(s) = \max_\pi \mathbb{E}_{\tau \sim \pi} [R(\tau) | s_0 = s]\)
- Optimal action-value function: \(Q^*(s, a) = \max_\pi \mathbb{E}_{\tau \sim \pi} [R(\tau) | s_0 = s, a_0 = a]\)

**Optimal action in a state:**\(a^*(s) = \arg \max_a Q^*(s, a)\)- This is somewhat how Deep-Q networks work, by improving the Q-function via satisfying Bellman equations, ultimately using the function to find the best action in a state

**Bellman equations:**- Self-consistency equations, formulating that the value of the current state is equal to the current reward, plus the value of your next state
- On-policy Bellman equations:
- \(V^\pi(s) = \mathbb{E}_{a \sim \pi, s' \sim P}[r(a, s) + \gamma V^\pi (s')]\)
- \(Q^\pi(s, a) = \mathbb{E}_{s' \sim P}[r(s, a) + \mathbb{E}_{a' \sim \pi}[Q^\pi(s', a')]\)

- Optimal Bellman equations:
- \(V^*(s) = \max_a \mathbb{E}_{s' \sim P}[r(a, s) + \gamma V^\pi(s')]\)
- \(Q^*(s, a) = \mathbb{E}_{s' \sim P}[r(s, a) + \gamma \max_{a'} Q^*(s', a')]\)

- Main difference is that optimal functions must pick the best action, where on-policy takes the expected action via the policy
- The “Bellman Backup” for a state (action pair) is the right side of the Bellman equations

**Advantage functions:***Advantage functions*measure how good an action is relative to other actions in the state- “How much better is it to take a certain action vs. the expectation action from the policy”
- \(A^\pi(s, a) = Q^\pi(s, a) - V^\pi(s)\)

**Markov Decision Processes (MDPs):**- Formal way of describing a setting/problem
- Shows the Markov property, the future is independent of the past (besides current state/action)
- MDPs are 5-tuples: \(\langle S, A, R, P,
\rho_0 \rangle\)
- \(S\): set of valid states
- \(A\): set of valid actions
- \(R\): reward function, \(r_t = R(s_t, a_t, s_{t+1})\)
- \(P\): state transition function, \(s' = P(s'|s,a)\)
- \(\rho_0\): starting state distribution

The goal of RL is to maximize the expected return over a trajectory from agent actions given by a policy. This can be expressed as finding the optimal policy, \(\pi^*\), where \(\pi^* = \arg \max_\pi J(\pi)\)

One of the biggest differentiators in RL is if the algorithm models the environement.

- Model-based RL can either learn or be given a model which predicts state transitions and rewards
- This is advantageous because it allows the agent to search and plan ahead
- These models can either be learned or given:
**Learned models:**- Learning a model can be difficult, due to bias the agent being able to exploit the biases in the model

**Given models:**- This is common for games, such as go or chess

**Explicit Policies:**- They learn a function \(\pi_\theta(a|s)\)
- Optimization is (almost) always on-policy (must use data collected from the most recent version of the policy)
- Often need to learn approximator for the value function \(V^\pi(s)\)
- A2C, PPO do this

**Q-Learning:**- Learn a Q-function by satisfying the Bellman equation
- Off-policy, can use old data
- Actions are decided via the action which has the highest value for the Q-function
- Deep-Q-networks use this!

**Model-based:**- Pure planning involves not representing the policy, instead use search to select best actions
- Expert iteration involves search along with a policy, which generates possible actions to search from

**Deriving policy gradients**

*Five helpful facts*

- Prob of trajectory: \[P(\tau | \theta) = \rho_0(s_0) \prod_{t=0}^T P(s_{t+1} | s_t, a_t) \pi_\theta(a_t | s_t)\]
- Log-derivative trick: \[\nabla_\theta P(\tau | \theta) = P(\tau | \theta) \nabla_\theta \log P(\tau | \theta)\]
- Log-prob of a trajectory: \[\log P(\tau | \theta) = \log \rho_0(s_0) + \sum_{t=0}^T [\log P(s_{t+1} | s_t, a_t) + \log \pi_\theta(a_t | s_t)]\]
- Gradients of env functions: \[\rho_0(s_0) = P(s_{t+1} | s_t, a_t) = R(\tau) = 0\]
- Grad-log-prob of trajectory: \[\nabla_\theta \log P(\tau | \theta) = \sum_{t=1}^T \nabla_\theta \log \pi_\theta(a_t | s_t)\]

**Basic policy gradient:**

\[\nabla_\theta J(\pi_\theta) = \mathbb{E}_{\tau \sim \pi_\theta}[\sum_{t=0}^T \nabla_\theta \log \pi_\theta(a_t | s_t) R(\tau)]\]

It can be estimated via: \(\nabla_\theta J(\pi_\theta) \sim \frac{1}{|D|} \sum_{\tau \in D} \sum_{t=0}^T \nabla_\theta \log \pi_\theta (a_t | s_t) R(\tau)\)

*“Expected Grad-Log-Prob Lemma”:*

\[\mathbb{E}_{x \sim P_\theta}[\nabla_\theta \log P_\theta(x)] = 0\]

The simple gradient includes rewards earned before the action was taken, which doesn’t make sense. We can equivalently express the gradient as:

\[\nabla_\theta J(\pi_\theta) = \mathbb{E}_{\tau \sim \pi_\theta}[\sum_{t=0}^T \nabla_\theta \log \pi_\theta (a_t | s_t) \sum_{t' = t}^T R(s_{t'}, a_{t'}, s_{t'+1})]\]

The “reward-to-go” is \(\hat{R}_t = \sum_{t'=t}^T R(s_{t'}, a_{t'}, s_{t'+1})\)

Because of EGLP, we can multiply by functions \(b(s_t)\) which only depend on state:

\[\mathbb{E}_{a_t \sim \pi_\theta}[\nabla_\theta \log \pi_\theta(a_t | s_t) b(s_t)] = 0\]

Therefore, we can add or subtract terms from the “reward” part of the policy gradient without changing the expectation. We can subtract (our prediction of) the value function from the reward (\(b(s_t) = V^\pi(s_t)\)).

As we don’t actually know \(V^\pi(s_t)\), we will typically estimate it via another neural network. The neural network typically minimizes a mean-squared loss function:

\[V_\phi = \arg \min_{V_\phi} \mathbb{E}_{s_t, \hat{R}_t, \sim \pi_k}[(V_\phi(s_t) - \hat{R}_t)^2]\]

Note that the states and rewards are from the current epoch, \(k\).

The *general form* of the policy gradient is:

\[\nabla_\theta J(\pi_\theta) = \mathbb{E}_{\tau \sim \pi_\theta}[\sum_{t=0}^T \nabla_\theta \log \pi_\theta(a_t | s_t) \Phi_t]\]

\(\Phi_t\) can be any of the following and still have the same expectation:

- \(\Phi_t = R(\tau)\)
- \(\Phi_t = \sum_{t'=t}^T R(s_{t'}, a_{t'}, s_{t'+1})\)
- \(\Phi_t = \sum_{t'=t}^T R(s_{t'}, a_{t'}, s_{t'+1}) - b(s_t)\)
- \(\Phi_t = Q^{\pi_\theta}(s_t, a_t)\): (on-policy action-value function)
- \(\Phi_t = A^{\pi_\theta}(s_t, a_t)\): (advantage function)

- VPG uses the basic policy gradient to push up the probabilities of good actions using an advantage function
- VPG is
**on-policy** - VPG has a stochastic policy
- Policy gradient: \[\nabla_\theta J(\pi_\theta) = \mathbb{E}_{\tau \sim \pi_\theta}[\sum_{t=0}^T \nabla_\theta \log \pi_\theta(a_t | s_t) A^{\pi_\theta}(s_t, a_t)]\]
- Policy update: \[\theta_{k+1} = \theta_k + \alpha \nabla_\theta J(\pi_{\theta_k})\]
- Exploration happens via randomness in stochastic policy