# Monte Carlo Methods — Learning from Complete Episodes

> Dynamic programming needs a model. Monte Carlo needs nothing but episodes. Run the policy, watch the returns, average them. The simplest idea in RL — and the one that unlocks everything downstream.

**Type:** Build
**Languages:** Python
**Prerequisites:** Phase 9 · 01 (MDPs), Phase 9 · 02 (Dynamic Programming)
**Time:** ~75 minutes

## The Problem

Dynamic programming is elegant, but it assumes you can query $P(s' \mid s, a)$ for every state and action. Almost nothing in the real world works that way. A robot cannot analytically compute the distribution over camera pixels after a joint torque. A pricing algorithm cannot integrate over every possible customer reaction. An LLM cannot enumerate all possible continuations after a token.

You need a method that only needs the ability to *sample* from the environment. Run the policy. Get a trajectory `s_0, a_0, r_1, s_1, a_1, r_2, …, s_T`. Use it to estimate values. That is Monte Carlo.

The shift from DP to MC is philosophically important: we move from *known model + exact backup* to *sampled rollouts + averaged return*. The variance jumps, but the applicability explodes. Every RL algorithm after this lesson — TD, Q-learning, REINFORCE, PPO, GRPO — is a Monte Carlo estimator at heart, sometimes with bootstrapping layered on top.

## The Concept

![Monte Carlo: rollout, compute returns, average; first-visit vs every-visit](../assets/monte-carlo.svg)

**The core idea, in one line:** $V^\pi(s) = \mathbb E_\pi[G_t \mid s_t = s] \approx (1/N) \sum_i G^{(i)}(s)$ where $G^{(i)}(s)$ are observed returns following visits to $s$ under policy $\pi$.

**First-visit vs every-visit MC.** Given an episode that visits state $s$ multiple times, first-visit MC only counts the return from the first visit; every-visit MC counts all visits. First-visit MC is unbiased. Every-visit MC can be slightly biased after a finite number of episodes, because it averages a random number of returns per episode and that number can correlate with the returns, but both converge to $V^\pi(s)$ as the visit count goes to infinity. First-visit is simpler to analyze (iid samples). Every-visit uses more data per episode and typically converges faster in practice.

**Incremental mean.** Instead of storing all returns, update the running average:

```math
V_n(s) = V_{n-1}(s) + (1/n) [G_n - V_{n-1}(s)]
```

Reorganize: $V_{\text{new}} = V_{\text{old}} + \alpha \cdot (\text{target} - V_{\text{old}})$ with $\alpha = 1/n$. Swap $1/n$ for a constant step-size $\alpha \in (0, 1)$ and you get a non-stationary MC estimator that tracks changes in $\pi$. That move is the entire jump from MC to TD to every modern RL algorithm.

**Exploration is now a problem.** DP touched every state by enumeration. MC only sees states the policy visits. If $\pi$ is deterministic, whole regions of the state space never get sampled, and their value estimates stay at zero forever. Three fixes, in historical order:

1. **Exploring starts.** Start each episode from a random (s, a) pair. Guarantees coverage; unrealistic in practice (you cannot "reset" a robot into an arbitrary state).
2. **ε-greedy.** Act greedy w.r.t. current Q, but with probability $\epsilon$ pick a random action. All state-action pairs get sampled asymptotically.
3. **Off-policy MC.** Collect data under a behavior policy $\mu$, learn about target policy $\pi$ via importance sampling. High variance, but it's the bridge to replay-buffer methods like DQN.

**Monte Carlo Control.** Evaluate → improve → evaluate, just like policy iteration, but evaluation is sampling-based:

1. Run $\pi$, get an episode.
2. Update $Q(s, a)$ from observed returns.
3. Make $\pi$ ε-greedy w.r.t. $Q$.
4. Repeat.

Converges to $Q^{\ast}$ and $\pi^{\ast}$ with probability 1 under mild conditions (every pair visited infinitely often, $\alpha$ satisfies Robbins-Monro).

```figure
epsilon-greedy
```

## Build It

### Step 1: rollout → list of (s, a, r)

```python
def rollout(env, policy, max_steps=200):
    trajectory = []
    s = env.reset()
    for _ in range(max_steps):
        a = policy(s)
        s_next, r, done = env.step(s, a)
        trajectory.append((s, a, r))
        s = s_next
        if done:
            break
    return trajectory
```

No model, only `env.reset()` and `env.step(s, a)`. Same interface as a gym environment but stripped down.

### Step 2: compute returns (reverse sweep)

```python
def returns_from(trajectory, gamma):
    returns = []
    G = 0.0
    for _, _, r in reversed(trajectory):
        G = r + gamma * G
        returns.append(G)
    return list(reversed(returns))
```

One pass, $O(T)$. The backward recurrence $G_t = r_{t+1} + \gamma G_{t+1}$ avoids re-summing.

### Step 3: first-visit MC evaluation

```python
def mc_policy_evaluation(env, policy, episodes, gamma=0.99):
    V = defaultdict(float)
    counts = defaultdict(int)
    for _ in range(episodes):
        trajectory = rollout(env, policy)
        returns = returns_from(trajectory, gamma)
        seen = set()
        for t, ((s, _, _), G) in enumerate(zip(trajectory, returns)):
            if s in seen:
                continue
            seen.add(s)
            counts[s] += 1
            V[s] += (G - V[s]) / counts[s]
    return V
```

Three lines do the work: mark state as seen on first visit, increment count, update running mean.

### Step 4: ε-greedy MC control (on-policy)

```python
def mc_control(env, episodes, gamma=0.99, epsilon=0.1):
    Q = defaultdict(lambda: {a: 0.0 for a in ACTIONS})
    counts = defaultdict(lambda: {a: 0 for a in ACTIONS})

    def policy(s):
        if random() < epsilon:
            return choice(ACTIONS)
        return max(Q[s], key=Q[s].get)

    for _ in range(episodes):
        trajectory = rollout(env, policy)
        returns = returns_from(trajectory, gamma)
        seen = set()
        for (s, a, _), G in zip(trajectory, returns):
            if (s, a) in seen:
                continue
            seen.add((s, a))
            counts[s][a] += 1
            Q[s][a] += (G - Q[s][a]) / counts[s][a]
    return Q, policy
```

### Step 5: compare to DP gold standard

Your MC estimate of $V^\pi$ should agree with the DP result from Lesson 02 as episodes → ∞. In practice: 50,000 episodes on 4×4 GridWorld gets you within `~0.1` of the DP answer.

## Pitfalls

- **Infinite episodes.** MC requires episodes to *terminate*. If your policy can loop forever, cap `max_steps` and treat the cap as implicit failure. GridWorld with a random policy routinely times out — that is normal, just make sure you count it correctly.
- **Variance.** MC uses full returns. On long episodes, variance is huge — one unlucky reward at the end shifts $V(s_0)$ by the same amount. TD methods (Lesson 04) cut this by bootstrapping.
- **State coverage.** Greedy MC on a fresh Q with ties will only ever try one action. You *must* explore (ε-greedy, exploring starts, UCB).
- **Non-stationary policies.** If $\pi$ changes (as in MC control), old returns are from a different policy. Constant-α MC handles this; sample-average MC does not.
- **Off-policy importance sampling.** The weights $\pi(a \mid s)/\mu(a \mid s)$ multiply across a trajectory. Variance explodes with horizon. Cap with per-decision weighted IS or switch to TD.

## Use It

The 2026 role of Monte Carlo methods:

| Use case | Why MC |
|----------|--------|
| Short-horizon games (blackjack, poker) | Episodes terminate naturally; returns are clean. |
| Offline evaluation of a logged policy | Average discounted returns over stored trajectories. |
| Monte Carlo Tree Search (AlphaZero) | MC rollouts from tree leaves guide selection. |
| LLM RL evaluation | Compute average reward over sampled completions for a given policy. |
| Baseline estimation in PPO | The advantage target $A_t = G_t - V(s_t)$ uses an MC $G_t$. |
| Teaching RL | Simplest algorithm that actually works — strip bootstrapping to see the core. |

Modern deep-RL algorithms (PPO, SAC) interpolate between pure MC (full returns) and pure TD (one-step bootstrap) via $n$-step returns or GAE. Both endpoints are instances of the same estimator.

## Ship It

Save as `outputs/skill-mc-evaluator.md`:

```markdown
---
name: mc-evaluator
description: Evaluate a policy via Monte Carlo rollouts and produce a convergence report with DP-comparison if available.
version: 1.0.0
phase: 9
lesson: 3
tags: [rl, monte-carlo, evaluation]
---

Given an environment (episodic, with reset+step API) and a policy, output:

1. Method. First-visit vs every-visit MC. Reason.
2. Episode budget. Target number, variance diagnostic, expected standard error.
3. Exploration plan. ε schedule (if needed) or exploring starts.
4. Gold-standard comparison. DP-optimal V* if tabular; otherwise a bound from a Q-learning / PPO baseline.
5. Termination check. Max-step cap, timeouts, handling of non-terminating trajectories.

Refuse to run MC on non-episodic tasks without a finite horizon cap. Refuse to report V^π estimates from fewer than 100 episodes per state for tabular tasks. Flag any policy with zero-variance actions as an exploration risk.
```

## Exercises

1. **Easy.** Implement first-visit MC evaluation of the uniform-random policy on 4×4 GridWorld. Run 10,000 episodes. Plot $V(0,0)$ as a function of episode count against the DP answer.
2. **Medium.** Implement ε-greedy MC control with $\epsilon \in \lbrace 0.01, 0.1, 0.3 \rbrace$. Compare mean return after 20,000 episodes. What does the curve look like? Where does the bias-variance tradeoff live?
3. **Hard.** Implement *off-policy* MC with importance sampling: collect data under uniform-random policy $\mu$, estimate $V^\pi$ for the deterministic optimal policy $\pi$. Compare plain IS vs per-decision IS vs weighted IS. Which has lowest variance?

## Key Terms

| Term | What people say | What it actually means |
|------|-----------------|-----------------------|
| Monte Carlo | "Random sampling" | Estimate expectations by averaging over iid samples from the distribution. |
| Return $G_t$ | "Future reward" | Sum of discounted rewards from step $t$ to episode end: $\sum_{k\ge 0} \gamma^k r_{t+k+1}$. |
| First-visit MC | "Count each state once" | Only the first visit in an episode contributes to the value estimate. |
| Every-visit MC | "Use all visits" | Every visit contributes; slightly biased for finite samples, consistent in the limit, and more sample-efficient. |
| ε-greedy | "Exploration noise" | Pick greedy action with prob $1-\epsilon$; random action with prob $\epsilon$. |
| Importance sampling | "Correcting for sampling from the wrong distribution" | Reweight returns by $\pi(a \mid s)/\mu(a \mid s)$ products to estimate $V^\pi$ from $\mu$ data. |
| On-policy | "Learn from my own data" | Target policy = behavior policy. Vanilla MC, PPO, SARSA. |
| Off-policy | "Learn from someone else's data" | Target policy ≠ behavior policy. Importance-sampled MC, Q-learning, DQN. |

## Further Reading

- [Sutton & Barto (2018). Ch. 5 — Monte Carlo Methods](http://incompleteideas.net/book/RLbook2020.pdf) — the canonical treatment.
- [Singh & Sutton (1996). Reinforcement Learning with Replacing Eligibility Traces](https://link.springer.com/article/10.1007/BF00114726) — first-visit vs every-visit analysis.
- [Precup, Sutton, Singh (2000). Eligibility Traces for Off-Policy Policy Evaluation](http://incompleteideas.net/papers/PSS-00.pdf) — off-policy MC and variance control.
- [Mahmood et al. (2014). Weighted Importance Sampling for Off-Policy Learning](https://arxiv.org/abs/1404.6362) — modern low-variance IS estimators.
- [Tesauro (1995). TD-Gammon, A Self-Teaching Backgammon Program](https://dl.acm.org/doi/10.1145/203330.203343) — the first large-scale empirical demonstration of MC/TD self-play converging to superhuman play; conceptual precursor to every lesson in the second half of this phase.