# Proximal Policy Optimization (PPO)

> A2C throws away each rollout after one update. PPO wraps the policy gradient in a clipped importance ratio so you can do 10+ epochs on the same data without the policy exploding. Schulman et al. (2017). Still the default policy-gradient algorithm in 2026.

**Type:** Build
**Languages:** Python
**Prerequisites:** Phase 9 · 06 (REINFORCE), Phase 9 · 07 (Actor-Critic)
**Time:** ~75 minutes

## The Problem

A2C (Lesson 07) is on-policy: the gradient $\mathbb E_{\pi_\theta}[A \cdot \nabla \log \pi_\theta]$ requires data sampled from the *current* $\pi_\theta$. Take one update, and $\pi_\theta$ changes; the data you used is now off-policy. Re-use it and your gradient is biased.

Rollouts are expensive. On Atari, one rollout across 8 envs × 128 steps = 1024 transitions and a dozen seconds of environment time. Throwing that away after one gradient step is wasteful.

Trust Region Policy Optimization (TRPO, Schulman 2015) was the first fix: constrain each update so the KL divergence between old and new policy stays below $\delta$. Theoretically clean, but requires a conjugate-gradient solve per update. Nobody runs TRPO in 2026.

PPO (Schulman et al. 2017) replaces the hard trust-region constraint with a simple clipped objective. One extra line of code. Ten epochs per rollout. No conjugate gradients. Good-enough theoretical guarantees. Nine years later it is still the default policy-gradient algorithm for everything from MuJoCo to RLHF.

## The Concept

![PPO clipped surrogate objective: ratio clipping at 1 ± ε](../assets/ppo.svg)

**The importance ratio.**

```math
r_t(\theta) = \frac{\pi_\theta(a_t \mid s_t)}{\pi_{\theta_{\text{old}}}(a_t \mid s_t)}
```

This is the likelihood ratio of the new policy vs the policy that collected the data. $r_t = 1$ means no change. $r_t = 2$ means the new policy is twice as likely to take $a_t$ as the old.

**The clipped surrogate.**

```math
L^{\text{CLIP}}(\theta) = \mathbb{E}_t \left[ \min\left( r_t(\theta) A_t, \mathop{\mathrm{clip}}(r_t(\theta), 1-\epsilon, 1+\epsilon) A_t \right) \right]
```

Two terms:

- If the advantage $A_t > 0$ and the ratio tries to grow past $1 + \epsilon$, the clip flattens the gradient — don't push a good action further than $+\epsilon$ above old probability.
- If the advantage $A_t < 0$ and the ratio tries to fall below $1 - \epsilon$ (meaning we would make a bad action much less likely than the old policy did), the clip flattens the gradient: don't push a bad action further than $-\epsilon$ below old probability.

The `min` handles the other direction: if the ratio has moved in the *beneficial* direction, you still get the gradient (no clipping on the side that would hurt you).

Typical $\epsilon = 0.2$. Plot the objective as a function of $r_t$: a piecewise-linear function with a flat roof on the "good side" and a flat floor on the "bad side."

**The full PPO loss.**

```math
L(\theta, \phi) = L^{\text{CLIP}}(\theta) - c_v \cdot (V_\phi(s_t) - V_t^{\text{target}})^2 + c_e \cdot H(\pi_\theta(\cdot \mid s_t))
```

Same actor-critic structure as A2C. Three coefficients, usually $c_v = 0.5$, $c_e = 0.01$, $\epsilon = 0.2$.

**The training loop.**

1. Collect $N \times T$ transitions across $N$ parallel envs for $T$ steps each.
2. Compute advantages (GAE), freeze them as constants.
3. Freeze $\pi_{\theta_{\text{old}}}$ as a snapshot of current $\pi_\theta$.
4. For $K$ epochs, for each minibatch of `(s, a, A, V_target, log π_old(a|s))`:
   - Compute $r_t(\theta) = \exp(\log \pi_\theta(a \mid s) - \log \pi_{\text{old}}(a \mid s))$.
   - Apply $L^{\text{CLIP}}$ + value loss + entropy.
   - Gradient step.
5. Discard the rollout. Return to step 1.

$K = 10$ and minibatches of 64 is a standard hyperparameter set. PPO is robust: the exact numbers rarely matter within ±50%.

**KL-penalty variant.** The original paper proposed an alternative using an adaptive KL penalty: $L = L^{\text{PG}} - \beta \cdot \mathrm{KL}(\pi_\theta \Vert \pi_{\text{old}})$ with $\beta$ adjusted based on observed KL. The clipping version became dominant; the KL variant survives in RLHF (where KL to the reference policy is a separate constraint you always want anyway).

```figure
ppo-clip
```

## Build It

### Step 1: capture `log π_old(a | s)` at rollout time

```python
for step in range(T):
    probs = softmax(logits(theta, state_features(s)))
    a = sample(probs, rng)
    s_next, r, done = env.step(s, a)
    buffer.append({
        "s": s, "a": a, "r": r, "done": done,
        "v_old": value(w, state_features(s)),
        "log_pi_old": log(probs[a] + 1e-12),
    })
    s = s_next
```

The snapshot is taken once, at rollout time. It does not change during the update epochs.

### Step 2: compute GAE advantages (Lesson 07)

Same as A2C. Normalize across the batch.

### Step 3: clipped surrogate update

```python
for _ in range(K_EPOCHS):
    for mb in minibatches(buffer, size=64):
        for rec in mb:
            x = state_features(rec["s"])
            probs = softmax(logits(theta, x))
            logp = log(probs[rec["a"]] + 1e-12)
            ratio = exp(logp - rec["log_pi_old"])
            adv = rec["advantage"]
            surrogate = min(
                ratio * adv,
                clamp(ratio, 1 - EPS, 1 + EPS) * adv,
            )
            # backprop -surrogate, add value loss, subtract entropy
            grad_logpi = onehot(rec["a"]) - probs
            if (adv > 0 and ratio >= 1 + EPS) or (adv < 0 and ratio <= 1 - EPS):
                pg_grad = 0.0  # clipped
            else:
                pg_grad = ratio * adv
            for i in range(N_ACTIONS):
                for j in range(N_FEAT):
                    theta[i][j] += LR * pg_grad * grad_logpi[i] * x[j]
```

The "clipped → zero gradient" pattern is the heart of PPO. If the new policy has already drifted too far in the beneficial direction, the update stops.

### Step 4: value and entropy

Add standard MSE to the critic target and an entropy bonus on the actor, same as A2C.

### Step 5: diagnostics

Three things to watch every update:

- **Mean KL** $\mathbb{E}[\log \pi_{\text{old}} - \log \pi_\theta]$. Should stay in `[0, 0.02]`. If it blows past `0.1`, reduce `K_EPOCHS` or `LR`.
- **Clip fraction** — the fraction of samples whose ratio lies outside $[1-\epsilon, 1+\epsilon]$. Should be `~0.1-0.3`. If `~0`, the clip never triggers → raise `LR` or `K_EPOCHS`. If `~0.5+`, you are over-fitting the rollout → lower them.
- **Explained variance** $1 - \text{Var}(V_{\text{target}} - V_{\text{pred}}) / \text{Var}(V_{\text{target}})$. Critic quality metric. Should climb toward 1 as the critic learns.

## Pitfalls

- **Clip coefficient mistuned.** $\epsilon = 0.2$ is the de-facto standard. Going to `0.1` makes updates too timid; `0.3+` invites instability.
- **Too many epochs.** $K > 20$ routinely destabilizes because the policy drifts far from $\pi_{\text{old}}$. Cap epochs, especially for large networks.
- **No reward normalization.** Large reward scales eat into the clip range. Normalize rewards (running std) before computing advantages.
- **Forgetting advantage normalization.** Per-batch zero-mean/unit-std normalization is standard. Skipping it wrecks PPO on most benchmarks.
- **Learning rate not decayed.** PPO benefits from linear LR decay to zero. Constant LR is often worse.
- **Importance ratio math errors.** Always `exp(log_new - log_old)` for numerical stability, not `new / old`.
- **Wrong gradient sign.** Maximize the surrogate = *minimize* $-L^{\text{CLIP}}$. A flipped sign is the most common PPO bug.

## Use It

PPO is 2026's default RL algorithm across a surprising number of domains:

| Use case | PPO variant |
|----------|-------------|
| MuJoCo / robotics control | PPO with Gaussian policy, GAE(0.95) |
| Atari / discrete games | PPO with categorical policy, rolling 128-step rollouts |
| RLHF for LLMs | PPO with KL penalty to reference model, reward from RM at end of response |
| Large-scale game agents | IMPALA + PPO (AlphaStar, OpenAI Five) |
| Reasoning LLMs | GRPO (Lesson 12) — PPO variant without critic |
| Preference-only data | DPO — closed-form collapsing of PPO+KL, no online sampling |

The PPO *loss shape* — clipped surrogate + value + entropy — is the scaffolding for DPO, GRPO, and nearly every RLHF pipeline.

## Ship It

Save as `outputs/skill-ppo-trainer.md`:

```markdown
---
name: ppo-trainer
description: Produce a PPO training config and a diagnostic plan for a given environment.
version: 1.0.0
phase: 9
lesson: 8
tags: [rl, ppo, policy-gradient]
---

Given an environment and training budget, output:

1. Rollout size. `N` envs × `T` steps.
2. Update schedule. `K` epochs, minibatch size, LR schedule.
3. Surrogate params. `ε` (clip), `c_v`, `c_e`, advantage normalization on.
4. Advantage. GAE(`λ`) with explicit `γ` and `λ`.
5. Diagnostics plan. KL, clip fraction, explained variance thresholds with alerts.

Refuse `K > 30` or `ε > 0.3` (unsafe trust region). Refuse any PPO run without advantage normalization or KL/clip monitoring. Flag clip fraction sustained above 0.4 as drift.
```

## Exercises

1. **Easy.** Run PPO on 4×4 GridWorld with $\epsilon=0.2, K=4$. Compare sample efficiency to A2C (one epoch per rollout) at matched env steps.
2. **Medium.** Sweep $K \in \lbrace 1, 4, 10, 30 \rbrace$. Plot return vs env steps and track mean KL per update. At what $K$ does KL explode on this task?
3. **Hard.** Replace the clipped surrogate with an adaptive KL penalty ($\beta$ doubled if $\mathrm{KL} > 2 \cdot \text{target}$, halved if $\mathrm{KL} < \text{target}/2$). Compare final return, stability, and clip-free-ness.

## Key Terms

| Term | What people say | What it actually means |
|------|-----------------|-----------------------|
| Importance ratio | "r_t(θ)" | $\pi_\theta(a \mid s) / \pi_{\text{old}}(a \mid s)$; deviation from the policy that collected the data. |
| Clipped surrogate | "PPO's main trick" | $\min(r \cdot A, \mathop{\mathrm{clip}}(r, 1-\epsilon, 1+\epsilon) \cdot A)$; flat gradient past the clip on beneficial side. |
| Trust region | "TRPO / PPO intent" | Limit each update's KL to guarantee monotone improvement. |
| KL penalty | "Soft trust region" | Alternative PPO: $L - \beta \cdot \mathrm{KL}(\pi_\theta \Vert \pi_{\text{old}})$. Adaptive $\beta$. |
| Clip fraction | "How often clipping triggers" | Diagnostic — should be 0.1-0.3; outside means mistuned. |
| Multi-epoch training | "Data reuse" | K epochs on each rollout; variance cost traded for sample efficiency. |
| On-policy-ish | "Mostly on-policy" | PPO is nominally on-policy but K>1 epochs uses slightly-off-policy data safely. |
| PPO-KL | "The other PPO" | KL-penalty variant; used in RLHF where KL-to-reference is already a constraint. |

## Further Reading

- [Schulman et al. (2017). Proximal Policy Optimization Algorithms](https://arxiv.org/abs/1707.06347) — the paper.
- [Schulman et al. (2015). Trust Region Policy Optimization](https://arxiv.org/abs/1502.05477) — TRPO, PPO's predecessor.
- [Andrychowicz et al. (2021). What Matters In On-Policy RL? A Large-Scale Empirical Study](https://arxiv.org/abs/2006.05990) — every PPO hyperparameter ablated.
- [Ouyang et al. (2022). Training language models to follow instructions with human feedback](https://arxiv.org/abs/2203.02155) — InstructGPT; the PPO-in-RLHF recipe.
- [OpenAI Spinning Up — PPO](https://spinningup.openai.com/en/latest/algorithms/ppo.html) — clean modern exposition with PyTorch.
- [CleanRL PPO implementation](https://github.com/vwxyzjn/cleanrl) — reference single-file PPO used by many papers.
- [Hugging Face TRL — PPOTrainer](https://huggingface.co/docs/trl/main/en/ppo_trainer) — the production recipe for PPO on language models; read alongside Lesson 09 (RLHF).
- [Engstrom et al. (2020). Implementation Matters in Deep Policy Gradients](https://arxiv.org/abs/2005.12729) — the "37 code-level optimizations" paper; which PPO tricks are load-bearing and which are folklore.