---
id: wiki-2026-0508-eligibility-traces
title: Eligibility Traces
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [eligibility trace, lambda return, TD-lambda, n-step bootstrapping, GAE]
duplicate_of: none
source_trust_level: A
confidence_score: 0.96
verification_status: applied
tags: [reinforcement-learning, eligibility-traces, td-learning, credit-assignment, gae, ppo]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: PyTorch / Stable-Baselines3 / CleanRL
---

# Eligibility Traces

## 매 한 줄
> **"매 TD(0) 와 Monte Carlo 의 가운데"**. 매 λ ∈ [0, 1] 의 trade-off bias-variance. 매 Sutton-Barto canonical 알고리즘. 매 modern: 매 GAE (Generalized Advantage Estimation) — PPO 의 standard. 매 credit assignment 의 efficient.

## 매 핵심

### 매 motivation
- **TD(0)**: 매 1-step bootstrap (low variance, high bias).
- **Monte Carlo**: 매 full return (high variance, no bias).
- **TD(λ)**: 매 λ-weighted average (sweet spot).

### 매 forward view
```
G_t^λ = (1 - λ) Σ_n λ^(n-1) G_t^(n)
```
매 n-step return 의 geometric weighting.

### 매 backward view (eligibility trace)
- 매 매 state 의 trace e(s).
- 매 visit → 매 trace ↑.
- 매 decay (γλ) 매 step.
- 매 TD error δ 의 trace 의 weight 의 update.

```
e_t(s) = γλ e_{t-1}(s) + 1[S_t = s]   (replacing or accumulating)
V(s) ← V(s) + α δ_t e_t(s)
```

### 매 variant
- **TD(0)**: λ=0.
- **TD(1)**: ≈ Monte Carlo.
- **TD(λ)**: 매 in between.
- **Watkins Q(λ)**: 매 off-policy 의 reset on exploration.
- **GAE(γ, λ)**: 매 modern policy gradient.

### 매 modern: GAE
```
A_t^GAE = Σ_l (γλ)^l δ_{t+l}
δ_t = r_t + γV(s_{t+1}) - V(s_t)
```

### 매 응용
1. **TD(λ) prediction**: 매 value learning.
2. **Sarsa(λ)**: 매 on-policy control.
3. **Q(λ)**: 매 off-policy.
4. **GAE in PPO/A2C**: 매 modern actor-critic.
5. **Replay buffer**: 매 trace replay.

## 💻 패턴

### TD(λ) (Sutton-Barto, accumulating trace)
```python
import numpy as np

class TDLambda:
    def __init__(self, n_states, alpha=0.1, gamma=0.99, lam=0.9):
        self.V = np.zeros(n_states)
        self.E = np.zeros(n_states)
        self.alpha, self.gamma, self.lam = alpha, gamma, lam
    
    def reset_trace(self):
        self.E[:] = 0
    
    def step(self, s, r, s_next, done):
        delta = r + (0 if done else self.gamma * self.V[s_next]) - self.V[s]
        self.E[s] += 1  # 매 accumulating
        self.V += self.alpha * delta * self.E
        self.E *= self.gamma * self.lam
        if done: self.reset_trace()
```

### Replacing trace
```python
def replacing_trace_update(self, s, r, s_next, done):
    delta = r + (0 if done else self.gamma * self.V[s_next]) - self.V[s]
    self.E *= self.gamma * self.lam
    self.E[s] = 1  # 매 replace, not accumulate
    self.V += self.alpha * delta * self.E
```

### Sarsa(λ)
```python
class SarsaLambda:
    def __init__(self, n_s, n_a, alpha=0.1, gamma=0.99, lam=0.9, eps=0.1):
        self.Q = np.zeros((n_s, n_a))
        self.E = np.zeros((n_s, n_a))
        self.alpha, self.gamma, self.lam, self.eps = alpha, gamma, lam, eps
    
    def act(self, s):
        if np.random.rand() < self.eps: return np.random.randint(self.Q.shape[1])
        return self.Q[s].argmax()
    
    def update(self, s, a, r, s_next, a_next, done):
        delta = r + (0 if done else self.gamma * self.Q[s_next, a_next]) - self.Q[s, a]
        self.E[s, a] += 1
        self.Q += self.alpha * delta * self.E
        self.E *= self.gamma * self.lam
        if done: self.E[:] = 0
```

### Watkins Q(λ)
```python
def q_lambda_update(self, s, a, r, s_next, done):
    a_next = self.Q[s_next].argmax()
    delta = r + (0 if done else self.gamma * self.Q[s_next, a_next]) - self.Q[s, a]
    self.E[s, a] += 1
    self.Q += self.alpha * delta * self.E
    
    # 매 if action was exploratory, reset trace
    if exploratory: self.E[:] = 0
    else: self.E *= self.gamma * self.lam
```

### GAE (PyTorch)
```python
import torch

def compute_gae(rewards, values, dones, gamma=0.99, lam=0.95):
    """매 PPO standard advantage estimation."""
    advantages = torch.zeros_like(rewards)
    last_gae = 0
    for t in reversed(range(len(rewards))):
        if t == len(rewards) - 1:
            next_value = 0  # 매 bootstrap = 0 at end (or value of last state)
        else:
            next_value = values[t + 1]
        
        delta = rewards[t] + gamma * next_value * (1 - dones[t]) - values[t]
        last_gae = delta + gamma * lam * (1 - dones[t]) * last_gae
        advantages[t] = last_gae
    
    returns = advantages + values
    return advantages, returns
```

### Lambda choice (typical)
```python
# 매 GAE
LAM_CONSERVATIVE = 0.95  # 매 PPO default — 매 stable
LAM_AGGRESSIVE = 0.99    # 매 closer to MC, more variance
LAM_BIASED = 0.9         # 매 closer to TD(0), more bias

# 매 task-dependent
def choose_lambda(task):
    if task.episodes_short: return 0.95
    if task.sparse_reward: return 0.99  # 매 long credit
    if task.dense_reward: return 0.9
```

### N-step return
```python
def n_step_return(rewards, values, n, gamma):
    """매 forward-view n-step."""
    returns = np.zeros_like(rewards)
    for t in range(len(rewards)):
        G = 0
        for k in range(n):
            if t + k < len(rewards):
                G += gamma**k * rewards[t + k]
        if t + n < len(values):
            G += gamma**n * values[t + n]
        returns[t] = G
    return returns
```

### True online TD(λ)
```python
# 매 dutch trace (van Seijen)
def true_online_step(self, s, r, s_next, done):
    delta = r + (0 if done else self.gamma * self.V[s_next]) - self.V[s]
    e_dot_phi = self.E[s]
    self.E *= self.gamma * self.lam
    self.E[s] += self.alpha * (1 - self.gamma * self.lam * e_dot_phi)
    self.V += (delta + self.V[s] - self.V_old) * self.E
    self.V[s] -= self.alpha * (self.V[s] - self.V_old)
    self.V_old = self.V[s_next] if not done else 0
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Tabular RL | TD(λ) replacing |
| Linear function approx | True online TD(λ) |
| DRL actor-critic | GAE λ=0.95 |
| Sparse reward | λ → 1 (Monte Carlo-like) |
| Dense reward | λ → 0 (TD-like) |
| Off-policy | Watkins Q(λ) or V-trace |

**기본값**: 매 modern DRL = GAE(γ=0.99, λ=0.95). 매 tabular = TD(λ) replacing trace.

## 🔗 Graph
- 부모: [[Reinforcement-Learning]] · [[TD-Learning]]
- 변형: [[TD-Lambda]] · [[GAE]]
- 응용: [[PPO]] · [[A2C]] · [[Actor-Critic]]
- Adjacent: [[Bias-Variance-Trade-off]] · [[Credit-Assignment]]

## 🤖 LLM 활용
**언제**: 매 RL credit assignment. 매 actor-critic. 매 sparse reward.
**언제 X**: 매 deterministic supervised. 매 1-step bandit.

## ❌ 안티패턴
- **λ=1 always**: 매 high variance.
- **λ=0 always**: 매 high bias 의 long-horizon 의 fail.
- **Forget trace reset**: 매 episode boundary.
- **GAE without value baseline**: 매 advantage 의 wrong.
- **Wrong direction loop**: 매 forward 의 do (must reverse).

## 🧪 검증 / 중복
- Verified (Sutton-Barto Ch12, Schulman GAE 2016, PPO 2017).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-04-26 | RL-ELIG auto |
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — TD(λ) + GAE + 매 forward / backward / Sarsa / Watkins / true online code |
