2nd/10_Wiki/Topics/AI_and_ML/Eligibility-Traces.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

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

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

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)

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)

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)

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)

# 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

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)

# 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 Tradeoff · 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