---
id: wiki-2026-0508-reward-prediction-error
title: Reward Prediction Error
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [RPE, TD Error, Dopamine Prediction Error]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [neuroscience, reinforcement-learning, dopamine, td-learning]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: PyTorch/Gymnasium
---

# Reward Prediction Error

## 매 한 줄
> **"매 actual reward minus predicted reward — 매 학습 의 driver."**. Wolfram Schultz 의 1997 dopamine experiments 가 monkey VTA neuron 의 firing 가 TD-error δ = r + γV(s') − V(s) 와 매 same signature 의 보임. 매 neuroscience 와 RL 의 connect 의 historic moment, 매 modern dopamine RL theory 의 foundation.

## 매 핵심

### 매 RPE definition
- **δ_t = r_t + γ·V(s_{t+1}) − V(s_t)** — temporal difference error.
- δ > 0: better than expected → strengthen association.
- δ = 0: as predicted → no learning needed.
- δ < 0: worse than expected → weaken / extinct.

### 매 dopamine signature (Schultz 1997)
1. Untrained: dopamine neurons fire at reward delivery.
2. After conditioning: fire at predictive cue, 의 X reward.
3. Cue followed by no reward: firing dips below baseline (negative RPE).
4. 매 exact pattern 의 TD-error δ 와 match.

### 매 응용
1. Classical/operant conditioning models.
2. RL algorithms (TD, Q-learning, Actor-Critic).
3. Addiction theory (drugs hijack RPE signal).
4. RLHF reward model interpretation (LLM training).

## 💻 패턴

### TD(0) Update
```python
def td_update(V, s, r, s_next, alpha=0.1, gamma=0.99):
    rpe = r + gamma * V[s_next] - V[s]   # 매 RPE
    V[s] += alpha * rpe
    return V, rpe
```

### Q-Learning RPE
```python
import numpy as np

def q_learning_step(Q, s, a, r, s_next, alpha=0.1, gamma=0.99):
    target = r + gamma * np.max(Q[s_next])
    rpe = target - Q[s, a]
    Q[s, a] += alpha * rpe
    return Q, rpe
```

### Actor-Critic with RPE as Advantage
```python
import torch

def actor_critic_step(actor, critic, opt_a, opt_c, s, a, r, s_next, gamma=0.99):
    v_s, v_next = critic(s), critic(s_next).detach()
    rpe = r + gamma * v_next - v_s        # 매 RPE = advantage

    critic_loss = rpe.pow(2)
    log_prob = actor(s).log_prob(a)
    actor_loss = -(log_prob * rpe.detach())

    opt_c.zero_grad(); critic_loss.backward(); opt_c.step()
    opt_a.zero_grad(); actor_loss.backward(); opt_a.step()
    return rpe.item()
```

### Distributional RPE (C51-style)
```python
# 매 modern: scalar RPE 의 X, 의 reward distribution.
def distributional_td_target(r, p_next, support, gamma=0.99):
    """p_next: prob over atoms; support: atom values."""
    Tz = r + gamma * support      # shifted support
    return Tz, p_next             # project onto original support next
```

### RLHF reward model RPE
```python
def rlhf_advantage(rewards, values, gamma=1.0, lam=0.95):
    """GAE — generalized advantage estimation. Each step δ_t = RPE."""
    advantages = []
    gae = 0
    for t in reversed(range(len(rewards))):
        v_next = values[t + 1] if t + 1 < len(values) else 0
        delta = rewards[t] + gamma * v_next - values[t]   # 매 RPE
        gae = delta + gamma * lam * gae
        advantages.insert(0, gae)
    return advantages
```

### Phasic vs Tonic Dopamine simulation
```python
def simulate_dopamine(trial, cue_time, reward_time, predicted=True):
    """Phasic burst at predictive cue (after learning); dip at omitted reward."""
    signal = []
    for t in range(trial):
        if t == cue_time and predicted: signal.append(+1.0)   # phasic burst
        elif t == reward_time and not predicted: signal.append(+1.0)
        elif t == reward_time and predicted: signal.append(0.0)
        else: signal.append(0.05)   # tonic baseline
    return signal
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Tabular small state space | TD(0) / Q-learning |
| Continuous state, value-based | DQN (RPE = TD target − Q) |
| Policy + value | Actor-Critic with RPE as advantage |
| Distribution matters | Distributional RL (C51, QR-DQN) |
| LLM RLHF | PPO with GAE — RPE summed |

**기본값**: PPO + GAE — 매 modern RPE 의 actor-critic instantiation.

## 🔗 Graph
- 부모: [[Reinforcement Learning]]
- 변형: [[TD Learning]] · [[Distributional RL]]
- 응용: [[Actor-Critic]] · [[RLHF]]
- Adjacent: [[Dopamine]] · [[데이터 사이언스 및 ML 엔지니어링|Bellman Equation]]

## 🤖 LLM 활용
**언제**: RLHF/DPO/GRPO 의 advantage computation 의 understand, 의 reward model debugging.
**언제 X**: LLM 의 의 RPE 의 conceptual explanation 의 helpful 의 X — 의 raw neural data 의 X.

## ❌ 안티패턴
- **Confusing reward and RPE**: r 의 X RPE — RPE = r − prediction.
- **Always positive RPE**: 의 X — negative RPE (omission) 의 critical for extinction learning.
- **Ignoring discount**: γ 의 omit 의 X — temporal credit assignment 의 broken.
- **Dopamine = pleasure**: 의 X — dopamine 의 reward signal 의 X, 의 prediction error 의.

## 🧪 검증 / 중복
- Verified (Schultz, Dayan, Montague 1997 Science; Sutton & Barto Ch 6).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — RPE neuroscience + RL bridge |