---
id: wiki-2026-0508-joint-optimization
title: Joint Optimization
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Multi-Objective Optimization, Co-Optimization, End-to-End Optimization]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [optimization, ML, multi-objective]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: pytorch-jax
---

# Joint Optimization

## 매 한 줄
> **"매 multiple objectives / variables 를 동시에 optimize"**. 매 separate / sequential optimization 보다 매 globally better solution 도달 가능 — 매 cost: 매 higher complexity, 매 risk: 매 conflicting gradients. 매 modern DL (end-to-end training), 매 RL (actor-critic), 매 chip design (DSE) 의 매 핵심.

## 매 핵심

### 매 왜 jointly?
- **Coupling**: 매 variables 의 interaction 강 → 매 separate solve 매 suboptimal.
- **Information sharing**: 매 shared representation / gradient → 매 mutual benefit.
- **End-to-end**: 매 pipeline 의 손실 누적 X.

### 매 challenges
- **Conflicting gradients**: 매 objectives 매 push opposite directions.
- **Scaling**: 매 loss magnitudes 매 mismatched → 매 dominant loss problem.
- **Local minima**: 매 joint landscape 매 더 rugged.
- **Compute**: 매 N variables 매 jointly → search space exponential.

### 매 응용
1. **Multi-task learning**: 매 shared encoder + 매 multiple heads.
2. **Actor-critic RL**: 매 policy + value 매 jointly.
3. **HW/SW co-design**: 매 chip floorplan + scheduler 매 jointly.
4. **Pareto front**: 매 cost vs latency 매 frontier.

## 💻 패턴

### Weighted sum (simplest)
```python
import torch

def joint_loss(pred1, pred2, y1, y2, w=(0.5, 0.5)):
    l1 = torch.nn.functional.cross_entropy(pred1, y1)
    l2 = torch.nn.functional.mse_loss(pred2, y2)
    return w[0] * l1 + w[1] * l2
```

### GradNorm (auto-balance)
```python
# Chen et al 2018 — 매 dynamic loss weighting
class GradNorm:
    def __init__(self, n_tasks, alpha=1.5):
        self.weights = torch.ones(n_tasks, requires_grad=True)
        self.alpha = alpha
    def update(self, losses, shared_params):
        # 매 normalize 매 gradient magnitudes across tasks
        grads = [torch.autograd.grad(l, shared_params, retain_graph=True)
                 for l in losses]
        norms = torch.stack([g[0].norm() for g in grads])
        target = norms.mean() * (losses / losses.mean()) ** self.alpha
        gradnorm_loss = (norms - target.detach()).abs().sum()
        return gradnorm_loss
```

### MGDA (Multi-Gradient Descent)
```python
# Sener & Koltun 2018 — 매 Pareto-optimal direction 찾기
import numpy as np

def mgda_solver(grads):
    """grads: list of gradient vectors per task."""
    # 매 minimum-norm point in convex hull
    G = np.stack([g.flatten() for g in grads])
    # solve min ||sum α_i g_i||² s.t. α≥0, sum α=1
    from scipy.optimize import minimize
    def obj(a): return np.linalg.norm(a @ G) ** 2
    a0 = np.ones(len(grads)) / len(grads)
    cons = [{"type": "eq", "fun": lambda a: a.sum() - 1}]
    bnds = [(0, 1)] * len(grads)
    res = minimize(obj, a0, constraints=cons, bounds=bnds)
    return res.x  # 매 Pareto direction
```

### Actor-critic joint update
```python
# PPO-style joint optimization
def actor_critic_loss(states, actions, advantages, returns, policy, value):
    log_p = policy.log_prob(states, actions)
    actor_loss = -(log_p * advantages).mean()
    critic_loss = (value(states) - returns).pow(2).mean()
    entropy = policy.entropy(states).mean()
    return actor_loss + 0.5 * critic_loss - 0.01 * entropy
```

### Pareto frontier sampling
```python
# 매 multi-objective 의 frontier 발견
def pareto_front(solutions):
    """solutions: list of (obj1, obj2) tuples (minimize both)."""
    front = []
    for s in solutions:
        dominated = any(
            s2[0] <= s[0] and s2[1] <= s[1] and s2 != s
            for s2 in solutions
        )
        if not dominated:
            front.append(s)
    return front
```

## 매 결정 기준
| 상황 | Strategy |
|---|---|
| 매 objectives 매 aligned | Weighted sum (simple) |
| 매 objectives 매 conflicting | MGDA / PCGrad |
| 매 magnitude 매 mismatched | GradNorm |
| 매 trade-off 매 explore 필요 | Pareto frontier sweep |
| 매 RL actor + critic | Joint PPO/SAC |

**기본값**: Weighted sum 시작 → 매 imbalance 발견시 GradNorm 도입.

## 🔗 Graph
- 부모: [[Optimization]]
- 응용: [[Actor-Critic]]

## 🤖 LLM 활용
**언제**: 매 loss function design 매 multi-objective, 매 gradient conflict diagnosis, 매 Pareto analysis explanation.
**언제 X**: 매 single-objective optimization — over-complication.

## ❌ 안티패턴
- **Random weight tuning**: 매 grid search w/o GradNorm → 매 unstable.
- **Ignore gradient conflict**: 매 cosine(g1,g2) < 0 무시 → 매 destructive interference.
- **Premature joint**: 매 separate pretrain → joint finetune 매 더 좋은 경우 많음.

## 🧪 검증 / 중복
- Verified (Chen 2018 GradNorm; Sener & Koltun 2018 MGDA; Yu 2020 PCGrad; Schulman 2017 PPO).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — multi-objective optimization patterns + Pareto |
