⚖️ 第23课：多目标优化

🎯 学习目标

🗂️ 课程定位

📖 核心理论

推荐系统需要同时优化多个目标：点击率、转化率、用户时长、多样性、商业收入等。这些目标往往相互冲突，需要精心设计融合策略。

📐 数学公式

💻 代码实现

import numpy as np
np.random.seed(42)
print("="*60+"\n多目标优化推荐\n"+"="*60)
n_samples=500;n_features=8
X=np.random.randn(n_samples,n_features)
# 三个目标
click=(X@np.random.randn(n_features)+np.random.randn(n_samples)*0.5>0).astype(float)
duration=np.abs(X@np.random.randn(n_features)+np.random.randn(n_samples)*0.3)+1
diversity=np.random.random(n_samples)
print(f"样本:{n_samples} 点击率:{click.mean():.4f} 平均时长:{duration.mean():.2f} 平均多样性:{diversity.mean():.4f}")
# 单目标模型
def sigmoid(x):return 1/(1+np.exp(-np.clip(x,-500,500)))
def train_single(X,y,task='click',lr=0.01,epochs=80):
    w=np.random.randn(X.shape[1])*0.01;b=0
    for ep in range(epochs):
        pred=X@w+b
        if task=='click':
            pred=sigmoid(pred);err=pred-y
            loss=-np.mean(y*np.log(pred+1e-10)+(1-y)*np.log(1-pred+1e-10))
        elif task=='duration':
            err=pred-y;loss=np.mean(err**2)
        else:
            pred=sigmoid(pred);err=pred-y
            loss=-np.mean(y*np.log(pred+1e-10)+(1-y)*np.log(1-pred+1e-10))
        w-=lr*(X.T@err/len(y)+0.01*w);b-=lr*np.mean(err)
    return w,b
w_c,b_c=train_single(X,click,'click')
w_d,b_d=train_single(X,duration,'duration')
w_v,b_v=train_single(X,diversity,'diversity')
# 加权融合
print("\n--- 加权融合 ---")
scores_c=sigmoid(X@w_c+b_c)
scores_d=sigmoid((X@w_d+b_d-duration.mean())/duration.std())
scores_v=sigmoid(X@w_v+b_v)
for alpha,beta,gamma in [(0.5,0.3,0.2),(0.7,0.2,0.1),(0.3,0.4,0.3)]:
    fused=alpha*scores_c+beta*scores_d+gamma*scores_v
    top10=np.argsort(-fused)[:50]
    print(f"权重({alpha},{beta},{gamma}):点击率={click[top10].mean():.4f} 时长={duration[top10].mean():.2f} 多样性={diversity[top10].mean():.4f}")
# 多任务学习(Shared-Bottom)
print("\n--- 多任务学习(Shared-Bottom) ---")
h_dim=16
W_shared=np.random.randn(n_features,h_dim)*0.01
b_shared=np.zeros(h_dim)
W_click=np.random.randn(h_dim,1)*0.01;W_dur=np.random.randn(h_dim,1)*0.01;W_div=np.random.randn(h_dim,1)*0.01
b_click=0;b_dur=0;b_div=0
lr_mt=0.005
for ep in range(80):
    shared=np.maximum(0,X@W_shared+b_shared)
    pred_c=sigmoid((shared@W_click+b_click).flatten())
    pred_d=shared@W_dur+b_dur
    pred_v=sigmoid((shared@W_div+b_div).flatten())
    err_c=pred_c-click;err_d=pred_d.flatten()-duration;err_v=pred_v-diversity
    loss_c=-np.mean(click*np.log(pred_c+1e-10)+(1-click)*np.log(1-pred_c+1e-10))
    loss_d=np.mean(err_d**2);loss_v=-np.mean(diversity*np.log(pred_v+1e-10)+(1-diversity)*np.log(1-pred_v+1e-10))
    total_loss=0.5*loss_c+0.3*loss_d+0.2*loss_v
    # 更新task-specific
    W_click-=lr_mt*(shared.T@err_c[:,None]/n_samples+0.01*W_click)
    W_dur-=lr_mt*(shared.T@err_d[:,None]/n_samples+0.01*W_dur)
    W_div-=lr_mt*(shared.T@err_v[:,None]/n_samples+0.01*W_div)
    # 更新shared(梯度汇总)
    d_shared=err_c[:,None]@W_click.T+err_d[:,None]@W_dur.T+err_v[:,None]@W_div.T
    d_shared*=((X@W_shared+b_shared)>0).astype(float)
    W_shared-=lr_mt*(X.T@d_shared/n_samples+0.01*W_shared)
    if(ep+1)%20==0:print(f"  Epoch{ep+1}:TotalLoss={total_loss:.4f}")
# 帕累托前沿
print("\n--- 帕累托分析 ---")
pareto_points=[]
for a in np.arange(0,1.1,0.2):
    for b in np.arange(0,1.1-a,0.2):
        c=1-a-b
        fused=a*scores_c+b*scores_d+c*scores_v
        top20=np.argsort(-fused)[:50]
        pareto_points.append((click[top20].mean(),duration[top20].mean(),diversity[top20].mean()))
print("帕累托前沿(点击率,时长,多样性):")
for p in pareto_points[:5]:print(f"  ({p[0]:.4f},{p[1]:.2f},{p[2]:.4f})")

🔴 运行结果

✅ 验证通过

🔍 代码解读

💡 关键洞察

📝 本章小结

🧪 练习

📚 延伸阅读

💼 面试考点

🏭 工业案例