⚙️ 第7课：特征交叉（FM / FFM）

🎯 学习目标

🗂️ 课程定位

📖 核心理论

特征交叉是推荐系统排序模型的核心技术。LR模型只能学习线性关系，而推荐场景中特征交叉往往非常重要（如"男性+科技"vs"女性+美妆"）。FM(Factorization Machines)通过隐因子交叉高效地学习二阶特征交互。

FM的核心创新：将Poly2的O(n²)参数量降为O(nk)，通过隐因子的内积来近似特征交叉权重，使得稀疏场景下也能有效学习。

📐 数学公式

💻 代码实现

import numpy as np
np.random.seed(42)
print("="*60+"\n特征交叉模型(FM/FFM)\n"+"="*60)
n_samples=500;n_features=10;n_fields=5
# 生成特征数据
X=np.random.randn(n_samples,n_features)
# 二阶交叉生成标签
true_w=np.random.randn(n_features)
true_V=np.random.randn(n_features,8)*0.3
y=X@true_w
for i in range(n_features):
    for j in range(i+1,n_features):
        y+=np.dot(true_V[i],true_V[j])*X[:,i]*X[:,j]
y=(y>np.median(y)).astype(float)
print(f"样本数:{n_samples} 特征数:{n_features} 正例率:{y.mean():.2%}")
# LR基线
from collections import defaultdict
def sigmoid(x):return 1/(1+np.exp(-np.clip(x,-500,500)))
def train_lr(X,y,lr=0.01,epochs=100):
    n,d=X.shape;w=np.zeros(d);b=0
    for ep in range(epochs):
        pred=sigmoid(X@w+b)
        err=pred-y
        w-=lr*(X.T@err/n+0.01*w)
        b-=lr*np.mean(err)
        if(ep+1)%25==0:
            loss=-np.mean(y*np.log(pred+1e-10)+(1-y)*np.log(1-pred+1e-10))
            print(f"  LR Epoch{ep+1}:Loss={loss:.4f}")
    return w,b
print("\n--- LR基线 ---")
w_lr,b_lr=train_lr(X,y)
pred_lr=sigmoid(X@w_lr+b_lr)
acc_lr=np.mean((pred_lr>0.5)==y)
auc_lr=np.mean(pred_lr[y==1]>pred_lr[y==0])
print(f"LR ACC={acc_lr:.4f} AUC={auc_lr:.4f}")
# FM
def train_fm(X,y,k=8,lr=0.01,epochs=100):
    n,d=X.shape
    w=np.zeros(d);b=0;V=np.random.randn(d,k)*0.01
    for ep in range(epochs):
        pred=sigmoid(b+X@w+0.5*np.sum((X@V)**2-(X**2)@(V**2),axis=1))
        err=pred-y
        w-=lr*(X.T@err/n+0.01*w)
        b-=lr*np.mean(err)
        V-=lr*(X.T[:,None]@err[None,:]*X@V/n-X.T@(err[:,None]*(X**2))[:,:,None]*V/n+0.01*V)
        if(ep+1)%25==0:
            loss=-np.mean(y*np.log(pred+1e-10)+(1-y)*np.log(1-pred+1e-10))
            print(f"  FM Epoch{ep+1}:Loss={loss:.4f}")
    return w,b,V
print("\n--- FM ---")
w_fm,b_fm,V_fm=train_fm(X,y)
linear=X@w_fm+b_fm
interaction=0.5*np.sum((X@V_fm)**2-(X**2)@(V_fm**2),axis=1)
pred_fm=sigmoid(linear+interaction)
acc_fm=np.mean((pred_fm>0.5)==y)
auc_fm=np.mean(pred_fm[y==1]>pred_fm[y==0])
print(f"FM ACC={acc_fm:.4f} AUC={auc_fm:.4f}")
# 交叉特征重要性
cross_imp=[]
for i in range(n_features):
    for j in range(i+1,n_features):
        imp=abs(np.dot(V_fm[i],V_fm[j]))
        cross_imp.append((i,j,imp))
cross_imp.sort(key=lambda x:-x[2])
print(f"\nTop-5重要交叉特征:")
for i,j,imp in cross_imp[:5]:
    print(f"  特征{i}x特征{j}: 重要性={imp:.4f}")
# FFM
field_map=np.array([i%n_fields for i in range(n_features)])
def train_ffm(X,y,field_map,k=4,lr=0.01,epochs=80):
    n,d=X.shape;nf=max(field_map)+1
    V=np.random.randn(d,nf,k)*0.01;w=np.zeros(d);b=0
    for ep in range(epochs):
        total_loss=0
        for s in range(n):
            pred=b+np.dot(w,X[s])
            for i in range(d):
                if X[s,i]==0:continue
                for j in range(i+1,d):
                    if X[s,j]==0:continue
                    pred+=np.dot(V[i,field_map[j]],V[j,field_map[i]])*X[s,i]*X[s,j]
            pred=sigmoid(pred)
            err=pred-y[s]
            total_loss+=-y[s]*np.log(pred+1e-10)-(1-y[s])*np.log(1-pred+1e-10)
            w-=lr*(err*X[s]/n+0.01*w);b-=lr*err/n
            for i in range(d):
                if X[s,i]==0:continue
                for j in range(i+1,d):
                    if X[s,j]==0:continue
                    gi=err*V[j,field_map[i]]*X[s,i]*X[s,j]/n
                    gj=err*V[i,field_map[j]]*X[s,i]*X[s,j]/n
                    V[i,field_map[j]]-=lr*(gi+0.01*V[i,field_map[j]])
                    V[j,field_map[i]]-=lr*(gj+0.01*V[j,field_map[i]])
        if(ep+1)%20==0:print(f"  FFM Epoch{ep+1}:Loss={total_loss/n:.4f}")
    return w,b,V
print("\n--- FFM ---")
w_ffm,b_ffm,V_ffm=train_ffm(X[:100],y[:100],field_map)  # 小样本
print(f"FFM训练完成(100样本)")
# 对比
print(f"\n{'模型':<10}{'ACC':<10}{'AUC':<10}")
print("-"*30)
print(f"{'LR':<10}{acc_lr:<10.4f}{auc_lr:<10.4f}")
print(f"{'FM':<10}{acc_fm:<10.4f}{auc_fm:<10.4f}")

🔴 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