🕸️ 第10课：图神经网络推荐（LightGCN）

🎯 学习目标

🗂️ 课程定位

📖 核心理论

图神经网络（GNN）将推荐系统建模为用户-物品二部图，通过消息传递在图上传播信息。LightGCN是2020年提出的轻量级GCN推荐模型，去掉了传统GCN中的非线性变换和权重矩阵，仅保留邻域聚合，效果却更好。

📐 数学公式

💻 代码实现

import numpy as np
np.random.seed(42)
print("="*60+"\n图神经网络推荐(LightGCN)\n"+"="*60)
n_users=30;n_items=20;n_nodes=n_users+n_items
# 构建用户-物品二部图
n_interactions=120
edges=[]
for _ in range(n_interactions):
    u=np.random.randint(n_users)
    i=np.random.randint(n_items)
    edges.append((u,n_users+i))
edges=list(set(edges))
print(f"用户:{n_users} 物品:{n_items} 边:{len(edges)}")
# 邻接矩阵
A=np.zeros((n_nodes,n_nodes))
for u,i in edges:A[u,i]=A[i,u]=1
# 度归一化
D=np.diag(A.sum(axis=1)**(-0.5))
D[np.isinf(D)]=0
A_norm=D@A@D
print(f"图密度:{len(edges)/(n_users*n_items):.4f}")
# 初始embedding
emb_size=16
E=np.random.randn(n_nodes,emb_size)*0.1
# LightGCN传播
def lightgcn_propagate(A_norm,E,n_layers=3):
    all_embs=[E]
    E_cur=E.copy()
    for l in range(n_layers):
        E_cur=A_norm@E_cur
        all_embs.append(E_cur)
    # 层组合(平均)
    final=np.mean(all_embs,axis=0)
    return final,all_embs
E_final,all_embs=lightgcn_propagate(A_norm,E)
print(f"\nLightGCN传播后embedding形状:{E_final.shape}")
# 推荐评分
user_embs=E_final[:n_users]
item_embs=E_final[n_users:]
scores=user_embs@item_embs.T
# 为用户0推荐
target=0;interacted=set(i for u,i in edges if u==target)
rec_scores=[(i,scores[target,i])for i in range(n_items)if i not in interacted]
rec_sorted=sorted(rec_scores,key=lambda x:-x[1])[:5]
print(f"\n用户0推荐Top-5:")
for i,s in rec_sorted:print(f"  物品{i}:分数={s:.4f}")
# 评估BPR
print("\n--- BPR训练LightGCN ---")
E_train=np.random.randn(n_nodes,emb_size)*0.01
lr=0.01;reg=0.001
for ep in range(50):
    for _ in range(200):
        u=np.random.randint(n_users)
        pos=[i-n_users for uu,i in edges if uu==u]
        neg=[i for i in range(n_items)if i not in pos]
        if not pos or not neg:continue
        i=np.random.choice(pos);j=np.random.choice(neg)
        E_f,_=lightgcn_propagate(A_norm,E_train)
        u_emb=E_f[u];i_emb=E_f[n_users+i];j_emb=E_f[n_users+j]
        x_uij=np.dot(u_emb,i_emb)-np.dot(u_emb,j_emb)
        grad=1/(1+np.exp(x_uij))
        # 简化梯度更新embedding
        E_train[u]-=lr*(-grad*(i_emb-j_emb)+reg*E_train[u])
        E_train[n_users+i]-=lr*(-grad*u_emb+reg*E_train[n_users+i])
        E_train[n_users+j]-=lr*(grad*u_emb+reg*E_train[n_users+j])
    if(ep+1)%10==0:
        E_f,_=lightgcn_propagate(A_norm,E_train)
        u_embs=E_f[:n_users];i_embs=E_f[n_users:]
        auc_vals=[]
        for u in range(min(15,n_users)):
            pos=[i-n_users for uu,i in edges if uu==u]
            neg=[i for i in range(n_items)if i not in pos]
            if not pos or not neg:continue
            sp=u_embs[u]@i_embs[pos].T
            sn=u_embs[u]@i_embs[neg].T
            auc_vals.append(np.mean(sp[:,None]>sn[None,:]))
        print(f"  Epoch{ep+1}:AUC={np.mean(auc_vals):.4f}")
# 层数影响
print("\nGCN层数影响:")
for n_layers in [1,2,3,4]:
    E_f,_=lightgcn_propagate(A_norm,E_train,n_layers)
    u_e=E_f[:n_users];i_e=E_f[n_users:]
    auc_v=[]
    for u in range(min(15,n_users)):
        pos=[i-n_users for uu,i in edges if uu==u]
        neg=[i for i in range(n_items)if i not in pos]
        if not pos or not neg:continue
        sp=u_e[u]@i_e[pos].T;sn=u_e[u]@i_e[neg].T
        auc_v.append(np.mean(sp[:,None]>sn[None,:]))
    print(f"  layers={n_layers}:AUC={np.mean(auc_v):.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