📊 第13课：排序模型（LambdaMART）

🎯 学习目标

🗂️ 课程定位

📖 核心理论

排序模型（Learning to Rank, LTR）是推荐系统精排阶段的核心技术。与传统的回归/分类不同，LTR直接优化排序质量指标（NDCG/MRR），更符合推荐场景的需求。

LambdaMART是LTR的集大成者，结合了LambdaRank的lambda梯度和MART(GBDT)的梯度提升框架。它通过lambda梯度直接优化NDCG，同时利用GBDT强大的非线性建模能力。

📐 数学公式

💻 代码实现

import numpy as np
np.random.seed(42)
print("="*60+"\n排序模型(LambdaMART)\n"+"="*60)
n_queries=20;n_docs_per_query=10;n_features=8
# 生成排序数据
X=[];y=[];qids=[]
for q in range(n_queries):
    q_features=np.random.randn(n_features)*0.5
    for d in range(n_docs_per_query):
        features=np.random.randn(n_features)*0.3+q_features
        relevance=np.dot(features,np.random.randn(n_features))
        relevance=int(np.clip(relevance+2,0,4))
        X.append(features);y.append(relevance);qids.append(q)
X=np.array(X);y=np.array(y);qids=np.array(qids)
print(f"查询数:{n_queries} 文档数:{len(X)} 特征数:{n_features}")
print(f"相关性分布: {dict(zip(*np.unique(y,return_counts=True)))}")
# Pointwise方法(MSE回归)
print("\n--- Pointwise(MSE) ---")
from collections import defaultdict
w=np.random.randn(n_features)*0.01;b=0
lr=0.01
for ep in range(100):
    pred=X@w+b
    err=pred-y
    w-=lr*(X.T@err/len(X)+0.01*w)
    b-=lr*np.mean(err)
    if(ep+1)%25==0:
        mse=np.mean(err**2)
        print(f"  Epoch{ep+1}:MSE={mse:.4f}")
pred_pointwise=X@w+b
# Pairwise方法(RankNet简化)
print("\n--- Pairwise(RankNet) ---")
w_pair=np.random.randn(n_features)*0.01;b_pair=0
def sigmoid(x):return 1/(1+np.exp(-np.clip(x,-500,500)))
for ep in range(50):
    total_loss=0;n_pairs=0
    for q in range(n_queries):
        mask=qids==q
        q_X=X[mask];q_y=y[mask]
        idx=np.random.choice(len(q_y),2,replace=False)
        i,j=idx
        if q_y[i]==q_y[j]:continue
        s_i=q_X[i]@w_pair+b_pair
        s_j=q_X[j]@w_pair+b_pair
        target=1 if q_y[i]>q_y[j] else 0
        pred=sigmoid(s_i-s_j)
        loss=-target*np.log(pred+1e-10)-(1-target)*np.log(1-pred+1e-10)
        total_loss+=loss;n_pairs+=1
        grad=sig_grad=pred-target
        w_pair-=lr*(grad*(q_X[i]-q_X[j])+0.01*w_pair)
        b_pair-=lr*grad
    if(ep+1)%10==0:print(f"  Epoch{ep+1}:Loss={total_loss/max(n_pairs,1):.4f}")
# LambdaMART简化(用lambda梯度)
print("\n--- LambdaMART简化 ---")
w_lm=np.random.randn(n_features)*0.01;b_lm=0
def ndcg_at_k(rel,k):
    dcg=sum((2**r-1)/np.log2(i+2)for i,r in enumerate(sorted(rel,reverse=True)[:k]))
    idcg=sum((2**r-1)/np.log2(i+2)for i,r in enumerate(sorted(rel,reverse=True)[:k]))
    return dcg/idcg if idcg>0 else 0
for ep in range(50):
    for q in range(n_queries):
        mask=qids==q;q_X=X[mask];q_y=y[mask]
        pred=q_X@w_lm+b_lm
        order=np.argsort(-pred)
        # 计算lambda梯度
        lambda_grad=np.zeros(len(q_y))
        for i in range(len(q_y)):
            for j in range(len(q_y)):
                if q_y[i]<=q_y[j]:continue
                delta_ndcg=abs(ndcg_at_k(list(q_y),5)-ndcg_at_k(list(q_y[[j if k==i else i if k==j else k for k in range(len(q_y))]]),5))
                rho=1/(1+np.exp(pred[i]-pred[j]))
                lambda_grad[i]+=delta_ndcg*rho
                lambda_grad[j]-=delta_ndcg*rho
        w_lm-=lr*(-q_X.T@lambda_grad/len(q_y)+0.01*w_lm)
        b_lm-=lr*(-np.mean(lambda_grad))
    if(ep+1)%10==0:
        # 计算NDCG@5
        ndcgs=[]
        for q in range(n_queries):
            mask=qids==q;q_X=X[mask];q_y=y[mask]
            pred=q_X@w_lm+b_lm
            top5=np.argsort(-pred)[:5]
            ndcgs.append(ndcg_at_k(list(q_y[top5]),5))
        print(f"  Epoch{ep+1}:NDCG@5={np.mean(ndcgs):.4f}")
# 最终评估
methods={"Pointwise":pred_pointwise,"Pairwise":X@w_pair+b_pair,"LambdaMART":X@w_lm+b_lm}
print(f"\n{'方法':<15}{'NDCG@5':<10}{'MRR':<10}")
print("-"*35)
for name,pred in methods.items():
    ndcgs=[];mrrs=[]
    for q in range(n_queries):
        mask=qids==q;q_y=y[mask];q_pred=pred[mask]
        top5=np.argsort(-q_pred)[:5]
        ndcgs.append(ndcg_at_k(list(q_y[top5]),5))
        for rank,idx in enumerate(np.argsort(-q_pred)):
            if q_y[idx]==max(q_y):mrrs.append(1/(rank+1));break
    print(f"{name:<15}{np.mean(ndcgs):<10.4f}{np.mean(mrrs):<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