📈 第14课：学习排序（LTR）

🎯 学习目标

🗂️ 课程定位

📖 核心理论

学习排序（Learning to Rank）是信息检索和推荐系统的核心技术。Listwise方法直接优化整个排序列表的质量，比Pointwise和Pairwise更符合排序的本质需求。

📐 数学公式

💻 代码实现

import numpy as np
np.random.seed(42)
print("="*60+"\n学习排序(LTR)深入\n"+"="*60)
n_queries=15;n_docs=8;n_features=6
# 生成数据
all_X=[];all_y=[];all_qids=[]
for q in range(n_queries):
    q_feat=np.random.randn(n_features)*0.3
    for d in range(n_docs):
        feat=np.random.randn(n_features)*0.2+q_feat
        rel=max(0,int(np.dot(feat,np.random.randn(n_features)*0.5)+2))
        all_X.append(feat);all_y.append(min(rel,4));all_qids.append(q)
X=np.array(all_X);y=np.array(all_y);qids=np.array(all_qids)
print(f"查询:{n_queries} 文档:{len(X)} 特征:{n_features}")
# ListNet
print("\n--- ListNet ---")
def softmax(x):e=np.exp(x-x.max());return e/e.sum()
w_list=np.random.randn(n_features)*0.01;b_list=0
lr_list=0.01
for ep in range(80):
    total_loss=0
    for q in range(n_queries):
        mask=qids==q;q_X=X[mask];q_y=y[mask]
        scores=q_X@w_list+b_list
        pred_probs=softmax(scores)
        true_probs=softmax(q_y.astype(float))
        loss=-np.sum(true_probs*np.log(pred_probs+1e-10))
        total_loss+=loss
        grad=pred_probs-true_probs
        w_list-=lr_list*(q_X.T@grad/len(q_y)+0.01*w_list)
        b_list-=lr_list*np.mean(grad)
    if(ep+1)%20==0:print(f"  Epoch{ep+1}:Loss={total_loss/n_queries:.4f}")
# ListMLE
print("\n--- ListMLE ---")
w_mle=np.random.randn(n_features)*0.01;b_mle=0
lr_mle=0.01
for ep in range(80):
    total_loss=0
    for q in range(n_queries):
        mask=qids==q;q_X=X[mask];q_y=y[mask]
        scores=q_X@w_mle+b_mle
        # 按真实相关性排序
        true_order=np.argsort(-q_y)
        # ListMLE: 在真实排序下的概率
        log_prob=0
        remaining=scores.copy()
        for pos in range(len(true_order)):
            probs=softmax(remaining)
            chosen=true_order[pos]
            log_prob+=np.log(probs[chosen]+1e-10)
            remaining[chosen]=-999
        loss=-log_prob
        total_loss+=loss
        # 简化梯度
        grad_probs=softmax(scores)
        w_mle-=lr_list*(q_X.T@(grad_probs-softmax(q_y.astype(float)))/len(q_y)+0.01*w_mle)
        b_mle-=lr_list*np.mean(grad_probs-softmax(q_y.astype(float)))
    if(ep+1)%20==0:print(f"  Epoch{ep+1}:Loss={total_loss/n_queries:.4f}")
# 评估
def eval_ltr(name,w,b):
    ndcgs=[];map_vals=[]
    for q in range(n_queries):
        mask=qids==q;q_X=X[mask];q_y=y[mask]
        scores=q_X@w+b
        order=np.argsort(-scores)
        # NDCG@5
        dcg=sum((2**q_y[order[i]]-1)/np.log2(i+2)for i in range(min(5,len(order))))
        idcg=sum((2**r-1)/np.log2(i+2)for i,r in enumerate(sorted(q_y,reverse=True)[:5]))
        ndcgs.append(dcg/idcg if idcg>0 else 0)
        # MAP
        hits=0;prec_sum=0
        for i,idx in enumerate(order):
            if q_y[idx]>0:
                hits+=1;prec_sum+=hits/(i+1)
        map_vals.append(prec_sum/max(hits,1))
    return np.mean(ndcgs),np.mean(map_vals)
print(f"\n{'方法':<15}{'NDCG@5':<10}{'MAP':<10}")
print("-"*35)
for name,w,b in [("ListNet",w_list,b_list),("ListMLE",w_mle,b_mle)]:
    ndcg,map_v=eval_ltr(name,w,b)
    print(f"{name:<15}{ndcg:<10.4f}{map_v:<10.4f}")
# 排序一致性分析
print("\n排序一致性:")
for q in range(3):
    mask=qids==q;q_y=y[mask]
    s1=X[mask]@w_list+b_list
    s2=X[mask]@w_mle+b_mle
    o1=np.argsort(-s1);o2=np.argsort(-s2)
    print(f"  Query{q}:ListNet={o1} ListMLE={o2} 真实={np.argsort(-q_y)}")

🔴 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