Logistic Regression
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introduce logistic regression, inference with maximize likelihood with gradient descent, compare L1 and L2 regularization, generalized linear model
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Logistic Regression
- 1. Machine learning workshop guodong@hulu.com Machine learning introduc7on Logis&c regression Feature selec7on Boos7ng, tree boos7ng See more machine learning post: h>p://dongguo.me
- 2. Overview of machine learning Machine Learning Unsupervised Learning Supervised Learning Classifica7on Logis7c regression Semi-‐supervised Learning Regression
- 3. How to choose a suitable model? Characteris&c Naïve Bayes Trees K Nearest neighbor Logis&c regression Neural SVM Networks Computa7onal scalability 3 3 1 3 1 1 Interpretability 2 2 1 2 1 1 Predic7ve power 1 1 3 2 3 3 Natural handling data of “mixed” type 1 3 1 1 1 1 Robustness to outliers in input space 3 3 3 3 1 1 <Elements of Sta-s-cal Learning> II P351
- 4. Why model can’t perform perfectly on unseen data • Expected risk • Empirical risk • Choose func7on family for predic7on func7ons • Error
- 6. Outline • • • • • Introduc7on Inference Regulariza7on Experiments More – Mul7-‐nominal LR – Generalized linear model • Applica7on
- 7. Logit func7on and logis7c func7on • Logit func7on • logis7c func7on: Inversed logit
- 8. Logis7c regression • Predic7on func7on
- 9. Inference with maximize likelihood (1) • Likelihood • Inference
- 10. Inference with maximize likelihood (2) • Inference cont. • Use gradient descent • Stochas7c gradient descent
- 11. Regulariza7on • Penalize large weight to avoid overfi`ng – L2 regulariza7on – L1 regulariza7on
- 12. Regulariza7on: Maximum a posteriori • MAP
- 13. L2 regulariza7on : Gaussian Prior • Gaussian prior • MAP • Gradient descent step
- 14. L1 regulariza7on : Laplace Prior • Laplace prior • MAP • Gradient descent step
- 15. Implementa7on • L2 LR _weightOfFeatures[fea] += step * (feaValue * error - reguParam * _weightOfFeatures[fea]); • L1 LR if (_weightOfFeatures[fea] > 0) { _weightOfFeatures[fea] += step * (feaValue * error) - step * reguParam; if (_weightOfFeatures[fea] < 0) _weightOfFeatures[fea] = 0; }else if (_weightOfFeatures[fea] < 0) { _weightOfFeatures[fea] += step * (feaValue * error) + step * reguParam; if (_weightOfFeatures[fea] > 0) _weightOfFeatures[fea] = 0; }else{ _weightOfFeatures[fea] += step * (feaValue * error); }
- 16. L2 VS. L1 • L2 regulariza7on – Almost all weights are not equal to zero – Not suitable when training samples are scarce • L1 regulariza7on – Produces sparse parameter vectors – More suitable when most features are irrelevant – Could handle scarce training samples be>er
- 17. Experiments • Dataset – Goal: gender predic7on – Dataset: train samples (431k), test samples (167k) • Comparison algorithms – A: gradient descent with L1 regulariza7on – B: gradient descent with L2 regulariza7on – C: OWL-‐QN (L-‐BFGS based op7miza7on with L1 regulariza7on) • Parameters choice – – – – Regulariza7on value Step(learning speed) Decay ra7o Itera7on over condi7on • Max itera7on 7mes(50) || AUC change <=0.0005
- 18. Experiments (cont.) • Experiments results Parameters and metrics gradient descent with gradient descent with L1 L2 OWL-‐QN ‘best’ regulariza7on 0.001~0.005 term 0.0002~0.001 1 Best step 0.05 0.02~0.05 -‐ Best decay ra7o 0.85 0.85 -‐ Itera7on 7mes 26 20~26 48 Not zero feature / all feature 10492/10938 10938/10938 6629/10938 AUC 0.8470 0.8463 0.8467
- 19. Mul7-‐nominal logis7c regression • Predic7on func7on • Inference with maximize likelihood • Gradient descent step (L2)
- 20. More Link func7ons • Inference with maximize likelihood • Link func7on • Link func7ons for binomial distribu7on – Logit func7on – Probit func7on – Log-‐log func7on
- 21. Generalized linear model • What is GLM – Generaliza7on of linear regression – Connect linear model with response variable by link func7on – More distribu7on for response variable • Typical GLM • Overview – Linear regression , Logis7c regression, Poisson regression
- 22. Applica7on • Yahoo – <Personalized Click Predic7on in Sponsored Search> WSDM’10 • Microsoq – <Scalable Training of L1-‐Regularized Log-‐Linear Models> ICML’07 • Baidu – Contextual ads CTR predic7on • h>p://www.docin.com/p-‐376254439.html • Hulu – – – – Demographic targe7ng Other ad-‐targe7ng project Custom churn predic7on More…
- 23. Reference • ‘Scalable Training of L1-‐Regularized Log-‐Linear Models’ ICML’07 – h>p://www.docin.com/p-‐376254439.html# • ‘Genera-ve and discrimina-ve classifiers: Naïve Bayes and logis-c regression’ by Mitchell • ‘Feature selec-on, L1 vs. L2 regulariza-on, and rota-onal invariance’ ICML’04
- 24. Recommended resources • Machine Learning open class – by Andrew Ng – //10.20.0.130/TempShare/Machine-‐Learning Open Class • h>p://www.cnblogs.com/vivounicorn/archive/ 2012/02/24/2365328.html • logis7c regression Implementa7on[link] – //10.20.0.130/TempShare/guodong/Logis7c regression Implementa7on/ – Support binomial and mul7nominal LR with L1 and L2 regulariza7on • OWL-‐QN – //10.20.0.130/TempShare/guodong/OWL-‐QN/
- 25. Thanks
Hinweis der Redaktion
- Unsupervised learning(聚类，降维（topic model）): learn structure from unlabeled data. Closely related with density estimation; summarize the dataSemi-supervised learning: use both labeled and unlabeled samples for training; It’s cost to collect lots of labels sometimes, use both
- 除此之外，你对模型的熟悉程度。
- Expected risk: 定义好loss function，选择一个预估函数，有一个输入变量和response value的联合分布，在该联合分布上对损失函数求积分，即为期望风险；通过最小化该期望风险，我们找到一个最优的预估函数。但是实际上，我们并不知道该联合分布，我们有的是从该联合分布中有偏或无偏采样得到的有限样本，可能还有一些noise点。我们转为最小化在该有限样本上的最小化loss function寻找预估函数。 即我们转为最小化经验风险另一方面，我们往往给目标函数指定function family，该function family极有可能没有包含最优或者较优的那些目标函数。误差的大小： 第一部分：函数family F中的预估函数有多接近真正最优的预估函数；第二部分：我们选择优化经验而不是经验风险
- Logistic regression is one of the most popular classifier.Advantage: 1. easy understand and implement; 2. not bad performance; 3. light weight and less time taken for training and prediction;(can handle large dataset) 4. easy parallelizationValue to attendances:Know about what is logistic regression, what’s the advantages and disadvantage. what kind of problems are suitable apply to.L1 and L2 regularizationHow to inference through maximize likelihood with gradient descent. And know how to implement it
- 对于generalized linear model，如果response variable是binomial或者multinomial分布，且选择了logit function作为link function 就是logistic regressionLogistic function 是logit function的反函数
- Binary(binomial) logistic regression
- 负梯度方向是使得函数值下降最快的方向
- 在重新计算likelihood前，我们看一下这2种正则化背后的理论基础
- 假设全部的weight服从一致的分布
- Laplace 分布一阶倒数不连续假设全部的weight服从一致的分布（均值为0，Laplace参数也一样）W_k在一次更新中不能变换正负号
- L1拟合得到的weight通常较稀疏，带来2点好处： 帮我们做特征选择，工程上更有利
- 增加了decay ratio：AUC稍有提高（0.845 -> 0.847）在不同step时，适合的decay ratio也不一样Iteration times： 与样本量的大小有关
- 例子：今天是高考第一天，高考选专业，每个人有多个候选，但是仅能选择一门专业（计算机，金融，化学，数学，物理，生物）和binomial分布应用的区别多类问题，可以转化为多个两类问题，如果我们的问题是“找出每门课成绩前10%的学生”，我们可以用两类logistic regression来做如果问题是“对于每个学生找出其成绩最好的课，或者最好的几门课”，两类问题就不是很适合 （每一类上的预估概率之和不等于1，无法比较不同类上的概率）Multi-nominal适用于response value为category的情况，不太适合ordinal的情况。我实现了。
- Link function: (1) generalized linear model的重要组成部分：将linear regression拓展到generalized linear model；（2）link function的反函数的自变量介于(-无穷，+无穷)，若y服从binominal分布，应变量介于【0,1】区间The inverse of any continuous cumulative distribution function (CDF) can be used for the link since the CDF’s range is [0,1]
- Generalized linear model 广义上的线性模型，都有一个基本的线性单元W*X（linear regression），通过各种link function建立该线性单元和各种分布的response variable的关系。包含linear regression （normal distribution），logistic regression （binominal/multi-nominal distribution）, Poisson regression (Poisson distribution)对于binominal/multi-nominal distribution，我们也可以选择除logit link function之外的link function （广义的logistic regression）