Machine Learning Lecture 17 "Regularization / Review" -Cornell CS4780 SP17

Midterm Reminder and Regularization Concepts

Introduction to Midterm

• The speaker reminds students about the upcoming midterm, emphasizing the importance of starting preparations if they haven't already.

Recap on Regularization

• The discussion revisits L2 regularization, explaining it as a method that constrains solutions within a "ball" around the origin to control complexity.

• The goal is to avoid overly complex solutions with extreme values; instead, regularization encourages simpler solutions closer to zero for better generalization.

Understanding L1 Norm

• The speaker introduces L1 regularization, characterized by minimizing the absolute value of weights (W), which forms an "L1 ball" resembling a diamond shape.

• This shape allows for unique properties in optimization, particularly promoting sparsity in weight vectors.

Advantages of L1 Regularization

• One key advantage of using the L1 norm is its ability to increase sparsity in models, effectively setting some weights to zero.

• By enforcing this constraint, optimal solutions can be found at corners of the L1 ball, allowing certain features (weights) to be completely disregarded.

Implications for Predictive Modeling

• Sparse representations are beneficial in fields like biology where understanding feature contributions is crucial; zero weights indicate non-contributing features.

Understanding L1 and Elastic Net Regularization

L1 Regularization Insights

• The discussion begins with the concept of L1 regularization, highlighting that it allows for closed-form solutions to determine values of lambda where features become nonzero sequentially.

• When the lunder (lambda) is very large, only one feature will be nonzero, indicating that the model prioritizes the most significant predictor for the label.

• A common practice involves gradually decreasing lambda to observe which features are selected first by the model.

Limitations of L1 Regularization

• One downside of L1 regularization is its lack of strict convexity; identical features can lead to multiple equivalent solutions, complicating interpretation.

• To address this issue, practitioners often use Elastic Net, which combines both L1 and a small amount of L2 regularization (mu), ensuring a unique solution due to the strict convexity of L2.

Optimization Problems and Constraints

• The speaker explains how minimizing loss subject to constraints can be represented in different forms, including using a budget constraint related to lambda.

• The relationship between norms and budgets is clarified: minimizing weights while adhering to constraints leads to simpler models.

Complexity Definitions in Regularization

• The absolute value representation penalizes each weight equally, aiming for simplicity in solutions rather than complexity through distribution across dimensions.

• A distinction is made between putting all weight on one dimension versus spreading it out; squaring weights increases penalties for larger values disproportionately compared to smaller ones.

Robustness vs. Brittleness in Models

• While models using L1 may focus heavily on few features leading to brittleness (e.g., reliance on specific sensors), Elastic Net offers more robustness by distributing weights across multiple features.

• This robustness helps prevent drastic prediction errors if certain inputs fail or drop out during operation (e.g., self-driving cars).

Deterministic Solutions with Elastic Net

• The speaker emphasizes that Elastic Net provides deterministic answers even when dealing with correlated features, unlike pure L1 regularization which can yield varying results based on implementation.

Understanding Feature Selection and Regularization in Predictive Modeling

The Conflict Between Feature Importance and Prediction Accuracy

• A biologist's focus on identifying useful features for predictions can conflict with the need for a robust predictor, highlighting a tension between feature selection and model performance.

• The challenge lies in balancing the classifier's complexity within a constrained area while ensuring that selected features are not overly penalized during regularization.

Regularization Techniques: Lasso and Elastic Net

• The discussion introduces Lasso (L1 regularization) and Elastic Net as popular methods for feature selection, emphasizing their practical application in predictive modeling.

• Lasso is characterized by squared loss combined with an L1 regularizer, while Elastic Net incorporates both L1 and L2 regularizers to enhance flexibility in feature selection.

Demonstrating Regularization Effects on Weights

• A demonstration using a small dataset illustrates how varying the lambda parameter affects feature weights; higher lambda values lead to more features being set to zero.

• As lambda decreases, additional features are unlocked, allowing the model to assign weights progressively across relevant features.

Analyzing Training vs. Test Error

• The training error decreases consistently as lambda lowers, indicating improved fit; however, test error initially decreases before rising again after reaching an optimal point with five key features.

• This finding underscores the importance of identifying which specific features contribute most effectively to generalizing results beyond the training data.

Implications for Biological Research

Understanding Regularization and Feature Selection

The Role of Lambda in Feature Selection

• Discussion on the importance of lambda in feature selection, where decreasing lambda leads to nonzero features.

• Introduction of least angular regression as an algorithm that can determine the next value for lambda effectively.

Relationship Between Features and Loss

• Explanation of how the number of nonzero features correlates with the increase in B (the model's performance metric), emphasizing that this relationship is not linear.

• Clarification that while minimizing square error is important, training error must also be monitored to avoid overfitting.

Importance of Regularization

• Emphasis on regularization even when focusing solely on prediction accuracy; it helps prevent fitting noise and ensures generalization.

Elastic Net and SVM Connection

• Introduction to elastic net and its relationship with support vector machines (SVM), highlighting a discovery by a student regarding their equivalence in certain contexts.

• Mention that SVM can solve problems faster than traditional methods used for elastic net.

Classroom Dynamics: Engaging Students Through Competition

Team Formation and Participation

• Overview of team formation between undergraduates and graduate students, setting up a competitive environment for learning.

Game Mechanics Explained

• Description of how points are awarded based on correct answers to questions about machine learning concepts, including SVM.

Questioning Strategy

• Explanation of how questions will be posed, allowing any student to answer after raising their hand once the question is read aloud.

Key Concepts in Machine Learning Discussed During Q&A

Understanding Classifiers

• Discussion around the performance comparison between k-nearest neighbors (KNN) classifiers and Bayes optimal classifiers, emphasizing KNN's limitations.

Logistic Regression vs. Decision Boundaries

• Inquiry into when a KNN decision boundary aligns with logistic regression, showcasing an important concept in classification algorithms.

Bias Incorporation in SVM

• Examination of why bias should not be treated as a constant feature within SVM frameworks due to implications on margin maximization.

Understanding SVM and K-Nearest Neighbors

Support Vector Machines (SVM) and Decision Boundaries

• The discussion highlights that maximizing the margin in SVM does not necessarily mean bringing the decision boundary close to zero; rather, it emphasizes that the bias term B should remain free.

Modifying K-Nearest Neighbors for Regression

• A question arises about modifying the K-nearest neighbor algorithm for regression. The response suggests taking an average of the K nearest neighbors to derive a value.

• An improvement can be made by using a weighted average, where weights are assigned based on distance, allowing closer points to have more influence on the prediction.

Assumptions of Different Algorithms

• Each algorithm has specific assumptions:

• KNN assumes similar points have similar labels.

• Naive Bayes assumes features are conditionally independent given the class.

• Logistic Regression is less strict regarding distribution but generally expects some form of linearity.

• SVM assumes data is linearly separable or uses slack variables when it's not.

Loss Functions in Machine Learning

• The discussion shifts to loss functions used in different scenarios, including squared hinge loss with small C , large C , and exponential loss. Understanding these helps clarify their impact on model performance.

Margin Calculation in SVM

Understanding Margin Maximization in Machine Learning

The Relationship Between W and Margin

• Minimizing the squared norm of weight vector W maximizes the margin in classification tasks, as the margin is inversely proportional to the norm of W .

• The discussion includes a playful interaction about betting points on questions, emphasizing engagement among students.

Betting Strategy in Class

• Students can bet points on their answers; if correct, they double their points, but incorrect answers result in losing those points.

• The optimal strategy suggested is to go "all-in" when betting on answers to maximize potential gains.

Challenges with Newton's Method

Limitations of Newton's Method

• Two distinct reasons for not using Newton's method include:

• If the function is not twice differentiable.

• High dimensionality may complicate computations or make Hessians non-invertible.

High Dimensional Data and KNN Performance

Understanding KNN in High Dimensions

• In high-dimensional spaces, K-nearest neighbors (KNN) performs well because it can effectively classify data that appears high-dimensional due to its intrinsic low-dimensional nature.

IBM Watson vs. Human Players: A Case Study

Strategies for Success

• IBM Watson utilized strategies based on analyzing past games to optimize betting during Daily Doubles, highlighting human players' weaknesses in identifying these opportunities.

Comparing SVM and KNN Performance

Algorithm Effectiveness Based on Data Characteristics

• Despite high dimensionality being present, effective classification depends more on data characteristics than mere dimensions; thus, KNN remains robust even when data appears complex.

Choosing Between Algorithms

• For text classification tasks with high dimensionality (like topic prediction), linear classifiers like SVM outperform KNN due to better handling of sparse features.

Discussion on Grad Student Points and K-Nearest Neighbors

Grad Student Points Allocation

• The speaker discusses the allocation of points to graduate students, indicating a competitive environment where 30 points are given, but there is some confusion about the distribution.

• There is an emphasis on wanting a "tight race" among participants, suggesting that the competition should be close and engaging.

• Applause follows a statement made by the speaker, indicating audience engagement or approval regarding the point distribution discussion.

K-Nearest Neighbors Computation

• A question is posed regarding whether it is true or false that K-nearest neighbors (KNN) on a dataset with n training points and D dimensions takes O(nD) computation time. The answer is hinted to be complex.

• The speaker clarifies that typically, KNN involves O(K * D), but if all points are considered as neighbors, it indeed results in O(n * D).