Lecture 6 "Perceptron Convergence Proof" -Cornell CS4780 SP17

Project Updates and Perceptron Algorithm Overview

Project Announcements

• The instructor reminds students that Project 2 has been released, while Project 1 is due this Friday. Students are encouraged to start immediately if they haven't already.

• Each project allows for two late days, emphasizing the importance of timely submissions.

Introduction to Perceptron

• The discussion shifts to the perceptron algorithm, with a focus on its foundational assumption: data must be linearly separable.

• The instructor explains that the dataset consists of two labels (minus one and plus one), which can be separated by a hyperplane in higher dimensions.

Understanding Hyperplanes and Classification

• A question arises about classification versus regression; the instructor clarifies that regression does not involve separating hyperplanes as there are no distinct classes.

• The perceptron algorithm's simplicity is highlighted, where the hyperplane is defined by vector W. The equation for the hyperplane is presented as W^T X + P = 0.

Algorithm Mechanics

• Students learn how to check if data points lie on the correct side of the hyperplane based on their labels. If conditions aren't met, updates to W are made accordingly.

• The condition checked during iterations is whether Y cdot (W^T X) is greater than zero for each data point.

Convergence Proof of Perceptron Algorithm

• The instructor aims to prove that if a separating hyperplane exists, the algorithm will converge towards it without looping indefinitely.

• It’s established that upon stopping, a valid hyperplane must have been found since all data points satisfy separation conditions.

Assumptions and Infinite Solutions

• An assumption states there exists a weight vector W^* such that all data points satisfy Y cdot (W^* cdot X)^T > 0.

• It's noted that infinitely many solutions exist because scaling W^* by any positive constant yields another valid solution.

Understanding Data Rescaling and Vector Updates

Introduction to Rescaling Data

• The initial step involves rescaling data points, specifically focusing on the vector W and its norm. The process allows for adjustments without altering the fundamental relationships within the dataset.

• It is emphasized that all data points can be scaled by a positive scalar, ensuring that each point maintains its relative position in relation to others.

Geometric Intuition Behind Rescaling

• The geometric intuition illustrates how rescaling transforms the dataset into a unit circle of radius one, with some points lying on the edge representing maximum distance.

• The vector W , which has a norm of one, aligns with this transformation, allowing for easier manipulation of the original dataset after scaling.

Conceptual Analogy

• An analogy is drawn to the movie "Honey I Shrunk the Kids," where shrinking everything uniformly preserves relationships between data points while simplifying calculations.

• This uniform scaling ensures that even after transformations, properties such as orientation remain unchanged.

Understanding Vector Updates

• A critical aspect discussed is how updates to vector W bring it closer to an optimal vector W^* . This relationship is explored through inner product measures.

• The goal is to demonstrate that as updates occur, W 's alignment with W^* improves over time.

Challenges in Proving Convergence

• While showing that W^T W^* increases with updates suggests improvement, it does not guarantee proximity unless additional conditions are met.

• A potential issue arises if W 's magnitude increases without actual progress towards W^* , necessitating further analysis of their relationship.

Key Insights on Inner Products

• To address concerns about mere growth in norms without convergence, attention shifts to how both vectors' inner products change during updates.

• If growth occurs in alignment but not in magnitude alone, it indicates that W 's direction tilts toward W^* , leading them toward convergence.

Update Mechanism Overview

• When updating vector W , it becomes evident that misclassification triggers these changes. Thus, understanding when and why updates occur is crucial for grasping overall dynamics.

Understanding Updates in Weight Vectors and Margins in Machine Learning

The Impact of Updating Weight Vector W

• The discussion begins with the importance of understanding how updates to the weight vector W affect its relationship with the optimal weight vector W^* .

• By substituting the new W into the equation, it is noted that this change affects terms related to W^* , specifically focusing on how these relationships evolve.

• The speaker emphasizes clarity by breaking down complex expressions involving W^* , ensuring participants grasp the significance of these changes.

• It is highlighted that since W^* is a critical component for separating data effectively, any update must maintain positivity in certain terms to ensure effective learning.

• A brief introduction to the concept of "margin" is provided, which refers to the distance from data points to a hyperplane.

Defining Margin and Its Importance

• The margin is defined as the distance from data points (denoted as X's) to their closest hyperplane, emphasizing its role in classification tasks.

• The minimum inner product with respect to the hyperplane gives rise to this margin, which becomes crucial for evaluating model performance.

• It’s clarified that margins are strictly greater than zero; this distinction plays a vital role in machine learning applications where hyperplanes are defined.

• Participants are prompted to consider preferences between small and large margins when defining hyperplanes, leading them toward an understanding of generalization capabilities.

• A larger margin is generally preferred because it reduces vulnerability to misclassification when new test points are introduced.

Implications of Margin on Model Performance

• The speaker reiterates that knowing whether a margin exists helps gauge how close data points are relative to their respective classifications.

• It’s established that if W^T X geq gamma , then there exists a minimum threshold (gamma), reinforcing confidence in predictions made by models using such margins.

• This leads into discussions about how updates can increase inner products associated with hyperplanes by at least gamma, indicating consistent improvement through iterations.

• Emphasizing consistency, every update guarantees an increase by at least gamma rather than diminishing returns over time—an important aspect for model training stability.

Analyzing Changes in Norm of Weight Vector

• Transitioning into another key point: what happens when we analyze changes in W^T W . This involves plugging updated definitions back into existing equations for clarity and accuracy.

• As updates occur, expanding expressions reveals insights about squared norms associated with weight vectors and their implications for overall model performance.

Understanding the Update Mechanism in Machine Learning

Analyzing the Update Formula

• The discussion begins with an examination of the update formula, specifically focusing on W^T X + y^2 cdot X^T X . The speaker emphasizes that this formula includes both old terms and new components.

• A question is posed regarding the term W^T X , which is identified as negative. This negativity indicates that X lies on the wrong side of the hyperplane, prompting an update.

Understanding Key Terms

• The value of y^2 is confirmed to be 1, while X^T X is acknowledged as a positive quantity due to prior definitions made by the speaker.

• By substituting known values into the equation, it’s established that certain terms can be simplified or bounded, leading to inequalities involving W^T W + 1 .

Establishing Bounds and Convergence

• The speaker explains how lower and upper bounds are established for changes in inner products over iterations. This leads to a conclusion about convergence rates between two quantities.

• After making m updates, it’s stated that m times gamma leq W^T W^* , indicating a relationship between updates and growth in weight norms.

Growth Analysis of Weight Norm

• It’s clarified that since W^* starts from zero, every update increases its norm by at least gamma. Thus after multiple updates, it must reflect cumulative growth proportional to updates made.

• The relationship between inner products and norms is discussed further; each update contributes positively to this relationship.

Application of Cauchy-Schwarz Inequality

• The Cauchy-Schwarz inequality is introduced as a critical mathematical tool for bounding inner products. This inequality states that the product of norms cannot exceed their respective inner product.

• It’s noted that since one vector's norm equals one (specifically referring to W^* ), simplifications can be made leading towards conclusions about overall system behavior.

Final Steps Towards Conclusion

• As discussions progress towards final conclusions about weight norms post-updates, it's reiterated that after several updates, certain bounds hold true based on previous calculations.

Understanding the Convergence of Algorithms

Proving Inequalities in Algorithm Updates

• The discussion begins with proving an inequality related to the number of updates M made by an algorithm, showing that M times gamma leq sqrtM .

• By manipulating this inequality, it can be derived that Mgamma leq 1 , leading to a significant conclusion about the maximum number of mistakes allowed by the algorithm.

Implications of Update Limits

• The result indicates that after M updates, one cannot exceed 1/gamma^2 mistakes, emphasizing a finite limit on errors based on the value of gamma .

• For example, if gamma = 0.1, then only up to 100 updates are permissible before convergence is reached.

Understanding Hyperplanes and Margins

• A question arises regarding multiple optimal solutions ( w^* ), with emphasis on selecting the hyperplane with the largest margin for better convergence.

• The optimal hyperplane is described as being centrally located between data classes, maximizing distance from both sides.

Data Separation and Convergence Speed

• If data points are well-separated (large margins), convergence occurs quickly; conversely, closely packed data leads to slower convergence due to smaller margins.

• This relationship intuitively makes sense as tighter clusters require more adjustments for accurate classification.

Support Vector Machines and Margin Considerations

• The conversation transitions into support vector machines (SVM), which aim to find optimal hyperplanes through different optimization techniques developed later than perceptron algorithms.

• It’s noted that for maximum margin hyperplanes, equal distances must exist on both sides; otherwise, adjustments could yield better margins.

Rescaling and Its Effects

• Rescaling does not fundamentally alter results; inequalities still hold true under rescaled conditions.

• An upper bound is established for how much weight vector differences can grow over iterations while maintaining certain constraints.

Exploring Existence of Solutions

• A hypothetical scenario questions what happens if no solution exists; running the perceptron algorithm without convergence indicates non-existence.

Understanding Convergence in Perceptron Learning

Convergence and Updates

• The existence of a solution W^* is crucial for convergence; without it, the algorithm cannot guarantee convergence.

• After a certain number of updates, mistakes can no longer occur, indicating that the learning process must converge.

• A large value of gamma accelerates convergence, which is beneficial for the learning process.

High Dimensional Data Challenges

• The effectiveness of perceptrons diminishes with high-dimensional data; finding a hyperplane to separate classes becomes unlikely.

• The separability of data points significantly influences the ability to classify them correctly.

Scaling Data Points

• Data points need to be scaled down to fit within a unit circle for effective processing; this scaling can bring distant points closer together.

• The margin between classes is less important than how closely different labeled points are situated after scaling.

The Historical Context of Perceptrons

Introduction of the Perceptron

• The perceptron was introduced in 1957 and generated significant excitement as an early model of artificial neurons.

• It was celebrated as a breakthrough in AI, likening its function to that of biological neurons firing based on input sums.

Public Reception and Disillusionment

• Early media coverage raised questions about the limitations of perceptrons, including their inability to understand complex human emotions or concepts like love.

• Marvin Minsky's work highlighted fundamental limitations by presenting problems (like XOR), demonstrating scenarios where perceptrons fail.

Impact on AI Funding and Research