Bell Laboratories Data Complexity Analysis: Linkage between Context and Solution in Classification Tin Kam Ho With contributions from Mitra Basu, Ester Bernado-Mansilla, Richard Baumgartner, Martin Law, Erinija Pranckeviciene, Albert Orriols-Puig, Nuria Macia

All Rights Reserved © Alcatel-Lucent Pattern Recognition: Research vs. Practice Steps to solve a practical pattern recognition problem Feature Extraction Classifier Training Classification Sensory Data Decision Feature Vectors Classifier Data Collection Study of the Problem Context Study of the Mathematical Solution Practical Focus Research Focus Danger of Disconnection

All Rights Reserved © Alcatel-Lucent Reconnecting Context and Solution Feature Vectors Study of the Problem Context Study of the Mathematical Solution To understand how such properties may impact the classification solution To understand how changes in the problem set-up and data collection procedures may affect such properties Data Complexity Analysis: Analysis of the properties of feature vectors Improvements Limitations Expectations

All Rights Reserved © Alcatel-Lucent Kolmogorov complexity Boundary length can be exponential in dimensionality A trivial description is to list all points & class labels Is there a shorter description? Focus is on Boundary Complexity

All Rights Reserved © Alcatel-Lucent Early Discoveries Problems distribute in a continuum in complexity space Several key measures provide independent characterization There exist identifiable domains of classifier’s dominant competency Feature selection and transformation induce variability in complexity estimates

All Rights Reserved © Alcatel-Lucent Parameterization of Data Complexity

All Rights Reserved © Alcatel-Lucent Complexity Classes vs. Complexity Scales Study is driven by observed limits in classifier accuracy, even with new, sophisticated methods (e.g., ensembles, SVM, …) Analysis is needed for each instance of a classification problem, not just the worst case of a family of problems Linear separability: the earliest attempt to address classification complexity Observed in real-world problems: different degrees of linear non-separability Continuous scale is needed

All Rights Reserved © Alcatel-Lucent Some Useful Measures of Geometric Complexity Classical measure of class separability Maximize over all features to find the most discriminating Fisher’s Discriminant RatioDegree of Linear Separability Find separating hyper- plane by linear programming Error counts and distances to plane measure separability Length of Class Boundary Compute minimum spanning tree Count class-crossing edges Shapes of Class Manifolds Cover same-class pts with maximal balls Ball counts describe shape of class manifold

All Rights Reserved © Alcatel-Lucent Real-World Data Sets: Benchmarking data from UC-Irvine archive 844 two-class problems 452 are linearly separable, 392 non-separable Synthetic Data Sets: Random labeling of randomly located points 100 problems in dimensions Continuous Distributions in Complexity Space Random labeling Linearly separable real-world data Linearly non- separable real- world data Complexity Metric 1 Metric 2

All Rights Reserved © Alcatel-Lucent Measures of Geometrical Complexity

All Rights Reserved © Alcatel-Lucent The First 6 Principal Components

All Rights Reserved © Alcatel-Lucent Interpretation of the First 4 PCs PC 1: 50% of variance: Linearity of boundary and proximity of opposite class neighbor PC 2: 12% of variance: Balance between within-class scatter and between-class distance PC 3: 11% of variance: Concentration & orientation of intrusion into opposite class PC 4: 9% of variance: Within-class scatter

All Rights Reserved © Alcatel-Lucent Continuous distribution Known easy & difficult problems occupy opposite ends Few outliers Empty regions Random labels Linearly separable Problem Distribution in 1 st & 2 nd Principal Components

All Rights Reserved © Alcatel-Lucent Apparent vs. True Complexity: Uncertainty in Measures due to Sampling Density 2 points10 points 100 points500 points1000 points Problem may appear deceptively simple or complex with small samples

All Rights Reserved © Alcatel-Lucent Observations Problems distribute in a continuum in complexity space Several key measures/dimensions provide independent characterization Need further analysis on uncertainty in complexity estimates due to small sample size effects

All Rights Reserved © Alcatel-Lucent Relating Classifier Behavior to Data Complexity

All Rights Reserved © Alcatel-Lucent Class Boundaries Inferred by Different Classifiers XCS: a genetic algorithm Nearest neighbor classifier Linear classifier

All Rights Reserved © Alcatel-Lucent Accuracy Depends on the Goodness of Match between Classifiers and Problems NNXCS error= 0.06% error= 1.9% Better ! Problem A Problem B error= 0.6% error= 0.7% XCS NN Better !

All Rights Reserved © Alcatel-Lucent Domains of Competence of Classifiers Given a classification problem, we want determine which classifier is the best for it. Can data complexity give us a hint? Complexity metric 1 Metric 2 NN LC XCS Decision Forest ? Here is my problem !

All Rights Reserved © Alcatel-Lucent Domain of Competence Experiment Use a set of 9 complexity measures Boundary, Pretop, IntraInter, NonLinNN, NonLinLP, Fisher, MaxEff, VolumeOverlap, Npts/Ndim Characterize 392 two-class problems from UCI data, all shown to be linearly non-separable Evaluate 6 classifiers NN (1-nearest neighbor) LP (linear classifier by linear programming) Odt (oblique decision tree) Pdfc (random subspace decision forest) Bdfc (bagging based decision forest) XCS (a genetic-algorithm based classifier) ensemble methods

All Rights Reserved © Alcatel-Lucent Identifiable Domains of Competence by NN and LP Best Classifier for Benchmarking Data

All Rights Reserved © Alcatel-Lucent Regions in complexity space where the best classifier is (nn,lp, or odt) vs. an ensemble technique Boundary-NonLinNN IntraInter-Pretop MaxEff-VolumeOverlap  ensemble + nn,lp,odt Less Identifiable Domains of Competence

All Rights Reserved © Alcatel-Lucent Uncertainty of Estimates at Two Levels Sparse training data in each problem & complex geometry cause ill-posedness of class boundaries (uncertainty in feature space) Sparse sample of problems causes difficulty in identifying regions of dominant competence (uncertainty in complexity space)

All Rights Reserved © Alcatel-Lucent Complexity and Data Dimensionality: Class Separability after Dimensionality Reduction Feature selection/transformation may change the difficulty of a classification problem: Widening the gap between classes Compressing the discriminatory information Removing irrelevant dimensions It is often unclear to what extent these happen We seek quantitative description of such changes Feature selection Discrimination

All Rights Reserved © Alcatel-Lucent Spread of classification accuracy and geometrical complexity due to forward feature selection

All Rights Reserved © Alcatel-Lucent Designing a Strategy for Classifier Evaluation

All Rights Reserved © Alcatel-Lucent A Complete Platform for Evaluating Learning Algorithms To facilitate progress on learning algorithms: Need a way to systematically create learning problems Provide a complete coverage of the complexity space Be representative of all the known problems i.e., every classification problem arising in the real-world should have a close neighbor representing it in the complexity space. Is this possible?

All Rights Reserved © Alcatel-Lucent Ways to Synthesize Classification Problems Synthesizing data with targeted levels of complexity e.g. compute MST over a uniform point distribution, then assign class-crossing edges randomly [Macia et al. 2008] or, create partitions with increasing resolution can create continuous cover of complexity space but, are the data similar to those arising from reality?

All Rights Reserved © Alcatel-Lucent Ways to Synthesize Classification Problems Synthesizing data to simulate natural processes e.g. Neyman-Scott process how many such processes have explicit models? how many are needed to cover all real-world problems? Systematically degrade real-world datasets increase noise, reduce image resolution, …

All Rights Reserved © Alcatel-Lucent Simplification of Class Geometry

All Rights Reserved © Alcatel-Lucent Manifold Learning and Dimensionality Reduction Manifold learning techniques that highlight intrinsic dimensions But the class boundary may not follow the intrinsic dimensions

All Rights Reserved © Alcatel-Lucent Manifold Learning and Dimensionality Reduction Supervised manifold learning – seek mappings that exaggerate class separation [de Ridder et al., 2003] Best, the mapping should be sought to directly minimize some measures of data complexity

All Rights Reserved © Alcatel-Lucent Seeking Optimizations Upstream Back to the application context: Use data complexity measures for guidance Change the setup, definition of the classification problem Collect more samples, in finer resolution, extract more features … Alternative representations: dissimilarity-based? [Pekalska & Duin 2005] Data complexity gives an operational definition of learnability Optimization in the upstream: formalize the intuition of seeking invariance, systematically optimize the problem setup and data acquisition scenario to reduce data complexity

All Rights Reserved © Alcatel-Lucent Recent Examples from the Internet

All Rights Reserved © Alcatel-Lucent CAPTCHA: Completely Automated Public Turing test to tell Computers and Humans Apart Also known as Reverse Turing Test Human Interactive Proofs [von Ahn et al., CMU 2000] Exploit limitations in accuracy of machine pattern recognition

All Rights Reserved © Alcatel-Lucent The Netflix Challenge $1 Million Prize for the first team to improve 10% over the company’s own recommender system But, is the goal achievable? Do the training data support such possibility?

All Rights Reserved © Alcatel-Lucent Amazon’s Mechanical Turk “Crowd-sourcing” tedious human intelligence (pattern recognition) tasks Which ones are doable by machines?

All Rights Reserved © Alcatel-Lucent Conclusions

All Rights Reserved © Alcatel-Lucent Summary Automatic classification is useful, but can be very difficult. We know the key steps and many promising methods. But we have not fully understood how they work, what else is needed. We found measures for geometric complexity that are useful to characterize difficulties of classification problems and classifier domains of competence. Better understanding of how data and classifiers interact can guide practice, and re-establish the linkage between context and solution.

All Rights Reserved © Alcatel-Lucent For the Future Further progress in statistical and machine learning will need systematic, scientific evaluation of the algorithms with problems that are difficult for different reasons. A “problem synthesizer” will be useful to provide a complete evaluation platform, and reveal the “blind spots” of current learning algorithms. Rigorous statistical characterization of complexity estimates from limited training data will help gauge the uncertainty, and determine applicability of data complexity methods.