Meta-Learning and Learning in Highly Non-separable Cases Włodzisław Duch Department of Informatics, Nicolaus Copernicus University, Toruń, Poland Google: W. Duch

Plan Problems with Computational intelligence (CI) What can we learn? Why solid foundations are needed. Similarity based framework. Transformations and heterogeneous systems. Meta-learning. Hard problems. Beyond pattern recognition. Scaling up intelligent systems to human level competence?

What is Computational Intelligence? The Field of Interest of the Society shall be the theory, design, application, and development of biologically and linguistically motivated computational paradigms emphasizing neural networks, connectionist systems, genetic algorithms, evolutionary programming, fuzzy systems, and hybrid intelligent systems in which these paradigms are contained. Artificial Intelligence (AI) was established in 1956! AI Magazine 2005, Alan Mackworth: In AI's youth, we worked hard to establish our paradigm by vigorously attacking and excluding apparent pretenders to the throne of intelligence, pretenders such as pattern recognition, behaviorism, neural networks, and even probability theory. Now that we are established, such ideological purity is no longer a concern. We are more catholic, focusing on problems, not on hammers. Given that we do have a comprehensive toolbox, issues of architecture and integration emerge as central.

CI definition Computational Intelligence. An International Journal (1984) + 10 other journals with “Computational Intelligence”, D. Poole, A. Mackworth & R. Goebel, Computational Intelligence - A Logical Approach. (OUP 1998), GOFAI book, logic and reasoning. CI should: be problem-oriented, not method oriented; cover all that CI community is doing now, and is likely to do in future; include AI – they also think they are CI ... CI: science of solving (effectively) non-algorithmizable problems. Problem-oriented definition, firmly anchored in computer sci/engineering. AI: focused problems requiring higher-level cognition, the rest of CI is more focused on problems related to perception/action/control.

The future of computational intelligence ...

What can we learn? Good part of CI is about learning. Neural networks are universal approximators and evolutionary algorithms solve global optimization problems – so everything can be learned? Not quite ... Duda, Hart & Stork, Ch. 9, No Free Lunch + Ugly Duckling Theorems: Uniformly averaged over all target functions the expected error for all learning algorithms is the same. Averaged over all target functions no learning algorithm yields generalization error that is superior to any other. There is no problem-independent or “best” set of features. “Experience with a broad range of techniques is the best insurance for solving arbitrary new classification problems.”

What is there to learn? Brains ... what is in EEG? What happens in the brain? Industry: what happens? Genetics, proteins ...

Are we really so good?

Data mining packages GhostMiner, data mining tools from our lab + Fujitsu: http://www.fqspl.com.pl/ghostminer/ Separate the process of model building (hackers) and knowledge discovery, from model use (lamers) => GM Developer & Analyzer No free lunch => provide different type of tools for knowledge discovery: decision tree, neural, neurofuzzy, similarity-based, SVM, committees, tools for visualization of data. Support the process of knowledge discovery/model building and evaluating, organizing it into projects. Many other interesting DM packages of this sort exists: Weka, Yale, Orange, Knime ... 168 packages on the-data-mine.com list! We are building Intemi, completely new tools. Surprise! Almost nothing can be learned using such tools!

What DM packages do? Hundreds of components ... transforming, visualizing ... Visual “knowledge flow” to link components, or script languages (XML) to define complex experiments. Yale 3.3: type # components Data preprocessing 74 Experiment operations 35 Learning methods 114 Metaoptimization schemes 17 Postprocessing 5 Performance validation 14 Visualization, presentation, plugin extensions ...

Why solid foundations are needed Hundreds of components ... thousands of combinations ... Our treasure box is full! We can publish forever! But what would we really like to have? Press the button and wait for the truth! Computer power is with us, meta-learning should find all interesting data models = sequences of transformations/procedures. Many considerations: optimal cost solutions, various costs of using feature subsets; models that are simple & easy to understand; various representation of knowledge: crisp, fuzzy or prototype rules, visualization, confidence in predictions ...

Computational learning approach: let there be light!

Making things easy: principles

Principles: information compression Neural information processing in perception and cognition: information compression, or algorithmic complexity. In computing: minimum length (message, description) encoding. Wolff (2006): cognition and computation as compression by multiple alignment, unification and search. Analysis and production of natural language, fuzzy pattern recognition, probabilistic reasoning and unsupervised inductive learning. So far only models for sequential data and 1D alignment. Information compression: encoding new information in terms of old has been used to define the measure of syntactic and semantic information (Duch, Jankowski 1994); based on the size of the minimal graph representing a given data structure or knowledge-base specification, thus it goes beyond alignment.

Graphs of consistent concepts Learn new concepts in terms of old; using large semantic network and add new concepts linking them to known. Disambiguate concepts by spreading activation and selecting those that are consistent with already active subnetworks.

Similarity-based framework (Dis)similarity: more general than feature-based description, no need for vector spaces (structured objects), more general than fuzzy approach (F-rules are reduced to P-rules), includes nearest neighbor algorithms, MLPs, RBFs, separable function networks, SVMs, kernel methods and many others. Similarity-Based Methods (SBMs) are organized in a framework: p(Ci|X;M) posterior classification probability or y(X;M) approximators, models M are parameterized in increasingly sophisticated way. A systematic search (greedy, beam, evolutionary) in the space of all SBM models is used to select optimal combination of parameters and procedures, opening different types of optimization channels, trying to discover appropriate bias for a given problem. Results: several candidate models, very limited version gives best results in 7 out of 12 Stalog problems.

SBM framework Pre-processing: from objects (cases) O to features X or directly to (diss)similarities D(O,O’). Calculation of similarity between features d(xi,yi) and objects D(X,Y). Reference (or prototype) vector R selection/creation/optimization. Weighted influence of references vectors G(D(Ri,X)), i=1..k. Functions/procedures to estimate p(C|X;M) or y(X;M). Cost functions E[DT;M] and model selection/validation procedures. Optimization procedures for the whole model Ma. Search control procedures to create more complex models Ma+1. Creation of ensembles of (local, competent) models. M={X(O), d(.,.), D(.,.), k, G(D), {R}, {pi(R)}, E[.], K(.), S(.,.)}, where: S(Ci,Cj) is a matrix evaluating similarity of the classes; a vector of observed probabilities pi(X) instead of hard labels. The kNN model p(Ci|X;kNN) = p(Ci|X;k,D(.),{DT}); the RBF model: p(Ci|X;RBF) = p(Ci|X;D(.),G(D),{R}), etc.

Meta-learning in SBM scheme k-NN 67.5/76.6% +d(x,y); Canberra 89.9/90.7 % + si=(0,0,1,0,1,1); 71.6/64.4 % +selection, 67.5/76.6 % +k opt; 67.5/76.6 % +d(x,y) + si=(1,0,1,0.6,0.9,1); Canberra 74.6/72.9 % +d(x,y) + selection; Canberra 89.9/90.7 % k-NN 67.5/76.6% +d(x,y); Canberra 89.9/90.7 % + si=(0,0,1,0,1,1); 71.6/64.4 % +selection, 67.5/76.6 % +k opt; 67.5/76.6 % +d(x,y) + si=(1,0,1,0.6,0.9,1); Canberra 74.6/72.9 % +d(x,y) + sel. or opt k; Canberra 89.9/90.7 % Start from kNN, k=1, all data & features, Euclidean distance, end with a model that is a novel combination of procedures and parameterizations.

What NN components really do? Vector mappings from the input space to hidden space(s) and to the output space + adapt parameters to improve cost functions. Hidden-Output mapping done by MLPs: T = {Xi} training data, N-dimensional. H = {hj(T)} X image in the hidden space, j =1 .. NH-dim. ... more transformations in hidden layers Y = {yk(H )} X image in the output space, k =1 .. NC-dim. ANN goal: data image H in the last hidden space should be linearly separable; internal representations will determine network generalization. But we never look at them!

Transformation-based framework Extend SBM adding fine granulation of methods and relations between them to enable meta-learning by search in the model space. For example, first transformation (layer) after pre-processing: PCA networks, with each node computing principal component. LDA networks, each node computes LDA direction (including FDA). ICA networks, nodes computing independent components. KL, or Kullback-Leibler networks with orthogonal or non-orthogonal components; max. of mutual information is a special case c2 and other statistical tests for dependency to aggregate features. Factor analysis networks, computing common and unique factors. Matching pursuit networks for signal decomposition. Evolving Transformation Systems (Goldfarb 1990-2006), unified paradigm for inductive learning and structural representations.

Heterogeneous systems Problems requiring different scales (multiresolution). 2-class problems, two situations: C1 inside the sphere, C2 outside. MLP: at least N+1 hyperplanes, O(N2) parameters. RBF: 1 Gaussian, O(N) parameters. C1 in the corner defined by (1,1 ... 1) hyperplane, C2 outside. MLP: 1 hyperplane, O(N) parameters. RBF: many Gaussians, O(N2) parameters, poor approx. Combination: needs both hyperplane and hypersphere! Logical rule: IF x1>0 & x2>0 THEN C1 Else C2 is not represented properly neither by MLP nor RBF! Different types of functions in one model, first step beyond inspirations from single neurons => heterogeneous models.

Heterogeneous everything Homogenous systems: one type of “building blocks”, same type of decision borders, ex: neural networks, SVMs, decision trees, kNNs Committees combine many models together, but lead to complex models that are difficult to understand. Ockham razor: simpler systems are better. Discovering simplest class structures, inductive bias of the data, requires Heterogeneous Adaptive Systems (HAS). HAS examples: NN with different types of neuron transfer functions. k-NN with different distance functions for each prototype. Decision Trees with different types of test criteria. 1. Start from large networks, use regularization to prune. 2. Construct network adding nodes selected from a candidate pool. 3. Use very flexible functions, force them to specialize.

Taxonomy of NN activation functions

Taxonomy of NN output functions Perceptron: implements logical rule x>q for x with Gaussian uncertainty.

Taxonomy - TF

HAS decision trees Decision trees select the best feature/threshold value for univariate and multivariate trees: Decision borders: hyperplanes. Introducing tests based on La Minkovsky metric. For L2 spherical decision border are produced. For L∞ rectangular border are produced. Many choices, for example Fisher Linear Discrimination decision trees. For large databases first clusterize data to get candidate references R.

SSV HAS DT example SSV HAS tree in GhostMiner 3.0, Wisconsin breast cancer (UCI) 699 cases, 9 features (cell parameters, 1..10) Classes: benign 458 (65.5%) & malignant 241 (34.5%). Single rule gives simplest known description of this data: IF ||X-R303|| < 20.27 then malignant else benign coming most often in 10xCV 97.4% accuracy (18 errors); good prototype for malignant! Simple thresholds, that’s what MDs like the most! Best 10CV around 97.5±1.8% (Naïve Bayes + kernel, or SVM) SSV without distances: 96.4±2.1% C 4.5 gives 94.7±2.0% Several simple rules of similar accuracy but different specificity or sensitivity may be created using HAS DT. Need to select or weight features and select good prototypes.

More meta-learning Meta-learning: learning how to learn, replace experts who search for best models making a lot of experiments. Search space of models is too large to explore it exhaustively, design system architecture to support knowledge-based search. Abstract view, uniform I/O, uniform results management. Directed acyclic graphs (DAG) of boxes representing scheme placeholders and particular models, interconnected according to the input–output connections. Configuration level for meta-schemes, expanded at runtime level.

Intemi, Intelligent Miner Meta-schemes: templates with placeholders May be nested; the role decided by the input/output types. Machine learning generators based on meta-schemes. Granulation level allows to create novel methods. Complexity control: Length + log(time) A unified meta-parameters description, defining the range of sensible values and the type of the parameter changes.

Advanced meta-learning Finding the correlations of occurring different items in most accurate results, and different machine structures of similar behavior in an area of the model space. Finding new successful complex structures and converting them into meta-schemes (which we call meta abstraction) by replacing proper substructures by placeholders. Extracting meta-rules, describing interesting search directions. Depositing the knowledge they gain in a reusable meta-knowledge repository (for meta-learning experience exchange between different meta-learners). A uniform representation of the meta-knowledge should enable extending expert knowledge, adjusting the prior knowledge according to performed tests, etc. Beyond transformations & feature spaces: actively search for info. Intemi software incorporating these ideas coming “soon” ...

How much can we learn? Linearly separable or almost separable problems are relatively simple – deform or add dimensions to make data separable. How to define “slightly non-separable”? There is only separable and the vast realm of the rest.

Difficult case: complex logic For n bits there are 2n nodes; in extreme cases such as parity all neighbors are from the wrong class, so localized networks will fail. Achieving linear separability without special architecture may be impossible. Projection on 111 ... 111 gives clusters with 0, 1, 2 ... n bits.

Easy and difficult problems Linear separation: good goal if simple topological deformation of decision borders is sufficient. Linear separation of such data is possible in higher dimensional spaces; this is frequently the case in pattern recognition problems. RBF/MLP networks with one hidden layer solve such problems. Difficult problems: disjoint clusters, complex logic. Continuous deformation is not sufficient; networks with localized functions need exponentially large number of nodes. Boolean functions: for n bits there are K=2n binary vectors that can be represented as vertices of n-dimensional hypercube. Each Boolean function is identified by K bits. BoolF(Bi) = 0 or 1 for i=1..K, for 2K Boolean functions. Ex: n=2 functions, vectors {00,01,10,11}, Boolean functions {0000, 0001 ... 1111}, decimal numbers 0 to 15.

Boolean functions n=2, 16 functions, 12 separable, 4 not separable. n=3, 256 f, 104 separable (41%), 152 not separable. n=4, 64K=65536, only 1880 separable (3%) n=5, 4G, but << 1% separable ... bad news! Existing methods may learn some non-separable functions, but most functions cannot be learned ! Example: n-bit parity problem; many papers in top journals. No off-the-shelf systems are able to solve such problems. For all parity problems SVM is below base rate! Such problems are solved only by special neural architectures or special classifiers – if the type of function is known. But parity is still trivial ... solved by

Learning trajectories Take weights Wi from iterations i =1..K; PCA on Wi covariance matrix usually captures 95-98% variance, so error function in 2D shows realistic learning trajectories. M. Kordos & W. Duch Instead of local minima large flat valleys are seen – why? Data far from decision borders has almost no influence, the main reduction of MSE is achieved by increasing ||W||, sharpening sigmoidal functions.

RBF for XOR Is RBF solution with 2 hidden Gaussians nodes possible? Typical architecture: 2 input – 2 Gaussians – 1 linear output, EM training 50% errors, but there is perfect separation - not a linear separation! Network knows the answer, but cannot say it ... Single Gaussian output node may solve the problem. Output weights provide reference hyperplanes (red and green lines), not the separating hyperplanes like in case of MLP.

3-bit parity For RBF parity problems are difficult; 8 nodes solution: 1) Output activity; 2) reduced output, summing activity of 4 nodes. 3) Hidden 8D space activity, near ends of coordinate versors. 4) Parallel coordinate representation. 8 nodes solution has zero generalization, 50% errors in L1O.

3-bit parity in 2D and 3D Output is mixed, errors are at base level (50%), but in the hidden space ... Conclusion: separability in the hidden space is perhaps too much to desire ... inspection of clusters is sufficient for perfect classification; add second Gaussian layer to capture this activity; train second RBF on the data (stacking), reducing number of clusters.

Goal of learning If simple topological deformation of decision borders is sufficient linear separation is possible in higher dimensional spaces, “flattening” non-linear decision borders; this is frequently the case in pattern recognition problems. RBF/MLP networks with one hidden layer solve the problem. For complex logic this is not sufficient; networks with localized functions need exponentially large number of nodes. Such situations arise in AI problems, real perception, object recognition, text analysis, bioinformatics ... Linear separation is too difficult, set an easier goal. Linear separation: projection on 2 half-lines in the kernel space: line y=WX, with y<0 for class – and y>0 for class +. Simplest extension: separation into k-intervals. For parity: find direction W with minimum # of intervals, y=W.X

k-separability Can one learn all Boolean functions? Problems may be classified as 2-separable (linear separability); non separable problems may be broken into k-separable, k>2. X1 X2 X3 X4 y=W.X +1 -1 s(by+q1) s(by+q2) s(by+q4) Neural architecture for k=4 intervals, or 4-separable problems. Blue: sigmoidal neurons with threshold, brown – linear neurons.

k-sep learning Try to find lowest k with good solution, start from k=2. Assume k=2 (linear separability), try to find good solution; if k=2 is not sufficient, try k=3; two possibilities are C+,C-,C+ and C-, C+, C- this requires only one interval for the middle class; if k<4 is not sufficient, try k=4; two possibilities are C+, C-, C+, C- and C-, C+, C-, C+ this requires one closed and one open interval. Network solution is equivalent to optimization of specific cost function. Simple backpropagation solved almost all n=4 problems for k=2-5 finding lowest k with such architecture! Parity-like problems up to n=8 work fine.

A better solution? What is needed to learn Boolean functions? cluster non-local areas in the X space, use W.X capture local clusters after transformation, use G(W.X-q) SVMs fail because the number of directions W that should be considered grows exponentially with the size of the problem n. A class of constructive neural network solution: Train the first neuron using G(W.X-q) transfer function on whole data T, capture the largest pure cluster TC . Train next neuron on reduced data T 1=T-TC Repeat until all data is handled; they creates transform. X=>H Use linear transformation H => Y for classification.

Strong reduction?

Learning hard functions Training almost perfect for parity, with linear growth in the number of vectors for k-sep. solution created by the constructive neural algorithm.

Real data Simple data – similar results, but much simpler models.

Its good to have wide interests ... You know, Christopher, I am fed up with honey

Beyond pattern recognition A step towards problems requiring combinatorial reasoning: learning from partial observations in robotics or reasoning. We have observed unicorns, and know/guess/infer that: If the unicorn is mythical, then it is immortal. But if it is not mythical, then it is a mortal mammal. If the unicorn is either immortal or a mammal, then it is horned. The unicorn is magical if it is horned. Can you draw any firm conclusions about unicorns? Variables: mythical, mortal, mammal, horned, magical. Using intuitive computing – search based on neural heuristics – the answer may be found quickly ...

Mental models Easy reasoning A=>B, B=>C, so A=>C All mammals suck milk. Humans are mammals. => Humans suck milk. ... but almost no-one can draw conclusion from: All academics are scientist. No wise men is an academic. What can we say about wise men and scientists? Surprisingly only ~10% of students get it right. No simulations explaining why some mental models are difficult?

Scaling-up to human level ... Recent discussions: Roadmap to human level intelligence + session – WCCI 2006/08 Cognitive Systems, IJCNN panel 2007; AGI conference 2008 ... Books: Challenges to CI (with J Mandziuk); Roadmap (with J Taylor); Meta-learning (with Jankowski/Grabczewski); neurocognitive informatics. Neuromorphic, mesoscopic, hybrid neuro-symbolic architectures? Designs for artificial brains, blueprints for billion neuron cortex models, scalable neuromorphic approaches. What is the role of CI in brain-like computing systems? Is human-style creativity using CI possible? We are not far from cognitive/affective architectures, artificial minds with human characteristics integrating perception, affect and cognition, large-scale semantic memories, implementing control/attention.

Work like a horse and never loose your enthusiasm!

Summary CI is a branch of science dealing with problems for which effective algorithms do not exist; it includes AI, machine learning and all the rest. Similarity-based framework enables meta-learning as search in the model space, heterogeneous systems add fine granularity. Transformation-based learning extends that, formalizing component-based approach to DM, automating discovery of interesting models. Many known and new learning methods result from such framework, but neurocognitive inspirations extend it much further. No off-shelf classifiers are able to learn difficult Boolean functions. Visualization of activity of the hidden neurons shows that frequently perfect but non-separable solutions are found despite base-rate outputs. Linear separability is not the best goal of learning, other targets that allow for easy handling of final non-linearities should be defined. Simplest extension is to isolate non-linearity in form of k intervals, breaking the non-separable class of problems into k-separable classes. Many interesting new ideas arise from this line of thinking.

Thank you for lending your ears ... Google: W. Duch => Papers & presentations