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Data Modeling using Symbolic Regression

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Patrick Nicolas
Patrick Nicolas

This is an introduction to the concept of symbolic regression for managing effectively data stream. Symbolic regression combines genetic algorithm, reinforcement learning and flexible policies to extract meaning or knowledge from data, in an ever changing environment. As the knowledge extracted from real-time data is human readable and consumable, decision makers can validate the findings of the algorithm and act appropriately. Symbolic regression is used in signal processing, process monitoring and adaptive caching in data centers.

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Patrick Nicolas http://patricknicolas.blogspot.com 07/13/2013
Need for reliability Copyright 2013 Patrick Nicolas 2 Existing algorithms used in recommendation, predictive behavior of consumers or target advertising do not have to be very accurate: the negative impact of recommending a book, movie incorrectly or failing to detect the interest of a consumer is very limited. However, some problems requires a far more reliable solution: failure to preserve large amount of data, detect security intrusion or predict the progress of a disease have grave consequence.
Options Copyright 2013 Patrick Nicolas 3 Traditional data mining approaches such as clustering (Unsupervised learning), generative or discriminative supervised learning algorithm failed to capture the evolutionary nature of a system with its states and underlying data.
Supervised learning Copyright 2013 Patrick Nicolas 4 Supervised learning is effective for problems with a large training compared to the dimension of the model. However it suffers from the following limitations: • Over-fitting: A supervised learning algorithm needs a large training to account for bias in the training set • No descriptive (human) knowledge representation • Role of domain expert is limited to providing labeled data and validate the results. • The model has to be retrained in case of false positive or false negative
Unsupervised learning Copyright 2013 Patrick Nicolas 5 Unsupervised learning methods such as Spectral Clustering, Kernel-based K-Means are used for anomaly detections or dimension reduction but have drawbacks: • Poor classification, in case of mix discrete & continuous variables • No descriptive knowledge representation • Limited leverage of domain expertise: Role of the domain expert is limited to validating the cluster • Clusters have to be rebuilt if number of outliers increases
Symbolic Regression Copyright 2013 Patrick Nicolas 6 Symbolic Regression addresses the key limitations of unsupervised and supervised learning methods. It combines evolutionary computation with reinforcement learning to provide domain experts a tool to create, evaluate and modify rules, policies or models. The most commonly used algorithms in Symbolic Regression •Genetic programming •Learning Classifiers System
Symbolic Regression Copyright 2013 Patrick Nicolas 7 • Optimization of data archiving • Intelligent data and instrumentation streaming • Predicting behavior of ecommerce site during “flash” or holiday sales • Monitoring and predicting security vulnerabilities in data centers • Distribution of network traffic and flow in public cloud Symbolic Regression is used in very different applications such as
Symbolic representation Copyright 2013 Patrick Nicolas 8 The goal is to extract knowledge from data (numerical, textual, events…) as symbolic or human readable representation using primitives or operators • Boolean operators OR, AND, XOR,.. • Numerical functions Sin, Exp, Sigmoid,…. • Numerical operators +, *, o, … • Differentiable operators derivative, integral,. • Logical operators: Predicate, rules,.. Domain ExpertDomain Expert Data MiningData Mining DataData sinIf _ then _ _ has a _ If _ then _ exp _ * _
Knowledge Extraction Copyright 2013 Patrick Nicolas 9 Knowledge extraction is the process of selecting, combining the appropriate symbolic primitives or operators to describe and predict states of a system. Expertise Model Expertise Model sinIf _ then _ _ has a _ If _ then _ exp _ * _ f” SystemSystem State/DataState/Data PredictionPrediction
Knowledge Primitives Copyright 2013 Patrick Nicolas 10 The generation of knowledge from a set of symbolic primitives to represent underlying state of a system is a NP problem (combinatorial explosion). Moreover computers process data in binary format (theory of information). Value Binary Encoding The solution is to represent knowledge as symbolic primitives in binary format.
Knowledge Encoding Copyright 2013 Patrick Nicolas 11 The most common representation is to encode symbolic primitives as sequences 0 & 1’s f(x) = 2.sin(x) – exp(x*x) - ( * (sin,2), o (exp, sqr)) - * o sin 2 exp sqr long long long Binary data 0101001001110111011101110111011101111111000111111011101101000001001000101010
Data Modeling using Genetic Algorithm Copyright 2013 Patrick Nicolas 12 For a given state of a system we need to find the optimal model (combination of primitives) to describe the current state using a Genetic Algorithm. The (0,1) encoding is associated to a chromosome with selection, cross-over, transposition and mutation operators 100100111011101110111011101110oo 10000010111100001010010011011 1001010111011101110100100111011 100000101111000010011011101110 Cross-over Parents Off-springs 10010011101110111000111011101110 100100111010111101110111111100110 Mutation 10010011101110111000111011101110 Transposition 101110100100111011011101110111011 s e se
Computation Flow of Genetic Algorithm Copyright 2013 Patrick Nicolas 13 Initial Pool of Models Initial Pool of Models EncodingEncoding Initial Chromosomes Initial Chromosomes New population New population SelectionSelectionFitnessFitness Cross-overCross-over MutationMutation Fittest Chromosome Fittest ChromosomeDecodingDecoding Best ModelBest Model Once the initial set of chromosomes is randomly generated the algorithm iterates until fittest chromosome emerges TranspositionTransposition
Limitation of Genetic Algorithm Copyright 2013 Patrick Nicolas 14 The selection of the best chromosome representing the best classifier (or model) relies on the computation of a fitness value under the assumption that the objective does not change over time. As most system evolves over-time, so does the objective. Reinforcement learning is used to adjust the objective using a reward/credit assignment mechanism.
EncodingEncoding Concept of Reinforcement Learning Copyright 2013 Patrick Nicolas 15 As the state of the system evolves over-time, it rewards or punishes the fittest classifier which action has been executed. The rewards or punishment is used to adjust the objective and fitness function. System State/DataState/Data ProbesProbes EffectorsEffectors RewardReward Best Action Best Action Reward AssignmentReward Assignment DecodingDecoding Genetic Algorithm Genetic Algorithm PrimitivesPrimitives Best classifier Best classifier
Elements of Reinforcement Learning Copyright 2013 Patrick Nicolas 16 The main challenge of reinforcement learning is to predict the impact of each action An on the global state. We need … •Actions (or classifiers) that support logic, IF/THEN, numerical, y=f(x1, … xn) and discrete {ai} classifiers to predict the impact of a remedial action on the security of the system 1.A metric to measure the security of the overall system (distance between the current state and the baseline) 1.An actions discovery & adaptation mechanism 1.An efficient optimizer to select the best action at any state: Stochastic Descent Gradient for continuous variables {xi} only or Genetic Algorithm for mix of Boolean, Integer and Double
Putting All Together Copyright 2013 Patrick Nicolas 17 EnvironmentInitial Knowledge Initial Knowledge EncodingEncoding Expert Supervised Learning Classifiers Population Classifiers Population State/DataState/Data SelectSelect Cross- over Cross- over MutateMutate ProbesProbes EffectorsEffectors RewardReward Best Classifiers Best Classifiers Actions Predictor Actions Predictor ActionAction Q-LearningQ-LearningReward AssignmentReward Assignment Genetic AlgorithmReinforcement Learning MatchMatch TransposeTranspose
References Copyright 2013 Patrick Nicolas 18 • Genetic Programming: On the Programming of Computers by Means of Natural Selection - J. Koza • Reinforcement Learning: An Introduction (Adaptive Computation and Machine Learning) – R. Sutton, A. Barto • http://www.mendeley.com/catalog/symbolic-regression-via-genetic- programming/

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Data Modeling using Symbolic Regression

  • 2. Need for reliability Copyright 2013 Patrick Nicolas 2 Existing algorithms used in recommendation, predictive behavior of consumers or target advertising do not have to be very accurate: the negative impact of recommending a book, movie incorrectly or failing to detect the interest of a consumer is very limited. However, some problems requires a far more reliable solution: failure to preserve large amount of data, detect security intrusion or predict the progress of a disease have grave consequence.
  • 3. Options Copyright 2013 Patrick Nicolas 3 Traditional data mining approaches such as clustering (Unsupervised learning), generative or discriminative supervised learning algorithm failed to capture the evolutionary nature of a system with its states and underlying data.
  • 4. Supervised learning Copyright 2013 Patrick Nicolas 4 Supervised learning is effective for problems with a large training compared to the dimension of the model. However it suffers from the following limitations: • Over-fitting: A supervised learning algorithm needs a large training to account for bias in the training set • No descriptive (human) knowledge representation • Role of domain expert is limited to providing labeled data and validate the results. • The model has to be retrained in case of false positive or false negative
  • 5. Unsupervised learning Copyright 2013 Patrick Nicolas 5 Unsupervised learning methods such as Spectral Clustering, Kernel-based K-Means are used for anomaly detections or dimension reduction but have drawbacks: • Poor classification, in case of mix discrete & continuous variables • No descriptive knowledge representation • Limited leverage of domain expertise: Role of the domain expert is limited to validating the cluster • Clusters have to be rebuilt if number of outliers increases
  • 6. Symbolic Regression Copyright 2013 Patrick Nicolas 6 Symbolic Regression addresses the key limitations of unsupervised and supervised learning methods. It combines evolutionary computation with reinforcement learning to provide domain experts a tool to create, evaluate and modify rules, policies or models. The most commonly used algorithms in Symbolic Regression •Genetic programming •Learning Classifiers System
  • 7. Symbolic Regression Copyright 2013 Patrick Nicolas 7 • Optimization of data archiving • Intelligent data and instrumentation streaming • Predicting behavior of ecommerce site during “flash” or holiday sales • Monitoring and predicting security vulnerabilities in data centers • Distribution of network traffic and flow in public cloud Symbolic Regression is used in very different applications such as
  • 8. Symbolic representation Copyright 2013 Patrick Nicolas 8 The goal is to extract knowledge from data (numerical, textual, events…) as symbolic or human readable representation using primitives or operators • Boolean operators OR, AND, XOR,.. • Numerical functions Sin, Exp, Sigmoid,…. • Numerical operators +, *, o, … • Differentiable operators derivative, integral,. • Logical operators: Predicate, rules,.. Domain ExpertDomain Expert Data MiningData Mining DataData sinIf _ then _ _ has a _ If _ then _ exp _ * _
  • 9. Knowledge Extraction Copyright 2013 Patrick Nicolas 9 Knowledge extraction is the process of selecting, combining the appropriate symbolic primitives or operators to describe and predict states of a system. Expertise Model Expertise Model sinIf _ then _ _ has a _ If _ then _ exp _ * _ f” SystemSystem State/DataState/Data PredictionPrediction
  • 10. Knowledge Primitives Copyright 2013 Patrick Nicolas 10 The generation of knowledge from a set of symbolic primitives to represent underlying state of a system is a NP problem (combinatorial explosion). Moreover computers process data in binary format (theory of information). Value Binary Encoding The solution is to represent knowledge as symbolic primitives in binary format.
  • 11. Knowledge Encoding Copyright 2013 Patrick Nicolas 11 The most common representation is to encode symbolic primitives as sequences 0 & 1’s f(x) = 2.sin(x) – exp(x*x) - ( * (sin,2), o (exp, sqr)) - * o sin 2 exp sqr long long long Binary data 0101001001110111011101110111011101111111000111111011101101000001001000101010
  • 12. Data Modeling using Genetic Algorithm Copyright 2013 Patrick Nicolas 12 For a given state of a system we need to find the optimal model (combination of primitives) to describe the current state using a Genetic Algorithm. The (0,1) encoding is associated to a chromosome with selection, cross-over, transposition and mutation operators 100100111011101110111011101110oo 10000010111100001010010011011 1001010111011101110100100111011 100000101111000010011011101110 Cross-over Parents Off-springs 10010011101110111000111011101110 100100111010111101110111111100110 Mutation 10010011101110111000111011101110 Transposition 101110100100111011011101110111011 s e se
  • 13. Computation Flow of Genetic Algorithm Copyright 2013 Patrick Nicolas 13 Initial Pool of Models Initial Pool of Models EncodingEncoding Initial Chromosomes Initial Chromosomes New population New population SelectionSelectionFitnessFitness Cross-overCross-over MutationMutation Fittest Chromosome Fittest ChromosomeDecodingDecoding Best ModelBest Model Once the initial set of chromosomes is randomly generated the algorithm iterates until fittest chromosome emerges TranspositionTransposition
  • 14. Limitation of Genetic Algorithm Copyright 2013 Patrick Nicolas 14 The selection of the best chromosome representing the best classifier (or model) relies on the computation of a fitness value under the assumption that the objective does not change over time. As most system evolves over-time, so does the objective. Reinforcement learning is used to adjust the objective using a reward/credit assignment mechanism.
  • 15. EncodingEncoding Concept of Reinforcement Learning Copyright 2013 Patrick Nicolas 15 As the state of the system evolves over-time, it rewards or punishes the fittest classifier which action has been executed. The rewards or punishment is used to adjust the objective and fitness function. System State/DataState/Data ProbesProbes EffectorsEffectors RewardReward Best Action Best Action Reward AssignmentReward Assignment DecodingDecoding Genetic Algorithm Genetic Algorithm PrimitivesPrimitives Best classifier Best classifier
  • 16. Elements of Reinforcement Learning Copyright 2013 Patrick Nicolas 16 The main challenge of reinforcement learning is to predict the impact of each action An on the global state. We need … •Actions (or classifiers) that support logic, IF/THEN, numerical, y=f(x1, … xn) and discrete {ai} classifiers to predict the impact of a remedial action on the security of the system 1.A metric to measure the security of the overall system (distance between the current state and the baseline) 1.An actions discovery & adaptation mechanism 1.An efficient optimizer to select the best action at any state: Stochastic Descent Gradient for continuous variables {xi} only or Genetic Algorithm for mix of Boolean, Integer and Double
  • 17. Putting All Together Copyright 2013 Patrick Nicolas 17 EnvironmentInitial Knowledge Initial Knowledge EncodingEncoding Expert Supervised Learning Classifiers Population Classifiers Population State/DataState/Data SelectSelect Cross- over Cross- over MutateMutate ProbesProbes EffectorsEffectors RewardReward Best Classifiers Best Classifiers Actions Predictor Actions Predictor ActionAction Q-LearningQ-LearningReward AssignmentReward Assignment Genetic AlgorithmReinforcement Learning MatchMatch TransposeTranspose
  • 18. References Copyright 2013 Patrick Nicolas 18 • Genetic Programming: On the Programming of Computers by Means of Natural Selection - J. Koza • Reinforcement Learning: An Introduction (Adaptive Computation and Machine Learning) – R. Sutton, A. Barto • http://www.mendeley.com/catalog/symbolic-regression-via-genetic- programming/