Learning from just one or a few examples, and mostly unlabeled examples ( semi-supervised learning ).

Transcription

1 Outline for lectures Introduction Cognition as probabilistic inference Learning concepts from examples (continued) Learning and using intuitive theories (more structured systems of knowledge)

2 tufa tufa tufa Learning from just one or a few examples, and mostly unlabeled examples ( semi-supervised learning ).

3 Simple model of concept learning This is a blicket. Can you show me the other blickets?

4 Learning to learn: what object features count for word learning? 24-month-olds show the shape bias with simple novel objects. 20-month-olds do not. (Landau, Smith, Jones 1988) This is a dax. Show me the dax Smith et al (2002) trained 17- month-olds on labels for 4 artificial categories: After 8 weeks of training (20 min/week), 19-month-olds show the shape bias. wib zup lug div

5 Transfer to real-world vocabulary The intuition: Learn that shape varies across categories but is relatively constant within nameable categories. The puzzle: The shape bias is a powerful inductive constraint, yet can be learned from very little data.

6 Learning about feature variability (Kemp, Perfors & Tenenbaum, Dev. Science 2007)?

7 Learning about feature variability (Kemp, Perfors & Tenenbaum, Dev. Science 2007)?

8 A hierarchical Bayesian model Level 3: Prior expectations on bags in general Level 2: Bags in general Level 1: Bag proportions Simultaneously infer Data

9 A hierarchical Bayesian model Level 3: Prior expectations on bags in general Level 2: Bags in general α = 0.1 β (within-bag variability) (overall population distribution) Level 1: Bag proportions Data

10 A hierarchical Bayesian model Level 3: Prior expectations on bags in general Level 2: Bags in general α = 5 β (within-bag variability) (overall population distribution) Level 1: Bag proportions Data

11 Learning the shape bias (Kemp, Perfors & Tenenbaum, Dev. Science 2007) wib lug Assuming independent Dirichletmultinomial models for each dimension zup div Training we learn that: Shape varies across categories but not within categories. Texture, color, size vary across and within categories.

12 Second-order generalization test (Kemp, Perfors & Tenenbaum, Dev. Science 2007) 1 st order gen 2 nd order gen This is a dax. Show me the dax Training Test blessing of abstraction

13 A more realistic model Prior expectations on categories in general Categories in general Individual categories Data????? ? ? ? ? ? ? ? ? ? ? ? ? ? (Perfors & Tenenbaum, Proc Cog Sci 2009)

14 A more realistic model Prior expectations on categories in general Categories in general Individual categories Data y z y z y z y z (Perfors & Tenenbaum, Proc Cog Sci 2009)

15 A more realistic model Prior expectations on categories in general Categories in general Individual categories Data y z y z y z y z y z (Perfors & Tenenbaum, Proc Cog Sci 2009) 2 nd order generalization: ? or ?

16 A more realistic model Prior expectations on categories in general Learning the base distribution of a DP mixture Categories in general Individual categories Data y z y z y z y z y z (Perfors & Tenenbaum, Proc Cog Sci 2009) 2 nd order generalization: ? or ?

17 Model vs. People % correct generalization 1 st order generalization 2 nd order generalization % correct generalization 1 st order generalization 2 nd order generalization Category labels absent Category labels present Category labels absent Category labels present % coherence: % coherence: % correct generalization % correct generalization (Perfors & Tenenbaum, Proc Cog Sci 2009)

18 Towards more natural concepts

19 Towards more natural concepts CRP mixture:

20 How many different ways to structure a domain? (Shafto, Kemp, Mansingka, Tenenbaum, 2006; submitted) CrossCat : nonparametric clustering over features, with a different clustering of objects for each feature-cluster.

21 How many different ways to structure a domain? (Shafto, Kemp, Mansingka, Tenenbaum, 2006; submitted) CrossCat : nonparametric clustering over features, with a different clustering of objects for each feature-cluster. Evidence for CrossCat-like learning in humans: - Sorting natural categories - Sorting artificial categories - Predicting values for novel features and novel objects

22 Learning relational concepts CONCEPTS Professors Graduate students Undergraduates RELATIONSHIPS Professors give advice to Grad students and Undergrads. Grad students give advice to Undergrads. Undergrads give advice to no one.

23 Learning relational concepts CONCEPTS Magnets Magnetic objects Non-magnetic objects RELATIONSHIPS Magnets interact with each other. Magnets and Magnetic objects interact. Magnetic objects do not interact with each other. Non-magnetic objects do not interact with anything.

24 people Learning relational concepts people person 1 gives advice to person 9

25 Learning relational concepts gives advice to people gives advice to Prof Grads Ug people Profs Grads Ugrads

26 Infinite Relational Model (IRM) (Kemp, Griffiths, Tenenbaum, Yamada, & Ueda, 2006)

27 Learning algorithm Continuous parameters (weights/probabilities) integrated out analytically. Gibbs sampling + split-merge moves: Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5 Iteration 6

28 The causal blocks world (Tenenbaum and Niyogi, 2003) G F O W C L C A

29 The causal blocks world (Tenenbaum and Niyogi, 2003) G F O W C L A C

30 Training Test A O G F W? X B C A C L? Learning curves 9 Probability of lighting up Likelihood of Lighting Up Stage (# of objects / 3) # of objects observed Pre Post Same Group Post Different Group Model predictions

31 Constructing semantic networks (Collins & Quillian, 1969)

32 Upper level medical ontology

33 Upper level medical ontology Biomedical predicate data used to construct ontology in UMLS (McCrae et al.): 134 concepts: enzyme, hormone, organ, disease, cell function predicates: affects(hormone, organ), complicates(enzyme, cell function), treats(drug, disease), diagnoses(procedure, disease)

34 Learning semantic networks with IRM concept concept predicate Biomedical predicate data from UMLS (McCrae et al.): 134 concepts: enzyme, hormone, organ, disease, cell function predicates: affects(hormone, organ), complicates(enzyme, cell function), treats(drug, disease), diagnoses(procedure, disease)

35 Learning semantic networks with IRM e.g., Diseases affect Organisms Chemicals interact with Chemicals Chemicals cause Diseases

36 Learning semantic networks with IRM Relations between clusters: e.g., Diseases affect Organisms Chemicals interact with Chemicals Chemicals cause Diseases

37 Extracting semantic networks from text via relational clustering (Kok & Domingos 2008) Tested several algorithms for relational clustering on TextRunner data: ~ 2 million triples of the form R(x, y): e.g., upheld(court, ruling), named_after(jupiter, Roman_god). ~ 10,214 R symbols, 8942 x symbols, 7995 y symbols (each appears >25 times).

38 Annotated hierarchies model (Roy, Kemp, Mansinghka & Tenenbaum, 2007)

39 Annotated hierarchies model (Roy, Kemp, Mansinghka & Tenenbaum, 2007)

40 The Mondrian Process (Roy & Teh, 2008; in prep)

41 The Mondrian Process (Roy & Teh, 2008; in prep)

42 tufa tufa tufa Learning from just one or a few examples, and mostly unlabeled examples ( semi-supervised learning ).

43 Learning words for objects tufa tufa tufa (Collins & Quillian, 1969) (IT population responses Hung et al., 2005; c.f. Kiani et al. 2007)

44 Learning words for objects Bayesian inference over treestructured hypothesis space: (Xu & Tenenbaum, Psych. Review 2007; Schmidt & Tenenbaum, in prep) tufa tufa tufa

45 Learning words for objects Bayesian inference over treestructured hypothesis space: (Xu & Tenenbaum, Psych. Review 2007; Schmidt & Tenenbaum, in prep) People Model tufa tufa tufa

46 Hierarchical Bayesian framework F: form Tree S: structure tufa tufa tufa F1 F2 F3 F4 D: data

47 Learning to learn: what is the right form of structure for the domain? F: form Tree Space Order S: structure X1 X2 X3 X4 X5 X6 X2 X5 X4 X1 X3 X6 X1 X2 X3 X4 X5 X6 Features D: data X1 X2 X3 X4 X5 X6

48 The value of structural form knowledge: inductive constraints (bias) Mystery city - average annual temperature: 66 F / 19 C - voted 60% for George W Bush in popular foods are fried and BBQ

49 Property induction (Kemp & Tenenbaum, Psych. Review 2009; Shafto et al., Cognition 2008) Given that {X 1,, X n } have property P, how likely is it that Y does? 2D Tree 2D Tree Horses All mammals Minneapolis Houston

50 Property induction (Kemp & Tenenbaum, Psych. Review 2009) Given that {X 1,, X n } have property P, how likely is it that Y does? 2D Tree 2D Tree ( ) 1 1 Property ~ N 0,( Δ + 2 I) σ (Zhu, Lafferty & Ghahramani, 2003) Horses All mammals N ( ) ) 1 0,exp( σ (c.f. Lawrence, 2004) x i x j Minneapolis Houston

51 Property induction (Kemp & Tenenbaum, Psych. Review 2009; Shafto et al., Cognition 2008) Given that {X 1,, X n } have property P, how likely is it that Y does? Tree Smooth: P(D S) high Not smooth: P(D S) low 2D Tree 2D ( ) 1 1 Property ~ N 0,( Δ + 2 I) σ (Zhu, Lafferty & Ghahramani, 2003) Horses All mammals Minneapolis Houston

52 People can discover structural forms Scientists Learning structural forms Linnaeus Kingdom Animalia Phylum Chordata Class Mammalia Order Primates Family Hominidae Genus Homo Species Homo sapiens Darwin Mendeleev Children e.g., hierarchical structure of category labels, cyclical structure of seasons or days of the week, clique structure of social networks. but standard learning algorithms assume fixed forms. Principal components analysis: low-dimensional spatial structure Hierarchical clustering: tree structure k-means clustering, mixture models: flat partition.

53 Hypothesis space of structural forms (Kemp & Tenenbaum, PNAS 2008) Form Process Form Process

54 A hierarchical Bayesian approach P(F) F: form S: structure D: data (Kemp & Tenenbaum, PNAS 2008) P(S F) Simplicity P(D S) Smoothness (Fit to data) X1 X2 X3 X4 X5 X6 x X1 X2 X1 X2 X6 X3 X4 X5 X6 X3 X5 Features X4 X1 X2 X3 X4 X5 X6

55 judges animals features cases

56 objects objects similarities

57 Development of structural forms as more data are observed blessing of abstraction

58 Causal learning and reasoning Causal model Event data (Griffiths & Tenenbaum; Kemp, Goodman, Tenenbaum)

59 Causal learning and reasoning Causal schema Behaviors Diseases Symptoms high-fat diet working in factory heart disease lung cancer coughing chest pain Causal model Event data Cut down hypothesis space from size 521,939,651,343,829, 405,020,504,063 to 131,072 (Griffiths & Tenenbaum; Kemp, Goodman, Tenenbaum)

60 Causal schema Causal model Event data Patient 1 Patient 2 Patient 3 Patient 4 Patient 5 Laws Classes B D S B η z 1 2 B D S D S Infinite relational model Belief net with Dirichletmultinomial parameterization (Mansinghka, Kemp, Tenenbaum, Griffiths, UAI 2006)

61 Ground-truth causal network Causal model Event data Causal schema c c 2 # samples recovered model blessing of abstraction recovered schema Causal model Event data recovered model (Mansinghka, Kemp, Tenenbaum, Griffiths, UAI 2006)

62 Conclusions How does the mind get so much from so little, in learning about objects, classes, causes, scenes, sentences, thoughts, social systems? A toolkit for studying the nature, use and acquisition of abstract knowledge: Bayesian inference in probabilistic generative models. Probabilistic models defined over a range of structured representations: spaces, graphs, grammars, predicate logic, schemas, programs. Hierarchical models, with inference at multiple levels of abstraction. Nonparametric models, adapting their complexity to the data and balancing constraint with flexibility. An alternative to classic either-or dichotomies: Nature versus Nurture, Logic (Structure, Rules, Symbols) versus Probability (Statistics). How can domain-general mechanisms of learning and representation build domain-specific abstract knowledge? How can structured symbolic knowledge be acquired by statistical learning? A different way to think about the development of a cognitive system. Powerful abstractions can be learned surprisingly quickly, together with or prior to learning the more concrete knowledge they constrain. Structured representations need not be rigid, static, hand-wired, brittle. Embedded in a probabilistic framework, they can grow dynamically and robustly in response to the sparse, noisy data of experience.

63 Open directions and challenges More precise relation to psychology How does human cognitive processing perform approximate probabilistic inference (i.e., approximately implement rational methods of approximate inference)? Relation to the brain How to implement structured probabilistic models in neural architectures? Probabilistic models for richly structured knowledge How to formalize an intuitive theory of physics or psychology? Effective learning of structured probabilistic models How to balance expressiveness/learnability tradeoff?

64 Goal-directed action (production and comprehension) (Wolpert et al., 2003)

65 Goal inference as inverse probabilistic planning (Baker, Tenenbaum & Saxe, Cognition, in press) constraints goals rational planning (MDP) Gergely, Csibra et al.: Agent actions

66 Inferring social goals constraints goals (Baker, Goodman & Tenenbaum, Cog Sci 2008; Ullman, Baker, Evans, Macindoe & Tenenbaum, submitted) Hamlin, Kuhlmeier, Wynn & Bloom: constraints goals rational planning (MDP) Agent rational planning (MDP) actions Agent actions Model prediction Subject ratings Model prediction Subject ratings

67 The really big questions Where does it all end? Across different domains and tasks, and different levels of abstraction, our probabilistic models are starting to look increasingly complex and to differ in almost arbitrarily many ways. This seems unsatisfying The brain appears to have a uniform circuitry. Other cognitive modeling paradigms adopt a single unifying representational primitive (production rules, predicate logic, synaptic strengths, tensors). Is there a single universal Bayesian primitive? How does it all begin? Can all these different kinds of representations be learned? What is the ultimate hypothesis space of innate primitives or is it simply turtles all the way up? Could a universal hypothesis space for all probabilistic models possibly be searched effectively? (C.f. Kolmogorov complexity theory, Chater & Vitanyi)

68 Church: a universal probabilistic language (Goodman et al., UAI 2008)

69 Church: a universal probabilistic language (Goodman et al., UAI 2008) Causal networks

70 Church: a universal probabilistic language (Goodman et al., UAI 2008) Causal networks Relational schema

71 Church: a universal probabilistic language (Goodman et al., UAI 2008) Causal networks Relational schema Physical objects

72 Church: a universal probabilistic language (Goodman et al., UAI 2008) Causal networks Relational schema Physical objects Rational agents

73 Open directions and challenges More precise relation to psychology How does human cognitive processing perform approximate probabilistic inference (i.e., approximately implement rational methods of approximate inference)? Relation to the brain How to implement structured probabilistic models in neural architectures? Probabilistic models for richly structured knowledge How to formalize an intuitive theory of physics or psychology? Effective learning of structured probabilistic models How to balance expressiveness/learnability tradeoff?