1. Learning, testing, and approximating halfspaces Rocco Servedio Columbia University DIMACS-RUTCOR Jan 2009

2. Overview + + + + + + + + + + + + - - -- - - - - - - - Halfspaces over testing learning approximation

3. Joint work with: Ronitt Rubinfeld Kevin Matulef Ryan O’Donnell Ilias Diakonikolas

4. Approximation Given a function goal is to obtain a “simpler” function such that Measure distance between functions under uniform distribution.

5. Approximating classes of functions Interested in statements of the form: “Every function in class has a simple approximator.” Every -size decision tree can be -approximated by a decision tree of depth Example statement: 01010 1 0 1 1 01 1 1

6. Testing Goal: infer “global” property of function via few “local” inspections Tester makes black-box queries to arbitrary Tester must output “yes” whp if “no” whp if is -far from every Any answer OK if is -close to some distance Usual focus: information-theoretic # queries required oracle for

7. Some known property testing results parity functions [BLR93] deg- polynomials [AKK+03] literals [PRS02] conjunctions [PRS02] -juntas [FKRSS04] -term monotone DNF [PRS02] -term DNF [DLM+07] size- decision trees [DLM+07] -sparse polynomials [DLM+07] Class of functions over # of queries

8. We’ll get to learning later

9. Halfspaces A function is a halfspace if such that for all Also called linear threshold functions (LTFs), threshold gates, etc. Fundamental to learning theory –Halfspaces are at the heart of many learning algorithms: Perceptron, Winnow, boosting, Support Vector Machines,… Well studied in complexity theory

10. Some examples of halfspaces Weights can be all the same… (decision list) …but don’t have to be…

11. What’s a “simple” halfspace? Every halfspace has a representation with integer weights: –finite domain, so can “nudge” weights to rational #s, scale to integers Some halfspaces over require integer weights [MTT61, H94] Low-weight halfspaces are nice for complexity, learning. is equivalent to

12. Approximating halfspaces using small weights? Let be an arbitrary halfspace. If is a halfspace which -approximates how large do the weights of need to be? Consider (view as n-bit binary numbers) This is a halfspace: but it’s easy to -approximate with weight Any halfspace for requires weight … Let’s warm up with a concrete example.

13. Approximating all halfspaces using small weights? So there are halfspaces that require weight but can be -approximated with weight Let be an arbitrary halfspace. If is a halfspace which -approximates how large do the weights of need to be? Can every halfspace be approximated by a small-weight halfspace? Yes

14. Every halfspace has a low-weight approximator Can’t do better in terms of ; may need some Dependence on must be [H94] Theorem: [S06] Let be any halfspace. For any there is an -approximator with integer weights that has How good is this bound?

15. Idea behind the approximation Let If weights decrease rapidly, then is well approximated by a junta WOLOG have Key idea: look at how these weights decrease. If weights decrease slowly, then is “nice” – can get a handle on distribution of

16. A few more details Def: Critical index of is the first index such that is “small relative to the remaining weights”: Let How do these weights decrease? critical index

17. Sketch of approximation: case 1 “First weights all decrease rapidly” – factor of Remaining weight after very small Can show is -close to, so can approximate just by truncating has relevant variables so can be expressed with integer weights each at most Critical index is first index such that First case:

18. Why does truncating work? Have only if either or each of these weights small, so unlikely by Hoeffding bound unlikely by more complicated argument (split up into blocks; symmetry argument on each block bounds prob by ½; use independence) Let’s write for

19. Sketch of approximation: case 2 Second case: Critical index is first index such that “weights are smooth” Intuition: behaves like Gaussian Can show it’s OK to round weights to small integers (at most )

20. Why does rounding work? Let so Haveonly if either or each small, so unlikely by Hoeffding bound unlikely since Gaussian is “anticoncentrated” }

21. Sketch of approximation: case 2 Second case: Critical index is first index such that “weights are smooth” Intuition: behaves like Gaussian Can show it’s OK to round weights to small integers (at most ) Need to deal with first weights, but at most many – they cost at most END OF SKETCH

22. Extensions Let be any halfspace. For any there is an -approximator with integer weights that has We saw: Recent improvement [DS09]: replace with For with bit flipped Standard fact: Every halfspace has (but can be much less)

23. Proof uses structural properties of halfspaces from testing & learning. Can be viewed as (exponential) sharpening of Friedgut’s theorem: Every Boolean is -close to a function on variables. We show: Every halfspace is -close to a function on variables. approximation Combines Littlewood-Offord type theorems on “anticoncentration” of delicate linear programming arguments Gives new proof of original bound that does not use the “critical index”

24. So halfspaces have low-weight approximators. What about testing? Use approximation viewpoint: two possibilities depending on critical index. First case: critical index large close to junta halfspace over variables Implicitly identify the junta variables (high influence) Do Occam-type “implicit learning” similar to [DLMORSW07] (building on [FKRSS02]): check every possible halfspace over the junta variables –If is a halfspace, it’ll be close to some function you check –If far from every halfspace, it’ll be close to no function you check

25. So halfspaces have low-weight approximators. What about testing? Second case: critical index small every restriction of high-influence vars makes “regular” –all weights & influences are small Low-influence halfspaces have nice Fourier properties Can use Fourier analysis to check that each restriction is close to a low-influence halfspace Also need to check: –cross-consistency of different restrictions (close to low-influence halfspaces with same weights)? – global consistency with a single set of high-influence weights most s

26. A taste of Fourier A helpful Fourier result about low-influence halfspaces: “Theorem”: [MORS07] Let be any Boolean function such that: all the degree-1 Fourier coefficients of are small the degree-0 Fourier coefficient synchs up with the degree-1 coeffs Then is close to a halfspace

27. A taste of Fourier A helpful Fourier result about low-influence halfspaces: “Theorem”: [MORS07] Let be any Boolean function such that: all the degree-1 Fourier coefficients of are small the degree-0 Fourier coefficient synchs up with the degree-1 coeffs Then is close to a halfspace – in fact, close to the halfspace Useful for soundness portion of test

28. Testing halfspaces When all the dust settles: Theorem: [MORS07] The class of halfspaces over is testable with queries. approximation testing

29. What about learning? Learning halfspaces from random labeled examples is easy using poly-time linear programming. 1.The RFA model 2.Agnostic learning under uniform distribution - + + + + + + + + + + + + + + + + - - - - - - -- - - - - - - + + + + + + + + + + + + + + + + + + + - - - - - - - -- -- - - - - - - - - - ? ! There are other harder learning models…

30. The RFA learning model Introduced by [BDD92]: “restricted focus of attention” For each labeled example the learner gets to choose one bit of the example that he can see (plus the label of course). Examples are drawn from uniform distribution over Goal is to construct -accurate hypothesis Question: [BDD92, ADJKS98, G01] Are halfspaces learnable in RFA model?

31. The RFA learning model in action learneroracle May I have a random example, please? Sure, which bit would you like to see? Oh, man…uh, x 7. Thanks, I guess Watch your manners Here’s your example:

32. Very brief Fourier interlude Every has a unique Fourier representation The coefficients are sometimes called the Chow parameters of

33. Another view of the RFA learning model Every has a unique Fourier representation The coefficients are sometimes called the Chow parameters of RFA model: learner gets Not hard to see: In the RFA model, all the learner can do is estimate the Chow parameters With examples, can estimate any given Chow parameter to additive accuracy

34. ( Approximately) reconstructing halfspaces from their (approximate) Chow parameters Theorem [C61]: If is a halfspace & has for all then Perfect information about Chow parameters suffices for halfspaces: To solve 1-RFA learning problem, need a version of Chow’s theorem which is both robust and effective robust: only get approximate Chow parameters (and only hope for approximation to ) effective: want an actual poly(n) time algorithm!

35. Previous results Theorem: Let be a weight- halfspace. Let be any Boolean function satisfying for all Then is an -approximator for [Goldberg01] proved: [ADJKS98] proved: Theorem: Let be any halfspace. Let be any function satisfying for all Then is an -approximator for Good for low-weight halfspaces, but could be Better bound for high-weight halfspaces, but superpolynomial in n. Neither of these results is algorithmic.

36. Robust, effective version of Chow’s theorem Theorem: [OS08] For any constant and any halfspace given accurate enough approximations of the Chow parameters of algorithm runs in time and w.h.p. outputs a halfspace that is -close to Fastest runtime dependence on of any algorithm for learning halfspaces, even in usual random-examples model –Previous best runtime: time for learning to constant accuracy –Any algorithm needs examples, i.e. bits of input Corollary: [OS08] Halfspaces are learnable to any constant accuracy in time in the RFA model.

37. A tool from testing halfspaces If itself is a low-influence halfspace, means we can plug in degree-1 Fourier coefficients as weights and get a good approximator. Also need to deal with high-influence case…a hassle, but doable. Recall helpful Fourier result about low-influence halfspaces: “Theorem”: Let be any function which is such that: all the degree-1 Fourier coefficients of are small the degree-0 Fourier coefficient synchs up with the degree-1 coeffs Then is close to We know (approximations to) these in the RFA setting! polynomial time!

38. Recap of whole talk + + + + + + + + + + + + - - -- - - - - - - - approximation testing learning 1.Every halfspace can be approximated to any constant accuracy with small integer weights. 2.Halfspaces can be tested with queries. 3.Halfspaces can be efficiently learned from (approximations of) their degree-0 and degree-1 Fourier coefficients. Halfspaces over

39. Future directions Better quantitative results (dependence on ?) –Testing: –Approximating: –Learning (from Chow parameters): What about {approximating, testing, learning} w.r.t. other distributions? –Rich theory of distribution-independent PAC learning –Less fully developed theory of distribution-independent testing [HK03,HK04,HK05,AC06] –Things are harder; what is doable? –[GS07] Any distribution-independent algorithm for testing whether is a halfspace requires queries.

40. Thank you for your attention

41. II. Learning a concept class Setup: Learner is given a sample of labeled examples Target function is unknown to learner Each example in sample is independent, uniform over Goal: For every, with probability learner should output a hypothesis such that “PAC learning concept class under the uniform distribution”