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

Overview Halfspaces over learning + - testing approximation

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

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

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

Testing Goal: infer “global” property of function via few “local” inspections Tester makes black-box queries to arbitrary oracle for 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

Some known property testing results Class of functions over # of queries 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]

We’ll get to learning later

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

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

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 is equivalent to Some halfspaces over require integer weights [MTT61, H94] Low-weight halfspaces are nice for complexity, learning.

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? Let’s warm up with a concrete example. Consider (view as n-bit binary numbers) This is a halfspace: Any halfspace for requires weight … but it’s easy to -approximate with weight

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

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

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

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

Sketch of approximation: case 1 Critical index is first index such that First case: “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

Why does truncating work? Let’s write for 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)

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

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

Sketch of approximation: case 2 Critical index is first index such that Second case: “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

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)

Can be viewed as (exponential) sharpening of Friedgut’s theorem: 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. 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” approximation

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

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 s most

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

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

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

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

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?

The RFA learning model in action May I have a random example, please? Sure, which bit would you like to see? Oh, man…uh, x7. Here’s your example: Thanks, I guess Watch your manners learner oracle

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

Another view of the RFA learning model RFA model: learner gets Every has a unique Fourier representation The coefficients are sometimes called the Chow parameters of 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

(Approximately) reconstructing halfspaces from their (approximate) Chow parameters Perfect information about Chow parameters suffices for halfspaces: Theorem [C61]: If is a halfspace & has for all then 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!

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

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 Corollary: [OS08] Halfspaces are learnable to any constant accuracy in time in the RFA model. 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

A tool from testing halfspaces 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 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. We know (approximations to) these in the RFA setting! polynomial time!

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

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.

Thank you for your attention

II. Learning a concept class “PAC learning concept class under the uniform distribution” 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