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Mathematical Theories of Interaction with Oracles Liu Yang Carnegie Mellon University 1© Liu Yang 2013

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Thesis Committee Avrim Blum (co-chair) Jaime Carbonell (co-chair) Manuel Blum Sanjoy Dasgupta (UC, San Diego) Yishay Mansour (Tel Aviv University) Joel Spencer (Courant Institute, NYU)

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Outline Active Property Testing -Do we need to imitate human to advance AI? -I see air planes can fly without flapping their wings. © Liu Yang 20133

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Property Testing Given access to massive dataset: want to quickly determine if a given fn f has some given property P or is far from having it Goal: test from very small num of queries. One motivation: preprocessing step before learning © Liu Yang 20134

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Property Testing © Liu Yang 20135 Instance space X = R n (Distri D over X) Tested function f : X->{-1,1} A property P of Boolean fn is a subset of all Boolean fns h : X -> {-1,1} (e.g ltf) dist D (f, P):=min g P P x~D [f(x) g(x)] Standard Type of query: membership query (ask for f(x) at arbitrary point x)

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Property Testing: An Example E.g. Union of d Intervals 0----++++----+++++++++-----++---+++--------1 - UINT 4 ? Accept! UINT 3 ? Depend on ε - Model selection: testing can tell us how big d need be to be close to target (double and guess, d = 2, 4, 8, 16, ….) If f P should accept w/ prob 2/3 If dist(f,P)>ε should reject w/ prob 2/3 6© Liu Yang 2013

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Property Testing and Learning : Motivation What is Property Testing for ? - Quickly tell if the right fn class to use - Estimate complexity of fn without actually learning Want to do it with fewer queries than learning 7© Liu Yang 2013

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Standard Model uses Membership Query Results of Testing basic Boolean fns using MQ: Constant QC for UINTd, dictator, ltf, … However … 8© Liu Yang 2013

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Membership Query is Unrealistic for ML Problems: An Object Recognition example Recognizing cat/dog ? MQ gives … Is this a dog or a cat? 9© Liu Yang 2013

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An example: movie reviews An example: movie reviews Is this a positive or negative review ? Typical representation in ML (bag-of-words): {fell, holding, interest, movie, my, of, short, this} The original review (human labelers see): This movie fell short of holding my interest. © Liu Yang 201310 - Object a human expert labels has more structure than internal representation used by alg. - MQs construct ex.s in internal representation. - Can be very difficult to order constructed examples words so a human can label the example (esp for long reviews)

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Passive : Waste Too Many Queries ML people move on Can we SAVE #queries ? Passive Model (sample from D) query samples exist in ; but quite wasteful (many examples uninformative) NATURE 11© Liu Yang 2013

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Active Testing 12© Liu Yang 2013 Alg can ask for labels but only pts in the pool Goal: small #queries Pool of unlabeled data (poly-size)

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Property Tester Definition.Definition. An s-sample, q-query ε-tester for P over the distribution D is a randomized algorithm A that draws s samples from D, sequentially queries for the value of f on q of those samples, and then 1. Accepts w.p. at least 2/3 when f P 2. Rejects w.p. at least 2/3 when dist D (f,P)>ε cheap expensive 13© Liu Yang 2013

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Definition.Definition. An s-sample, q-query ε-tester for P over the distribution D is a randomized algorithm A that draws s samples from D, sequentially queries for the value of f on q of those samples, and then 1. Accepts w.p. at least 2/3 when f P 2. Rejects w.p. at least 2/3 when dist D (f,P)>ε cheap expensive 14© Liu Yang 2013 Active tester: s = poly(n) Passive tester: s = q MQ tester: s = (D= Unif)

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Active Property Testing Testing as preprocessing step of learning Need an example? where Active testing - get same QC saving as MQ - better in QC than Passive - need fewer queries than Learning Union of d Intervals, active testing help! 0 ----++++----+++++++++-----++---+++-------- 1 - Testing tells how big d need to be close to target - #Label: Active Testing need O(1), Passive Testing need Θ(d), Active Learning need Θ(d) 15© Liu Yang 2013

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Our Results MQ-like on testing UINTd Passive-like on testing Dictator Active TestingPassive TestingActive Learning Union of d IntervalsO(1)Θ(d 1/2 )Θ(d) DictatorΘ(log n) Linear Threshold FnO(n 1/2 ) ~ Θ(n 1/2 )Θ(n) Cluster AssumptionO(1)Ω(N 1/2 )Θ(N) MQ-like Passive-like NEW !! 16© Liu Yang 2013

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Testing Unions of Intervals 0 Testing Unions of Intervals 0 ----++++----+++++++++-----++---+++-------- 1 Theorem. Testing UINTd in the active testing model can be done using O(1) queries. Recall: Learning requires Ω(d) examples. 17© Liu Yang 2013

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Testing Unions of Intervals (cont.) Suppose uniform distribution Definition: Fixδ>0. The localδ-noise sensitivity of fn f: [0, 1]->{0, 1} at x [0; 1] is. The noise sensitivity of f is Proposition: Fixδ>0. Let f: [0, 1] -> {0,1} be a union of d intervals. NS δ (f) dδ. easy har d Lemma: Fix δ= ε 2 /(32d). Let f : [0, 1] -> {0, 1} be a fn with noise sensitivity bounded by NS δ (f) dδ(1 + ε/4 ). Then f is ε-close to a union of d intervals. 18© Liu Yang 2013

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Easy Lemma Lemma. If f is a union of d intervals, NS δ (f) dδ. Proof sketch: - The probability that x lands within distance δ of any of the boundaries is at most 2d*2δ. -The probability that y crosses a boundary given that x is within distance δ of it is 1/4. -P(f(x)f(y)| |x-y|<δ) (2d*2δ)*(1/4) = dδ. © Liu Yang 201319

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Hard Lemma Lemma. Fix δ = ε 2 /(32d). If f is ε-far from a union of d intervals, then NS δ (f) > (1+ε/4)dδ. Proof strategy: If NS δ (f) is small, do self-correction. g(x) = E[f(y) | yÃ[x-δ,x+δ]], f(x) = round g(x) to 0 if ¿ or to 1 if 1-¿ © Liu Yang 201320

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Hard Lemma Lemma. Fix δ = ε 2 /(32d). If f is ε-far from a union of d intervals, then NS δ (f) > (1+ε/4)dδ. Proof strategy: - Argue dist(f,f) ε/2. - Show f is union of d(1 + ε/2) intervals. - Implies dist(f,P) ε/2. © Liu Yang 201321

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z z ----++++----+++++++++-----++---+++-------------++---+++-------- at δ Nr δ δ δ δ !!! δ δ 22© Liu Yang 2013 Uniform

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Testing Unions of Intervals © Liu Yang 201323 Theorem. Testing UINT d in the active testing model can be done using O(1) queries. If non-uniform distribution, use data to stretch/squash the axis, makes the distribution near-uniform Total num unlabeled samples: O(d 1/2 ).

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Testing Linear Threshold Fns 24© Liu Yang 2013 Linear Threshold Functions (LTF): f(x) = sign( ), for w,x 2 R n

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Testing Linear Threshold Fns Theorem. We can efficiently test LTFs under the Gaussian distribution with Õ(n 1/2 ) labeled examples in both active and passive testing models. We have lower bounds of ~ Ω (n 1/3 ) for active testing and ~ Ω (n 1/2 ) for passive testing. Learning LTFs need Ω(n) under Gaussian. So testing is better than learning in this case. 25© Liu Yang 2013

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Testing Linear Threshold Fns [MORS10] => suffices to estimate E[f(x) f(y) ] up to ± poly(ε). Intuition: LTF is characterized by a nice linear relation between angle ( ) and probability of having same label (f(x)f(y)=1). 26© Liu Yang 2013

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Testing Linear Threshold Fns [MORS10] => suffices to estimate E[f(x) f(y) ] up to ± poly(ε). Could take m random pairs and use empirical average. - But most pairs x,y would have n 1/2 (CLT) So would need m = Ω(n) to get within ± poly(ε). Solution: take O(n 1/2 ) random points and average f(x)f(y) over all O(n) pairs x,y. - Concentration inequalities for U-statistics [Arcones,95] imply this works. 27© Liu Yang 2013

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General Testing Dimension Testing dim characterize (up to constant factors) the intrinsic #label requests needed to test the given property w.r.t. the given distribution All our lower bounds are proved via testing dim 28© Liu Yang 2013

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Minimax Argument min Alg max f P(Alg mistaken) = max π 0 min Alg P(Alg mistaken) wolg, π 0 =α π + (1-α) π, π Π 0,π Π ε Let π S, π S be induced distributions on labels of S. For a given π 0, min Alg P(Alg makes mistake|S) 1-d S (π, π) 29© Liu Yang 2013

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Passive Testing Dim Define d passive largest q in N, s.t. Theorem: Sample Complexity of passive testing is Θ(d passive ). 30© Liu Yang 2013 Compare with VC-dimension: Want exists set S s.t. all labelings occur at least once.

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Active Testing Dim Fair(π,π,U): distri. of labeled (y; l): w.p.½ choose y~π U, l= 1; w.p.½ choose y~π U, l= 0. err*(H; P): err of optimal fn in H w.r.t data drawn from distri. P over labeled egs. Given u=poly(n) unlabeled egs, d active (u): largest q in N s.t. Theorem: Active testing w/ failure prob 1/8 using u unlabeled egs needs Ω(d active (u)) label queries; can be done w/ O(u) unlabeled egs and O(d active (u)) label queries 31© Liu Yang 2013

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Application: Dictator fns Theorem: For dictator functions under the uniform distribution, d active (u)=Θ(log n) (for any large-enough u=poly(n)). Corollary: Any class that contains dictator functions requires log(n) queries to test in the active model, including poly-size decision trees, functions of low Fourier degree, juntas, DNFs, etc. 32© Liu Yang 2013

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Application: Dictator fns Theorem: For dictator functions under the uniform distribution, d active (u)=Θ(log n) (for any large-enough u=poly(n)). π = unif over dictator fns π = unif over all Boolean fns 33© Liu Yang 2013

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Application: LTFs Theorem. For LTFs under the standard n-dim Gaussian distrib, d passive = Ω((n/logn) 1/2 ) and d active (u) = Ω((n/logn) 1/3 ) (for any u=poly(n)). - π: distrib over LTFs obtained by choosing w~N(0, I nxn ) and outputting f(x) = sgn(w x). - π: uniform distrib over all functions. 34© Liu Yang 2013 - Obtain d passive :bound tvd(distrib of Xw/n, N(0, I qxq )). - Obtain d active : similar to dictator LB but rely on strong concentration bounds on spectrum of random matrices

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Open Problem Matching lb/ub for active testing LTF: n? Tolerant Testing ε/2 vs. ε (UINTd, LTF) Testing LTF under general distrib. 35© Liu Yang 2013

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Outline Learnability of DNF with Representation Specific Queries - Liu: We do statistical learning for … - Marvin: but we haven't not done well at the fundamentals, e.g. knowledge representation. © Liu Yang 201336

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Learning DNF formulas Poly-sized DNF: # terms = n O(1) e.g. f=(x1 x2) (x1 x4) -Natural form of knowledge representation -PAC-learning DNF appears to be very hard. 37© Liu Yang 2013 Your ticket : n: number of var.s Concept space C: collection of fn h: {0, 1}^n -> {0,1} Unknown target fn f*: the true labeling fn Err(h) = Px~D[h(x) ~= f*(x)] (Distri. D over X) Best known alg in standard model is exponential over arbitrary distri; Over Unif, no known poly time alg

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New Models: Interaction with Oracles 38© Liu Yang 2013 - Boolean queries: K(x, y) = 1 if share some term - Numerical queries: K(x, y) = #terms share Hi, Tim, do x and y have some term in common ? Yes! Imagine …

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Query: Similarity about TYPE 39© Liu Yang 2013 Fraud Detection Type of Query: pair of POSITIVE ex.s from a random dataset, teacher says YES if they share some term; or report how many terms they share. Question: can we efficiently learn DNF with this type of query? Identity theft Stolen cards Stolen cards BIN attack x y What if have similarity info about TYPE ? Fraud detection: fraudulent of same type ? YES! x and y share a term Skimming Term 1 of xTerm 2 of xTerm 3 of x

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Warm Up: Disjoint DNF w/Boolean Queries Use similarity queries to partition positive ex.s into t buckets, one per term. Separately learn a conjunction for each bucket (intersect the pos ex.s in it) OR the results 40© Liu Yang 2013

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Pos Result 1: Weakly Disjoint DNF w/Boolean Queries -Distinguishing ex for T 1 : ex. sat. T 1 & no other term -Weakly disjoint: for each term, poly(n, 1/ε) fraction rand. ex.s sat. it & no other term. - Neighbor-method: get all its neighbors in the graph and learn a conjunc. - Neighbor-method w.p. 1-δ, produce an ε-accu. DNF if weakly disjoint. 41© Liu Yang 2013 T1T1 Graph: - Nodes: pos examples - Edge exists if K(.) = 1

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Hardness Results Boolean Queries Thm. Learning DNF from random data under arb. distri. w/ Boolean queries is as hard as learning DNF from random data under arb. distri. w/ only labels (no queries). 42© Liu Yang 2013 m K (giant 1, giant 2) = 1 - Group-learn: tell data from D+ or D- - Reduction from group-learn DNF in std. model to our model - How to use our alg A to group-learn ? - Simulate the oracle by always saying yes whenever there is a query made to two pos ex.s; Given the output of A, we give a group-learn alg for the original problem n var.s

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Hardness Results Approx Numerical Queries Thm. Learning DNF from random data under arbitrary distri. w/ approx-numerical-queries is as hard as learning DNF from random data under arb. distri. w/ only labels i.e. if C is #terms x i and x j sat in common, oracle returns a value in [(1 – τ )C, (1 + τ )C]. © Liu Yang 201343

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Pos Result 3: learn DNF w/ Numerical Queries -Sample m = O((t/ ε ) log(t/( εδ) )) landmark points -Landmark F i (x) is sum-of-monotone-terms fn (rm terms not sat by pos x i ). Fi(·) = K(x i, ·), K is numerical query -Use subroutine to learn hypo. h i (x) ε /(2m)-accu w.r.t. Fi. Subroutine: learn a sum of monotone t terms over unif., using time & samples poly(t, n, 1/ε). -Combine all hypo.s h i to h: h(x) = 0 if h i (x) = 0 for all i, else h(x) = 1. © Liu Yang 201344 Thm. Under unif distri., w/ numerical queries, can learn any poly(n)-term DNF. f(x) = T 1 (x)+T 2 (x)+ … +T t (x)

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Learn Sum of Monotone Terms Estimate Fourier coeffi. of S Inclusion check: mag. ε/(16t) ? otw x 1 | x 2 |x 3 |x 4 |x 5 |x 6 |x 7 |x 8 |x 9 S = {x 1 } Output x 1 x 3 x 4 x 9 S = {x 1, x 2 } S = {x 1, x 3 } S = {x 1, x 3 x 4 } YES S = {x 1, x 3 x 4, x 5 } S = {x 1, x 3 x 4, x 6 } S = {x 1, x 3 x 4, x 7 } S = {x 1, x 3 x 4, x 8 } S = {x 1, x 3 x 4, x 9 } - Greedy: - Inclusion Check: - Fourier coeffi. of S: © Liu Yang 201345

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Learn Sum of Monotone Terms : Greedy Alg Examine each parity fn of size 1 & est its Fourier coeffi. (up to θ/4 accu.). Set θ = ε/(8t) Place all coeffi. of mag. θ/2 into a list L 1. For j = 2, 3,... repeat: - For each parity fn Φ S in list L j-1 and each x i not in S, est Fourier coeffi. of - If est. is θ/2, add it to list L j (if not already in) - maintain list L j : size-j parity fns w/ coeffi. mag. θ. Construct fn g: weight sum of parities for identified coeff. Output fn h(x) = [g(x)] © Liu Yang 201346 Inclusion check

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Other Positive Results BinaryNumeric O(log(n)) terms DNF (any distrib.) 2-term DNF (any distrib.) DNF: each var in at most O(log(n)) terms (Unif) log(n)-Junta (Unif) log(n)-Junta (any distrib) DNF having 2 O(log(n)) terms (Unif.) Open problems: - learn arbitrary DNF (unif, Boolean queries)? - learn arbitrary DNF (any distri. numerical queries)?

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Outline Active Learning with a Drifting Distribution Active Learning with a Drifting Distribution - If not every poem has a proof, can we at least try to make every theorem proved beautiful like a poem? © Liu Yang 201348

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Active Learning with a Drifting Distrib: Model Scenario: - Unobservable seq. of distrib.s with each - Unobservable time-indep. regular cond. distrib. represent by fn - : an infinite seq. of indep. r. v., s.t., and cond. distrib. Of Y t given X t satisfies Active learning protocol At each time t, alg is presented with X t, and is required to predict a label, then it may optionally request to see true label value Y t Interested in cumulative #mistakes up to time T and total #labels requested up to time T © Liu Yang 201349

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x1x1 x2x2 xtxt x3x3 Data Space D2D2 D1D1 D3D3 DtDt x4x4 D4D4 © Liu Yang 201350 Distrib. Space

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Definition and Notations Instance space X = R n Distribution space of distributions on X Concept space C of classifiers h: X -> {-1,1} - Assume C has VC dimension vc < D t : Data distrib. on X at t Unknown target fn h*: true labeling fn Err t (h) = P x~D t [h(x) h*(x)] In realizable case, h* in C and err t (h*) = 0. For, © Liu Yang 201351

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Def: disagreement coefficient, tvd The disagreement coefficient of h* under a distri. P on X, is define as, (r > 0) Total variation distance of probability measures P and Q on a sigma-algebra of subsets of the sample space is defined via © Liu Yang 201352

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Assumptions Independence of the X t variables Vc-dim < Assumption 1 (totally bounded) : is totally bounded (i.e. satisfies ) - For each ε > 0, denote a minimal subset of s.t. s.t. (i.e. a minimal ε-cover of ) Assumption 2 (poly-covers) where c,m 0 are constants.

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Realizable-case Active Learning CAL © Liu Yang 201354

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Sublinear Result: Realizable Case © Liu Yang 201355 Theorem. If is totally bounded, then CAL, achieves an expected mistake bound And if, then CAL makes an E[#queries] [Proof Sketch]: Partition D into buckets of diam < eps. Pick a time T_eps past all indices from finite buckets and all the infinite bucket has at least

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Number of Mistakes Alternative scenario: - Let P i be in bucket i - Swap the L( ε ) samples for bucket i with L( ε ) samples from P i - L( ε ) large enough so E[diam(V)] alternative < sqrt{eps}. - Note: E[diam(V)] E[diam(V)] alternative + sum L( ε ) t values ||P_i – D_t|| < ε + L( ε )* ε. So E[diam] -> 0 as T -> - E[#mistake] - Since

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Number of Queries E[#queries] P(make query) = E[P(DIS(V t-1 ))] Let then and E[#queries] =>

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Explicit Bound: Realizable Case © Liu Yang 201358 Theorem. If poly-covers assumption is satisfied ( ) then CAL achieves an expected mistake bound and E[#queries] such that where [Proof Sketch] Fix any ε >0, and enumerate For t in N, let K(t) be the index k of the closest to D t. Alternative data sequence: Let be indep., with This way all samples corresp. to distrib.s in a given bucket all came from same distri. Let V t be the corresponding version spaces.

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E[#mistakes] Classic PAC bound => (#previous distrib.s in D t 's bucket) So (each bucket has at most T samples) So E[#mistakes] Take to get the stated theorem. To bound E[#queries], again it is just showed this is So Again, taking gives the stated result. © Liu Yang 201359

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Learning with Noise Noise conditions Strictly benign noise condition: Special case: Tsybakov's noise conditions η satisfies strictly benign noise condition and for some c > 0 and α0, Unif Tsybakov assumption: Tsybakov Assumption is satisfied for all with the same c and α values. © Liu Yang 201360 and

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Agnostic CAL [DHM] © Liu Yang 201361 Based on subroutine:

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Tsybakov Noise: Sublinear Results & Explicit Bound Theorem. If is totally bounded and η satisfies strictly benign noise condition, then ACAL achieves an excess expected mistake bound and if additionally, then ACAL makes an expected number of queries © Liu Yang 201362 Theorem. If poly-covers Assumption and Unif Tsybakov assumption are satisfied, then ACAL achieves an expected excess number of mistakes ACAL achieves expected #mistakes and expected #queries such that, for

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Outline Transfer Learning Transfer Learning - Do not ask what Bayesians can do for - Do not ask what Bayesians can do for Machine Learning, ask what Machine Machine Learning, ask what Machine Learning can do for Bayesians Learning can do for Bayesians

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Transfer Learning Principle: solving a new learning problem is easier given that weve solved several already ! How does it help? - New task directly ``related to previous task [e.g., Ben-David & Schuller 03; Evgeniou, Micchelli, & Pontil 2005] - Previous tasks give us useful sub-concepts [e.g., Thrun 96] - Can gather statistical info on the variety of concepts [e.g., Baxter 97; Ando & Zhang 04] Example: Speech Recognition - After training a few times, figured out the dialects. - Next time, just identify the dialect. - Much easier than training a recognizer from scratch

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prior h1*h1* x 1 1,y 1 1 … x 1 k,y 1 k Task 1 hT*hT* x T 1,y T 1 … x T k,y T k … Task T Model of Transfer Learning Motivation: Learners often Not Too Altruistic h2*h2* x 2 1,y 2 1 … x 2 k,y 2 k Task 2 Layer 1: draw task i.i.d. from unknown prior Layer 2: per task, draw data i.i.d. from target Better Estimate of Prior !!

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- Marvin: so you assume learning French is similar to learning English? - Marvin: so you assume learning French is similar to learning English? - Liu: It indeed seems many English words have a French counterpart … - Liu: It indeed seems many English words have a French counterpart …

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Identifiability of priors from joint distribs Let prior π be any distribution on C - example: (w, b) ~ multivariate normal Target h* π ~ π Data X = (X 1, X 2, …) i.i.d. D indep h* π Z(π) = ((X 1, h* π (X 1 ), (X 2, h* π (X 2 ), …). Let [m] = {1, …, m}. Denote X I = {X i } i I (I : subset of natural numbers) Z I (π) = {(X i, h* π (X i ))} i I Theorem: Z [VC] (π 1 ) = d Z [VC] (π 2 ) iff π 1 = π 2.

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Identifiability of priors by VC-dim joint distri. Threshold: - for two points x 1, x 2, if x 1 < x 2, then Pr(+,+)=Pr(+.), Pr(-,-)=Pr(.-), Pr(+,-)=0, So Pr(-,+)=Pr(.+)-Pr(++) = Pr(.+)-Pr(+.) - for any k > 1 points, can directly to reduce number of labels in the joint prob from k to 1 P(-----------(-+)+++++++++++++++++) = P( (-+) ) = P( (.+) ) - P( (++) ) = P( (.+) ) - P( (+.) ) + P( (+-) ) (unrealized labeling !!) = P( (.+) ) - P( (+.) ) --------------------- 01 ++++++++++++++++

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Theorem: Z [VC] (π 1 ) = d Z [VC] (π 2 ) iff π 1 = π 2. Proof Sketch Let ρ m (h,g) = 1/m Σ i=1 m II(h(X m ) g(X m )) Then vc < implies w.p.1 forall h, g C with h g lim m -> ρ m (h,g) = ρ(h,g) > 0 ρ is a metric on C by assumption, so w.p.1 each h in C labels -seq (X 1, X 2 …) distinctly (h(X 1 ), h(X 2 ), …) => w.p.1 conditional distribution of the label seq Z(π)|X identifies π => distrib of Z(π) identifies π i.e. Z (π 1 ) = d Z (π 2 ) implies π 1 = π 2

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Identifiability of Priors from Joint Distributions lower–dim cond distrib y closer to

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Identifiability of Priors from Joint Distributions

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Transfer Learning Setting Collection Π of distribs on C. (known) Target distrib π* Π. (unknown) Indep target fns h 1 *, …, h T * ~ π* (unknown) Indep i.i.d. D data sets X (t) = (X 1 (t), X 2 (t), …), t [T]. Define Z (t) = ((X 1 (t), h t *(X 1 (t) )), (X 2 (t), h t *(X 2 (t) )), …). Learning alg. gets Z (1), then produces ĥ 1, then gets Z (2), then produces ĥ 2, etc. in sequence. Interested in: values of ρ(ĥ t, h*(t)), and the number of h* t (X j (t) ) value alg. needs to access.

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Estimating the prior Principle: learning would be easier if know π* Fact: π* is identifiable by distrib of Z [VC] (t) Strategy: Take samples Z [VC] (i) from past tasks 1, …, t-1, use them to estimate distrib of Z [VC] (i), convert that into an estimate π t-1 of π*, Use π t-1 in a prior-dependent learning alg for new task h t * Assume Π is totally bounded in total variation Can estimate π* at a bounded rate: || π* - π t ||< δ t converges to 0 (holds whp)

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Transfer Learning Given a prior-dependent learning A(ε, π), with E[# labels accessed] =Λ(ε, π) and producing ĥ with E[ρ(ĥ, h*)]ε For t = 1,…, T If δ t-1 > ε/4, run prior-indep learning on Z [VC/ε] (t) to get ĥ t Else let π t = argmin π B(π t-1, δ t-1 ) Λ(ε/2, π) and run A(ε/2, π t ) on Z (t) to get ĥ t Theorem: Forall t, E[ρ(ĥ t, h t *)] ε, and limsup T -> E[#labels accessed]/T Λ(ε/2, π*) + vc.

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- Yonatan: Ill send you an email to summarize what we just discussed. - Yonatan: Ill send you an email to summarize what we just discussed. - Liu: Thank you but I now invented a model to transfer knowledge with provable guarantees; - Liu: Thank you but I now invented a model to transfer knowledge with provable guarantees; so I use that all the time. so I use that all the time. - Yonatan: But thats asymptotic guarantee. My life span is finite. So Im still gonna to send you an email. - Yonatan: But thats asymptotic guarantee. My life span is finite. So Im still gonna to send you an email.

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Outline Online Allocation and Pricing with Economies of Scale Online Allocation and Pricing with Economies of Scale © Liu Yang 201377 - Jamie Dimon: Economies of scale are a good thing. - Jamie Dimon: Economies of scale are a good thing. If we didn't have them, we'd still be living in tents and eating buffalo.

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Setting Christmas season - Nov : customer survey - Dec : purchasing and selling Buyers arrive online one at a time w/ val.s on items sampled iid from some unknown distri.

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Thrifty Santa Claus Each shopper wants only one item though it might prefer some items than others Minimize total cost to seller Buyers: binary valuation Goal of seller: sat. everyone

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Hardness: Set-Cover If costs much more rapidly, then even if all customers' val.s known up front, would be (roughly) a set-cover problem and could not hope to achieve cost o(log n) times optimal. Natural case : for each good, cost (to the seller) for ordering T copies is sublinear in T. Production cost Marginal cost #copies α = 1 α = 0 α in (0, 1) α = 1 α = 0 α in (0, 1)

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Thrifty Santa Claus : Results Mar-cost non-increa, exists optimal strategy? - order items by some perm.; give new buyer earliest item it desires in the perm. What if n (#buyers) >> k (#items) AND mar- cost not too rapidly? (rate 1/T α for 0α<1) - can efficiently perform allocation w/ cost a const. factor greater than OPT

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Algorithm Alg: use initial buyers to learn about distri. determine how best to allocate to new buyers. If cost fn c(x) = Σ i=1 x 1/i α, for α in [0,1) - run greedy weighted set cover => total cost 1/(1-α) {± OPT}. Essentially smooth variant of set-cover If ave-cost within some factor of mar-cost, have a greedy alg w/ const. approx ratio

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Sample Complexity Analysis How complicated the allocation rule needs to be to achieve good perf.? Theorem

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Outline Factor Models for Correlated Auctions Factor Models for Correlated Auctions © Liu Yang 201384

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The Problem Auctioneer sells good to a group of n buyers. Seller wants to maximize his revenue. Each buyer maximize his utility of getting good: val. - price Seller doesnt know exact val.s of players He knows distri D from which vec. of val.s (v1, …, vn) is drawn.

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Our Contribution When D is a product distri., - Myerson gives dominant strategy truthful auction General correlated distr.s, not known - how to create truthful auctions - how to use player js bid to capture info about player i. What if correlation between buyer val.s driven by common factors?

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Example Two firms produce same type of good Each firms value: production cost need to hire workers (W) & rent capital (Z) l i : #workers firm i needs to produce one unit K i : amount of capital firm needs ε i :fixed costs unique to firm i. firms costs: C i = l i W + k i Z + ε i firms costs correlated : hire workers & rent capital from the same pool.

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The Factor Model Factor model as V = F + U where - V: vec. of observations - λ : matrix of coefficients - F : vec. of factors - U: vec. of idiosyncratic components ind. of each other & ind. of the factors

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Discussions Possible that: - Designer & bidders might not know common factors - Bidders might only know their val. - seller only knows joint distri. of bidders val.s, Seller RECOVER factor model by making inferences over observed bids. Aggregate info.: common factors inferred from collective knowledge of all players.

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The Auction

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The Auction (cont.) Thm: When correlation follows this factor model, this auction is dominant strategy truthful, ex-post individually rational, and asymptotically optimal.

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Dominant Strategy Truthfulness Toss a coin & choose between: - 2nd price auction: truthful - mechanism M estimates factors from a random set of bidders S: bidders in S receive utility 0 regardless of allocation & price output by M Players S incentivized truthful for small incentive they get from participating in 2nd price auction.

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Dominant Strategy Truthfulness (Cont.) Remaining bidders set R = {1, …, n} - S receive incentives from both 2 nd price auction and mechanism M. M offers them allocation and price vec.s x(b R ), p(b R ) by running Myerson (b R,V R |^f) on players' bids, and on cond. distri.s estimated for these players. No player in R can influence the estimated conditional distri. V R |^f, and Myerson's optimal auction is truthful.

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Thanks ! 94© Liu Yang 2013 Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law.

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6.896: Topics in Algorithmic Game Theory Lecture 21 Yang Cai.

6.896: Topics in Algorithmic Game Theory Lecture 21 Yang Cai.

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