Learning From Satisfying Assignments Rocco A Servedio Columbia

Learning From Satisfying Assignments Rocco A. Servedio Columbia University Anindya De UC Berkeley/IAS Brown University Ilias Diakonikolas U. Edinburgh December 2013 1

Learning Probability Distributions • Big topic in statistics literature (“density estimation”) for decades • Exciting work in the last decade+ in TCS, largely on learning continuous distributions (mixtures of Gaussians & more) • This talk: distribution learning from a complexity theoretic perspective – What about distributions over the hypercube? – Can we formalize intuition that “simple distributions are easy to learn”? 2

What do we mean by “learn a distribution”? • Unknown target distribution • Algorithm gets i. i. d. draws from • With probability 9/10, must output (a sampler for a) distribution such that statistical distance between and is small: (Natural analogue of Boolean function learning. ) 3

Previous work: [KRRSS 94] • Looked at learning distributions over {0, 1}n in terms of n-output circuits that generate distributions: output x 1. . . xn distributed according to circuit input z 1. . . zm uniform over {0, 1}m • [AIK 04] showed it’s hard to learn even very simple distributions from this perspective: already hard even if each output bit is a 4 -junta of input bits. 4

This work: A different perspective Our notion of a “simple” distribution over {0, 1}n: uniform distribution over satisfying assignments of a “simple” Boolean function. What kinds of Boolean functions can we learn from their satisfying assignments? Want algorithms that have polynomial runtime and # of samples required. 5

What are “simple” functions? + Halfspaces: + + + - - - ++ + - -- - - + OR DNF formulas: AND _ x 2 x 3 _ x 5 x 6 AND _ x 3 AND x 5 x 1 x 6 _ x 7 6

Simple functions, cont. AND 3 -CNF formulas: OR _ x 2 x 3 OR _ x 5 OR _ x 3 x 7 x 5 x 1 x 6 _ x 7 AND Monotone 2 -CNF: OR x 2 x 3 OR OR x 2 x 3 x 5 x 6 x 7 7

Yet more simple functions Low-degree polynomial threshold functions: Intersections of k halfspaces: - ++ + + -- +- + + -- -- - + + ++ - - - + - -- - - -- 8

The model, more precisely • Let be a fixed class of Boolean functions over • There is some unknown. Learning algorithm sees samples drawn uniformly from. Target distribution: . • Goal : With probability 9/10, output a sampler for a hypothesis distribution such that We’ll call this a distribution learning algorithm for . 9

Relation to other learning problems Q: How is this different from learning (function learning) under the uniform distribution? A: Only get positive examples. At least two other ways: • (not so major) Want to output a hypothesis distribution rather than a hypothesis function • (really major) Much more demanding guarantee than usual uniform-distribution learning. 10

Example: Halfspaces Usual uniform-distribution model for learning functions: Hypothesis allowed to be wrong on points in . 1 n 0 n For highly biased target function like a fine hypothesis for any , constant-0 function is. 11

A stronger requirement Our distribution-learning model: “constant-0 hypothesis” is meaningless! For to be good hypothesis distribution, must be only a fraction of. 1 n 0 n Essentially, we require hypothesis function with multiplicative rather than additive -accuracy relative to. 12

Usual functionlearning setting Given: random labeled examples from , must Output: hypothesis such that If both regions are small, this is fine! Our setting Given: draws from , must Output: hypothesis with the following guarantee : must satisfy 13

Brief motivational digression into the real world: language learning People typically learn new languages by being exposed to correct utterances (positive examples), which are a sparse subset of all possible vocalizations (all examples). Goal is to be able to generate new correct utterances (generate draws from a distribution similar to the one the samples came from). 14

Our positive results Theorem 1: We give an efficient distribution learning algorithm for = { halfspaces }. + + + Runtime is - - - ++ - - + Theorem 2: We give a (pretty) efficient distribution learning algorithm for OR = { poly(n)-term DNFs }. AND AND Runtime is _ x 2 x 3 _ x 5 x 6 _ x 3 Both results obtained via a general approach, plus work. x 5 x 1 x 6 _ x 7 -specific 15

Our negative results Assuming crypto-hardness (essentially RSA), there are no efficient distribution learning algorithms for: o o - --- - - -- - + + ++ Intersections of two halfspaces --- -+ + -- + + + - -- -+ -+ + Degree-2 polynomial threshold functions AND o 3 – CNFs , or even _ x 2 OR x 3 OR _ x 5 OR _ x 3 x 7 x 5 x 1 x 6 _ x 7 AND o Monotone 2 -CNFs OR x 2 x 3 OR OR x 2 x 3 x 5 x 6 x 7 16

Rest of talk • Mostly positive results • Mostly halfspaces (and general approach) • Touch on DNFs, negative results 17

Learning halfspace distributions 1 n Given positive examples drawn uniformly from for some unknown halfspace , +++++ unknown 0 n We need to (whp) output a sampler for a distribution that’s close to. 18

Let’s fantasize 1 n Suppose somebody gave us. Even then, we need to output a sampler for a distribution close to uniform over. +++++ known 0 n Is this doable? Yes. 19

Approximate sampling for halfspaces Theorem: Given over can return a uniform point from in time (with failure probability ) • [Morris. Sinclair 99]: sophisticated MCMC analysis • [Dyer 03]: elementary randomized algorithm & analysis using “dart throwing” Of course, in our setting we are not given. But, we should expect to use (at least) this machinery for our general problem. , 20

A potentially easier case…? For approximate sampling problem (where we’re given ), problem is much easier if is large: sample uniformly & do rejection sampling. Maybe our problem is easier too in this case? In fact, yes. Let’s consider this case first. 21

Halfspaces: the high-density case • Let . • We will first consider the case that . • We’ll solve this case using Statistical Query learning & hypothesis testing for distributions. 22

First Ingredient for the high-density case: SQ Statistical Query (SQ) learning model: o SQ oracle : given poly-time computable outputs where. o An algorithm is said to be a SQ learner for (under distribution ) if can learn given access to. 23

SQ learning for halfspaces Good news: [Blum. Frieze. Kannan. Vempala 97] gave an efficient SQ learning algorithm for halfspaces. Outputs halfspace hypotheses! Of course, to run it, need access to oracle for the unknown halfspace. So, we need to simulate this given our examples from. 24

The high-density case: first step Lemma: Given access to uniform random samples from and such that , queries to can be simulated up to error in time. Proof sketch: Estimate using samples from 25

The high-density case: first step Lemma: Given access to uniform random samples from and such that , queries to can be simulated up to error in time. Recall promise: Additionally, we assume that we have = . A halfspace! Lemma lets us use the halfspace SQ-learner to get that such 26

Handling the high-density case • Since , have that o o • Hence using rejection sampling, we can easily sample. Caveat : We don’t actually have an estimate for. 27

Ingredient #2: Hypothesis testing • Try all possible values of multiplicative grid in a sufficiently fine • We will get a list of candidate distributions such that at least one of them is -close to. • Run a “distribution hypothesis tester” to return which is - close to. 28

Distribution hypothesis testing Theorem: Given • Sampler for target distribution • Approximate samplers for distributions • Approximate evaluation oracles for • Promise : Hypothesis tester guarantee: Outputs in time such that Having samplers & evaluators for hypotheses is crucial 29 for this.

Distribution hypothesis testing, cont. We need samplers & evaluators for our hypothesis distributions All our hypotheses are dense, so can do approximate counting easily (rejection sampling) to estimate Note that So we get the required (approximate) evaluators. Similarly, (approximate) samples are easy via rejection sampling. 30

Recap So we handled the high-density case using • SQ learning (for halfspaces) • Hypothesis testing (generic). (Also used approximate sampling & counting, but they were trivial because we were in the dense case. ) Now let’s consider the low-density case (the interesting case). 31

Low density case: A new ingredient New ingredient for the low-density case: A new kind of algorithm called a densifier. • Input: such that samples from • Output: A function , and such that: – – For simplicity, assume that (like ) 32

Densifier illustration : g Samples from Good estimate f : 33

Low-density case (cont. ) To solve the low-density case, we need approximate sampling and approximate counting algorithms for the class. This, plus previous ingredients (SQ learning, hypothesis testing, & densifier) suffices: given all these ingredients, we get a distribution learning algorithm for. 34

How does it work? The overall algorithm: (recall that ) Needs good estimate 1. Run densifier to get 2. Use approximate sampling algorithm for from 3. Run SQ-learner for under distribution hypothesis for 4. Sample from till get such that this. of to get samples to get ; output Repeat with different guesses for , & use hypothesis testing to choose that’s close to 35

A picture of one stage Note: This all assumed we have a good estimate g h 1. Using samples from , run densifier to get g f 2. Run approximate uniform generation algorithm to get uniform positive examples of g 3. Run SQ-learner on distribution to get high-accuracy hypothesis h for (under ) 4. Sample from till get point where , and output it. 36

How it works, cont. Recall that to carry out hypothesis testing, we need samplers & evaluators for our hypothesis distributions Now some hypotheses may be very sparse… • Use approximate counting to estimate As before, so we get (approximate) evaluator. • Use approximate sampling to get samples from . 37

Recap: a general method Theorem: Let be a class of Boolean functions such that: (i) is efficiently SQ-learnable; (ii) has a densifier with an output in ; and (iii) has efficient approximate counting and sampling algorithms. Then there is an efficient distribution learning algorithm for. 38

Back to halfspaces: what have we got? • Saw earlier we have SQ learning [Blum. Frieze. Kannan. Vempala 97] • [Morris. Sinclair 99, Dyer 03] give approximate counting and sampling. So we have all the necessary ingredients. …except a densifier. Reminiscent of [Dyer 03] “dart throwing” approach to approximate counting – but in that setting, we are given f Approximate counting setting: g Given , come up with f g Densifier setting: Can we come up with a suitable given only samples from 39 ?

A densifier for halfspaces Theorem: There is an algorithm running in time such that for any halfspace , if the algorithm gets as input such that and access to , it outputs a halfspace with the following properties : 1. , and 2. . 40

Getting a densifier for halfspaces Key ingredients: o Online learner of [Maass. Turan 90] o Approximate sampling for halfspaces [Morris. Sinclair, Dyer 03] 41

Towards a densifier for halfspaces Recall our goals: 1. 2. Fact: Let be of size. Then, with probability , condition (1) holds for any halfspace such that. Proof: If (1) fails for a halfspace , then Fact follows from union bound over all (at most So ensuring (1) is easy – choose consistent with. How to ensure (2)? . many) halfspaces . and ensure is 42

Online learning as a two-player game Imagine a two player game in which Alice has a halfspace and Bob wants to learn : i. Bob initializes to the empty set ii. Bob runs a (specific polytime) algorithm on the set and returns halfspace consistent with iii. Alice either says “yes, “ or else returns an such that iv. Bob adds to and returns to step (ii). 43

Guarantee of the game Theorem: [Maass. Turan 90] There is a specific algorithm that Bob can run so that the game terminates in at most rounds. At the end, either or Bob can certify that there is no halfspace meeting all the constraints. (Algorithm is essentially the ellipsoid algorithm. ) Q: How is this helpful for us ? A: Bob seems to have a powerful strategy We will exploit it. 44

Using the online learner • Choose as defined earlier. Start with. • “Bob” simulation: stage – Run Bob’s strategy and return consistent with. • “Alice” simulation: If for some , then return. – Else, if (approx counting) then we are done and return. – Else use approx sampling to randomly choose a point and return. 45

Why is the simulation correct? • If for , then the simulation step is indeed correct. • The other case in which Alice returns a point is that. This means that the simulation at every step is correct with probability. • Since the simulation lasts steps, all the steps are correct with probability. 46

Finishing the algorithm • Provided the simulation is correct, which gets returned always satisfies the conditions: 1. 2. So, we have a densifier – and a distribution learning algorithm – for halfspaces. 47

DNFs Recall general result: Theorem: Let be a class of Boolean functions such that: (i) is efficiently SQ-learnable; (ii) has a densifier with an output in ; and (iii) has efficient approximate counting and sampling algorithms. Then there is an efficient distribution learning algorithm for . Get (iii) from [Karp. Luby. Madras 89]. What about densifier and SQ learning? 48

Sketch of the densifier for DNFs • Consider a DNF suppose each • Key observation: for each i, So Pr[ . For concreteness, consecutive samples from satisfy same ] is all • If this happens, whp these samples completely identify • The densifier finds candidate terms in this way, outputs OR of all candidate terms. 49

SQ learning for DNFs • Unlike halfspaces, no efficient SQ algorithm for learning DNFs under arbitrary distributions is known; best known runtime is. • But: our densifier identifies “candidate terms” such that f is (essentially) an OR of at most of them. • Can use noise-tolerant SQ learner for sparse disjunctions, applied over “metavariables” (the candidate terms). • Running time is poly(# metavariables). 50

Hardness results 51

Secure signature schemes : (randomized) key generation algorithm; produces key pairs • : signing algorithm; is signature for message using secret key. • : verification algorithm; if • Security guarantee: Given signed messages no poly-time algorithm can produce for a new , such that 52

Connection with our problem Intuition: View messages. . as uniform distribution over signed If, given signed messages, you can (approximately) sample from , this means you can generate new signed messages – contradicts security guarantee! Need to work with a refinement of signature schemes – unique signature schemes [Micali. Rabin. Vadhan 99] – for intuition to go through. Unique signature schemes known to exist under various crypto assumptions (RSA’, Diffie-Hellman’, etc. ) 53

Signature schemes + Cook-Levin Lemma: For any secure signature scheme, there is a secure signature scheme with the same security where the verification algorithm is a 3 -CNF. corresponds to , so security of signature scheme no distribution learning algorithm for 3 -CNF. 54

More hardness Same approach yields hardness for intersections of 2 halfspaces & degree-2 PTFs. (Require parsimonious reductions, efficiently computable/invertible maps between sat. assignments of and sat. assignments of 3 -CNF. ) For monotone 2 CNFs: use the “Blow-up” reduction used in proving hardness of approximate counting for monotone-2 -SAT. Roughly, most sat. assignments of monotone-2 -CNF correspond to sat. assignments of 3 -CNF. 55

Summary of talk • New model: Learning distribution • “Multiplicative accuracy” learning + • Positive results: + + + - -- - ++ + OR AND • Negative results: OR OR 3 -CNFs OR + + ++ -- -- - Intersection of 2 halfspaces AND DNFs Halfspaces AND AND OR OR OR Monotone 2 -CNFs --- -+ + -- +- + ++ + ---- -+ + Degree-2 PTFs 56

Thank you! 57