Robust Quantum Minimum Finding with an application to hypothesis - PowerPoint PPT Presentation

Robust Quantum Minimum Finding with an application to hypothesis selection Yihui Quek (Joint work with Clement Canonne (IBM Research Almaden), Patrick Rebentrost (CQT-NUS)) Stanford University 21 Apr 2020 1/30

Problem: Hypothesis selection Problem (Hypothesis selection) Given ◮ Unknown probability distribution p 0 , sample access to it. ◮ N known candidate distributions: P = { p 1 , . . . p N } ; PDF comparator between every pair. Task: Output a distribution ˆ p ∈ P with small ℓ 1 -distance to p 0 with as few samples from p 0 as possible. Remark: Maximum likelihood does not work for ℓ 1 -distance. 2/30

A closely related problem Problem (Robust minimum finding) Given ◮ A list of N elements { x i } N i =1 ◮ A well-defined distance metric d ( x i , x j ) Task: Find the minimum using an imprecise pairwise comparator between elements. ◮ Comparator imprecision: Outputs correct answer if the elements are far enough apart; otherwise no guarantees. √ ◮ Result: can do this in ˜ O ( N ) comparator invocations. 3/30

Noisy Comparator Input to comparator: indices i , j . � argmin { x i , x j } if d ( x i , x j ) > 1 NoisyComp( i , j ) = unknown (possibly adversarial) otherwise. (1) Definition (Oracle notation) Will denote oracle implementing noisy comparator as ˆ O and noiseless comparator as ˆ O (0) . Will count the number of calls of either ˆ O or ˆ O (0) . 4/30

Classical noisy minimum selection Definition ( t -approximation) An element y ∈ L is a t-approximation of the true minimum y ∗ if it satisfies d ( y , y ∗ ) < t. Lemma (Optimal approximation guarantee) To get a t-approximation for t < 2 , P [ error ] ≥ 1 1 2 − 2N . Hence, will aim for a 2-approximation guarantee. 5/30

Classical noisy minimum selection, part 2 Run time dependence on N ? Classically, linear is optimal. Theorem ( COMB (Theorem 15 of [AFJOS’16])) A classical randomized algorithm, COMB ( δ, S ) , outputs a N log 1 � � 2 -approximation of the minimum w.p ≥ 1 − δ , using O δ queries to the noisy comparator. √ We will do this in sublinear – i.e. ˜ O ( N ) time. Assumption There exists ∆ ∈ [ N ] ′ such that at most 2∆ elements are contained in the fudge zone of any element in the list. ◮ Reasonable assumption in most cases, including hypothesis selection. 6/30

Recap: D¨ urr-Høyer Quantum Minimum Finding Figure: Durr-Hoyer ’96. Exponential search algorithm = BBHT ’98. Key point: quantum exponential search rapidly moves the pivot to lower ranks. 7/30

Some initial observations ◮ What happens if we naively run D¨ urr-Høyer with noisy comparator? Problem: we could in principle go back up the ranks! ◮ Will show that we still make (on expectation) positive progress down the ranks, if rank of pivot is Ω(1 + ∆). ◮ However, this stops working when pivot is o (1 + ∆) ranks from the minimum. ◮ V1 algorithm: stop iterating when pivot is, on expectation, ≤ O (1 + ∆) ranks from the minimum: already an improvement from O ( N ). 8/30

Subroutine: QSearchWithCutoff ◮ We add an explicit run time cutoff to exponential search and allow to use noisy oracle, ˆ O . Lemma Let the current pivot y be of rank r > ∆ . Then � QSearchWithCutoff ( ˆ N O , y , 9 r − ∆ ) succeeds in finding a marked element with probability at least 1 2 . Proof. Follows from running for twice the expected runtime. 9/30

Will present 3 algorithms. Algorithm 1: Pivot Counting QMF

Algorithm 1: Pivot-Counting QMF Differs from D¨ urr-Høyer Quantum Minimum Finding in 2 ways: ◮ At each iteration, we run QSearchWithCutoff with a � N finite cutoff at 9 1+∆ – i.e. may fail to change the pivot. ◮ Instead of cutting off total runtime, we cut off the number of runs of QSearchWithCutoff at N trials runs. ill differentiate between ‘attempted’ vs. ‘successful’ pivot change. 11/30

PCQMF pseudocode Algorithm 1 Pivot-Counting Quantum Minimum Finding Piv- otQMF ( ˆ O , ∆, N trials ) Input: comparison oracle ˆ O , ∆ ∈ [ N ] ′ , N trials ∈ Z + k ← 0. y ← Unif[ N ]. for k ∈ [ N trials ] do � ( y ′ , 0) ← QSearchWithCutoff ( ˆ N O , y , 9 1+∆ ). if O y ( y ′ ) = 1 then y ← y ′ . Output: y For the next few slides, we will pretend all pivot changes are successful. 12/30

Lemma: worst-case expected improvement ◮ For ‘high’ rank’: Lemma (Worst-case expected rank change for r i > 3(∆ + 1)) For r i > 3(∆ + 1) , r i +∆ 1 s = r i + ∆ + 1 < 2 r i � E [ r i +1 | r i ] ≤ 3 . r i + ∆ 2 s =1 ◮ For ‘low’ rank: Lemma For r i ≤ 3(∆ + 1) , E [ r i +1 | r i ] ≤ 4∆ + 3 . 13/30

Crucial intuition With noiseless comparator: E [ r i +1 | r i ] ≤ r i 2 . i.e. length of ‘remaining list’ halves with every pivot change. ◮ With every successful pivot change, we still make positive progress down the ranks with noisy comparator! – with worse factor: 2 / 3. ◮ Hence number of successful pivot changes necessary still logarithmic-in- N ∼ log 3 / 2 ( N ). 14/30

Completing the argument � � log ( 4∆+3 ) N Argument so far: with N p = successful pivot changes, log(3 / 2) expected final rank ≤ 4∆ + 3. To finish off: ◮ Each attempt succeeds with probability > 1 2 . Chernoff bound number of attempts to get N p successful pivot changes. ◮ Markov’s inequality bounds actual final rank (pay a multiplicative factor). Name Success prob. Final guarantee Run time � 3 ˜ N rank( y ) ≤ 16∆ + 16 O ( 1+∆ ) PivotQMF 4 15/30

Algorithm 2: Repeated PivotQMF

Algorithm 2: Repeated PivotQMF ◮ Use PivotQMF as a basic subroutine to find some element of “pretty-good” rank with constant probability ◮ Repeat log(1 /δ ) times to get a pool of indices. ◮ Use classical min selection on the pool. Algorithm 2 Repeated Pivot Quantum Minimum Finding Repeat- edPivotQMF ( ˆ O , δ, ∆) Input: ˆ O , δ, ∆ S ← ∅ . Stage I: Finding pretty-small element w.h.p. for i = 1 , . . . log 4 (2 /δ ) do � � O , ∆ , ⌈ 8 max( log( N / (4∆+3) ˆ y ← PivotQMF , 2 ln N ) ⌉ log(3 / 2) S ← S ∪ { y } . Stage II: Classical minimum selection with noisy comparator Perform COMB ( δ/ 2, S ). Output: Output of COMB ( δ/ 2, S ). 17/30

Guarantees Success Final guarantee Run time prob. � ˜ 1+∆ log( N ) + log 2 (1 /δ )) N 1 − δ rank( y ) ≤ 18∆ + 16 O ( Intuition: ◮ Use quick and dirty quantum subroutine to find a ‘representative’ element. Bootstrap with repetitions. ◮ Use slow and precise classical subroutine to select the best of the repetitions. 18/30

Algorithm 3: RobustQMF

RobustQMF overview ◮ Rank approximation guarantee of PivotQMF can be strengthened to a distance guarantee. ◮ Key idea of RobustQMF : ◮ Run PivotQMF , get a “pretty-good element” Y out ∼ O (∆ + 1) rank ◮ ∗ Get classical sublist of elements ranking below Y out ◮ Run classical minimum-selection algorithm. ◮ Final approximation guarantee: a 2-approximation of the minimum! ∼ optimal. ∗ This almost works, but our runtime cutoff for exponential search � N at 9 1+∆ now comes back to bite us... 20/30

Two refinements ◮ Getting the classical sublist of elements ranking below Y out ◮ Fix Y out as a pivot, then repeatedly apply � QSearchWithCutoff ( ˆ N O , Y out , 9 1+∆ ). ◮ The run time cutoff problem: For Y out of rank < 1 + 2∆, � N expected run time of exponential search is > O ( 1+∆ )), hence the above run time cutoff may be premature! ◮ Denominator of run time cutoff comes from number of marked elements. ◮ Append K dummy elements to the list that will always be marked ( K = 2∆ works). 21/30

RobustQMF Algorithm 3 Robust Quantum Minimum Finding Ro- bustQMF ( ˆ O , δ, ∆) Input: ˆ O , δ Stage I: Finding a “pretty-small” element with RepeatedPiv- otQMF Y out ← RepeatedPivotQMF ( ˆ O , δ/ 2) Stage II: Finding even smaller elements S ← { Y out } ˜ T ← 19∆ + 16 δ ) ˜ for i = 1 , . . . 2 ln 2 log( 4 T do � ( y k , g ) ← QSearchWithCutoff ( ˆ N O , Y out , 9 1+∆ , list = L ′ ). if O Y out ( y k ) = 1 , and y k is not a dummy element then S ← S ∪{ y k } Remove repeated elements from S . Stage III: Classical minimum-selection with noisy comparator Perform COMB ( δ/ 4, S ). Output: Output of COMB ( δ/ 4, S ). 22/30

Table of main algorithms Name Success Final guarantee Run time prob. � ˜ 3 N rank( y ) ≤ 16∆ + 16 O ( 1+∆ ) PivotQMF 4 � ˜ N 1 − δ rank( y ) ≤ 18∆ + 16 O ( 1+∆ ) Repeated PivotQMF ˜ � RobustQMF 1 − δ d ( y , y ∗ ) ≤ 2 (optimal) O ( N (1 + ∆)) Table: Comparison of quantum minimum-finding algorithms. y ∗ : true minimum. 23/30

Application: sublinear-time hypothesis selection

Scheff´ e test (algorithm) Unknown distribution q (sample access). Want to choose the closer of two hypotheses, p i , p j . Algorithm 4 Scheff´ e test Input: Access to distributions p i ∈ P and p j ∈ P , { x k } K k =1 i.i.d. samples from unknown distribution q . Compute µ S = 1 � K k =1 1 x k ∈S ij . K Output: If | p i ( S ij ) − µ S | ≤ | p j ( S ij ) − µ S | output p i , otherwise output p j 25/30

Algorithm by picture For a pair of hypotheses, p 1 , p 2 ... Figure: Scheffe test 26/30