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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

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Problem: Hypothesis selection

Problem (Hypothesis selection)

Given ◮ Unknown probability distribution p0, sample access to it. ◮ N known candidate distributions: P = {p1, . . . pN}; PDF comparator between every pair. Task: Output a distribution ˆ p ∈ P with small ℓ1-distance to p0 with as few samples from p0 as possible. Remark: Maximum likelihood does not work for ℓ1-distance.

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A closely related problem

Problem (Robust minimum finding)

Given ◮ A list of N elements {xi}N

i=1

◮ A well-defined distance metric d(xi, xj) 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.

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Noisy Comparator

Input to comparator: indices i, j. NoisyComp(i, j) = argmin {xi, xj} if d(xi, xj) > 1 unknown (possibly adversarial)

• therwise.

(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).

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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

2 − 1 2N.

Hence, will aim for a 2-approximation guarantee.

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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 2-approximation of the minimum w.p ≥ 1 − δ, using O

• N log 1

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.

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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.

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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).

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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 O, y, 9

• N

r−∆) succeeds in finding a

marked element with probability at least 1

2.

Proof.

Follows from running for twice the expected runtime.

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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 finite cutoff at 9

• N

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 Ntrials runs. ill differentiate between ‘attempted’ vs. ‘successful’ pivot change.

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PCQMF pseudocode

Algorithm 1 Pivot-Counting Quantum Minimum Finding Piv-

• tQMF( ˆ

O, ∆, Ntrials) Input: comparison oracle ˆ O, ∆ ∈ [N]′, Ntrials ∈ Z+ k ← 0. y ← Unif[N]. for k ∈ [Ntrials] do (y′, 0) ← QSearchWithCutoff( ˆ O, y, 9

• N

1+∆).

if Oy(y′) = 1 then y ← y′. Output: y For the next few slides, we will pretend all pivot changes are successful.

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Lemma: worst-case expected improvement

◮ For ‘high’ rank’:

Lemma (Worst-case expected rank change for ri > 3(∆ + 1))

For ri > 3(∆ + 1), E[ri+1 | ri] ≤ 1 ri + ∆

ri+∆

• s=1

s = ri + ∆ + 1 2 < 2ri 3 . ◮ For ‘low’ rank:

Lemma

For ri ≤ 3(∆ + 1), E[ri+1 | ri] ≤ 4∆ + 3.

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Crucial intuition

With noiseless comparator: E[ri+1 | ri] ≤ ri 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 ∼ log3/2(N).

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Completing the argument

Argument so far: with Np =

• log(

N 4∆+3)

log(3/2)

• successful pivot changes,

expected final rank ≤ 4∆ + 3. To finish off: ◮ Each attempt succeeds with probability > 1

• 2. Chernoff bound

number of attempts to get Np successful pivot changes. ◮ Markov’s inequality bounds actual final rank (pay a multiplicative factor). Name Success prob. Final guarantee Run time PivotQMF

3 4

rank(y) ≤ 16∆ + 16 ˜ O(

• N

1+∆)

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Algorithm 2: Repeated PivotQMF

Algorithm 2: Repeated PivotQMF

◮ Use PivotQMF as a basic subroutine to find some element

• f “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, . . . log4(2/δ) do y ← PivotQMF

• ˆ

O, ∆, ⌈8 max( log(N/(4∆+3)

log(3/2)

, 2 ln N)⌉

• S ← S ∪ {y}.

Stage II: Classical minimum selection with noisy comparator Perform COMB(δ/2, S). Output: Output of COMB(δ/2, S).

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Guarantees

Success prob. Final guarantee Run time 1 − δ rank(y) ≤ 18∆ + 16 ˜ O(

• N

1+∆ log(N) + log2(1/δ))

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.

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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” Yout ∼ O(∆ + 1) rank ◮ ∗Get classical sublist of elements ranking below Yout ◮ 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

at 9

• N

1+∆ now comes back to bite us...

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Two refinements

◮ Getting the classical sublist of elements ranking below Yout

◮ Fix Yout as a pivot, then repeatedly apply QSearchWithCutoff( ˆ O, Yout, 9

• N

1+∆).

◮ The run time cutoff problem: For Yout of rank < 1 + 2∆, expected run time of exponential search is > O(

• N

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).

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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-

• tQMF

Yout ← RepeatedPivotQMF( ˆ O, δ/2) Stage II: Finding even smaller elements S ← {Yout} ˜ T ← 19∆ + 16 for i = 1, . . . 2 ln 2 log( 4

δ) ˜

T do (yk, g) ← QSearchWithCutoff( ˆ O, Yout, 9

• N

1+∆, list = L′).

if OYout(yk) = 1, and yk is not a dummy element then S ← S∪{yk} 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).

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Table of main algorithms

Name Success prob. Final guarantee Run time PivotQMF

3 4

rank(y) ≤ 16∆ + 16 ˜ O(

• N

1+∆)

Repeated PivotQMF 1 − δ rank(y) ≤ 18∆ + 16 ˜ O(

• N

1+∆)

RobustQMF 1 − δ d(y, y ∗) ≤ 2 (optimal) ˜ O(

• N(1 + ∆))

Table: Comparison of quantum minimum-finding algorithms. y ∗: true minimum.

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Application: sublinear-time hypothesis selection

Scheff´ e test (algorithm)

Unknown distribution q (sample access). Want to choose the closer of two hypotheses, pi, pj. Algorithm 4 Scheff´ e test Input: Access to distributions pi ∈ P and pj ∈ P, {xk}K

k=1 i.i.d. samples from unknown distribution q.

Compute µS = 1

K

K

k=1 1xk∈Sij.

Output: If |pi(Sij) − µS| ≤ |pj(Sij) − µS| output pi, otherwise

• utput pj

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Algorithm by picture

For a pair of hypotheses, p1, p2 ...

Figure: Scheffe test

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Scheff´ e test guarantees

W.p. 1 − δ, the Scheff´ e test between two hypotheses outputs an index i ∈ [2] such that pi − q1 ≤ 3 min

j∈[2]pj − q1 + ε

(2) with ˜ O(log(1/δ)/ε2) samples. ◮ Scheff´ e test functions as a noisy comparator on two hypotheses’ ℓ1-distance to the unknown distribution p0. ◮ Sample-optimal: note no dependence on domain size.

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Map to Robust Quantum Minimum Finding

That is, w.h.p, output of Scheff´ e test satisfies    pi if pi − q1 < 1

3 pj − q1

pj if pj − q1 < 1

3 pi − q1

either pi or pj

• therwise.

(3) Taking xi = − log3 pi − p01, reduces to familiar noisy comparator. NComp(i, j) = argmin {xi, xj} if d(xi, xj) > 1 unknown (possibly adversarial)

• therwise.

(4)

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Theorem for sublinear time hypothesis selection

Theorem (Robust QMF for hypothesis selection)

Given Assumption 1, there exists a quantum algorithm with expected number of oracle queries O

• N

1 + ∆ ˜ R log 1 δ

• + (1 + ∆) log

1 δ

• ,

where ˜ R := max (O (log (N)) , O (∆ + 1)), that with probability at least 1 − δ outputs a hypothesis ˆ p that satisfies ˆ p − q1 ≤ 9 min

p∈Pδp − q1 + 4

• 10 log (N

2)

δ

k . (5)

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Concluding remarks

◮ We have exhibited an algorithm that preserves the quadratic speedup of Durr-Hoyer even with a noisy comparator and

• btains optimal approximation guarantees.

◮ Can do hypothesis selection on N hypotheses in sublinear-in-N time with optimal sample complexity. ◮ Take-home message:

◮ Use a quick and dirty quantum subroutine to reduce ‘problem size’ (here, find a pretty-good element); finish off with a slow and precise classical subroutine.

◮ Open questions:

◮ In which other contexts is this comparator model valid? ◮ Can we do without the assumption that the fudge zone is upper-bounded by ∆?

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