Strengths and weaknesses of quantum examples Srinivasan Arunachalam - - PowerPoint PPT Presentation

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Strengths and weaknesses

• f quantum examples

Srinivasan Arunachalam (MIT)

joint with Ronald de Wolf (CWI, Amsterdam) and others

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Machine learning

Classical machine learning

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Machine learning

Classical machine learning

Grand goal: enable AI systems to improve themselves

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Machine learning

Classical machine learning

Grand goal: enable AI systems to improve themselves Practical goal: learn“something” from given data

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Machine learning

Classical machine learning

Grand goal: enable AI systems to improve themselves Practical goal: learn“something” from given data Recent success: deep learning is extremely good at image recognition, natural language processing, even the game of Go

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Machine learning

Classical machine learning

Grand goal: enable AI systems to improve themselves Practical goal: learn“something” from given data Recent success: deep learning is extremely good at image recognition, natural language processing, even the game of Go Why the recent interest? Flood of available data, increasing computational power, growing progress in algorithms

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Machine learning

Classical machine learning

Grand goal: enable AI systems to improve themselves Practical goal: learn“something” from given data Recent success: deep learning is extremely good at image recognition, natural language processing, even the game of Go Why the recent interest? Flood of available data, increasing computational power, growing progress in algorithms

Quantum machine learning What can quantum computing do for machine learning?

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Machine learning

Classical machine learning

Grand goal: enable AI systems to improve themselves Practical goal: learn“something” from given data Recent success: deep learning is extremely good at image recognition, natural language processing, even the game of Go Why the recent interest? Flood of available data, increasing computational power, growing progress in algorithms

Quantum machine learning What can quantum computing do for machine learning? The learner will be quantum, the data may be quantum

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Machine learning

Classical machine learning

Grand goal: enable AI systems to improve themselves Practical goal: learn“something” from given data Recent success: deep learning is extremely good at image recognition, natural language processing, even the game of Go Why the recent interest? Flood of available data, increasing computational power, growing progress in algorithms

Quantum machine learning What can quantum computing do for machine learning? The learner will be quantum, the data may be quantum Some examples are known of reduction in time complexity:

clustering (A¨ ımeur et al. ’13) Principal component analysis (Lloyd et al. ’13) perceptron learning (Wiebe et al. ’16) recommendation systems (Kerenidis & Prakash ’16)

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Learning using classical examples

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Learning using classical examples

Basic definitions Concept class C: collection of Boolean functions on n bits (Known)

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Learning using classical examples

Basic definitions Concept class C: collection of Boolean functions on n bits (Known) Target concept c: some function c ∈ C (Unknown)

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Learning using classical examples

Basic definitions Concept class C: collection of Boolean functions on n bits (Known) Target concept c: some function c ∈ C (Unknown) Distribution D : {0, 1}n → [0, 1]

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Learning using classical examples

Basic definitions Concept class C: collection of Boolean functions on n bits (Known) Target concept c: some function c ∈ C (Unknown) Distribution D : {0, 1}n → [0, 1] Labeled example for c ∈ C: (x, c(x)) where x ∼ D

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Classical learner using classical examples

Basic definitions Concept class C: collection of Boolean functions on n bits (Known) Target concept c: some function c ∈ C. (Unknown) Distribution D : {0, 1}n → [0, 1] Labeled example for c ∈ C: (x, c(x)) where x ∼ D Learner is trying to learn c

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Classical learner using classical examples

Basic definitions Concept class C: collection of Boolean functions on n bits (Known) Target concept c: some function c ∈ C. (Unknown) Distribution D : {0, 1}n → [0, 1] Labeled example for c ∈ C: (x, c(x)) where x ∼ D Learner is trying to learn c

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Classical learner using classical examples

Basic definitions Concept class C: collection of Boolean functions on n bits (Known) Target concept c: some function c ∈ C. (Unknown) Distribution D : {0, 1}n → [0, 1] Labeled example for c ∈ C: (x, c(x)) where x ∼ D Learner is trying to learn c

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Classical learner using classical examples

Basic definitions Concept class C: collection of Boolean functions on n bits (Known) Target concept c: some function c ∈ C. (Unknown) Distribution D : {0, 1}n → [0, 1] Labeled example for c ∈ C: (x, c(x)) where x ∼ D Learner is trying to learn c

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Quantum learning using quantum examples

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Quantum learning using quantum examples

Learner is quantum:

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Quantum learning using quantum examples

Learner is quantum: Data is quantum: Bshouty-Jackson’95 introduced a quantum example as a superposition

• x∈{0,1}n
• D(x) |x, c(x)

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Quantum learning using quantum examples

Learner is quantum: Data is quantum: Bshouty-Jackson’95 introduced a quantum example as a superposition

• x∈{0,1}n
• D(x) |x, c(x)

Measuring this state gives a (x, c(x)) with probability D(x),

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Quantum learning using quantum examples

Learner is quantum: Data is quantum: Bshouty-Jackson’95 introduced a quantum example as a superposition

• x∈{0,1}n
• D(x) |x, c(x)

Measuring this state gives a (x, c(x)) with probability D(x), so quantum examples are at least as powerful as classical

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Motivating question for this talk

Fix a concept class C, distribution D : {0, 1}n → [0, 1]

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Motivating question for this talk

Fix a concept class C, distribution D : {0, 1}n → [0, 1]

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Motivating question for this talk

Fix a concept class C, distribution D : {0, 1}n → [0, 1] Question Understanding the concept classes C and distributions D where fewer quantum examples suffice for a quantum learner

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Distribution (in)dependent PAC learning

Focus on Probably Approximately Correct (PAC) model of learning

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Distribution (in)dependent PAC learning

Focus on Probably Approximately Correct (PAC) model of learning Fix C ⊆ {c : {0, 1}n → {0, 1}} and D : {0, 1}n → [0, 1]

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Distribution (in)dependent PAC learning

Focus on Probably Approximately Correct (PAC) model of learning Fix C ⊆ {c : {0, 1}n → {0, 1}} and D : {0, 1}n → [0, 1] Using i.i.d. labeled examples, learner for C should output hypothesis h that is close to c w.r.t. D,

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Distribution (in)dependent PAC learning

Focus on Probably Approximately Correct (PAC) model of learning Fix C ⊆ {c : {0, 1}n → {0, 1}} and D : {0, 1}n → [0, 1] Using i.i.d. labeled examples, learner for C should output hypothesis h that is close to c w.r.t. D, i.e., errD(c, h) = Prx∼D[c(x) = h(x)] should be small

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Distribution (in)dependent PAC learning

Focus on Probably Approximately Correct (PAC) model of learning Fix C ⊆ {c : {0, 1}n → {0, 1}} and D : {0, 1}n → [0, 1] Using i.i.d. labeled examples, learner for C should output hypothesis h that is close to c w.r.t. D, i.e., errD(c, h) = Prx∼D[c(x) = h(x)] should be small Distribution-dependent learning (for a fixed D) An algorithm (ε, δ)-learns C under D if: ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

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Distribution (in)dependent PAC learning

Focus on Probably Approximately Correct (PAC) model of learning Fix C ⊆ {c : {0, 1}n → {0, 1}} and D : {0, 1}n → [0, 1] Using i.i.d. labeled examples, learner for C should output hypothesis h that is close to c w.r.t. D, i.e., errD(c, h) = Prx∼D[c(x) = h(x)] should be small Distribution-dependent learning (for a fixed D) An algorithm (ε, δ)-learns C under D if: ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

PAC learning (Distribution-independent learning for every D)

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Distribution (in)dependent PAC learning

Focus on Probably Approximately Correct (PAC) model of learning Fix C ⊆ {c : {0, 1}n → {0, 1}} and D : {0, 1}n → [0, 1] Using i.i.d. labeled examples, learner for C should output hypothesis h that is close to c w.r.t. D, i.e., errD(c, h) = Prx∼D[c(x) = h(x)] should be small Distribution-dependent learning (for a fixed D) An algorithm (ε, δ)-learns C under D if: ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

PAC learning (Distribution-independent learning for every D) An algorithm (ε, δ)-PAC-learns C if: ∀D ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

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Complexity of learning

How to measure the efficiency of the classical or quantum learner?

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Complexity of learning

How to measure the efficiency of the classical or quantum learner? Sample complexity: number of labeled examples used by learner

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Complexity of learning

How to measure the efficiency of the classical or quantum learner? Sample complexity: number of labeled examples used by learner Time complexity: number of time-steps used by learner

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Complexity of learning

How to measure the efficiency of the classical or quantum learner? Sample complexity: number of labeled examples used by learner Time complexity: number of time-steps used by learner In this talk Strengths of quantum examples

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Complexity of learning

How to measure the efficiency of the classical or quantum learner? Sample complexity: number of labeled examples used by learner Time complexity: number of time-steps used by learner In this talk Strengths of quantum examples ACLW’18: Sample complexity of learning Fourier-sparse Boolean functions under uniform D

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Complexity of learning

How to measure the efficiency of the classical or quantum learner? Sample complexity: number of labeled examples used by learner Time complexity: number of time-steps used by learner In this talk Strengths of quantum examples ACLW’18: Sample complexity of learning Fourier-sparse Boolean functions under uniform D Bshouty-Jackson’95: Quantum polynomial time learnability of DNFs under uniform D

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Complexity of learning

How to measure the efficiency of the classical or quantum learner? Sample complexity: number of labeled examples used by learner Time complexity: number of time-steps used by learner In this talk Strengths of quantum examples ACLW’18: Sample complexity of learning Fourier-sparse Boolean functions under uniform D Bshouty-Jackson’95: Quantum polynomial time learnability of DNFs under uniform D ACKW’18: Quantum examples can help the coupon collector

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Complexity of learning

How to measure the efficiency of the classical or quantum learner? Sample complexity: number of labeled examples used by learner Time complexity: number of time-steps used by learner In this talk Strengths of quantum examples ACLW’18: Sample complexity of learning Fourier-sparse Boolean functions under uniform D Bshouty-Jackson’95: Quantum polynomial time learnability of DNFs under uniform D ACKW’18: Quantum examples can help the coupon collector Weaknesses of quantum examples AW’17: Quantum examples are not more powerful than classical examples for PAC learning

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Fourier sampling: a useful trick under uniform D

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Fourier sampling: a useful trick under uniform D

Let c : {0, 1}n → {−1, 1}.

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Fourier sampling: a useful trick under uniform D

Let c : {0, 1}n → {−1, 1}. Then the Fourier coefficients are

• c(S) = 1

2n

• x∈{0,1}n

c(x)(−1)S·x for all S ∈ {0, 1}n

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Fourier sampling: a useful trick under uniform D

Let c : {0, 1}n → {−1, 1}. Then the Fourier coefficients are

• c(S) = 1

2n

• x∈{0,1}n

c(x)(−1)S·x for all S ∈ {0, 1}n Parseval’s identity:

S

c(S)2 =

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Fourier sampling: a useful trick under uniform D

Let c : {0, 1}n → {−1, 1}. Then the Fourier coefficients are

• c(S) = 1

2n

• x∈{0,1}n

c(x)(−1)S·x for all S ∈ {0, 1}n Parseval’s identity:

S

c(S)2 = Ex[c(x)2]

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Fourier sampling: a useful trick under uniform D

Let c : {0, 1}n → {−1, 1}. Then the Fourier coefficients are

• c(S) = 1

2n

• x∈{0,1}n

c(x)(−1)S·x for all S ∈ {0, 1}n Parseval’s identity:

S

c(S)2 = Ex[c(x)2] = 1

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Fourier sampling: a useful trick under uniform D

Let c : {0, 1}n → {−1, 1}. Then the Fourier coefficients are

• c(S) = 1

2n

• x∈{0,1}n

c(x)(−1)S·x for all S ∈ {0, 1}n Parseval’s identity:

S

c(S)2 = Ex[c(x)2] = 1 So { c(S)2}S forms a probability distribution

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Fourier sampling: a useful trick under uniform D

Let c : {0, 1}n → {−1, 1}. Then the Fourier coefficients are

• c(S) = 1

2n

• x∈{0,1}n

c(x)(−1)S·x for all S ∈ {0, 1}n Parseval’s identity:

S

c(S)2 = Ex[c(x)2] = 1 So { c(S)2}S forms a probability distribution Given quantum example under uniform D: 1 √ 2n

• x

|x, c(x)

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Fourier sampling: a useful trick under uniform D

Let c : {0, 1}n → {−1, 1}. Then the Fourier coefficients are

• c(S) = 1

2n

• x∈{0,1}n

c(x)(−1)S·x for all S ∈ {0, 1}n Parseval’s identity:

S

c(S)2 = Ex[c(x)2] = 1 So { c(S)2}S forms a probability distribution Given quantum example under uniform D: 1 √ 2n

• x

|x, c(x)

Hadamard

− →

• S
• c(S) |S

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Fourier sampling: a useful trick under uniform D

Let c : {0, 1}n → {−1, 1}. Then the Fourier coefficients are

• c(S) = 1

2n

• x∈{0,1}n

c(x)(−1)S·x for all S ∈ {0, 1}n Parseval’s identity:

S

c(S)2 = Ex[c(x)2] = 1 So { c(S)2}S forms a probability distribution Given quantum example under uniform D: 1 √ 2n

• x

|x, c(x)

Hadamard

− →

• S
• c(S) |S

Measuring allows to sample from the Fourier distribution { c(S)2}S

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Applications of Fourier sampling

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n Classical: Ω(n) classical examples needed

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n Classical: Ω(n) classical examples needed Quantum: 1 quantum example suffices to learn C1 (Bernstein-Vazirani’93)

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n Classical: Ω(n) classical examples needed Quantum: 1 quantum example suffices to learn C1 (Bernstein-Vazirani’93) Consider C2 = {c is a ℓ-junta}, i.e., c(x) depends only on ℓ bits of x

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n Classical: Ω(n) classical examples needed Quantum: 1 quantum example suffices to learn C1 (Bernstein-Vazirani’93) Consider C2 = {c is a ℓ-junta}, i.e., c(x) depends only on ℓ bits of x Classical: Efficient learning is notoriously hard for ℓ = O(log n) and uniform D

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n Classical: Ω(n) classical examples needed Quantum: 1 quantum example suffices to learn C1 (Bernstein-Vazirani’93) Consider C2 = {c is a ℓ-junta}, i.e., c(x) depends only on ℓ bits of x Classical: Efficient learning is notoriously hard for ℓ = O(log n) and uniform D Quantum: C2 can be exactly learnt using O(2ℓ) quantum examples and in time

• O(n2ℓ + 22ℓ) (Atıcı-Servedio’09)

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n Classical: Ω(n) classical examples needed Quantum: 1 quantum example suffices to learn C1 (Bernstein-Vazirani’93) Consider C2 = {c is a ℓ-junta}, i.e., c(x) depends only on ℓ bits of x Classical: Efficient learning is notoriously hard for ℓ = O(log n) and uniform D Quantum: C2 can be exactly learnt using O(2ℓ) quantum examples and in time

• O(n2ℓ + 22ℓ) (Atıcı-Servedio’09)

Generalizing both these concept classes? Definition: We say c is k-Fourier sparse if |{S : c(S) = 0}| ≤ k.

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n Classical: Ω(n) classical examples needed Quantum: 1 quantum example suffices to learn C1 (Bernstein-Vazirani’93) Consider C2 = {c is a ℓ-junta}, i.e., c(x) depends only on ℓ bits of x Classical: Efficient learning is notoriously hard for ℓ = O(log n) and uniform D Quantum: C2 can be exactly learnt using O(2ℓ) quantum examples and in time

• O(n2ℓ + 22ℓ) (Atıcı-Servedio’09)

Generalizing both these concept classes? Definition: We say c is k-Fourier sparse if |{S : c(S) = 0}| ≤ k. Note that C1 is 1-Fourier sparse and C2 is 2ℓ-Fourier sparse

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n Classical: Ω(n) classical examples needed Quantum: 1 quantum example suffices to learn C1 (Bernstein-Vazirani’93) Consider C2 = {c is a ℓ-junta}, i.e., c(x) depends only on ℓ bits of x Classical: Efficient learning is notoriously hard for ℓ = O(log n) and uniform D Quantum: C2 can be exactly learnt using O(2ℓ) quantum examples and in time

• O(n2ℓ + 22ℓ) (Atıcı-Servedio’09)

Generalizing both these concept classes? Definition: We say c is k-Fourier sparse if |{S : c(S) = 0}| ≤ k. Note that C1 is 1-Fourier sparse and C2 is 2ℓ-Fourier sparse Consider the concept class C = {c : {0, 1}n → {−1, 1} : c is k-Fourier sparse}

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n Classical: Ω(n) classical examples needed Quantum: 1 quantum example suffices to learn C1 (Bernstein-Vazirani’93) Consider C2 = {c is a ℓ-junta}, i.e., c(x) depends only on ℓ bits of x Classical: Efficient learning is notoriously hard for ℓ = O(log n) and uniform D Quantum: C2 can be exactly learnt using O(2ℓ) quantum examples and in time

• O(n2ℓ + 22ℓ) (Atıcı-Servedio’09)

Generalizing both these concept classes? Definition: We say c is k-Fourier sparse if |{S : c(S) = 0}| ≤ k. Note that C1 is 1-Fourier sparse and C2 is 2ℓ-Fourier sparse Consider the concept class C = {c : {0, 1}n → {−1, 1} : c is k-Fourier sparse} Observe that C1 ⊆ C. C contains linear functions

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Applications of Fourier sampling

Consider the concept class of linear functions C1 = {cS(x) = S · x}S∈{0,1}n Classical: Ω(n) classical examples needed Quantum: 1 quantum example suffices to learn C1 (Bernstein-Vazirani’93) Consider C2 = {c is a ℓ-junta}, i.e., c(x) depends only on ℓ bits of x Classical: Efficient learning is notoriously hard for ℓ = O(log n) and uniform D Quantum: C2 can be exactly learnt using O(2ℓ) quantum examples and in time

• O(n2ℓ + 22ℓ) (Atıcı-Servedio’09)

Generalizing both these concept classes? Definition: We say c is k-Fourier sparse if |{S : c(S) = 0}| ≤ k. Note that C1 is 1-Fourier sparse and C2 is 2ℓ-Fourier sparse Consider the concept class C = {c : {0, 1}n → {−1, 1} : c is k-Fourier sparse} Observe that C1 ⊆ C. C contains linear functions Observe that C2 ⊆ C. C contains (log k)-juntas

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Learning C = {c is k-Fourier sparse}

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Learning C = {c is k-Fourier sparse}

Exact learning C under the uniform distribution D

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Learning C = {c is k-Fourier sparse}

Exact learning C under the uniform distribution D Classically (Haviv-Regev’15): Θ(nk) classical examples (x, c(x)) are necessary and sufficient to learn the concept class C

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Learning C = {c is k-Fourier sparse}

Exact learning C under the uniform distribution D Classically (Haviv-Regev’15): Θ(nk) classical examples (x, c(x)) are necessary and sufficient to learn the concept class C Quantumly (ACLW’18): O(k1.5) quantum examples

1 √ 2n

• x |x, c(x) are

sufficient to learn C (independent of the universe size n)

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Learning C = {c is k-Fourier sparse}

Exact learning C under the uniform distribution D Classically (Haviv-Regev’15): Θ(nk) classical examples (x, c(x)) are necessary and sufficient to learn the concept class C Quantumly (ACLW’18): O(k1.5) quantum examples

1 √ 2n

• x |x, c(x) are

sufficient to learn C (independent of the universe size n)

• Ω(k) examples are necessary to learn C

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Learning C = {c is k-Fourier sparse}

Exact learning C under the uniform distribution D Classically (Haviv-Regev’15): Θ(nk) classical examples (x, c(x)) are necessary and sufficient to learn the concept class C Quantumly (ACLW’18): O(k1.5) quantum examples

1 √ 2n

• x |x, c(x) are

sufficient to learn C (independent of the universe size n)

• Ω(k) examples are necessary to learn C

Sketch of upper bound

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Learning C = {c is k-Fourier sparse}

Exact learning C under the uniform distribution D Classically (Haviv-Regev’15): Θ(nk) classical examples (x, c(x)) are necessary and sufficient to learn the concept class C Quantumly (ACLW’18): O(k1.5) quantum examples

1 √ 2n

• x |x, c(x) are

sufficient to learn C (independent of the universe size n)

• Ω(k) examples are necessary to learn C

Sketch of upper bound Use Fourier sampling to sample S ∼ { c(S)2}S

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Learning C = {c is k-Fourier sparse}

Exact learning C under the uniform distribution D Classically (Haviv-Regev’15): Θ(nk) classical examples (x, c(x)) are necessary and sufficient to learn the concept class C Quantumly (ACLW’18): O(k1.5) quantum examples

1 √ 2n

• x |x, c(x) are

sufficient to learn C (independent of the universe size n)

• Ω(k) examples are necessary to learn C

Sketch of upper bound Use Fourier sampling to sample S ∼ { c(S)2}S Collect Ss until the learner learns the Fourier span of c, V = span{S : c(S) = 0}

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Learning C = {c is k-Fourier sparse}

Exact learning C under the uniform distribution D Classically (Haviv-Regev’15): Θ(nk) classical examples (x, c(x)) are necessary and sufficient to learn the concept class C Quantumly (ACLW’18): O(k1.5) quantum examples

1 √ 2n

• x |x, c(x) are

sufficient to learn C (independent of the universe size n)

• Ω(k) examples are necessary to learn C

Sketch of upper bound Use Fourier sampling to sample S ∼ { c(S)2}S Collect Ss until the learner learns the Fourier span of c, V = span{S : c(S) = 0} Suppose dim(V) = r, then O(rk) quantum examples suffice to find V

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Learning C = {c is k-Fourier sparse}

Exact learning C under the uniform distribution D Classically (Haviv-Regev’15): Θ(nk) classical examples (x, c(x)) are necessary and sufficient to learn the concept class C Quantumly (ACLW’18): O(k1.5) quantum examples

1 √ 2n

• x |x, c(x) are

sufficient to learn C (independent of the universe size n)

• Ω(k) examples are necessary to learn C

Sketch of upper bound Use Fourier sampling to sample S ∼ { c(S)2}S Collect Ss until the learner learns the Fourier span of c, V = span{S : c(S) = 0} Suppose dim(V) = r, then O(rk) quantum examples suffice to find V Use the result of [HR’15] to learn c′ completely using O(rk) classical examples

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Learning C = {c is k-Fourier sparse}

Exact learning C under the uniform distribution D Classically (Haviv-Regev’15): Θ(nk) classical examples (x, c(x)) are necessary and sufficient to learn the concept class C Quantumly (ACLW’18): O(k1.5) quantum examples

1 √ 2n

• x |x, c(x) are

sufficient to learn C (independent of the universe size n)

• Ω(k) examples are necessary to learn C

Sketch of upper bound Use Fourier sampling to sample S ∼ { c(S)2}S Collect Ss until the learner learns the Fourier span of c, V = span{S : c(S) = 0} Suppose dim(V) = r, then O(rk) quantum examples suffice to find V Use the result of [HR’15] to learn c′ completely using O(rk) classical examples Since r ≤ O( √ k) for every c ∈ C [Sanyal’15], we get O(k1.5) upper bound

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Learning Disjunctive normal Forms (DNF)

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Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables.

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Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8)

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Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8) We say a DNF on n variables is an s-term DNF if number of clauses is ≤ s

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Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8) We say a DNF on n variables is an s-term DNF if number of clauses is ≤ s Learning C = {c is an s-term DNF in n variables} under uniform D

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Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8) We say a DNF on n variables is an s-term DNF if number of clauses is ≤ s Learning C = {c is an s-term DNF in n variables} under uniform D Classically: Efficient learning using examples is a longstanding open question. Best known upper bound is nO(log n) [Verbeurgt’90]

12/ 18

Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8) We say a DNF on n variables is an s-term DNF if number of clauses is ≤ s Learning C = {c is an s-term DNF in n variables} under uniform D Classically: Efficient learning using examples is a longstanding open question. Best known upper bound is nO(log n) [Verbeurgt’90] Quantumly: Bshouty-Jackson’95 gave a polynomial-time quantum algorithm!

12/ 18

Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8) We say a DNF on n variables is an s-term DNF if number of clauses is ≤ s Learning C = {c is an s-term DNF in n variables} under uniform D Classically: Efficient learning using examples is a longstanding open question. Best known upper bound is nO(log n) [Verbeurgt’90] Quantumly: Bshouty-Jackson’95 gave a polynomial-time quantum algorithm! Proof sketch of quantum upper bound

12/ 18

Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8) We say a DNF on n variables is an s-term DNF if number of clauses is ≤ s Learning C = {c is an s-term DNF in n variables} under uniform D Classically: Efficient learning using examples is a longstanding open question. Best known upper bound is nO(log n) [Verbeurgt’90] Quantumly: Bshouty-Jackson’95 gave a polynomial-time quantum algorithm! Proof sketch of quantum upper bound Structural property: if c is an s-term DNF, then there exists U s.t. | c(U)| ≥ 1

s

12/ 18

Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8) We say a DNF on n variables is an s-term DNF if number of clauses is ≤ s Learning C = {c is an s-term DNF in n variables} under uniform D Classically: Efficient learning using examples is a longstanding open question. Best known upper bound is nO(log n) [Verbeurgt’90] Quantumly: Bshouty-Jackson’95 gave a polynomial-time quantum algorithm! Proof sketch of quantum upper bound Structural property: if c is an s-term DNF, then there exists U s.t. | c(U)| ≥ 1

s

Fourier sampling! Sample T ∼ { c(T)2}T , poly(s) many times to see such a U

12/ 18

Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8) We say a DNF on n variables is an s-term DNF if number of clauses is ≤ s Learning C = {c is an s-term DNF in n variables} under uniform D Classically: Efficient learning using examples is a longstanding open question. Best known upper bound is nO(log n) [Verbeurgt’90] Quantumly: Bshouty-Jackson’95 gave a polynomial-time quantum algorithm! Proof sketch of quantum upper bound Structural property: if c is an s-term DNF, then there exists U s.t. | c(U)| ≥ 1

s

Fourier sampling! Sample T ∼ { c(T)2}T , poly(s) many times to see such a U Construct a “weak learner” who outputs χU s.t. Pr[χU(x) = c(x)] = 1

2 + 1 s

12/ 18

Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8) We say a DNF on n variables is an s-term DNF if number of clauses is ≤ s Learning C = {c is an s-term DNF in n variables} under uniform D Classically: Efficient learning using examples is a longstanding open question. Best known upper bound is nO(log n) [Verbeurgt’90] Quantumly: Bshouty-Jackson’95 gave a polynomial-time quantum algorithm! Proof sketch of quantum upper bound Structural property: if c is an s-term DNF, then there exists U s.t. | c(U)| ≥ 1

s

Fourier sampling! Sample T ∼ { c(T)2}T , poly(s) many times to see such a U Construct a “weak learner” who outputs χU s.t. Pr[χU(x) = c(x)] = 1

2 + 1 s

Not good enough! Want an hypothesis that agrees with c on most inputs x’s

12/ 18

Learning Disjunctive normal Forms (DNF)

DNFs Simply an OR of AND of variables. For example, (x1 ∧ x4 ∧ x3) ∨ (x4 ∧ x6 ∧ x7 ∧ x8) We say a DNF on n variables is an s-term DNF if number of clauses is ≤ s Learning C = {c is an s-term DNF in n variables} under uniform D Classically: Efficient learning using examples is a longstanding open question. Best known upper bound is nO(log n) [Verbeurgt’90] Quantumly: Bshouty-Jackson’95 gave a polynomial-time quantum algorithm! Proof sketch of quantum upper bound Structural property: if c is an s-term DNF, then there exists U s.t. | c(U)| ≥ 1

s

Fourier sampling! Sample T ∼ { c(T)2}T , poly(s) many times to see such a U Construct a “weak learner” who outputs χU s.t. Pr[χU(x) = c(x)] = 1

2 + 1 s

Not good enough! Want an hypothesis that agrees with c on most inputs x’s Boosting: Run weak learner many times in some manner to obtain a strong learner who outputs h satisfying Pr[h(x) = c(x)] ≥ 2/3

13/ 18

Pretty good measurement for state identification

13/ 18

Pretty good measurement for state identification

Consider a concept class C consisting of n-bit Boolean functions. Let D : {0, 1}n → [0, 1] be a distribution

13/ 18

Pretty good measurement for state identification

Consider a concept class C consisting of n-bit Boolean functions. Let D : {0, 1}n → [0, 1] be a distribution For c ∈ C, a quantum example is |ψc =

x∈{0,1}n

• D(x) |x, c(x)

13/ 18

Pretty good measurement for state identification

Consider a concept class C consisting of n-bit Boolean functions. Let D : {0, 1}n → [0, 1] be a distribution For c ∈ C, a quantum example is |ψc =

x∈{0,1}n

• D(x) |x, c(x)

State identification: For uniform c ∈ C (unknown), given |ψc ⊗T , identify c

13/ 18

Pretty good measurement for state identification

Consider a concept class C consisting of n-bit Boolean functions. Let D : {0, 1}n → [0, 1] be a distribution For c ∈ C, a quantum example is |ψc =

x∈{0,1}n

• D(x) |x, c(x)

State identification: For uniform c ∈ C (unknown), given |ψc ⊗T , identify c Optimal measurement could be quite complicated,

13/ 18

Pretty good measurement for state identification

Consider a concept class C consisting of n-bit Boolean functions. Let D : {0, 1}n → [0, 1] be a distribution For c ∈ C, a quantum example is |ψc =

x∈{0,1}n

• D(x) |x, c(x)

State identification: For uniform c ∈ C (unknown), given |ψc ⊗T , identify c Optimal measurement could be quite complicated, but we can always use the Pretty Good Measurement (PGM)

13/ 18

Pretty good measurement for state identification

Consider a concept class C consisting of n-bit Boolean functions. Let D : {0, 1}n → [0, 1] be a distribution For c ∈ C, a quantum example is |ψc =

x∈{0,1}n

• D(x) |x, c(x)

State identification: For uniform c ∈ C (unknown), given |ψc ⊗T , identify c Optimal measurement could be quite complicated, but we can always use the Pretty Good Measurement (PGM) If Popt is the success probability of the optimal measurement,

13/ 18

Pretty good measurement for state identification

Consider a concept class C consisting of n-bit Boolean functions. Let D : {0, 1}n → [0, 1] be a distribution For c ∈ C, a quantum example is |ψc =

x∈{0,1}n

• D(x) |x, c(x)

State identification: For uniform c ∈ C (unknown), given |ψc ⊗T , identify c Optimal measurement could be quite complicated, but we can always use the Pretty Good Measurement (PGM) If Popt is the success probability of the optimal measurement, Ppgm is the success probability of the PGM,

13/ 18

Pretty good measurement for state identification

Consider a concept class C consisting of n-bit Boolean functions. Let D : {0, 1}n → [0, 1] be a distribution For c ∈ C, a quantum example is |ψc =

x∈{0,1}n

• D(x) |x, c(x)

State identification: For uniform c ∈ C (unknown), given |ψc ⊗T , identify c Optimal measurement could be quite complicated, but we can always use the Pretty Good Measurement (PGM) If Popt is the success probability of the optimal measurement, Ppgm is the success probability of the PGM, then Popt ≥ Ppgm

13/ 18

Pretty good measurement for state identification

Consider a concept class C consisting of n-bit Boolean functions. Let D : {0, 1}n → [0, 1] be a distribution For c ∈ C, a quantum example is |ψc =

x∈{0,1}n

• D(x) |x, c(x)

State identification: For uniform c ∈ C (unknown), given |ψc ⊗T , identify c Optimal measurement could be quite complicated, but we can always use the Pretty Good Measurement (PGM) If Popt is the success probability of the optimal measurement, Ppgm is the success probability of the PGM, then Popt ≥ Ppgm ≥ P2

• pt (Barnum-Knill’02)

14/ 18

Quantum examples help the coupon collector

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons.

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once?

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N)

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons.

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons. Fix unknown i∗ ∈ {1, . . . , N}.

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons. Fix unknown i∗ ∈ {1, . . . , N}. How many coupons to draw (with replacement) from {1, . . . , N}\{i∗}

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons. Fix unknown i∗ ∈ {1, . . . , N}. How many coupons to draw (with replacement) from {1, . . . , N}\{i∗} before learning i∗?

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons. Fix unknown i∗ ∈ {1, . . . , N}. How many coupons to draw (with replacement) from {1, . . . , N}\{i∗} before learning i∗? Answer: Same analysis as earlier shows Θ(N log N)

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons. Fix unknown i∗ ∈ {1, . . . , N}. How many coupons to draw (with replacement) from {1, . . . , N}\{i∗} before learning i∗? Answer: Same analysis as earlier shows Θ(N log N) What if we are given “quantum examples”

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons. Fix unknown i∗ ∈ {1, . . . , N}. How many coupons to draw (with replacement) from {1, . . . , N}\{i∗} before learning i∗? Answer: Same analysis as earlier shows Θ(N log N) What if we are given “quantum examples” Suppose a quantum learner obtains quantum examples

1 √N−1

• i∈({1,...,N}\{i∗}) |i.

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons. Fix unknown i∗ ∈ {1, . . . , N}. How many coupons to draw (with replacement) from {1, . . . , N}\{i∗} before learning i∗? Answer: Same analysis as earlier shows Θ(N log N) What if we are given “quantum examples” Suppose a quantum learner obtains quantum examples

1 √N−1

• i∈({1,...,N}\{i∗}) |i.

How many quantum examples before learning i∗?

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons. Fix unknown i∗ ∈ {1, . . . , N}. How many coupons to draw (with replacement) from {1, . . . , N}\{i∗} before learning i∗? Answer: Same analysis as earlier shows Θ(N log N) What if we are given “quantum examples” Suppose a quantum learner obtains quantum examples

1 √N−1

• i∈({1,...,N}\{i∗}) |i.

How many quantum examples before learning i∗? Answer [ACKW’..]: Can learn i∗ using Θ(N) quantum examples

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons. Fix unknown i∗ ∈ {1, . . . , N}. How many coupons to draw (with replacement) from {1, . . . , N}\{i∗} before learning i∗? Answer: Same analysis as earlier shows Θ(N log N) What if we are given “quantum examples” Suppose a quantum learner obtains quantum examples

1 √N−1

• i∈({1,...,N}\{i∗}) |i.

How many quantum examples before learning i∗? Answer [ACKW’..]: Can learn i∗ using Θ(N) quantum examples Proof idea: Analyze the success probability using the pretty good measurement.

14/ 18

Quantum examples help the coupon collector

Standard coupon collector Problem: Suppose there are N coupons. How many coupons to draw (with replacement) before having seen each coupon at least once? Answer: Simple probability analysis shows Θ(N log N) Variation to coupon collector Problem: Suppose there are N coupons. Fix unknown i∗ ∈ {1, . . . , N}. How many coupons to draw (with replacement) from {1, . . . , N}\{i∗} before learning i∗? Answer: Same analysis as earlier shows Θ(N log N) What if we are given “quantum examples” Suppose a quantum learner obtains quantum examples

1 √N−1

• i∈({1,...,N}\{i∗}) |i.

How many quantum examples before learning i∗? Answer [ACKW’..]: Can learn i∗ using Θ(N) quantum examples Proof idea: Analyze the success probability using the pretty good measurement. If T = O(N), then Popt ≥ Ppgm ≥ 2/3

15/ 18

Distribution-independent learning

Recall: PAC learning

15/ 18

Distribution-independent learning

Recall: PAC learning Given (x, c(x)) examples where x ∼ D, a learner (ε, δ)-PAC-learns C if: ∀D ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

15/ 18

Distribution-independent learning

Recall: PAC learning Given (x, c(x)) examples where x ∼ D, a learner (ε, δ)-PAC-learns C if: ∀D ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

Complexity measure: Number of labelled examples

15/ 18

Distribution-independent learning

Recall: PAC learning Given (x, c(x)) examples where x ∼ D, a learner (ε, δ)-PAC-learns C if: ∀D ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

Complexity measure: Number of labelled examples For a concept class C, associate a combinatorial parameter called VC-dimension of C.

15/ 18

Distribution-independent learning

Recall: PAC learning Given (x, c(x)) examples where x ∼ D, a learner (ε, δ)-PAC-learns C if: ∀D ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

Complexity measure: Number of labelled examples For a concept class C, associate a combinatorial parameter called VC-dimension of C. Classical PAC learning sample complexity is characterized by the VC-dimension of C

15/ 18

Distribution-independent learning

Recall: PAC learning Given (x, c(x)) examples where x ∼ D, a learner (ε, δ)-PAC-learns C if: ∀D ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

Complexity measure: Number of labelled examples For a concept class C, associate a combinatorial parameter called VC-dimension of C. Classical PAC learning sample complexity is characterized by the VC-dimension of C Fundamental theorem of PAC learning

15/ 18

Distribution-independent learning

Recall: PAC learning Given (x, c(x)) examples where x ∼ D, a learner (ε, δ)-PAC-learns C if: ∀D ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

Complexity measure: Number of labelled examples For a concept class C, associate a combinatorial parameter called VC-dimension of C. Classical PAC learning sample complexity is characterized by the VC-dimension of C Fundamental theorem of PAC learning Suppose VC-dim(C) = d

15/ 18

Distribution-independent learning

Recall: PAC learning Given (x, c(x)) examples where x ∼ D, a learner (ε, δ)-PAC-learns C if: ∀D ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

Complexity measure: Number of labelled examples For a concept class C, associate a combinatorial parameter called VC-dimension of C. Classical PAC learning sample complexity is characterized by the VC-dimension of C Fundamental theorem of PAC learning Suppose VC-dim(C) = d Blumer-Ehrenfeucht-Haussler-Warmuth’86: every (ε, δ)-PAC learner for C needs Ω

• d

ε + log(1/δ) ε

• examples

15/ 18

Distribution-independent learning

Recall: PAC learning Given (x, c(x)) examples where x ∼ D, a learner (ε, δ)-PAC-learns C if: ∀D ∀c ∈ C : Pr[ errD(c, h) ≤ ε

• Approximately Correct

] ≥ 1 − δ

Probably

Complexity measure: Number of labelled examples For a concept class C, associate a combinatorial parameter called VC-dimension of C. Classical PAC learning sample complexity is characterized by the VC-dimension of C Fundamental theorem of PAC learning Suppose VC-dim(C) = d Blumer-Ehrenfeucht-Haussler-Warmuth’86: every (ε, δ)-PAC learner for C needs Ω

• d

ε + log(1/δ) ε

• examples

Hanneke’16: exists an (ε, δ)-PAC learner for C using O

• d

ε + log(1/δ) ε

• examples

16/ 18

VC-dimension and quantum sample complexity

16/ 18

VC-dimension and quantum sample complexity

Quantum bounds Classical upper bound O

• d

ε + log(1/δ) ε

• carries over to quantum

16/ 18

VC-dimension and quantum sample complexity

Quantum bounds Classical upper bound O

• d

ε + log(1/δ) ε

• carries over to quantum

Atıcı-Servedio’04: lower bound Ω √

d ε + log(1/δ) ε

16/ 18

VC-dimension and quantum sample complexity

Quantum bounds Classical upper bound O

• d

ε + log(1/δ) ε

• carries over to quantum

Atıcı-Servedio’04: lower bound Ω √

d ε + log(1/δ) ε

• AW’17: Showed Ω
• d

ε + log(1/δ) ε

• quantum examples are necessary

16/ 18

VC-dimension and quantum sample complexity

Quantum bounds Classical upper bound O

• d

ε + log(1/δ) ε

• carries over to quantum

Atıcı-Servedio’04: lower bound Ω √

d ε + log(1/δ) ε

• AW’17: Showed Ω
• d

ε + log(1/δ) ε

• quantum examples are necessary

Proof idea: Reduce to state identification.

16/ 18

VC-dimension and quantum sample complexity

Quantum bounds Classical upper bound O

• d

ε + log(1/δ) ε

• carries over to quantum

Atıcı-Servedio’04: lower bound Ω √

d ε + log(1/δ) ε

• AW’17: Showed Ω
• d

ε + log(1/δ) ε

• quantum examples are necessary

Proof idea: Reduce to state identification. For a good learner Popt ≥ 2/3, so Ppgm ≥ P2

• pt ≥ 4/9.

16/ 18

VC-dimension and quantum sample complexity

Quantum bounds Classical upper bound O

• d

ε + log(1/δ) ε

• carries over to quantum

Atıcı-Servedio’04: lower bound Ω √

d ε + log(1/δ) ε

• AW’17: Showed Ω
• d

ε + log(1/δ) ε

• quantum examples are necessary

Proof idea: Reduce to state identification. For a good learner Popt ≥ 2/3, so Ppgm ≥ P2

• pt ≥ 4/9. If Ppgm ≥ 4/9, then T = Ω
• d

ε

16/ 18

VC-dimension and quantum sample complexity

Quantum bounds Classical upper bound O

• d

ε + log(1/δ) ε

• carries over to quantum

Atıcı-Servedio’04: lower bound Ω √

d ε + log(1/δ) ε

• AW’17: Showed Ω
• d

ε + log(1/δ) ε

• quantum examples are necessary

Proof idea: Reduce to state identification. For a good learner Popt ≥ 2/3, so Ppgm ≥ P2

• pt ≥ 4/9. If Ppgm ≥ 4/9, then T = Ω
• d

ε

• Quantum examples are no better than classical examples for PAC learning

16/ 18

VC-dimension and quantum sample complexity

Quantum bounds Classical upper bound O

• d

ε + log(1/δ) ε

• carries over to quantum

Atıcı-Servedio’04: lower bound Ω √

d ε + log(1/δ) ε

• AW’17: Showed Ω
• d

ε + log(1/δ) ε

• quantum examples are necessary

Proof idea: Reduce to state identification. For a good learner Popt ≥ 2/3, so Ppgm ≥ P2

• pt ≥ 4/9. If Ppgm ≥ 4/9, then T = Ω
• d

ε

• Quantum examples are no better than classical examples for PAC learning

Let’s get real! In computational learning theory, agnostic learning and learning under classification noise is a theoretical way to model noise in data

16/ 18

VC-dimension and quantum sample complexity

Quantum bounds Classical upper bound O

• d

ε + log(1/δ) ε

• carries over to quantum

Atıcı-Servedio’04: lower bound Ω √

d ε + log(1/δ) ε

• AW’17: Showed Ω
• d

ε + log(1/δ) ε

• quantum examples are necessary

Proof idea: Reduce to state identification. For a good learner Popt ≥ 2/3, so Ppgm ≥ P2

• pt ≥ 4/9. If Ppgm ≥ 4/9, then T = Ω
• d

ε

• Quantum examples are no better than classical examples for PAC learning

Let’s get real! In computational learning theory, agnostic learning and learning under classification noise is a theoretical way to model noise in data Again, in these realistic models we show that quantum sample complexity equals classical sample complexity

17/ 18

Future directions

17/ 18

Future directions

More mileage out of Fourier sampling?

17/ 18

Future directions

More mileage out of Fourier sampling? Extend result of Bshouty-Jackson from depth-2 circuits (i.e., DNFs) to depth-3?

17/ 18

Future directions

More mileage out of Fourier sampling? Extend result of Bshouty-Jackson from depth-2 circuits (i.e., DNFs) to depth-3? Can we PAC-learn DNFs? If so, then we could possibly learn depth-3 circuits under the uniform distribution

17/ 18

Future directions

More mileage out of Fourier sampling? Extend result of Bshouty-Jackson from depth-2 circuits (i.e., DNFs) to depth-3? Can we PAC-learn DNFs? If so, then we could possibly learn depth-3 circuits under the uniform distribution Scott Aaronson: Can AC0 be learnt in quantum polynomial time? (One of his ten semi-grand challenges for quantum computing!)

17/ 18

Future directions

More mileage out of Fourier sampling? Extend result of Bshouty-Jackson from depth-2 circuits (i.e., DNFs) to depth-3? Can we PAC-learn DNFs? If so, then we could possibly learn depth-3 circuits under the uniform distribution Scott Aaronson: Can AC0 be learnt in quantum polynomial time? (One of his ten semi-grand challenges for quantum computing!) Can TC0 be learnt in quantum polynomial time? A theoretical way to understand neural networks

17/ 18

Future directions

More mileage out of Fourier sampling? Extend result of Bshouty-Jackson from depth-2 circuits (i.e., DNFs) to depth-3? Can we PAC-learn DNFs? If so, then we could possibly learn depth-3 circuits under the uniform distribution Scott Aaronson: Can AC0 be learnt in quantum polynomial time? (One of his ten semi-grand challenges for quantum computing!) Can TC0 be learnt in quantum polynomial time? A theoretical way to understand neural networks Can we learn constant-depth quantum circuits?

17/ 18

Future directions

More mileage out of Fourier sampling? Extend result of Bshouty-Jackson from depth-2 circuits (i.e., DNFs) to depth-3? Can we PAC-learn DNFs? If so, then we could possibly learn depth-3 circuits under the uniform distribution Scott Aaronson: Can AC0 be learnt in quantum polynomial time? (One of his ten semi-grand challenges for quantum computing!) Can TC0 be learnt in quantum polynomial time? A theoretical way to understand neural networks Can we learn constant-depth quantum circuits? More open questions!

17/ 18

Future directions

More mileage out of Fourier sampling? Extend result of Bshouty-Jackson from depth-2 circuits (i.e., DNFs) to depth-3? Can we PAC-learn DNFs? If so, then we could possibly learn depth-3 circuits under the uniform distribution Scott Aaronson: Can AC0 be learnt in quantum polynomial time? (One of his ten semi-grand challenges for quantum computing!) Can TC0 be learnt in quantum polynomial time? A theoretical way to understand neural networks Can we learn constant-depth quantum circuits? More open questions! Can we learn the concept class of k-Fourier sparse Boolean functions using O(k log k) samples matching our lower bound?

17/ 18

Future directions

More mileage out of Fourier sampling? Extend result of Bshouty-Jackson from depth-2 circuits (i.e., DNFs) to depth-3? Can we PAC-learn DNFs? If so, then we could possibly learn depth-3 circuits under the uniform distribution Scott Aaronson: Can AC0 be learnt in quantum polynomial time? (One of his ten semi-grand challenges for quantum computing!) Can TC0 be learnt in quantum polynomial time? A theoretical way to understand neural networks Can we learn constant-depth quantum circuits? More open questions! Can we learn the concept class of k-Fourier sparse Boolean functions using O(k log k) samples matching our lower bound? Theoretically, one could consider more optimistic PAC-like models where learner need not succeed ∀c ∈ C and ∀D

17/ 18

Future directions

More mileage out of Fourier sampling? Extend result of Bshouty-Jackson from depth-2 circuits (i.e., DNFs) to depth-3? Can we PAC-learn DNFs? If so, then we could possibly learn depth-3 circuits under the uniform distribution Scott Aaronson: Can AC0 be learnt in quantum polynomial time? (One of his ten semi-grand challenges for quantum computing!) Can TC0 be learnt in quantum polynomial time? A theoretical way to understand neural networks Can we learn constant-depth quantum circuits? More open questions! Can we learn the concept class of k-Fourier sparse Boolean functions using O(k log k) samples matching our lower bound? Theoretically, one could consider more optimistic PAC-like models where learner need not succeed ∀c ∈ C and ∀D Find more distributions (other than uniform) where quantum provides a speedup

18/ 18

Conclusion

For PAC learning, quantum examples are no better than classical examples

18/ 18

Conclusion

For PAC learning, quantum examples are no better than classical examples Under uniform D, quantum examples seem to help tremendously in some cases

18/ 18

Conclusion

For PAC learning, quantum examples are no better than classical examples Under uniform D, quantum examples seem to help tremendously in some cases Quantum machine learning is still in its infancy! Not many strong examples where quantum significantly improves ML

18/ 18

Conclusion

For PAC learning, quantum examples are no better than classical examples Under uniform D, quantum examples seem to help tremendously in some cases Quantum machine learning is still in its infancy! Not many strong examples where quantum significantly improves ML Many recent surveys on quantum machine learning.