Randomness in Computing L ECTURE 2 Last time Verifying polynomial - - PowerPoint PPT Presentation

1/22/2020

Randomness in Computing

LECTURE 2

Last time

• Verifying polynomial identities
• Probability amplification
• Probability review

Discussions

• Law of Total Probability
• Bayes’law

Today

• More probability amplification
• Verifying matrix multiplication

Sofya Raskhodnikova;Randomness in Computing

Review question

Toss a fair coin three times. Let 𝐹𝑗 be the event that the 𝑗-th toss is HEADS. Let 𝐹 = 𝐹1 ∩ 𝐹2 ∩ 𝐹3. What is the probability of E? A. Pr(𝐹1) ⋅ Pr(𝐹2|𝐹1) ⋅ Pr(𝐹3|𝐹1 ∩ 𝐹2) B. Pr(𝐹1) ⋅ Pr(𝐹2) ⋅ Pr(𝐹3) C. Both A and B are correct. D. Neither A nor B is correct.

1/22/2020

Sofya Raskhodnikova; Randomness in Computing

Review question

Toss a coin that is biased with heads probability 𝑞 three times. Let 𝐹𝑗 be the event that the 𝑗-th toss is HEADS. Let 𝐹 = 𝐹1 ∩ 𝐹2 ∩ 𝐹3. What is the probability of E? A. Pr(𝐹1) ⋅ Pr(𝐹2|𝐹1) ⋅ Pr(𝐹3|𝐹1 ∩ 𝐹2) B. Pr(𝐹1) ⋅ Pr(𝐹2) ⋅ Pr(𝐹3) C. Both A and B are correct. D. Neither A nor B is correct.

1/22/2020

Sofya Raskhodnikova; Randomness in Computing

Probability Amplification

• Our algorithm for verifying polynomial identities accepts

incorrectly with probability ≤

𝑒 100𝑒 = 1 100

Idea: Repeat the algorithm and accept if all iterations accept.

1/23/2020

Sofya Raskhodnikova; Randomness in Computing

Pr error in all 𝑙 iterations ≤ 1 100

𝑙

Sampling without replacement

• Let 𝐹𝑗 be the event that we choose a root in iteration 𝑗
• It is 0 if 𝑙 > 𝑒.
• If 𝑙 ≤ 𝑒, then

1/22/2020

Sofya Raskhodnikova; Randomness in Computing

Pr 𝐹

𝑘 𝐹1 ∩ ⋯ ∩ 𝐹 𝑘−1 =

𝑒 − (𝑘 − 1) 100𝑒 − (𝑘 − 1) Pr error in all 𝑙 iterations = Pr 𝐹1 ∩ ⋯ ∩ 𝐹𝑙 = Pr 𝐹1 ⋅ Pr 𝐹2 𝐹1 ⋅ … ⋅ Pr[𝐹𝑙|𝐹1 ∩ ⋯ ∩ 𝐹𝑙−1] Pr error in all 𝑙 iterations ≤ 1 100

𝑙

§1.3 (MU) Verifying Matrix Multiplication Task: Given three 𝑜 × 𝑜 matrices 𝐵, 𝐶, 𝐷, verify if 𝐵 ⋅ 𝐶 = 𝐷. Matrix multiplication algorithms:

• Naïve

𝑃(𝑜3) time

• Strassen

𝑃 𝑜log2 7 ≈ 𝑃(𝑜2.81) time

• World record

𝑃(𝑜2.373…) time

[Coppersmith-Winograd `87, Vassilevska Williams `13, LeGall `14]

Verification:

• Fastest known deterministic algorithm is as above.
• Randomized algorithm [Freivalds `79]

𝑃(𝑜2) time

1/22/2020

Sofya Raskhodnikova; Randomness in Computing

Task: Given three 𝑜 × 𝑜 matrices 𝐵, 𝐶, 𝐷, verify if 𝐵 ⋅ 𝐶 = 𝐷. Idea: Pick a random vector 𝒔 and check if 𝐵 ⋅ 𝐶 ⋅ 𝒔 = 𝐷 ⋅ 𝒔. Running time: Three matrix-vector multiplications: O 𝑜2 time. Correctness: If 𝐵 ⋅ 𝐶 = 𝐷, the algorithm always accepts. Probability Amplification: With 𝑙 repetitions, error probability ≤ 2−𝑙

1/22/2020

Sofya Raskhodnikova; Randomness in Computing

O(𝒐𝟑) multiplications for each matrix-vector product

§1.3 (MU) Verifying Matrix Multiplication 1. Choose a random 𝑜-bit vector 𝑠 by making each bit 𝑠

𝑗

independently 0 or 1 with probability 1/2 each. 2. 𝐁𝐝𝐝𝐟𝐪𝐮 if 𝐵 ⋅ (𝐶 ⋅ 𝒔) = 𝐷 ⋅ 𝒔; o. w. 𝐬𝐟𝐤𝐟𝐝𝐮.

(input: 𝑜 × 𝑜 matrices A, B, C) Algorithm Basic Frievalds

Theorem

If 𝐵 ⋅ 𝐶 ≠ 𝐷, Basic-Frievalds accepts with probability ≤ 1/2.

Law of Total Probability

1/22/2020

Sofya Raskhodnikova; Randomness in Computing

For any two events 𝐵 and 𝐹,

Pr 𝐵 = Pr(𝐵 ∩ 𝐹) + Pr(𝐵 ∩ 𝐹) = Pr(𝐵|𝐹) ⋅ Pr 𝐹 + Pr(𝐵|𝐹) ⋅ Pr 𝐹 Let A be an event and let 𝐹1, … , 𝐹𝑜 be mutually disjoint events whose union is Ω. Pr(𝐵) =

𝑗∈ 𝑜

Pr(𝐵 ∩ 𝐹𝑗) =

𝑗∈ 𝑜

Pr(𝐵 ∣ 𝐹𝑗) ⋅ Pr(𝐹𝑗) .