New Algorithms for Learning in Presence of Errors

We give new algorithms for a variety of randomly-generated instances of computational problems using a linearization technique that reduces to solving a system of linear equations. These algorithms are derived in the context of learning with structured noise, a notion introduced in this paper. This notion is best illustrated with the learning parities with noise (LPN) problem —well-studied in learning theory and cryptography. In the standard version, we have access to an oracle that, each time we press a button, returns a random vector a ∈ GF(2)n together with a bit b ∈ GF(2) that was computed as a · u + η, where u ∈ GF(2)n is a secret vector, and η ∈ GF(2) is a noise bit that is 1 with some probability p. Say p = 1/3. The goal is to recover u. This task is conjectured to be intractable. In the structured noise setting we introduce a slight (?) variation of the model: upon pressing a button, we receive (say) 10 random vectors a1, a2, . . . , a10 ∈ GF(2)n , and correspon...

• Type: Journal
• Year: 2011
• Where: ICALP
• Authors: Sanjeev Arora, Rong Ge
• Keywords: ICALP 2011, Polynomial Time Algorithm, Polynomial Time Algorithms, Programming Languages, Pseudorandom Generator