Abstract
In this work, we show, for the wellstudied problem of learning parity under noise, where a learner tries to learn x = (x₁,…,x_n) ∈ {0,1}ⁿ from a stream of random linear equations over 𝔽₂ that are correct with probability 1/2+ε and flipped with probability 1/2ε (0 < ε < 1/2), that any learning algorithm requires either a memory of size Ω(n²/ε) or an exponential number of samples.
In fact, we study memorysample lower bounds for a large class of learning problems, as characterized by [Garg et al., 2018], when the samples are noisy. A matrix M: A × X → {1,1} corresponds to the following learning problem with error parameter ε: an unknown element x ∈ X is chosen uniformly at random. A learner tries to learn x from a stream of samples, (a₁, b₁), (a₂, b₂) …, where for every i, a_i ∈ A is chosen uniformly at random and b_i = M(a_i,x) with probability 1/2+ε and b_i = M(a_i,x) with probability 1/2ε (0 < ε < 1/2). Assume that k,𝓁, r are such that any submatrix of M of at least 2^{k} ⋅ A rows and at least 2^{𝓁} ⋅ X columns, has a bias of at most 2^{r}. We show that any learning algorithm for the learning problem corresponding to M, with error parameter ε, requires either a memory of size at least Ω((k⋅𝓁)/ε), or at least 2^{Ω(r)} samples. The result holds even if the learner has an exponentially small success probability (of 2^{Ω(r)}). In particular, this shows that for a large class of learning problems, same as those in [Garg et al., 2018], any learning algorithm requires either a memory of size at least Ω(((logX)⋅(logA))/ε) or an exponential number of noisy samples.
Our proof is based on adapting the arguments in [Ran Raz, 2017; Garg et al., 2018] to the noisy case.
BibTeX  Entry
@InProceedings{garg_et_al:LIPIcs.APPROX/RANDOM.2021.60,
author = {Garg, Sumegha and Kothari, Pravesh K. and Liu, Pengda and Raz, Ran},
title = {{MemorySample Lower Bounds for Learning Parity with Noise}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
pages = {60:160:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772075},
ISSN = {18688969},
year = {2021},
volume = {207},
editor = {Wootters, Mary and Sanit\`{a}, Laura},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14753},
URN = {urn:nbn:de:0030drops147534},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.60},
annote = {Keywords: memorysample tradeoffs, learning parity under noise, space lower bound, branching program}
}
Keywords: 

memorysample tradeoffs, learning parity under noise, space lower bound, branching program 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021) 
Issue Date: 

2021 
Date of publication: 

15.09.2021 