Core Concepts
Stochastic rounding implicitly regularizes tall and thin matrices, increasing their smallest singular value.
Abstract
The content discusses how stochastic rounding impacts the regularization of tall-and-thin matrices. It provides theoretical evidence and experimental evaluations supporting the idea that stochastic rounding ensures a well-bounded smallest singular value for rounded matrices, regardless of rank deficiency. The paper leverages Random Matrix Theory to explain the regularization effects of stochastic rounding on machine learning applications.
Abstract:
- Stochastic rounding in machine learning context.
- Novel theoretical evidence and experimental evaluation.
- Smallest singular value bounded away from zero.
Introduction:
- History and resurgence of stochastic rounding.
- Applications in low-precision arithmetic for machine learning.
Results:
- Theoretical bounds on smallest singular values after stochastic rounding.
- Importance of randomness in perturbations for regularization effects.
Background:
- Notation definitions and review of stochastic rounding properties.
Random Matrix Theory Bound:
- Theorem 4 explanation for bounding the smallest singular value.
Proof of Theorem 2:
- Outline and step-by-step proof explanation.
Experiments:
Rank Deficient Matrices (Figure 1):
- Construction and results showing increase in smallest singular value with aspect ratio n/d.
Controlled ν Matrices (Figure 2):
- Manipulation of parameter ν to observe its influence on smallest singular value.
Full Rank Matrices (Figure 3):
- Behavior of smallest singular value with full-rank matrices under different precisions.
Stats
With high probability, the smallest singular value of eA is well bounded away from zero - regardless of rank deficiency or closeness to it.
The elements eAhij and eAlij belong to F{1}, thus VAhijW − TAhijU ≤ 10−1 and VAlijW − TAlijU ≤ 10−1.
The elements eAhij and eAlij belong to F{1}, thus VAhijW − TAhijU ≤ 10−2 and VAlijW − TAlijU ≤ 10−2.
The elements eAhij and eAlij belong to F{1}, thus VAhijW − TAhijU ≤ 10−3 and VAlijW − TAlijU ≤ 10−3.
The elements eAij belong to F{p}, thus VAijW − TAijU ≤ p^-1.
Quotes
"SR implicitly regularizes tall-and-thin matrices."
"SR could serve as an implicit regularizer in modern ML applications."
"SR guarantees bounded smallest singular values post-rounding."