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Implicit Gradient Regularization
TLDR
This work indicates that backward error analysis is a useful theoretical approach to the perennial question of how learning rate, model size, and parameter regularization interact to determine the properties of overparameterized models optimized with gradient descent.
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On the Origin of Implicit Regularization in Stochastic Gradient Descent
TLDR
It is proved that for SGD with random shuffling, the mean SGD iterate also stays close to the path of gradient flow if the learning rate is small and finite, but on a modified loss.
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Symplectic microgeometry I: micromorphisms
We introduce the notion of symplectic microfolds and symplectic micromorphisms between them. They form a symmetric monoidal category, which is a version of the “category” of symplectic manifolds and…
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Symplectic microgeometry II: generating functions
We adapt the notion of generating functions for lagrangian submanifolds to symplectic microgeometry. We show that a symplectic micromorphism always admits a global generating function. As an…
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Formal Symplectic Groupoid
The multiplicative structure of the trivial symplectic groupoid over ℝd associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of…
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Integration of Lie algebroid comorphisms
We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the of integrable Lie algebroids and complete Lie algebroid comorphisms to the of…
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Symplectic microgeometry III: monoids
We show that the category of Poisson manifolds and Poisson maps, the category of symplectic microgroupoids and Lagrangian submicrogroupoids (as morphisms), and the category of monoids and monoid…
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Deformation quantization of Leibniz algebras
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Tensor products of representations up to homotopy
We study the construction of tensor products of representations up to homotopy, which are the A-infinity version of ordinary representations. We provide formulas for the construction of tensor…
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Symplectic microgeometry, IV: Quantization
We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the…
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