
Algorithmic Regularization in Overparameterized Matrix Recovery
We study the problem of recovering a lowrank matrix X^ from linear meas...
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Implicit Regularization in Matrix Factorization
We study implicit regularization when optimizing an underdetermined quad...
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Learning Onehiddenlayer neural networks via Provable Gradient Descent with Random Initialization
Although deep learning has shown its powerful performance in many applic...
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Implicit Regularization in ReLU Networks with the Square Loss
Understanding the implicit regularization (or implicit bias) of gradient...
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On the Convex Behavior of Deep Neural Networks in Relation to the Layers' Width
The Hessian of neural networks can be decomposed into a sum of two matri...
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Gradient descent with identity initialization efficiently learns positive definite linear transformations by deep residual networks
We analyze algorithms for approximating a function f(x) = Φ x mapping ^d...
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Model Repair: Robust Recovery of OverParameterized Statistical Models
A new type of robust estimation problem is introduced where the goal is ...
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Algorithmic Regularization in Overparameterized Matrix Sensing and Neural Networks with Quadratic Activations
We show that the (stochastic) gradient descent algorithm provides an implicit regularization effect in the learning of overparameterized matrix factorization models and onehiddenlayer neural networks with quadratic activations. Concretely, we show that given Õ(dr^2) random linear measurements of a rank r positive semidefinite matrix X^, we can recover X^ by parameterizing it by UU^ with U∈R^d× d and minimizing the squared loss, even if r ≪ d. We prove that starting from a small initialization, gradient descent recovers X^ in Õ(√(r)) iterations approximately. The results solve the conjecture of Gunasekar et al.'17 under the restricted isometry property. The technique can be applied to analyzing neural networks with quadratic activations with some technical modifications.
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