Research map / Optimization / Implicit bias and generalization
Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use gradient flow with infinitesimal initialization, which follows greedy low-rank learning rather than selecting by nuclear norm | Matrix factorization on the matrix completion counterexample · Matrix completion counterexample | nuclear norm: 400 -> 20002 | Hurt |
| Use GLRL instead of gradient descent with not-so-small initialization, nuclear norm minimization, or R1MP | Low-rank matrix sensing · Matrix sensing, ground truth rank 3 | ground-truth recovery | Helped |
| Use depth greater than 2 rather than depth 2 | Deep matrix factorization · Matrix sensing, ground truth rank 3 | low-rankness | Helped |
| Increase initialization scale | Depth-2 matrix factorization · Matrix sensing | distance to ground truth | Hurt |
| Use depth greater than 2 instead of depth 2 | Matrix factorization · Matrix sensing | initialization scale needed to approach the ground truth | Helped |
| Increase depth beyond 16 | Deep matrix factorization · Matrix sensing | convergence speed | No clear effect |
| Initialize GD along the top eigenvector direction instead of with a random full-rank matrix | Gradient descent · First-phase GLRL trajectory | distance to GLRL reference trajectory | Helped |
| Use GLRL trajectories instead of randomly initialized GD trajectories to approximate infinitesimal-initialization GF | Gradient-flow trajectory simulation · First-phase GLRL trajectory | distance to reference trajectory | Helped |
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