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Local and Global Convergence of General Burer-Monteiro Tensor Optimizations
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Burer-Monteiro factorization with gradient descent from a neighborhood of the true factors | General convex, restricted strongly convex and smooth tensor objectives · General Burer-Monteiro tensor optimization | local convergence rate: 1-0.26ηm\underline{c}^4 | Helped |
| Use good initialization and constant stepsize η₀ = 1/(21.6M‖U⁰‖⁴) | Vanilla gradient descent on the factored tensor objective · General Burer-Monteiro tensor optimization | local convergence rate: 1-0.26η_0 m\underline{c}^4 | Helped |
| Initialize with perturbation αD for α≤2 | Vanilla gradient descent · Tensor decomposition, r=n/2, n, and 3n/2 | success rate: 100% | Helped |
| Increase initialization perturbation to α=8 | Vanilla gradient descent · Tensor decomposition, n=64, r=3n/2, η=0.02 | success rate: 100% -> 5% | Hurt |
| Increase initialization perturbation to α=16 | Vanilla gradient descent · Tensor decomposition, n=64, r=n/2, η=0.02 | success rate: 100% -> 0% | Hurt |
| Increase rank from n/2 to 3n/2 at α=8 | Vanilla gradient descent with η=0.02 · Tensor decomposition, n=64 | success rate: 100% -> 5% | Hurt |
| Increase stepsize from η=0.02 to η=0.06 | Vanilla gradient descent · Tensor decomposition, n=64, r=n/2, α=8 | success rate: 100% -> 38% | Hurt |
| Use best rank-one approximation objective for an orthogonally decomposable tensor | Tensor factors in the orthogonal decomposition · Orthogonally decomposable tensor, best rank-one approximation | spurious local minima | Helped |
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