Low-rank factorization drops the global smoothness constant: Burer–Monteiro objectives are 2/3-generalized smooth, so step sizes adapt online

Factorized low-rank matrix objectives no longer need a global Lipschitz smoothness constant to set the step size. Bregman-type methods such as Dyn-NoLips must fix kernel parameters from global worst-case bounds, and fall back to grid search when those bounds are unknown. On TDT2, basic adaptive gradient descent needed 1014 iterations, and adding adaptive backtracking cut that to 206 with final objective 0.516670; in the same table Dyn-NoLips reached 0.519804 in 210 iterations.

The method separates step length from search direction, so smoothness constants can be estimated on the fly. In the Bregman framework the kernel parameters define the geometry of the space, and changing them mid-iteration breaks the convergence theory. On CBCL face images, SafeAdam reached objective 0.4186 in 6764 iterations and 5.46 seconds, while Dyn-NoLips reached 0.4188 in 9115 iterations and 6.04 seconds.

The result covers Burer–Monteiro factorizations of quadratic objectives that are strongly convex only on the symmetric-matrix subspace, with separate small-gradient and large-gradient rates. Experiments run on the SymNMF benchmarks TDT2, Reuters and CBCL against Dyn-NoLips. What to watch next is whether online estimation of the smoothness constants keeps this advantage on low-rank problems with unknown kernel parameters or larger condition numbers.

For the projected version, active nonnegativity constraints prevent splitting the guarantee into small-gradient and large-gradient regimes, leaving a single square-root-type estimate. The stopping criteria differ between methods: Dyn-NoLips uses the projected gradient norm while the proposed methods use the gradient-mapping norm, and the two are only locally equivalent when the projection is inactive, a condition not explicitly checked in the experiments.