Non Gradient Based Optimization
%20under%20the%20logistic%20loss%20often%20involves%20two%20distinct%20phases%2C%20where%20the%20empirical%20risk%20oscillates%20in%20the%20first%20phase%20but%20decreases%20monotonically%20in%20the%20second%20phase.%20We%20investigate%20this%20phenomenon%20in%20two-layer%20networks%20that%20satisfy%20a%20near-homogeneity%20condition.%20We%20show%20that%20the%20second%20phase%20begins%20once%20the%20empirical%20risk%20falls%20below%20a%20certain%20threshold%2C%20dependent%20on%20the%20stepsize.%20Additionally%2C%20we%20show%20that%20the%20normalized%20margin%20grows%20nearly%20monotonically%20in%20the%20second%20phase%2C%20demonstrating%20an%20implicit%20bias%20of%20GD%20in%20training%20non-homogeneous%20predictors.%20If%20the%20dataset%20is%20linearly%20separable%20and%20the%20derivative%20of%20the%20activation%20function%20is%20bounded%20away%20from%20zero%2C%20we%20show%20that%20the%20average%20empirical%20risk%20decreases%2C%20implying%20that%20the%20first%20phase%20must%20stop%20in%20finite%20steps.%20Finally%2C%20we%20demonstrate%20that%20by%20choosing%20a%20suitably%20large%20stepsize%2C%20GD%20that%20undergoes%20this%20phase%20transition%20is%20more%20efficient%20than%20GD%20that%20monotonically%20decreases%20the%20risk.%20Our%20analysis%20applies%20to%20networks%20of%20any%20width%2C%20beyond%20the%20well-known%20neural%20tangent%20kernel%20and%20mean-field%20regimes.)
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