Revisiting Gradient Ascent: Machine Unlearning from a Geometric Perspective for Source-Free Scenarios

Machine unlearning is becoming increasingly indispensable for meeting ever-stringent data compliance requirements, such as the ``right to be forgotten''. While Gradient Ascent (GA) based approximate methods have drawn attention for their conceptual simplicity, their practical deployment is severely constrained by three fundamental issues: first, the lack of feasibility analysis leads to divergent update directions; second, gradient conflict inevitably degrades the performance maintenance on retained data; and third, heavy reliance on the retain set restricts their applicability in real-world settings. To address these challenges, we conduct an in-depth investigation from a geometric perspective into the underlying root causes of instability in conventional GA methods. Our analysis reveals that the feasibility of unlearning intrinsically depends on specific geometric constraint relationships within the parameter space. Building upon this theoretical insight, we formally establish the methods governing unlearning feasibility. The first is the direction feasibility, which constrains the feasible update trajectory and keeps the model within reasonable parameter regions during unlearning. The second is the step-size feasibility, which further restricts the update magnitude along the feasible direction to prevent model collapse caused by large steps.Notably, we find that the direction feasibility does not rely on any information from the retain set. In light of this, we further explore viable approaches for effectively estimating the step-size feasibility under strict source-free scenarios.