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You're training a 50-billion-parameter MLP and monitori...
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See plansThe intuition that a zero gradient means “stuck at a minimum” comes from low-dimensional loss surfaces — the kind you can actually plot and stare at — where a flat point with high loss usually is a genuine local minimum, because there are only a couple of directions in which curvature needs to be checked. That intuition breaks down completely at the scale of a modern network. For a critical point to be a true local minimum in a 50-billion-dimensional parameter space, the loss surface has to curve upward in every single one of those 50 billion orthogonal directions at once. The probability of a high-loss critical point satisfying that condition by chance is, for practical purposes, zero. What you’re almost certainly looking at is a saddle point: flat enough in most directions that the gradient reads as zero, but still sloping downward in at least one direction you haven’t found yet.
The rigorous test is the second-order one: compute (or estimate) the Hessian matrix at that point and examine the sign of its eigenvalues. All eigenvalues strictly positive means the surface curves upward in every direction — a genuine local minimum. All strictly negative means a local maximum. Any mix of positive and negative eigenvalues is a saddle point by definition, and in high-dimensional non-convex optimization, a mix is overwhelmingly the expected outcome at any high-loss critical point.
The practical implication changes the response. A real local minimum would need a structural change — different architecture, different data, accept the loss — because there’s nowhere better nearby. A saddle point just needs help escaping along whichever direction has negative curvature, which is exactly what targeted noise injection or the momentum and adaptive scaling in an optimizer like AdamW is suited for. It’s a controlled nudge along a known escape direction, not the undisciplined learning-rate spike that risks destabilizing a run that’s otherwise healthy.