Nonconvex optimization and saddles

At a differentiable unconstrained local minimum, the gradient vanishes. If the function is twice differentiable, its Hessian there is positive semidefinite. Neither condition alone is sufficient: $x^3$ has both zero first derivative and zero second derivative at zero, yet it is not a local minimum. A positive-definite Hessian at a stationary point is a sufficient local condition, while a negative eigenvalue supplies a direction of negative curvature.

A strict saddle is commonly understood here as a stationary point with at least one negative Hessian eigenvalue. Terminology varies across research papers, so check definitions before applying a result. A local minimum can still be worse than another minimum elsewhere. Even an algorithm guaranteed to avoid all strict saddles may finish at a suboptimal local minimum unless special landscape structure connects the remaining stationary points to global solutions.

Suppose the objective is globally L-smooth and bounded below by a finite value. Exact gradient descent with step $1/L$ decreases its value by at least the squared gradient norm divided by $2L$. Summing this inequality over iterations bounds the total squared gradient. Therefore the smallest gradient norm among the first T iterates becomes small at a rate proportional to the inverse square root of T. Convexity is not needed for that first-order guarantee.

The conclusion concerns the best stationarity residual among those iterates, not convergence of the entire sequence to one point and not proximity to a global minimizer. The starting objective gap and smoothness constant affect the bound, and a function without a finite global L needs a local or adaptive argument. This is a useful example of a theorem that remains meaningful after losing convexity without silently keeping the stronger convex optimality interpretation.

At an exact saddle, deterministic gradient descent started exactly there remains fixed because its gradient is zero. Small perturbations can reveal an unstable negative-curvature direction. Research guarantees for perturbed gradient methods specify smoothness of gradients and often Hessians, perturbation sizes, tolerances, probabilities, and objective-gap assumptions. The resulting approximate second-order stationarity is more informative than a small gradient, but degenerate stationary points and poor local minima still need careful interpretation.

The interactive model has a single convex quadratic minimum and no saddle at all. Increase η above two to observe divergence, then explain why this is step-size instability rather than evidence of a nonconvex landscape. The two-dimensional worked calculation supplies the actual saddle example. Real experiments should inspect multiple starts, held-out objectives where appropriate, gradient residuals, and curvature evidence; agreement among runs is useful empirical information, not a universal proof of global optimality.