When Are Nesterov Trajectories Eventually Locally Minimizing?

We study geometric moment contraction (GMC) of the constant-parameter stochastic Nesterov recursion $$Y_k=Θ_k+β(Θ_k-Θ_{k-1}),\qquad Θ_{k+1}=Y_k-γG(Y_k,X_{k+1}).$$ Under mean strong monotonicity and stochastic $L^p$ Lipschitz continuity, an explicit Perron comparison proves synchronous $L^p$ contraction when $βγL_p<(1-β)(1-q_{γ,p})$. This direct criterion includes infinite-variance gradients for $1<p<2$, but its small-step regime requires $β<μ/(μ+L_p)$. A complementary power-Lyapunov argument establishes a positive, generally much smaller, step-size interval for every fixed $β<1$ and every $p>1$, using only a finite $p$th gradient moment. At $p=2$, a simpler explicit certificate gives $$0<γ<\frac{2μ(1-β)^2}{L_2^2(1-β+2β^2)}.$$ Its quadratic high-momentum scaling is a limitation of the chosen metric, not a sharp stability boundary. We quantify this loss, provide a general mean-only quadratic $S$-procedure, and exploit endpoint Lyapunov inequalities under stronger samplewise sector information. Verified endpoint certificates can be orders of magnitude less conservative than the explicit metric.

Recent observations that deep learning models train at the edge of stability have renewed interest in understanding stability properties of classical optimization algorithms. For quadratic objectives, standard gradient descent has a sharp stability threshold: it converges from every initialization when $η<2/L$, diverges from almost every initialization when $η>2/L$, and diverges from every nonzero initialization when $η>2/μ$. For non-Euclidean gradient descent, the corresponding known guarantees are weaker: convergence holds for every initialization when $η<2/S$, where $S$ is the generalized sharpness, while every nonzero initialization diverges when $η>2/μ$. The behavior in the intermediate regime $2/S<η<2/μ$ is not fully characterized. We show that Hilbertian norms are the only ones for which $2/S$ is an almost-everywhere sharp threshold for every positive-definite quadratic objective. Using linear conjugacy between non-Euclidean and Euclidean gradient descent, we show that for general norms the intermediate regime can contain arbitrarily many edges of stability associated with distinct attracting directions. We identify a forward-invariant region of exponentially divergent directions beyond the generalized stability threshold. When the directions of minimum curvature are finite and nondegenerate, we prove that, sufficiently close to $2/μ$ from below, only their invariant directions remain convergent. We give smooth two-dimensional constructions with arbitrarily many distinct thresholds, arbitrarily large relative threshold separation and show these properties to be stable.

This paper considers the problem of understanding the behavior of a general class of accelerated gradient methods on smooth nonconvex functions. Motivated by some recent works that have proposed effective algorithms, based on Polyak's heavy ball method and the Nesterov accelerated gradient method, to achieve convergence to a local minimum of nonconvex functions, this work proposes a broad class of Nesterov-type accelerated methods and puts forth a rigorous study of these methods encompassing the escape from saddle points and convergence to local minima through both an asymptotic and a non-asymptotic analysis. In the asymptotic regime, this paper answers an open question of whether Nesterov's accelerated gradient method (NAG) with variable momentum parameter avoids strict saddle points almost surely. This work also develops two metrics of asymptotic rates of convergence and divergence, and evaluates these two metrics for several popular standard accelerated methods such as the NAG and Nesterov's accelerated gradient with constant momentum (NCM) near strict saddle points. In the non-asymptotic regime, this work provides an analysis that leads to the "linear" exit time estimates from strict saddle neighborhoods for trajectories of these accelerated methods as well the necessary conditions for the existence of such trajectories. Finally, this work studies a sub-class of accelerated methods that can converge in convex neighborhoods of nonconvex functions with a near optimal rate to a local minimum and at the same time this sub-class offers superior saddle-escape behavior compared to that of NAG.