Continuized Nesterov Acceleration for Non-Convex Optimization

TL;DR

This work extends the continuized Nesterov framework to non-convex optimization by incorporating non-smooth Lyapunov functions and strengthening trajectory-wise guarantees. The authors develop an Itô-style tool for non-smooth Lyapunov analysis, derive sharp high-probability and almost-sure bounds, and improve convergence rates for strongly quasar-convex functions while relaxing minimizer-uniqueness assumptions. The method combines continuous-time Lyapunov analysis with exact discrete realizations of a Nesterov momentum scheme driven by random jump times, enabling a tight transfer of convergence properties from the continuous to the discrete setting. Compared to classical ODE-based treatments, the continuized approach yields robust, trajectory-wise guarantees without requiring subroutines or strict convexity, thus offering practical and theoretical advances for non-convex first-order optimization with momentum. Overall, the results tighten constants in convergence rates and broaden applicability to non-smooth, non-convex landscapes, with explicit high-probability and almost-sure convergence statements grounded in stochastic calculus.

Abstract

In convex optimization, continuous-time counterparts have been a fruitful tool for analyzing momentum algorithms. Fewer such examples are available when the function to minimize is non-convex. In several cases, discrepancies arise between the existing discrete-time results, namely those obtained for momentum algorithms, and their continuous-time counterparts, with the latter typically yielding stronger guarantees. We argue that the continuized framework (Even et al., 2021), mixing continuous and discrete components, can tighten the gap between known continuous and discrete results. This framework relies on computations akin to standard Lyapunov analyses, from which are deduced convergence bounds for an algorithm that can be written as a Nesterov momentum algorithm with stochastic parameters. In this work, we extend the range of applicability of the continuized framework, e.g. by allowing it to handle non-smooth Lyapunov functions. We then strengthen its trajectory-wise guarantees for linear convergence rate, deriving finite time bounds with high probability and asymptotic almost sure bounds. We apply this framework to the non-convex class of strongly quasar convex functions. Adapting continuous-time results that have weaker discrete equivalents to the continuized method, we improve by a constant factor the known convergence rate, and relax the existing assumptions on the set of minimizers.

Figures (2)

• Figure 1: Left: Visualisation of a single trajectory of $(x_t)_{t\ge 0}$ satisfying \ref{['eq:nest_continuized']}, between $t_0$ and $t_{\text{fin}}$. Right: Visualisation of the average over $10^5$ such trajectories, providing an estimate of the mean process $(\mathbb{E}\left[x_t \right])_{t \ge 0}$ for which the jumps are smoothed out.
• Figure 2: Visualization of functions verifying Assumption \ref{['ass:sqc']}. Left: 1-dimensional example $f(t) = 0.5\cdot(t +0.15\sin(5t))^2$ on top, its derivative on the bottom. Right: 2-dimensional example of the form $h(x,y) = f(\left\lVert(x,y)\right\rVert)g\left(\frac{(x,y)}{\left\lVert(x,y)\right\rVert}\right)$, where $f$ verify Assumption \ref{['ass:sqc']} and $g \ge 1$. See details in Appendix \ref{['app:fig_detail']}. It clearly appears in both cases that this class of functions permits high oscillations.

Theorems & Definitions (28)

• Example 1: Finite-Sum Setting
• Lemma 1
• Remark 2
• Proposition 3: $C^1$-Itô Formula
• Remark 4
• Proposition 5
• Theorem 6: Stopping theorem
• Proposition 7
• Theorem 8
• Theorem 9: wang2023continuizedaccelerationquasarconvex