[Paper Review] An $O(s^r)$-Resolution ODE Framework for Understanding Discrete-Time Algorithms and Applications to the Linear Convergence of Minimax Problems

Haihao Lu|arXiv (Cornell University)|Jan 23, 2020

Sparse and Compressive Sensing Techniques25 references4 citations

TL;DR

This paper introduces an $O(s^r)$-resolution ODE framework to analyze discrete-time algorithms (DTAs), offering a systematic way to derive ODEs from DTAs and link their convergence to that of the corresponding ODEs. It establishes that linear convergence of the $O(s^r)$-resolution ODE with respect to a proper energy function guarantees linear convergence of the original DTA, and applies the framework to show broader linear convergence for PPM and EGM in nonconvex-nonconcave minimax problems.

ABSTRACT

There has been a long history of using ordinary differential equations (ODEs) to understand the dynamics of discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to obtain a \emph{suitable} ODE from a given DTA, and (ii) it is unclear the connection between the convergence of a DTA and its corresponding ODEs. In this paper, we propose a new machinery -- an $O(s^r)$-resolution ODE framework -- for analyzing the behavior of a generic DTA, which (partially) answers the above two questions. The framework contains three steps: 1. To obtain a suitable ODE from a given DTA, we define a hierarchy of $O(s^r)$-resolution ODEs of a DTA parameterized by the degree $r$, where $s$ is the step-size of the DTA. We present a principal approach to construct the unique $O(s^r)$-resolution ODEs from a DTA; 2. To analyze the resulting ODE, we propose the $O(s^r)$-linear-convergence condition of a DTA with respect to an energy function, under which the $O(s^r)$-resolution ODE converges linearly to an optimal solution; 3. To bridge the convergence properties of a DTA and its corresponding ODEs, we define the properness of an energy function and show that the linear convergence of the $O(s^r)$-resolution ODE with respect to a proper energy function can automatically guarantee the linear convergence of the DTA. To better illustrate this machinery, we utilize it to study three classic algorithms -- gradient descent ascent (GDA), proximal point method (PPM) and extra-gradient method (EGM) -- for solving the unconstrained minimax problem $\min_{x\in\RR^n} \max_{y\in \RR^m} L(x,y)$.

Motivation & Objective

• To resolve two fundamental unresolved issues in using ODEs to analyze DTAs: (1) how to systematically derive a suitable ODE from a given DTA, and (2) how to connect the convergence of the DTA to that of its corresponding ODE.
• To provide a principled, higher-order resolution ODE framework that captures the essential dynamics of DTAs by expanding the update rule to $O(s^r)$-order terms.
• To establish conditions under which the $O(s^r)$-resolution ODE converges linearly, and to show that such convergence implies linear convergence of the original DTA when the energy function is proper.
• To apply the framework to classic minimax algorithms—GDA, PPM, and EGM—offering new insights into their convergence behavior and extending known convergence regimes.

Proposed method

• Proposes an $r$-th degree ODE expansion of a DTA to construct a unique $O(s^r)$-resolution ODE, parameterized by step-size $s$ and resolution order $r$, enabling higher-order approximation of discrete dynamics.
• Defines the $O(s^r)$-linear-convergence condition with respect to a chosen energy function, ensuring the ODE converges linearly to an optimal solution.
• Introduces the concept of a 'proper' energy function, under which the linear convergence of the $O(s^r)$-resolution ODE implies linear convergence of the original DTA.
• Applies Taylor expansion and matrix analysis to derive energy decay bounds, showing that higher-order remainder terms do not disrupt linear convergence for small enough step-sizes.
• Uses generalized block skew-symmetric matrices to analyze remainder terms in the energy decay, particularly for fast-rate convergence with $s = O(1/ ho)$.
• Compares the $O(s)$-resolution ODEs of GDA, PPM, and EGM to reveal how interaction terms affect convergence—helping PPM/EGM but harming GDA in bilinear settings.

Experimental results

Research questions

• RQ1 How can one systematically derive a unique and meaningful ODE from a given discrete-time algorithm, avoiding ambiguity from different continuous limits?
• RQ2 What conditions on the ODE ensure linear convergence, and how can these be linked to the convergence behavior of the original discrete algorithm?
• RQ3 Why do PPM and EGM exhibit linear convergence in broader contexts—including nonconvex-nonconcave minimax problems—while GDA fails in bilinear cases?
• RQ4 Can the ODE framework be used to design new optimization algorithms by analyzing differences in ODE structures across existing methods?
• RQ5 Under what conditions does the convergence of the $O(s^r)$-resolution ODE guarantee the convergence of the original DTA?

Key findings

• The $O(s)$-resolution ODEs of GDA, PPM, and EGM explain their divergent or convergent behavior in bilinear minimax problems, showing that interaction terms in PPM and EGM promote convergence while harming GDA.
• The $O(s)$-linear-convergence condition unifies and extends known convergence regimes for PPM and EGM, proving linear convergence even in nonconvex-nonconcave settings.
• For quadratic minimax problems, EGM achieves a fast linear rate ($s = O(1/ ho)$), while for general problems, it achieves a slower rate due to more complex remainder terms in the Taylor expansion.
• The framework proves that if the $O(s^r)$-resolution ODE converges linearly with respect to a proper energy function, then the original DTA also converges linearly, providing a bridge between continuous and discrete analysis.
• The remainder terms in the energy decay analysis are shown to be bounded by $O(s^2)$, and under fast-rate conditions, they contribute at most a constant factor of 2 to the decay rate.
• The framework enables algorithm design by comparing ODE structures: the difference between the $O(s)$-resolution ODE of GDA and that of PPM/EGM reveals how to modify GDA to achieve better convergence behavior.