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Acceleration via Symplectic Discretization of High-Resolution Differential Equations
Bin Shi, Simon S. Du, Weijie J. Su, Michael I. Jordan
TL;DR
The paper asks how to systematically obtain accelerated methods by discretizing optimization ODEs. It compares three Euler schemes using Lyapunov analysis and finds that symplectic discretization accelerates high-resolution ODEs, unlike the low-resolution, explicit, or impractical implicit cases.
Problem
The transition from continuous-time ODEs to discrete algorithms remains incompletely explored, despite the question of whether discretization can systematically yield accelerated first-order methods.
Source: §1 · pp. 1–2Method
Results
Takeaways & Limitations
High-resolution ODEs and symplectic schemes are critical to discretization-based acceleration, while allowing relatively simple implementations and large stable step sizes.
Source: §1 · p. 3For NAG-SC, implicit discretization requires solving a nonlinear fixed-point equation for nonquadratic objectives, while explicit discretization’s O(µ/L^2) step size prevents acceleration.
Source: §3.1 · p. 6
Abstract
from arXiv · showhide
We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme. We show that the optimization algorithm generated by applying the symplectic scheme to a high-resolution ODE proposed by Shi et al. [2018] achieves an accelerated rate for minimizing smooth strongly convex functions. On the other hand, the resulting algorithm either fails to achieve acceleration or is impractical when the scheme is implicit, the ODE is low-resolution, or the scheme is explicit.
1 Introduction
The introduction frames discretization as an unresolved bridge between accelerated ODEs and discrete optimization methods. It presents a Lyapunov-based analysis of how discretization choices affect acceleration.
1 Introduction
- The paper addresses the incomplete study of transitions from continuous-time ODEs to discrete algorithms with a dynamical-systems framework based on Lyapunov theory.Its motivating question is whether numerical ODE discretization can systematically and provably produce accelerated methods.Source: §1 · pp. 2–3
- Symplectic discretization of the high-resolution NAG-SC ODE achieves acceleration, while implicit Euler is impractical and explicit Euler fails to accelerate.The authors contrast the schemes in their introduction.Source: §1 · p. 3
- Symplectic discretization does not achieve acceleration for the low-resolution heavy-ball ODE, which lacks a gradient-correction term.The paper also reports other low-resolution examples without acceleration in Appendix D.Source: §1 · p. 3
- For the high-resolution ODE of NAG-C, Lyapunov analysis indicates that symplectic discretization is superior to explicit and implicit Euler schemes.The comparison is addressed in Section 4.Source: §1 · p. 3
- The paper identifies high-resolution ODEs and symplectic schemes as critical to discretization-based acceleration, with relatively simple implementations and large stable steps.For low-resolution ODEs, the introduction says only implicit schemes remain stable at large step sizes because gradient correction is absent.Source: §1 · p. 3
- A discrete Lyapunov function also yields an O(L^2/k^2) best-iterate rate for minimizing GD’s squared gradient norm.The introduction presents this as a systematic use of Lyapunov analysis for optimization problems.Source: §1 · p. 3
2 Preliminaries
This section establishes function-class notation, reviews low- and high-resolution ODE models for NAG and heavy-ball, and compares Euler discretization choices. It identifies the high-resolution gradient-difference approximation as critical to acceleration.
2 Preliminaries
- The preliminaries establish notation and define smooth convex and strongly convex function classes, with x⋆ denoting a minimizer.Source: §2 · p. 3
2.1 Approximating ODEs
- High-resolution ODEs preserve O(√s) terms, distinguishing the heavy-ball and NAG-SC models.Source: §2.1 · p. 4
- The high-resolution heavy-ball ODE has essentially the same properties as its low-resolution counterpart because it lacks the ∇²f(X)Ẋ term.Source: §2.1 · p. 4
2.2 Discretization schemes
- Discretization replaces time derivatives with step differences, while different schemes use different approximations.Source: §2.2 · p. 4
- The implicit scheme is impractical because each next-step update requires the other next-step variable; the symplectic update of x_(k+1) requires only v_k.Source: §2.2 · p. 4
- The gradient-difference approximation used for the Hessian term in high-resolution ODEs is critical to acceleration.Source: §2.2 · p. 4
3 High-Resolution ODEs for Strongly Convex Functions
The section compares symplectic, explicit, and implicit Euler discretizations of high-resolution ODEs for NAG-SC and the heavy-ball method. Symplectic discretization achieves acceleration for NAG-SC but not for the heavy-ball ODE.
3 High-Resolution ODEs for Strongly Convex Functions
- The section compares convergence rates of the three Euler schemes applied to high-resolution ODEs for NAG-SC and the heavy-ball method.The associated symplectic schemes are described as surprisingly similar to their corresponding classical methods.Source: §3 · p. 5
- Symplectic discretization accelerates NAG-SC but not the heavy-ball method; the implicit scheme accelerates in both cases but is impractical.For NAG-SC, implicit updates require a nonlinear fixed-point solve for nonquadratic objectives; for heavy-ball, the implicit scheme is impractical except for quadratic objectives.Source: §3.1 · p. 6, §3.2 · p. 7
3.1 NAG-SC
- The symplectic Euler scheme is closest to NAG-SC, and both it and NAG-SC have accelerated convergence rates.The explicit scheme's smaller O(µ/L^2) step size prevents acceleration.Source: §3.1 · pp. 5–6
3.2 The heavy-ball method
- For the heavy-ball ODE, neither the heavy-ball method nor its symplectic discretization attains an accelerated rate.The implicit scheme achieves acceleration but is impractical except for quadratic objectives.Source: §3.2 · p. 7
4 High-Resolution ODEs for Convex Functions
This section studies discretizations of high-resolution ODEs for convex functions, including a modified NAG-C ODE. It reports convergence guarantees for implicit Euler, while noting that explicit Euler does not guarantee convergence.
4 High-Resolution ODEs for Convex Functions
- Direct discretization of ODE (2.2) leaves it unclear how to prove convergence for the simple Euler schemes, motivating a modified ODE.The modification changes the third term on the right-hand side of the second equation.Source: §4 · p. 8
- The symplectic Euler scheme of (4.3), identified as original NAG-C, is considered for 0 < s ≤ 1/(3L).The supplied statement does not include the conclusion following this condition.Source: §4 · p. 8
- For step size 0 < s ≤ 1/L, implicit Euler on ODE (4.3) satisfies a bound involving min_{0≤i≤k} ∥∇f(x_i)∥².The passage also gives a bound involving ∥x_0 − x⋆∥², but its full expression is not included.Source: §4 · p. 8
- The explicit Euler scheme does not guarantee convergence; the passage points to Appendix C.1 for the analysis and identifies this statement as Theorem 4.1(a), the same as Theorem 6 of Shi et al. [2018].Source: §4 · p. 8
5 Discussion
The discussion concludes that symplectic discretization achieves accelerated rates for high-resolution ODEs, but not their low-resolution counterparts; explicit Euler fails to accelerate, while implicit Euler is generally impractical. The appendices also compare Lyapunov analyses and convergence statements for gradient flow and Euler schemes.
5 Discussion
- Symplectic discretization achieves accelerated rates for high-resolution NAG-SC and slightly modified NAG-C ODEs, but not for low-resolution counterparts; explicit Euler fails to accelerate, while implicit Euler is generally impractical.Implicit Euler is practical in limited cases such as quadratic objectives.Source: §5 · p. 8
- The discussion leaves open the mechanism of gradient correction in stabilizing symplectic discretization and whether higher-order symplectic schemes preserve acceleration.It also proposes comparing the discretization properties of two high-resolution NAG ODEs and studying interpolations of NAG-SC and its symplectic scheme.Source: §5 · p. 9
A.1 Convergence rate of gradient flow
- A Lyapunov function can retain the same function-value convergence rate while capturing faster convergence of the squared gradient norm.The passage contrasts this with using E = f(X) − f(x⋆) as the Lyapunov function.Source: §A.1 · p. 12
A.2 Explicit Euler scheme
- The discrete Lyapunov function used for the explicit-scheme analysis differs from the standard function-value Lyapunov function, which yields O(L/k) for function value but only O(L²/k) for squared gradient norm.The passage attributes the latter limitation to the absence of gradient information in that function.Source: §A.2 · p. 13
A.3 Implicit Euler scheme
This appendix presents implicit Euler analyses for gradient flow and related convergence statements, using distance and Lyapunov functions and specifying step-size conditions for some results.
A.3 Implicit Euler scheme
- One result specifies the step size s = θ/μ, where θ > 0.The passage introducing the condition does not include the resulting bound.Source: §A.3 · p. 13
- Another result specifies the step size s = θ/L.The corresponding conclusion is not included in the supplied passage.Source: §A.3 · p. 14
- The implicit gradient-descent analysis tracks the distance to the minimizer and uses a Lyapunov function to derive a convergence rate.The supplied passages do not include the full rate expression.Source: §A.3 · p. 14
B Proofs for Section 3
Theorem B.1 states conclusions for symplectic and explicit Euler discretizations of the NAG-SC ODE under separate step-size conditions.
B Proofs for Section 3
- Theorem B.1 concerns discretization of the NAG-SC ODE for functions in the stated strongly convex smooth class.The supplied theorem statement introduces the conclusions for this setting.Source: B Proofs for Section 3 · p. 15
- For 0 < s ≤ 4/(9L), the passage states that the symplectic Euler scheme satisfies a condition, but the supplied excerpt does not specify what it satisfies.Source: B Proofs for Section 3 · p. 15
B.1 Proof of Theorem B.1
The proof bounds Lyapunov-function iterate differences under step-size restrictions and uses these estimates to complete its argument.
B.1 Proof of Theorem B.1
- The proof derives a Lyapunov iterate-difference estimate when s≤4/(9L).The passage does not include the estimate itself.Source: §B.1 · p. 18
- The proof uses Cauchy-Schwarz when calculating Lyapunov iterate differences.The supplied proof fragments conclude with the proof complete.Source: §B.1 · pp. 19–20
- A further estimate is derived under the restriction s≤µ/(100L^2).The displayed estimate is not included in the passage.Source: §B.1 · pp. 19–20
B.2 Proof of Theorem 3.2
The section proves discretization results for the heavy-ball ODE and gives technical Lyapunov estimates, including a case where the estimate cannot certify nonpositive change.
Proof of Theorem B.2
- The proof develops Lyapunov iterate-difference estimates using Cauchy-Schwarz.The proof of Theorem B.2 is presented as the route to completing Theorem 3.2.Source: §B.2 · p. 20, Proof of Theorem B.2 · p. 21
B.2 Proof of Theorem 3.2
- Theorem B.2 gives symplectic and explicit Euler cases with step-size restrictions for the heavy-ball ODE.The stated thresholds are s≤µ/(16L^2) for symplectic Euler and s≤µ/(36L^2) for explicit Euler.Source: §B.2 · pp. 20–21
C.1 Technical details for numerical scheme of ODE (4.3)
- For ODE (4.3), the technical estimate cannot guarantee a nonpositive right-hand side.The supplied passage does not identify a resulting convergence claim.Source: §C.1 · p. 24
C.2 Technical details for standard numerical schemes
The technical analysis evaluates standard Euler schemes through Lyapunov iterate differences, but its estimates do not certify nonpositive change for the considered cases.
C.2 Technical details for standard numerical schemes
- For the symplectic scheme of ODE (4.1), the inequality’s right-hand side cannot be guaranteed non-positive; consequently, convergence cannot be proved by the above description.Source: §C.2 · pp. 25–26
- For the explicit scheme, the estimates likewise cannot guarantee a nonpositive right-hand side.The supplied analysis constructs and evaluates a Lyapunov function.Source: §C.2 · pp. 26–27
- For the implicit scheme, the negative term's k and k^2 multipliers still do not ensure a nonpositive right-hand side.The passage reports a limitation of the estimate, not a proof of divergence.Source: §C.2 · p. 27
D.1 Low-resolution ODE for strongly convex functions
The section examines low-resolution ODE discretizations for strongly convex and convex functions, distinguishing their convergence-rate behavior and the limits of the available Lyapunov estimates.
D.1 Low-resolution ODE for strongly convex functions
- With the strongly convex low-resolution ODE, symplectic and explicit Euler retain the continuous version's convergence-rate order but do not provide acceleration.For s=O(µ/L^2), the implicit scheme has no step-size limitation but is generally impractical for non-quadratic objectives.Source: §D.1 · pp. 28, 31
D.2.1 Symplectic Euler scheme
- For the symplectic Euler scheme, the Lyapunov estimate cannot guarantee a nonpositive right-hand side.The passage therefore does not establish convergence through this estimate.Source: §D.2.1 · p. 32
D.2.2 Explicit Euler scheme
- For the explicit Euler scheme, the estimates also fail to guarantee a nonpositive right-hand side.The analysis gives this limitation for both considered inequalities.Source: §D.2.2 · p. 33
D.2.3 Implicit scheme
- For the implicit Euler scheme, the estimate cannot guarantee a nonpositive right-hand side despite negative terms containing k and k^2.The section states that standard low-resolution ODE discretization cannot maintain the continuous-time rate due to numerical error.Source: §D.2.3 · p. 34
D.2 Low-resolution ODE for convex functions
- Standard numerical discretization of the low-resolution convex ODE cannot maintain the continuous-time convergence rate because of numerical error.The section considers symplectic, explicit, and implicit Euler schemes.Source: §D.2 · p. 31, §D.2.3 · p. 34