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An optimal gradient method for smooth strongly convex minimization

Adrien Taylor, Yoel Drori

arXiv:2101.09741v3math.OCcs.NAmath.NA

TL;DR

Smooth strongly convex minimization calls for first-order methods with strong worst-case guarantees. The paper presents ITEM and a constructive route to its parameters. ITEM’s worst-case distance guarantee matches the lower bound on oracle complexity.

  • Problem

    The paper addresses the design of black-box first-order methods with good worst-case guarantees for smooth strongly convex minimization.

    Source: §1 · p. 1

  • Method

    The paper derives ITEM by casting worst-case performance as a minimax problem and obtaining an analytical solution through a tractable relaxation.

    Source: §1 · p. 1, §3 · p. 8

  • Results

    ITEM’s worst-case distance guarantee matches the lower bound for smooth strongly convex minimization, which no black-box first-order method can improve.

    Source: §1 · pp. 1–2

  • Takeaways & Limitations

    The paper shows that ITEM can be developed constructively through performance estimation and semidefinite programming.

    Source: §4 · p. 17

    Extending ITEM and related methods to more general settings, possibly with constraints, seems less straightforward than extending other acceleration schemes.

    Source: §4 · p. 17

Abstract

from arXiv · show

We present an optimal gradient method for smooth strongly convex optimization. The method is optimal in the sense that its worst-case bound on the distance to an optimal point exactly matches the lower bound on the oracle complexity for the class of problems, meaning that no black-box first-order method can have a better worst-case guarantee without further assumptions on the class of problems at hand. In addition, we provide a constructive recipe for obtaining the algorithmic parameters of the method and illustrate that it can be used for deriving methods for other optimality criteria as well.

1 Introduction

The introduction presents ITEM for smooth strongly convex minimization and states that its worst-case distance guarantee matches the oracle-complexity lower bound. It places the method alongside prior optimized gradient methods and previews the paper’s construction and notation.

1 Introduction

  • The method’s parameters are expressed through sequences β_k and δ_k, parameterized by A_k, with q = µ/L denoting the inverse condition number.Source: §1 · pp. 1–2
  • ITEM attains the lower bound on oracle complexity when performance is measured by distance to an optimal solution, so no black-box first-order method can improve this guarantee.The introduction also states that ITEM’s bound guarantees strict improvement over its starting point.Source: §1 · pp. 1–2

1.1 Related works

  • The introduction connects prior methods by noting that OGM is not optimal when µ > 0, while TMM is undefined when µ = 0.It also situates these methods within work on accelerated methods and performance estimation.Source: §1.1 · pp. 2–3

1.2 Organization

  • The paper analyzes ITEM’s worst-case performance, develops a constructive approach to the method, and concludes with implications.Source: §1.2 · p. 3

1.3 Preliminaries and notations

  • The preliminaries define q := µ/L as the inverse condition number for the class, with 0 ≤ q < 1, and omit the trivial case L = µ for readability.Source: §1.3 · p. 3

2 An optimal gradient method

ITEM’s worst-case guarantee matches the lower bound for smooth strongly convex minimization, so that guarantee cannot be improved. The section analyzes its convergence, relates its limit cases to other methods, and gives simple functions attaining its worst case.

2 An optimal gradient method

  • Theorem 3 bounds distance to the solution and function-value performance; zN measures distance convergence, while xN has a function-value guarantee when µ = 0.The theorem also identifies yN as having a small corresponding ψN.Source: §2 · p. 4

2.1 Worst-case analysis

  • A potential-function argument combines weighted smoothness and strong-convexity inequalities to establish φk+1 ≤ φk.The inequalities are applied between x⋆ and yk, and between yk−1 and yk.Source: §2.1 · p. 5

2.2 Limit cases

  • When µ = 0, ITEM relates to OGM; for µ > 0, its parameters tend toward those of TMM as k → ∞.The OGM and TMM relationships are described as limit cases of ITEM.Source: §2.2 · pp. 6–7

2.3 Lower bound and matching examples

  • ITEM’s worst-case guarantee matches the lower bound, so it cannot be improved.The lower bound is from [Drori and Taylor, 2022].Source: §2.3 · p. 7
  • Two base quadratics, fµ and fL, attain ITEM’s worst-case behavior.The section presents these as simple examples of functions attaining the worst case.Source: §2.3 · p. 8

3 A constructive approach to ITEM

This section develops a constructive approach to ITEM by turning a minimax search over first-order methods into a tractable optimization problem, and describes adapting the procedure to other optimality criteria. It also gives coefficient implications and a piecewise definition of α for solutions to Minimax-R under stated parameter conditions.

3 A constructive approach to ITEM

  • ITEM is obtained analytically from a tractable relaxation of the minimax search for the best first-order method.The section also reports numerically designing methods for alternative criteria, including normalized objective error.Source: §3 · pp. 8–9

3.1 Fixed-step first-order methods

  • Optimizing fixed-step methods directly leads to a bilinear/quadratic problem that is nonconvex and NP-hard in general.Reparametrization, relaxation, and linearization are used to obtain a tractable relaxation.Source: §3.1 · p. 10

3.2 A reparametrization of fixed-step first-order methods

  • An equivalent reparametrization replaces quadratic dependence on the step coefficients with linear dependence in the upper-bound formulation.The reformulation makes the resulting maximization only bilinear in method parameters and multipliers.Source: §3.2 · pp. 10–11

3.3 A performance estimation problem and its relaxation

  • A relaxation retaining selected quadratic inequalities upper-bounds worst-case performance and is tight for ITEM; its formulation is semidefinite-representable.Source: §3.3 · pp. 12–13

3.4 Tractable upper bounds using semidefinite programming (SDP)

  • A Gram-matrix SDP and weak duality yield the upper bound UBµ,L(M), but its matrix inequality remains bilinear in method and dual variables.The bilinear structure motivates the linearization step that follows.Source: §3.4 · pp. 13–15

3.5 An approximate minimax and its semidefinite representation

  • An invertible change of variables linearizes the bilinear matrix inequality, reformulating the approximate minimax as a standard linear semidefinite program.Source: §3.5 · pp. 15–16
  • Under N > 1 and 0 ≤ μ < L, zero λ entries force the corresponding β coefficients to vanish, and α is defined piecewise from β and λ.Specifically, λ_{k,k+1} = 0 implies β_{k,j} = 0, while λ_{N−1,*} = 0 implies β_{N−1,j} = 0; the α cases include ratios of β to λ and β_{N,j}.Source: §3.5 · p. 16

The output of the corresponding method of the form (8) satisfies

The section establishes that ITEM solves the stated minimax relaxation. The proof links the relaxation to ITEM’s lower-bound attainment and fixed-step structure.

The output of the corresponding method of the form (8) satisfies

4 Conclusion

The conclusion presents ITEM as matching the lower bound for smooth strongly convex minimization and describes a constructive design approach. It also identifies broader performance criteria as a possible direction and notes that constrained extensions are less straightforward.

4 Conclusion

  • ITEM’s worst-case guarantee exactly matches the lower bound for minimizing smooth strongly convex functions.The authors describe the constructive approach as using performance estimation problems and semidefinite programming.Source: §4 · p. 17
  • The authors interpret the iterates z_k as optimal for distance to an optimal solution and y_k as essentially optimal for function values.They present these interpretations as a perspective on accelerated methods.Source: §4 · p. 17
  • The design approach may inspire methods for other settings and performance criteria, as illustrated numerically in Appendix E.The conclusion also notes that extending OGM, TMM, and ITEM to settings such as constrained problems seems less straightforward than extending other acceleration schemes.Source: §4 · p. 17

A Alternate parametrization for first-order methods

This appendix shows how to reparametrize methods of form (7) as form (8), and vice versa, using identity (9). It establishes the equivalence iteratively and makes the change of variables reversible.

A Alternate parametrization for first-order methods

B Algebraic manipulations of (dual-SDP-R)

This appendix reformulates (dual-SDP-R) to optimize the α and λ variables simultaneously. A Schur complement yields a lifted linear matrix inequality with linear dependence on the final-iteration α variables.

B Algebraic manipulations of (dual-SDP-R)

  • The reformulation enables simultaneous optimization over α_i,j and λ_i,j.The appendix begins by rewriting (dual-SDP-R).Source: B Algebraic manipulations of… · p. 20
  • A Schur complement produces a lifted linear matrix inequality that depends linearly on the final-iteration α variables.The final-iteration coefficients appear through w_N, which is absent from S′.Source: B Algebraic manipulations of… · p. 20
  • For N = 1, (6) can be solved without the appendix’s simplifications.For N > 1, the expanded S′ remains bilinear in α_i,j and λ_i,j, though the expression can be simplified.Source: B Algebraic manipulations of… · pp. 20–21

C The Information-Theoretic Exact Method is a solution to (Minimax-R)

The proof rewrites ITEM as a fixed-step first-order method in an alternate parametrization and constructs a feasible solution to (Minimax-R). Its optimality follows because the resulting value matches the lower complexity bound.

C The Information-Theoretic Exact Method is a solution to (Minimax-R)

  • ITEM is treated as a fixed-step first-order method and expressed in the alternate parametrization using its y and z sequences.The proof associates w_i with y_i for i = 0, …, N−1, and w_N with z_N.Source: C The Information-Theoretic Exact… · p. 21
  • The constructed solution is feasible for (Minimax-R), and its optimality follows from matching the lower complexity bound.The proof also identifies ITEM as a solution to (Minimax-R).Source: C The Information-Theoretic Exact… · pp. 21–22
  • The proof establishes dual feasibility by relating (Minimax-R) to the Lagrangian of (R).It then uses the association between the alternate parametrization and the y and z sequences in the subsequent argument.Source: C The Information-Theoretic Exact… · pp. 21–22

D An SDP formulation for optimizing function values

The section adapts the design methodology to optimize function-value criteria, with changes to the selected inequalities and linearization. The resulting semidefinite programs can be solved numerically for specified problem parameters.

D An SDP formulation for optimizing function values

  • The design methodology extends to optimizing function-value criteria, including final suboptimality normalized by initial distance or initial suboptimality.The section presents numerical examples for these two criteria.Source: D An SDP formulation for optimizing… · p. 22
  • For alternate criteria, the selected inequalities differ from those used for distance minimization, which changes the linearization procedure.The derivation starts from the discrete method and uses a slightly different upper bound and constraint subset.Source: D An SDP formulation for optimizing… · pp. 22–23
  • The resulting design problem is an SDP that can be solved with standard packages, with numerical results for the two example criteria.The criteria can be approximated numerically for specific µ, L, and N, although closed-form solutions were not obtained.Source: D An SDP formulation for optimizing… · p. 24

E Numerical examples

The numerical examples are produced by solving the first-order method design problem as a linear semidefinite program using standard solvers.

E Numerical examples

  • The design relaxations can be numerically approximated for specific values of µ, L, and N, though closed-form solutions were not obtained for the alternative criteria.The approach adapts the relaxations used for (Minimax-R).Source: E Numerical examples · p. 24
  • The examples come from numerically solving the first-order method design problem, formulated as a linear SDP.The authors use standard solvers.Source: E Numerical examples · p. 24

E.1 Optimized methods for (f(wN) −f⋆)/∥w0 −w⋆∥2

For the function-value criterion, the examples provide optimized methods for several iteration counts and compare worst-case guarantees with a lower bound and other methods. For µ > 0, the resulting step sizes depend on the horizon N.

E.1 Optimized methods for (f(wN) −f⋆)/∥w0 −w⋆∥2

  • The examples report optimized methods and worst-case guarantees for N = 1 through 5 with L = 1 and µ = .1.The supplied passages give no complete guarantee values for these cases.Source: §E.1 · p. 25
  • For µ > 0, the optimized method’s step sizes critically depend on horizon N, which the authors describe as making it less practical.For µ = 0, the step-size policy recovers OGM.Source: §E.1 · p. 25
  • The numerical comparison includes the optimized method’s guarantee, a lower bound, and guarantees for other methods; the evaluated guarantees are described as tight.Figure 1 compares the method with the lower bound and alternatives, while the guarantees are computed using performance estimation problems.Source: §E.1 · p. 25, Figure 1 · p. 26

E.2 Optimized methods for (f(wN) −f⋆)/(f(w0) −f⋆)

The method is adapted to optimize step sizes for the function-value ratio across iteration horizons. Its guarantees appear slightly better than those for distance, but the asymptotic rate is the same.

E.2 Optimized methods for (f(wN) −f⋆)/(f(w0) −f⋆)

  • The optimization sets L=1 and µ=.1 and derives step sizes for different horizons N; the resulting step sizes are symmetric but depend critically on N, making the method less practical than ITEM.Source: §E.2 · p. 27
  • For a single iteration, N = 1, the passage states that a guarantee is obtained for the quantity f(w1)−f⋆, without specifying the guarantee’s value.Source: §E.2 · p. 27
  • For N=2, the method yields a guarantee for the function-value ratio.Source: §E.2 · p. 27
  • Function-value guarantees appear slightly better than distance-based guarantees, while sharing the same asymptotic rate under strong convexity.Source: §E.2 · p. 27

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