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Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities

Jelena Diakonikolas

arXiv:2002.08872v3math.OCcs.DScs.LG

TL;DR

Monotone inclusion is connected to variational inequalities and convex-concave min-max optimization, including settings where feasible sets are unbounded. The paper analyzes Halpern iteration with a potential-based proof and uses problem-class reductions; it reports parameter-free, near-optimal methods across operator classes, while leaving a gap between upper and lower bounds.

  • Problem

    Monotone inclusion and monotone-operator VIs model convex-concave min-max problems, but optimality-gap guarantees for those problems generally require bounded feasible sets.

    Source: §1 · p. 1

  • Method

    The paper develops a potential-based convergence analysis of Halpern iteration and uses connections and reductions among monotone inclusion, VIs, nonexpansive maps, and proximal methods.

    Source: §1 · p. 2, §1.1 · p. 3, §2.2 · p. 9

  • Results

    The paper reports parameter-free 1/k convergence for cocoercive operators and a 1/k rate up to logarithmic factors for monotone Lipschitz operators, with near-optimal guarantees across the studied settings.

    Source: §1.1 · p. 3

  • Takeaways & Limitations

    The methods yield guarantees for approximating strong VI solutions and convex-concave min-max problems, including gradient-norm stationarity in the unconstrained setting.

    Source: §1 · p. 1, §4 · p. 12

    The gap between the upper and lower bounds remains open; the paper conjectures optimal complexity rates for the stated operator settings.

    Source: §4 · p. 12

Abstract

from arXiv · show

We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving monotone inclusion problems. These results immediately translate into near-optimal guarantees for approximating strong solutions to variational inequality problems, approximating convex-concave min-max optimization problems, and minimizing the norm of the gradient in min-max optimization problems. Our analysis is based on a novel and simple potential-based proof of convergence of Halpern iteration, a classical iteration for finding fixed points of nonexpansive maps. Additionally, we provide a series of algorithmic reductions that highlight connections between different problem classes and lead to lower bounds that certify near-optimality of the studied methods.

1 Introduction

Monotone inclusion connects to variational inequalities and convex-concave min-max optimization, while remaining approximable on unbounded feasible sets where optimality-gap guarantees may fail. The paper uses Halpern iteration and reductions across problem classes to develop parameter-free, near-optimal methods for several operator settings.

1 Introduction

  • Monotone inclusion models variational inequalities and convex-concave min-max problems, and remains approximable on unbounded feasible sets where optimality-gap guarantees may be unavailable.In the unconstrained min-max setting, it corresponds to minimizing the norm of the gradient; its approximations also target strong VI solutions.Source: §1 · pp. 1–2

1.1 Contributions and Related Work

  • For cocoercive operators, a potential-based Halpern analysis yields a parameter-free method with a 1/k convergence rate, extended to constrained problems via a generalized gradient mapping.The proof uses more general step sizes than Lieder [2017] and is described as requiring only elementary algebra.Source: §1.1 · p. 3
  • For monotone Lipschitz operators, inexact proximal maps coupled with Halpern iteration give a parameter-free 1/k rate up to a logarithmic factor.The cited prior rate for the same setting was of order 1/√k [Dang and Lan, 2015, Ryu et al., 2019].Source: §1.1 · p. 3
  • For strongly monotone Lipschitz operators, restarting the monotone-Lipschitz method gives a parameter-free method that is iteration-complexity-optimal up to poly-logarithmic factors under mild assumptions.The authors describe it as the first near-optimal parameter-free method for this setting and its associated inclusion, VI, and min-max problems.Source: §1.1 · p. 3
  • Algorithmic reductions between problem classes yield lower bounds used to certify near-optimality of the analyzed methods.The lower bounds leverage Ouyang and Xu [2019] on approximating the optimality gap in convex-concave min-max optimization.Source: §1.1 · p. 3

1.2 Notation and Preliminaries

  • The preliminaries distinguish weak and strong VI solutions and relate monotone inclusion solutions to strong VI solutions.When the operator is monotone, the exact solution sets of the Stampacchia and Minty VIs coincide.Source: §1.2 · p. 4

2 Halpern Iteration for Monotone Inclusion and Variational Inequalities

This section applies Halpern iteration to monotone inclusion and variational inequality problems, presenting parameter-free methods for cocoercive and monotone Lipschitz operators and a restart method for strongly monotone operators. For strongly monotone and Lipschitz operators, it also bounds the F-query cost of producing an iterate with residual at most ε.

2 Halpern Iteration for Monotone Inclusion and Variational Inequalities

  • For 1/L-cocoercive F on E and u0∈U, Algorithm 1 returns uk with ∥F(uk)∥≤ε after at most max{2L,L0}∥u0−u*∥/ε + max{0, log2(2L/L0)} oracle queries to F.Source: §2 · p. 6

2.1 Constrained Setups with Cocoercive Operators

  • For constrained cocoercive problems, Algorithm 2 finds ∥GLk(uk)∥≤ε in at most max{4L,L0}∥u0−u*∥/ε + max{0, log2(4L/L0)} oracle queries.The operator mapping is 1/(2η)-cocoercive for η≥L, and a small mapping norm yields an approximate inclusion solution; all constructed iterates are feasible.Source: §2.1 · pp. 7–9

2.2 Setups with non-Cocoercive Lipschitz Operators

  • For monotone, L-Lipschitz F, the inexact Halpern iteration uses the 1/2-cocoercive resolvent residual P, with each iteration requiring O((L + 1) log((L + 1)∥u0−u∗∥/ǫ)) oracle queries to F.If ∥P(uk)∥≤ǫ, then uk is an ǫ-approximate solution to (MI); when the diameter D is bounded, ∥P(uk)∥≤ǫD implies an ǫ-approximate solution to (SVI).Source: §2.2 · pp. 9–11

2.3 Setups with Strongly Monotone and Lipschitz Operators

  • Restarting Algorithm 3 is proposed as a parameter-free method with near-optimal oracle complexity for strongly monotone, Lipschitz operators, assuming L=Ω(1).The stated algorithm invokes Algorithm 3 with a sequence of error parameters.Source: §2.3 · p. 11
  • The method outputs u_k with ∥P(u_k)∥≤ε after a bounded number of iterations and states an O(·) bound on F queries to reach that residual.The supplied passages truncate the iteration-count and query-complexity expressions, so their full dependencies cannot be reported.Source: §2.3 · p. 11
  • The per-call query analysis uses ∥P̃(u_k)∥=Θ(∥P(u_k)∥) and relates the cost to the iterate’s distance from u*; the supplied expression is truncated.The passage attributes this analysis to Theorem 2.10.Source: §2.3 · p. 11
  • The guarantee is framed for an m-strongly monotone operator F when u* is an MVI solution.Source: §2.3 · p. 11

3 Lower Bound Reductions

The section states lower bounds in the operator-oracle model and uses them to certify that the paper’s algorithms are optimal up to logarithmic factors in the stated settings.

3 Lower Bound Reductions

  • The lower-bound lemma considers deterministic algorithms in the operator-oracle model and constructs an L-Lipschitz operator with a closed convex feasible set of diameter D.The section defers details about the oracle model and proof to Appendix A.Source: §3 · p. 12
  • Algorithms 2 and 3 are optimal up to logarithmic factors for their respective settings, as certified by parts (a)–(c) of the lower-bound lemma.For Algorithm 3, the stated iteration bounds are O(LD^2/ε) for an O(ε) global VI gap and O(LD/ε) for an O(ε) gap over U∩B_{u_k}; Algorithm 2 is optimal up to log(D/ε).Source: §3 · p. 12
  • For strongly monotone operators, the restarting algorithm is optimal up to log(D/ε) log(L/m) when L=Ω(L/m).The text says scaling can ensure this condition without changing L/m, while affecting only the target error, which appears under a logarithm.Source: §3 · p. 12

4 Conclusion

The paper concludes that variants of Halpern iteration yield near-optimal methods for monotone inclusion with Lipschitz operators. The gap between upper and lower bounds remains open, and the paper conjectures optimal complexity bounds for different operator settings.

4 Conclusion

  • Variants of Halpern iteration give near-optimal methods for monotone inclusion problems with Lipschitz operators.The paper highlights connections to variational inequalities, fixed points of nonexpansive maps, and proximal-point-type algorithms.Source: §4 · p. 12
  • Closing the gap between the upper and lower bounds remains an open question.The paper conjectures optimal complexity bounds for operators that are Lipschitz or cocoercive, and for Lipschitz operators that are strongly monotone.Source: §4 · p. 12

A.1 Unconstrained Setting with a Cocoercive Operator

This appendix proof uses a potential sequence to control the iterates for an unconstrained problem with a cocoercive operator. Under the stated assumptions, it derives a bound on the operator norm at an iterate in terms of its distance from a solution.

A.1 Unconstrained Setting with a Cocoercive Operator

  • The assumptions imply C_1≤0 and then C_k≤0 for every k, since A_k is positive.The proof establishes C_1≤0 using the initial update and the solution condition.Source: §A.1 · p. 15
  • The resulting inequality bounds ∥F(u_k)∥ by L_k∥u_0−u*∥/(1−λ_k) for a solution u*.The proof divides by ∥F(u_k)∥ and notes that the bound holds for the solution closest to u_0.Source: §A.1 · p. 15
  • The proof uses a potential C_k whose weighted value A_kC_k is nonincreasing under the stated sequence conditions.The recurrence is A_{k+1}C_{k+1} ≤ A_kC_k, with A_1=1 and positive A_i.Source: §A.1 · pp. 15–16

A.2 Operator Mapping

The operator mapping proposition considers G_η for a cocoercive operator F and η≥L. Its proof invokes cocoercivity of the projection residual and Young’s inequality.

A.2 Operator Mapping

  • The proposition considers an 1/L-cocoercive operator F and defines G_η for η≥L.The supplied proposition states that G_η has a resulting operator property, but the property itself is not fully present in the passage.Source: §A.2 · p. 16
  • The proof uses that Id−Π_U is 1-cocoercive and then applies Young’s inequality to bound an inner product involving G_η and F.The inequality also uses a bound on ∥F(u)−F(v)∥².Source: §A.2 · p. 16
  • Algorithm 4 takes a_0, u_0∈U, m, and ε as inputs, setting a_0=1/m if it is omitted or exceeds 1/m.Source: §A.2 · p. 17

A.3 Approximating the Resolvent

The appendix analyzes an extragradient method that approximates the resolvent by solving a strongly monotone Lipschitz variational inequality. The method does not require the Lipschitz constant, and its backtracking loop has a logarithmic bound on its entries.

A.3 Approximating the Resolvent

  • Algorithm 4 is an extragradient method that does not require L but does require the strong monotonicity parameter m.When computing the resolvent, the passage specifies m=1.Source: §A.3 · p. 17
  • The analysis bounds the method’s iterations using a gap function and the three-point identity.It uses strong monotonicity to establish nonnegativity of the gap function.Source: §A.3 · pp. 17–18
  • The backtracking loop is entered at most log2(4L/a_0) times, with each accepted step size a_k>1/(4L).Source: §A.3 · p. 18
  • The oracle-query accounting counts two queries to F per inner iteration and two additional queries per outer iteration.Source: §A.3 · p. 19
  • The resolvent is approximated by solving a variational inequality for F+Id, which is 1-strongly monotone and (L+1)-Lipschitz.The passage gives a parameter-free query bound for producing an approximate resolvent solution.Source: §A.3 · p. 19

A.4 Inexact Halpern Iteration

The appendix analyzes inexact Halpern iteration by relating it to a nonexpansive fixed-point map and tracking the effect of errors. It also states a strongly monotone convergence result, though the supplied passages do not expose its full bound.

A.4 Inexact Halpern Iteration

  • Proposition A.2 considers iterates from any initial point when u∗ satisfies ∥P(u∗)∥=0.The proposition concerns the sequence defined by Eq. (2.3), with λk = 1… as stated in the supplied excerpt.Source: §A.4 · p. 19
  • Writing T = Id−P makes the iteration an inexact Halpern update of a nonexpansive map with fixed point u∗.The update combines the initial point, T(uk−1), and an error term.Source: §A.4 · p. 19
  • The proof uses the triangle inequality and nonexpansivity to recursively bound the distance between iterates and the fixed point.Source: §A.4 · p. 19
  • Lemma 2.9 studies the sequence with a 1/2-cocoercive operator, Lk = 2, and λk = 1/2.The excerpt identifies the lemma’s setting but does not include its conclusion.Source: §A.4 · p. 20

A.5 Strongly Monotone Lipschitz Operators

For L-Lipschitz, m-strongly monotone operators, Theorem 2.12 gives an iteration bound for reaching a point with residual at most ǫ. Its stated number of iterations depends logarithmically on the initial distance to a solution relative to ǫ.

A.5 Strongly Monotone Lipschitz Operators

  • Algorithm A invokes Algorithm 3 at each iteration with error parameter ǫk = 7… .The supplied excerpt truncates the parameter value.Source: §A.5 · p. 21
  • log2(∥u0 −u∗∥/ǫ) iterations suffice for Algorithm A to reach ∥P(uk)∥≤ǫ.The theorem states this for w.l.o.g. ǫ ≤ 1 and an L-Lipschitz, m-strongly monotone F.Source: §A.5 · p. 21

A.6 Lower Bounds

The paper derives lower bounds in deterministic first-order oracle models, using reductions among problem classes and a prior convex-concave min-max lower bound. The supplied passages describe existence of hard instances and some bound conditions, but several quantitative expressions are incomplete.

A.6 Lower Bounds

  • The lower-bound results use algorithmic reductions and Ouyang and Xu [2019]’s lower bound for convex-concave min-max optimality-gap approximation.The authors use these reductions to derive near-tight lower bounds for the problems considered in the paper.Source: §A.6 · p. 22
  • The lower bounds are formulated in a high-dimensional regime, with the cited min-max result requiring d = Ω(L(RX^2+RXRY)/ǫ).Here RX and RY are the diameters of the respective feasible sets.Source: §A.6 · p. 22
  • The lower bounds apply to deterministic algorithms in an operator-oracle model, where each iteration queries F at a point and receives F’s value.The algorithm’s query and candidate points depend on global parameters and prior oracle queries and answers.Source: §A.6 · p. 22
  • Lemma 3.1 asserts that for any L and D, some VI with an L-Lipschitz operator and feasible-set diameter D is hard for deterministic operator-oracle algorithms.Source: §A.6 · p. 22
  • The passage reports that the maximum over u∈U∩B_{u_k} of ⟨F(u_k),u_k−u⟩ is Ω(Lε), giving a local-gap lower bound.Source: §A.6 · p. 23

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