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On Generalized Forward-Reflected-Backward Method for Monotone Inclusion Problems

Santanu Soe, V. Vetrivel, Jen-Chih Yao

TL;DR

The paper develops and analyzes the generalized forward-reflected-backward ($GFRB$) method for monotone inclusions in real Hilbert spaces, introducing a non-decreasing stepsize rule that obviates the need for the Lipschitz constant of $B$. It proves weak convergence under this adaptive rule, provides rate analyses in simple settings (including a lower bound of $1/\sqrt{2}$ and conditions for improved linear rates), and derives an extended primal–dual twice-reflected algorithm (EPDTR) by embedding $GFRB$ into a three-operator framework. Additionally, it shows how EPDTR can be viewed as a $GFRB$ on a product space and establishes convergence under a specific step-size condition. Numerical experiments on LASSO and related problems demonstrate competitive performance and faster convergence relative to existing forward-backward-type schemes, highlighting the practical impact of a parameter-free, adaptive operator-splitting approach.

Abstract

We study the generalized forward-reflected-backward (GFRB) method, an extension of the forward-reflected-backward (FRB) scheme due to Malitsky and Tam, for solving monotone inclusion problems in real Hilbert spaces. We first analyze GFRB equipped with a non-decreasing step-size rule that does not require prior knowledge of the Lipschitz constant of the operator involved. We then present two illustrative examples: in the first, we show that the convergence rate of GFRB is bounded from below by that of FRB, and in the second, we obtain an improved convergence rate for GFRB via an appropriate choice of initial parameters. In the sequel, we propose an extended primal-dual twice-reflected (PDTR) algorithm and show that it can be recovered from GFRB under suitable metric selections. Finally, we validate the proposed approach on several state-of-the-art problems and demonstrate better numerical performance compared to the existing ones.

On Generalized Forward-Reflected-Backward Method for Monotone Inclusion Problems

TL;DR

The paper develops and analyzes the generalized forward-reflected-backward () method for monotone inclusions in real Hilbert spaces, introducing a non-decreasing stepsize rule that obviates the need for the Lipschitz constant of . It proves weak convergence under this adaptive rule, provides rate analyses in simple settings (including a lower bound of and conditions for improved linear rates), and derives an extended primal–dual twice-reflected algorithm (EPDTR) by embedding into a three-operator framework. Additionally, it shows how EPDTR can be viewed as a on a product space and establishes convergence under a specific step-size condition. Numerical experiments on LASSO and related problems demonstrate competitive performance and faster convergence relative to existing forward-backward-type schemes, highlighting the practical impact of a parameter-free, adaptive operator-splitting approach.

Abstract

We study the generalized forward-reflected-backward (GFRB) method, an extension of the forward-reflected-backward (FRB) scheme due to Malitsky and Tam, for solving monotone inclusion problems in real Hilbert spaces. We first analyze GFRB equipped with a non-decreasing step-size rule that does not require prior knowledge of the Lipschitz constant of the operator involved. We then present two illustrative examples: in the first, we show that the convergence rate of GFRB is bounded from below by that of FRB, and in the second, we obtain an improved convergence rate for GFRB via an appropriate choice of initial parameters. In the sequel, we propose an extended primal-dual twice-reflected (PDTR) algorithm and show that it can be recovered from GFRB under suitable metric selections. Finally, we validate the proposed approach on several state-of-the-art problems and demonstrate better numerical performance compared to the existing ones.

Paper Structure

This paper contains 10 sections, 9 theorems, 39 equations, 7 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. GFRB with a non-decreasing step-size strategy
  4. Rate of Convergence
  5. Improved Rate of Convergence
  6. An Extended Primal-Dual Twice Reflected Algorithm
  7. Derivation as a GFRB and Convergence
  8. Numerical Experiments
  9. LASSO signal recovery
  10. Conclusion

Key Result

Lemma 2.1

Let $a,b,c\in\mathbb{H}$. Then,

Figures (7)

  • Figure 1: Admissible $(\tau,\sigma)$ regions for Algorithm \ref{['alg:EPDTR_splitting']} across six values of $\alpha$, obtained by plotting $b(\tau,\sigma;\alpha)$.
  • Figure 2: Comparison of $\mathrm{err}_k$ for Example \ref{['example_1']} for $m\in\{200,500\}$. The parameter choices in Algorithm \ref{['alg:forb_incr']} are selected as: for (a) and (d), $(\alpha,\delta)=(10^{-2},10^{-2})$; for (b) and (e), $(\alpha,\delta)=(0,10^{-2})$; for (c) and (f) $(\alpha,\delta)=(0,0)$.
  • Figure 3: Comparison of $\mathrm{err}_k$ for Example \ref{['example_2']} for $m\in\{1000,5000\}$. The parameter choices in Algorithm \ref{['alg:forb_incr']} are selected as: for (a) and (d), $(\alpha,\delta)=(0,0)$; for (b) and (e), $(\alpha,\delta)=(10^{-2},10^{-2})$; for (c) and (f) $(\alpha,\delta)=(10^{-2},10^{-3})$.
  • Figure 4: Convergence plots for LASSO signal recovery: (a) iterate gap versus iterations, and (b) SNR recovery over iterations.
  • Figure 5: Comparison of the true sparse signal $x^*$ reconstructed by different algorithms.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Remark 5.1
  • Remark 5.2
  • ...and 3 more