AdaGrad Fixed Point Iterations

• The paper introduces adaptive fixed-point schemes that leverage AdaGrad methods to control convergence via regret minimization.
• Adaptive step-size and preconditioning methods enable per-coordinate scaling, improving convergence in applications like image processing and game theory.
• Empirical results demonstrate robust performance in ill-conditioned problems, outperforming traditional Krasnoselskii–Mann iterations with enhanced convergence constants.

AdaGrad-based fixed point iterations refer to iterative algorithms for finding fixed points of nonlinear operators in which classical fixed-point schemes—such as Krasnoselskii–Mann or Picard iterations—are replaced or augmented by adaptive step-size selection or preconditioning analogous to AdaGrad methods from online optimization. These adaptive algorithms leverage regret minimization and history-dependent scaling to control the convergence rate and robustness of fixed-point residuals. They have demonstrated advantages for nonexpansive, contractive, and monotone operators in practical applications spanning image processing, Markov chains, and game-theoretic settings.

1. Foundations: Fixed Point Iterations and Regret Minimization

Traditional fixed-point iterations for an operator $F$ on a convex set $\mathcal{X}$ take the form

$x_{t+1} = x_t + \gamma_t (F(x_t) - x_t),$

where $\gamma_t$ is a predetermined step-size. The Krasnoselskii–Mann (KM) scheme—typically with $\gamma_t = 1/2$—guarantees convergence of $\|F(x_t) - x_t\| \to 0$ at the rate $O(1/\sqrt{T})$ for $T$ iterations under nonexpansiveness.

A regret minimization framework (as in (Kwon, 25 Sep 2025)) establishes a direct correspondence between cumulative online regret bounds and fixed-point residuals. For an adaptive online algorithm, the regret

$\sum_{t=1}^T \langle u_t, x - x_t \rangle$

is related to the cumulative residuals $u_t = \gamma_t (F(x_t) - x_t)$. Converting regret bounds to residual bounds yields adaptive guarantees for convergence to a fixed point. This approach underpins the design of AdaGrad-based fixed-point iterations.

2. Adaptive Step-Size and Preconditioning via AdaGrad

AdaGrad algorithms adapt the learning rate based on the accumulated squared gradient (or residual in the fixed-point context). AdaGrad–Norm and its diagonal/full-matrix variants deploy history-dependent scaling:

• AdaGrad–Norm:

$$x_{t+1} = \Pi_{\mathcal{X}} \left[x_t + \frac{\eta}{\sqrt{\sum_{s=1}^t \|F(x_s) - x_s\|^2}} (F(x_t) - x_t) \right]$$

• AdaGrad–Diagonal:

$$x_{t+1} = \Pi_{\mathcal{X},A_t} \left[x_t + A_t^{-1}(F(x_t) - x_t)\right]$$

$$A_t = \frac{1}{\eta} \cdot \text{diag} \left(\sqrt{\epsilon^2 + \sum_{s=1}^t (F(x_s)_i - x_{s,i})^2} \right)$$

where $\Pi_{\mathcal{X}}$ denotes projection onto the constraint set (potentially with Mahalanobis norm induced by $A_t$).

This process adapts the effective step-size and conditioning matrix based on observed residual norms, leading to per-coordinate or full-matrix preconditioning. Unlike classical schemes, there is no need to select a fixed $\gamma$ a priori. The approach is sensitive to local anisotropy and geometric structure in residuals.

3. Theoretical Convergence Guarantees

For an operator $F$ that is scalar or matrix scaled nonexpansive (i.e., $$\|F(x) - x\|^2 \le 2 \langle F(x) - x, x^* - x \rangle$$ for fixed points $x^*$), conversion of AdaGrad's regret bounds yields:

$$\min_{t=1,\ldots,T} \|F(x_t) - x_t\| \leq \frac{L_T}{\sqrt{T}} \cdot \text{[additional factors]},$$

where $L_T$ depends on the adaptive geometry revealed by past iterates (see formula (1) in (Kwon, 25 Sep 2025)):

$$L_T := \inf \left\{ L > 0 : \forall t=1,\ldots,T,\, \|F(x_t) - x_t\|^2 \le 2L \langle F(x_t) - x_t, x^* - x_t\rangle \right\}.$$

If $F$ is $L$-nonexpansive, then $L_T \leq L$.

In contrast, fixed step-size KM iterations are bound by constants determined a priori and may be suboptimal in cases of geometric variability. AdaGrad-based methods adapt this constant dynamically, potentially yielding significantly faster convergence in practice.

4. Algorithmic Structure and Variants

AdaGrad-based fixed-point algorithms can be summarized in the table below:

Variant	Update Formula	Preconditioning
Norm	$$x_{t+1} = \Pi_{\mathcal{X}}[x_t + \frac{\eta}{\sqrt{G_t}}(F(x_t)-x_t)]$$	Scalar (history-dependent)
Diagonal	$$x_{t+1} = \Pi_{\mathcal{X},A_t}[x_t + A_t^{-1}(F(x_t)-x_t)]$$	Diagonal (coordinatewise)
Full	$$x_{t+1} = \Pi_{\mathcal{X},A_t}[x_t + A_t^{-1}(F(x_t)-x_t)]$$	Full matrix

Where $G_t = \sum_{s=1}^t \|F(x_s) - x_s\|^2$ and $A_t$ depends on the history of residuals.

Significance: These algorithmic variants allow for fine-grained control over the adaptation process and are robust to local geometry, scale heterogeneity, and ill-conditioning.

5. Practical Performance: Empirical Evidence

Experiments reported in (Kwon, 25 Sep 2025) demonstrate improved practical performance of AdaGrad-based fixed-point iterations:

• Markov Chain Stationary Distribution: AdaGrad–Norm and diagonal variants converged more rapidly and robustly than power iteration, even in cases where the standard method diverged.
• Image Denoising (Total Variation): AdaGrad schemes performed reliably across a wide range of step-sizes, whereas the classical KM/Chambolle–Pock required sensitive tuning and sometimes failed to converge.
• Zero-Sum Game Algorithms: AdaGrad-based fixed-point algorithms were more robust to untuned or large step-sizes, with convergence rates matching or surpassing standard mirror-prox or KM iterations.

Notably, in ill-conditioned or poorly scaled problems, AdaGrad-enabled methods maintained convergence where hand-tuned fixed step-size methods failed or were slow.

6. Comparison with Classical and Other Adaptive Schemes

While the Krasnoselskii–Mann iteration requires selection of a constant averaging parameter and lacks geometric sensitivity, AdaGrad-based fixed-point iterations automatically adapt step-sizes and preconditioners. This yields

• Data-dependent or coordinate-dependent convergence factors,
• Robustness to operator anisotropy and residual variance,
• Convergence rates matching classical $O(1/\sqrt{T})$ bounds but potentially with significantly improved constants.

AdaGrad preconditioning can be interpreted as online adaptation, similar to approaches in adaptive relaxation Anderson acceleration schemes (Lepage-Saucier, 29 Aug 2024), albeit with different residual minimization criteria.

This methodology is also closely linked to advanced theoretical treatments for fixed-point acceleration and optimality (Park et al., 2022), suggesting plausible further improvements by integrating meta-algorithmic acceleration (e.g., Halpern-type anchoring) with AdaGrad-based adaptation.

7. Context and Perspectives

The regret minimization approach to fixed-point iterations supplies a general framework by which existing online learning algorithms (especially AdaGrad family) can be converted into iterations with adaptive fixed-point guarantees (Kwon, 25 Sep 2025). This conversion applies to a wide class of operators—potentially non-self, nonexpansive, or monotone—far beyond the reach of traditional fixed-point theory.

A plausible implication is that AdaGrad-based fixed-point iterations may be further accelerated or modified for broader classes of problems (e.g., monotone inclusions, resolving variable metrics as in (Atenas et al., 10 Jul 2025)), and their robustness under moderate operator nonexpansiveness and nonstationarity provides a foundation for large-scale, streaming, or ill-posed problem settings common in modern applied mathematics and data science.

In summary, AdaGrad-based fixed point iterations leverage the adaptivity of online optimization to yield robust, geometry-aware fixed-point algorithms with practical and theoretical advantages over classical schemes.