A Geometric Framework for Stochastic Iterations

TL;DR

This work addresses stochastic iterations for finding a point in a closed solution set $Z$ within a Hilbert space by reinterpreting updates as random relaxed projections onto half-spaces that contain $Z$. It develops an abstract stochastic algorithm and two enhancements—super relaxations and random relaxations bounded by 2—providing rigorous almost-sure weak, strong, and linear convergence guarantees. The framework unifies stochastic fixed-point, stochastic gradient, and randomized extrapolated/parallel feasibility methods, enabling flexible, scalable designs for broad convex problems. Numerical experiments in signal and image restoration illustrate practical gains in convergence speed and robustness, underscoring the framework’s potential for real-world stochastic optimization and feasibility tasks.

Abstract

This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which existing solution methods can be recast and improved, and new ones can be designed. Almost sure weak, strong, and linear convergence results are established in particular for stochastic fixed point iterations, the stochastic gradient descent method, and stochastic extrapolated parallel algorithms for feasibility problems. In these areas, the proposed algorithms exceed the features and convergence guarantees of the state of the art. Numerical applications to signal and image recovery are provided.

Figures (5)

• Figure 1: Geometry of algorithms for finding a point in $\mathsf{Z}$ with $\uplambda_{{\mathsf{n}}}=1$. (a) Left: Iteration ${\mathsf{n}}$ of the deterministic Algorithm \ref{['algo:1']}. (b) Right: Iteration ${\mathsf{n}}$ of the stochastic Algorithm \ref{['algo:2']} with $\varepsilon_{{\mathsf{n}}}=0$.
• Figure 2: Experiment of Section \ref{['sec:61']}. (a): Original signal $\overline{\mathsf{x}}$. (b): Noisy observation $r_1$. (c): Solution produced by algorithm \ref{['e:p21']}.
• Figure 3: Experiment of Section \ref{['sec:61']}. Normalized error $20\log(\|x_{{\mathsf{n}}}-x_{\infty}\|/ \|x_{\mathsf{0}}-x_{\infty}\|)$ (dB) versus epoch count on the left-hand side and normalized error versus execution time (s) on single processor machine on the right-hand side. Green: $\lambda_{{\mathsf{n}}}\equiv 1$. Magenta: $\lambda_{{\mathsf{n}}}\equiv 1.9$. Blue: Super relaxations with $\mathsf{P}([\lambda_{{\mathsf{n}}}=1.5])=1/2$ and $\mathsf{P}([\lambda_{{\mathsf{n}}}=2.3])=1/2$. Brown: Super relaxations with $\lambda_{{\mathsf{n}}}\sim\text{\rmfamily uniform}([1.5,2.3])$. (a): $\mathsf{M}=1$. (b): $\mathsf{M}=16$.
• Figure 4: Experiment of Section \ref{['sec:62']}. (a) Original image $\bar{\mathsf{x}}$. (b) Noisy observation $r_1$. (c) Solution produced by algorithm \ref{['e:p21']}.
• Figure 5: Experiment of Section \ref{['sec:62']} using $\mathsf{M}=2$. Normalized error $20\log(\|x_{{\mathsf{n}}}-x_{\infty}\|/ \|x_{\mathsf{0}}-x_{\infty}\|)$ (dB) versus execution time (s) on a single processor machine. Green: $\lambda_{{\mathsf{n}}}\equiv 1$. Magenta: $\lambda_{{\mathsf{n}}}\equiv 1.9$. Blue: Super relaxations with $\mathsf{P}([\lambda_{{\mathsf{n}}}=1.8])=6/7$ and $\mathsf{P}([\lambda_{{\mathsf{n}}}=2.5])=1/7$. Brown: Super relaxations with $\lambda_{{\mathsf{n}}}\sim\text{\rmfamily uniform}([1.5,2.3])$.

Theorems & Definitions (50)

• Example 1.3
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Lemma 2.5
• proof
• Corollary 2.6
• proof