Abstract Proximal Operator

• Abstract proximal operator is a generalization that extends classical proximal mappings to abstract convexity frameworks and non-Euclidean spaces.
• It underpins unified methodologies for decomposing composite objectives and solving variational inequalities using operator-splitting techniques.
• Computational algorithms based on abstract proximal operators enable scalable solutions in machine learning and large-scale optimization applications.

An abstract proximal operator is a generalization of the classical proximal operator which arises in convex analysis, variational inequalities, and optimization algorithms. While the standard proximal operator is defined for proper, lower semicontinuous, convex (l.s.c.) functions in Hilbert and Banach spaces, abstract proximal operators broaden this framework to cover more general mathematical structures, the resolution of composite functionals, hierarchies of convexity (or abstract convexity), and non-Euclidean settings. This concept underpins a wide spectrum of operator-theoretic splitting methods, monotone inclusions, and optimization procedures in nonlinear, non-Hilbertian, or nonconvex contexts.

1. Classical Proximal Operator and Generalizations

The standard proximal operator for a proper, l.s.c., convex function $f: H \rightarrow (-\infty,+\infty]$ on a real Hilbert space is

$$\operatorname{prox}_f(x) := \mathop{\arg\min}_{y\in H} \left\{ f(y) + \frac{1}{2} \|y-x\|^2 \right\}$$

This admits equivalent characterizations as $(\text{Id} + \partial f)^{-1}$, and for each $x$ there exists a unique solution due to strong convexity of the objective in $y$. The operator is firmly nonexpansive, single-valued, and its fixed points coincide with the minimizers of $f$ (Polson et al., 2015).

Several generalizations exist:

• The Moreau envelope smooths $f$, with the envelope's gradient relating directly to the proximal operator.
• Bregman-proximal operators replace the squared norm with a Bregman divergence, supporting non-Euclidean geometries (Millán et al., 2024).
• Abstract proximal operators extend the proximal mapping to settings with generalized subdifferentials and convexity classes beyond standard linear or Hilbertian contexts (Bednarczuk et al., 2024, Millán et al., 2024).

2. Abstract Proximal Operators in Generalized Convexity

Abstract convexity frameworks, such as $L$-convexity or $lsc^R$-convexity, define convexity with respect to a family of “support functions” that may not be strictly linear or quadratic. In these contexts, the abstract proximal operator typically takes the form: $$\operatorname{Prox}_f^{D_\phi}(y, \lambda) = \arg\min_{x \in X} \left\{ f(x) + \frac{1}{\lambda} D_\phi^\lambda(x, y) \right\}$$ where $D_\phi^\lambda(x, y)$ is an abstract Bregman divergence associated to some kernel $\phi$ and abstract subdifferential $\lambda$ (Millán et al., 2024).

In the $lsc^R$-convex setting, the generalized subdifferential $\partial_{lsc}^R f(x)$ consists of quadratic minorants, and the proximal operator can be characterized as

$$\operatorname{prox}^{lsc,R}_{\gamma f}(x) = (J_\gamma + \partial_{lsc}^R f)^{-1}(J_\gamma(x))$$

recovering the classical proximal operator when the abstract structure specializes to usual convexity (Bednarczuk et al., 2024).

Such formulations allow global convergence guarantees even for nonconvex but abstractly convex functions, provided the support families satisfy minimal closure properties (such as closure under addition) (Millán et al., 2024).

3. Decomposition and Splitting via Abstract Proximality

A major application of abstract proximal operators is the decomposition of the prox of a sum of functions. For proper, l.s.c., convex $f, g$ in a Hilbert space, the “$f$-proximal operator of $g$” is defined by

$$\operatorname{prox}_g^f := (\text{Id} + \partial g \circ \operatorname{prox}_f)^{-1}$$

and, under the additivity of subdifferentials ($\partial(f+g)=\partial f + \partial g$), it satisfies the decomposition formula

$$\operatorname{prox}_{f+g} = \operatorname{prox}_f \circ \operatorname{prox}_g^f$$

This abstract construction generalizes the classical Douglas–Rachford and forward–backward splitting methods, and the operator $\operatorname{prox}_g^f$ itself can be interpreted as the fixed-point set of a generalized Douglas–Rachford operator

$$DR_{f,g}^{(x)}(y) := y - \operatorname{prox}_f(y) + \operatorname{prox}_g(x + \operatorname{prox}_f(y) - y)$$

with weakly convergent associated iterations (Adly et al., 2017). This approach unifies multiple algorithmic schemas at an abstract operator level and clarifies the nested structure of practical schemes for composite convex minimization and variational inequalities.

4. Proximal Algorithms and Abstract Metric Geometry

Abstract proximal operators are central to proximal point algorithms (PPA) in metric and geodesic spaces. In complete CAT(0) spaces (generalizing Hilbertian geometry), the resolvent (abstract proximal operator) of $f$ at $x$ is

$$J_{\gamma f}(x) = \arg\min_{y \in X} \left\{ f(y) + \frac{1}{2\gamma} d^2(y, x) \right\}$$

and is firmly nonexpansive in the metric sense (Leustean et al., 2017, Sipos, 2021). Weak (Δ-)convergence of PPA sequences is guaranteed under joint firm nonexpansivity. Strong convergence is achieved for Halpern-type and Tikhonov-type anchor modifications, with explicit (proof-mined) rates of metastability attainable in this abstract metric context (Sipos, 2021).

The theoretical unification under the abstract proximal point framework covers convex minimization, monotone inclusions, and fixed point computations for nonexpansive mappings in a single operator-theoretic and geometric paradigm (Leustean et al., 2017, Sipos, 2021).

5. Operator-Theoretic Properties and Functional Determination

Abstract proximal operators, when properly defined, inherit and generalize several key properties of the classical subdifferential and resolvent. For a proper, convex, l.s.c. function $f$, the classical proximal operator is the resolvent of the subdifferential, maximally monotone, and firmly nonexpansive. In the abstract framework, under suitable structure on the support family, these monotonicity and nonexpansivity properties extend (Millán et al., 2024, Bednarczuk et al., 2024).

A particularly notable result is that the norm profile of the proximal operator determines the underlying convex function up to an additive constant. Thus, knowing $$x \mapsto \|\operatorname{prox}_{\lambda f}(x) - x_0\|$$ for a fixed $x_0$ suffices to recover $f$ modulo a shift, reflecting a deep rigidity in the information carried by the abstract prox map (Vilches, 2020).

6. Computational Algorithms and Applications

Algorithmic implementation of abstract or generalized proximal operators typically involves solving auxiliary subproblems (possibly with Bregman-type or divergence penalties) or operator-splitting steps parameterized by subdifferentials or dual objects. Iterative algorithms based on the composition or fixed-point characterization of abstract prox operators enable efficient resolution of problems in large-scale, high-dimensional, or nonsmooth settings (Adly et al., 2017, Bednarczuk et al., 2024).

Applications include:

• Iterative splitting and decomposition for structured convex and variational inequalities (Adly et al., 2017).
• Large-scale machine learning via composite objectives, where closed-form or efficiently computable abstract prox mappings are crucial for scalability (Polson et al., 2015).
• Robust optimization and inverse recovery, where the operator-theoretic properties of the abstract prox support new modes of analysis and algorithmic design (Vilches, 2020).

7. Significance, Extensions, and Future Directions

The theory and computation of abstract proximal operators provide a structural foundation for generalized optimization, monotone inclusion, and variational regularization. The unification via abstract convexity, generalized subdifferentials, and metric (or geodesic) geometry permits systematic extension of operator-splitting methodologies beyond classical settings.

Potential research directions include:

• Identification of novel convexity classes admitting tractable abstract prox computations (Millán et al., 2024, Bednarczuk et al., 2024).
• Extension to stochastic, block-coordinate, or distributed frameworks.
• Applications in robust statistics, sensitivity analysis of variational inequalities, and machine learning for nonstandard regularizations (Adly et al., 2017).

A plausible implication is that abstract proximal operators will continue to serve as the theoretical core for next-generation splitting algorithms in nonconvex, nonsmooth, or non-Euclidean optimization frameworks.