Proximal-PL Inequality in Optimization

• Proximal-PL inequality is a condition defining a broader class of functions where proximal gradient methods achieve global linear convergence, even for nonconvex and nonsmooth problems.
• It generalizes the standard PL condition by replacing the gradient norm with the proximal gradient residual, effectively accommodating composite objective structures.
• The framework enables deriving sharp convergence rates via performance estimation problems, which guide optimal step-size selection and enhance practical algorithm performance.

The proximal-PL (proximal Polyak–Łojasiewicz) inequality is a fundamental analytical tool in modern optimization, generalizing the standard Polyak–Łojasiewicz (PL) condition to a composite objective framework. It characterizes a broad class of functions for which first-order methods, notably the Proximal Gradient Method (PGM), achieve global linear convergence rates. Unlike strong convexity, the proximal-PL inequality can hold for many practical non-strongly convex and even certain nonconvex problems, thus capturing a wider landscape relevant for machine learning, signal processing, and variational analysis.

1. Formal Definition and Structure

For the composite minimization problem

$\min_{x\in\mathbb{R}^d} F(x) = f(x) + h(x),$

where $f$ is differentiable with $L$-Lipschitz gradient and $h$ is closed, proper, convex, the proximal-PL inequality is formulated as follows. For fixed step size $\gamma > 0$, define the proximal gradient residual

$$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$

Then $F$ satisfies the proximal-PL inequality (alternatively, the RPL inequality) with constant $\mu > 0$ if

$$\|G_\gamma(x)\|^2 \geq 2\mu\left[F(x) - F^*\right], \quad \forall x \in X,$$

for a suitable subset $X$. In the unconstrained, smooth case ($f$0), this recovers the classical PL condition:

$f$1

In the presence of nonsmooth $f$2, the proximal-PL condition replaces $f$3 with the proximal residual $f$4, reflecting the geometry of the prox mapping and subdifferential calculus (Kong et al., 2024, Karimi et al., 2016, Zhang et al., 2019).

2. Relationship to Other Regularity Conditions

The proximal-PL inequality occupies a central position among regularity conditions for both convex and weakly convex problems. In the context of a proper, closed, $f$5-weakly convex function $f$6 (with $f$7 convex), the following implications hold on any sublevel set $f$8:

• Strong convexity $f$9 restricted secant inequality $L$0 error bound $L$1 proximal-PL $L$2 quadratic growth.

If $L$3 is convex or the quadratic growth constant exceeds half the weak convexity parameter, all four properties—restricted secant, error bound, proximal-PL, and quadratic growth—become equivalent. Thus, the proximal-PL property provides both a minimal sufficient and, in many cases, necessary condition for linear convergence of first-order algorithms in the absence of strong convexity (Liao et al., 2023, Karimi et al., 2016).

3. Linear Convergence of the Proximal Gradient Method

The proximal-PL inequality guarantees geometric decay in objective error for PGM. The iteration

$L$4

recursively satisfies

$L$5

where $L$6 depends explicitly on the problem class (convex, nonconvex), the constants $L$7 and $L$8, and the choice of step size $L$9 (Kong et al., 2024, Karimi et al., 2016, Zhang et al., 2019). Closed-form expressions are available for several regimes:

Regime	Contraction rate $h$0	Optimal $h$1
Convex, PL	$h$2 for $h$3	$h$4
Convex, RPL	$h$5 for $h$6	$h$7 (if $h$8)
Nonconvex, RPL	$h$9	$\gamma > 0$0

These rates, derived using the performance estimation problem (PEP), are exact and often sharper than previously established results, especially for larger step sizes (Kong et al., 2024, Zhang et al., 2019).

4. Methodological Framework: Performance Estimation Problem (PEP)

The PEP framework provides tight bounds for worst-case convergence rates by rephrasing the one-step contraction property as a semi-definite program (SDP) over a finite Gram matrix, exploiting precise interpolation conditions for the function classes under consideration. For a given regularity class and step size, one computes

$\gamma > 0$1

Solving this SDP, inspecting its optimal dual multipliers, and verifying closed-form certificates yields analytic expressions for the contraction factor $\gamma > 0$2 (Kong et al., 2024). This methodology delivers improved rates and guides optimal step-size selection in practice.

5. Extensions to Weakly Convex and Multiobjective Optimization

The proximal-PL inequality extends naturally to weakly convex objective functions using the Fréchet (or Clarke/limiting) subdifferential. For $\gamma > 0$3 proper, closed, and $\gamma > 0$4-weakly convex, the condition

$\gamma > 0$5

implies global linear convergence of the proximal point method and, under standard assumptions, of PGM as well (Liao et al., 2023).

In multiobjective optimization, the notion is generalized by introducing scalar merit functions $\gamma > 0$6 and $\gamma > 0$7, leading to a multiobjective proximal-PL condition:

$\gamma > 0$8

for some $\gamma > 0$9, yielding linear convergence rates for multiobjective variants of PGM (Tanabe et al., 2020).

6. Numerical Illustration and Rate Sharpness

Numerical experiments confirm that PEP-derived contraction rates for PGM under the proximal-PL condition outperform previous bounds across a wide range of step sizes. For typical parameter settings (e.g., $$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$0), the proximal-PL-based rate remains valid and favorable for $$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$1 up to $$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$2, with its minimum often noticeably better than the classical rates constrained to $$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$3 (Kong et al., 2024, Zhang et al., 2019).

Sharpened bounds are not only theoretically tight but also empirically match the observed decay of optimality gaps, reinforcing the practical utility of the proximal-PL inequality as a predictive modeling tool for first-order splitting methods.

7. Comparison with Standard PL and Practical Implications

Under standard PL, stationarity is measured via the distance to the subdifferential $$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$4, whereas the proximal-PL uses the PGM residual $$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$5, often better aligned with algorithmic updates for nonsmooth problems. In convex settings, both provide the same leading-order (quadratic in step size) convergence when $$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$6, but the proximal-PL rate retains validity for larger $$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$7 and yields globally improved rates. For nonconvex functions, the optimal step size shifts ($$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$8 vs. $$G_\gamma(x) := \frac{1}{\gamma}\left[x - \operatorname{prox}_{\gamma h}\left(x - \gamma \nabla f(x)\right)\right].$$9) depending on the choice of condition, emphasizing the operational relevance of distinguishing between these regularity properties (Kong et al., 2024).

A plausible implication is that, for a wide class of machine learning and variational problems, verifying the proximal-PL inequality can both guarantee and explain fast optimization in the absence of strong convexity, and that maximal practical efficiency can be attained by tuning the step size to exploit the full allowable regime determined via the PEP-based analysis.