Dangerous Liaisons of Convex Learning and Non-Affine Aggregation

Abstract: Last-iterate convergence and generalization guarantees in first-order convex learning hinge on the monotonicity of the update operator. While linear averaging preserves the monotonicity of gradient updates, this property is often violated when gradients are aggregated non-affinely, as in modern pipelines enforcing constraints like adaptivity, privacy, robustness or fairness. Whether it is possible to design non-affine aggregation rules that maintain monotonicity has remained an open question. We answer this question negatively: we prove that the monotonicity of aggregated gradients is preserved if and only if the aggregation rule is positively affine. Consequently, non-affine aggregation prevents steady convergence and substantially degrade algorithmic stability. We quantify these drawbacks and propose a path forward by identifying sufficient conditions under which monotonicity can be restored. Our results provide a unified theoretical framework explaining the disparate failure modes observed in modern learning systems.

• The paper proves that only positively affine gradient aggregation preserves monotonicity, rendering non-affine rules inherently unstable.
• It employs constructive algebraic techniques to show how non-affine aggregation disrupts last-iterate convergence in convex optimization.
• The study connects degraded algorithmic stability and generalization gaps directly to the intrinsic limitations of non-affine aggregation.

Convex Learning, Monotonicity, and the Limits of Non-Affine Aggregation

Overview

The paper "Dangerous Liaisons of Convex Learning and Non-Affine Aggregation" (2606.28123) systematically investigates the interplay between first-order optimization in convex learning and non-affine gradient aggregation rules typically introduced for adaptivity, robustness, privacy, or fairness. It addresses a long-standing open question of whether any non-affine aggregation can universally preserve the monotonicity of the update operator, a property foundational for last-iterate convergence and algorithmic stability.

The core result is a rigorous impossibility theorem: aggregation rules preserve monotonicity for all possible monotone input gradients if and only if they are positively affine. Thus, any non-affine aggregation entails an intrinsic loss of monotonicity, leading to algorithmic instability and failures of last-iterate convergence, even in the deterministic, convex setting. The paper proceeds to quantify and geometrically interpret these drawbacks, providing both negative and constructive insights.

Theoretical Foundations and Main Result

The authors formalize an extremely broad model: for gradients $g_i(\theta) = \nabla_\theta \ell(\theta; z_i)$ from $n$ samples or clients, aggregation is described as $$\theta_{t+1} = \theta_t - \gamma F(g_1(\theta_t), ..., g_n(\theta_t))$$ for an aggregation rule $F$. Importantly, $F$ subsumes not only averaging but also any nonlinear aggregation, such as coordinate-wise adaptivity, robust statistics, gradient clipping, fairness-based reweighting, or quantization.

A central property in convex optimization is monotonicity: $$\forall \theta, \omega \in \mathbb{R}^d, \langle \theta - \omega, F(\theta) - F(\omega) \rangle \geq 0.$$ Monotonicity is pivotal for last-iterate convergence and for nonexpansivity that underpins algorithmic stability [Bauschke & Combettes 2011]. Averaging ($F$ affine) inherits monotonicity from the underlying gradients, but non-affine $F$ may not.

The main theorem demonstrates that, except for positively affine $F$, there always exists a set of monotone gradients such that $F$ destroys monotonicity. The proof is constructive and algebraic, considering local geometric perturbations of the input gradients and showing that any nonlinear behavior of $n$0 leads to a violation. This settles, in the negative, a crucial open problem: no non-affine aggregation can universally guarantee monotonicity, nor the associated algorithmic benefits, in general convex learning.

Implications for Optimization Dynamics

Failures of Last-Iterate Convergence

Last-iterate convergence is practically significant for preserving model structure (e.g., sparsity), in contrast to convergence of averages. The absence of monotonicity with non-affine $n$1 obliterates the geometric descent properties necessary for last-iterate convergence. Explicitly, the paper demonstrates that for any stationary point $n$2 and any step size, there exists a region in parameter space where the update strictly increases the distance from $n$3. This occurs for all practical non-affine rules used in adaptive, robust, and private optimization (e.g., gradient clipping [Abadi et al. 2016], AdaGrad [Duchi et al. 2011]).

The geometric phenomenon is fundamentally different from bias, variance, or oscillatory behavior, and cannot be fixed by step-size tuning or variance reduction. Consequently, theoretical analyses for non-affine aggregation in the literature typically settle for weaker “averaged” convergence guarantees or require restrictive structural assumptions [preobrazhenskaia2026last-iterate-adagrad; koloskova2023revisitingclipping].

Degraded Algorithmic Stability

Algorithmic stability, quantified via uniform (argument) stability, is a pillar of generalization analysis. Key classical results require nonexpansivity of the update: this follows from monotonicity and, stronger, from co-coercivity when the loss is smooth. The paper extends its impossibility theorem to co-coercivity, demonstrating that non-affine $n$4 necessarily leads to strict expansivity for some trajectory pairs. The stability penalty induced by this expansivity cannot be generally offset and produces a uniform gap in stability between affine and non-affine aggregation, leading to provably weaker generalization guarantees. This conclusion is systematized through exact sensitivity analysis and appears in prior observed phenomena in DP- and robust-ML [hardt2016train; boudou2025generalization].

Restricted Positive Results and Algebraic Characterization

While universal monotonicity is impossible for non-affine aggregation, the proof unveils a recipe: if the class of convex losses is restricted so that their Hessians exclude arbitrary rank-1 PSD matrices, then there may exist non-affine $n$5 that preserve monotonicity. The paper gives a constructive algebraic condition for $n$6 on the set of possible Hessians.

As a concrete instance, when the loss is coordinate-wise separable (e.g., generalized linear models), the component-wise (coordinatewise) trimmed mean ($n$7)—a robust aggregation method—maintains monotonicity and even co-coercivity. This matches (and explains) the positive results for robust aggregation in highly structured regimes [yin2018byzantine], and recovers why GLMs evade the general impossibility.

Broad Connections to Related Literature

The paper provides a unifying explanation for previously disparate failure modes in adaptive, robust, and private learning, as well as momentum methods:

• Adaptive methods (AdaGrad, Adam): Instabilities and failures to converge [reddi2018convergenceadam; li2019convergence; preobrazhenskaia2026last-iterate-adagrad] are direct corollaries of lost monotonicity in their coordinate-wise adaptive rules.
• Gradient clipping and DP: Monotonicity violations elucidate why clipping (central in DP-SGD) cannot guarantee last-iterate convergence for general convex losses, and why favorable results are limited to specific structural settings [song2021evading; koloskova2023revisitingclipping].
• Robust aggregation: No robust aggregation (with breakdown point $n$8) can be monotonic for all convex loss gradients. This implies a universal tradeoff between outlier robustness and the theoretical guarantees enjoyed by basic averaging.
• Generalization under instability: The observed gap in generalization error for non-affine methods is grounded in the expansive dynamics demonstrated in the main results.

Practical and Theoretical Consequences

Practical Consequences

• Developers of robust, adaptive, or DP methods must accept a theoretical cost: using non-affine aggregation inevitably leads to a degradation in convergence and stability properties unless strong data or model restrictions are imposed.
• Performance guarantees for general convex (and even strongly convex) settings cannot rely upon monotonicity-based arguments when using non-affine $n$9; alternative analyses based on average convergence or scenario-specific properties must be developed.

Theoretical Directions

• The constructive characterization for cases where $$\theta_{t+1} = \theta_t - \gamma F(g_1(\theta_t), ..., g_n(\theta_t))$$0 may preserve monotonicity (e.g., highly structured gradient/Hessian sets) suggests a precise program for identifying allowable exceptions.
• A major direction is exploring new algorithmic invariants (e.g., saddle-point equilibrium, variational inequalities) that can accommodate the non-monotone vector fields induced by practical constraints.
• The gap in stability opens the path to alternative regularization—randomized or algorithmic—that could recover generalization without monotonicity.

Conclusion

This work establishes a rigorous boundary for the design space of scalable, constrained learning algorithms: affine aggregation rules are uniquely privileged in preserving monotonicity, last-iterate convergence, and algorithmic stability for general convex learning. Non-affine rules—ubiquitous in modern machine learning due to the demands of adaptivity, robustness, privacy, and fairness—unavoidably violate these properties unless the loss structure is highly restricted. This result explains and unifies diverse observed pathologies across robust, adaptive, private, and distributed optimization, and constitutes a theoretical foundation for future developments in structured learning theory and algorithm design.

Citation: "Dangerous Liaisons of Convex Learning and Non-Affine Aggregation" (2606.28123)