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Non-conflicting directions within the convex hull of gradients ensure convergence to Pareto stationarity in multi-objective optimization.

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2026-06-29 08:42 UTC pith:LCZ76F5U

load-bearing objection The paper unifies MOO gradient aggregation under an alignment condition for convergence rates and adds a capped MGDA variant, but the dual-cone projection step needs explicit confirmation that it stays inside the convex hull. the 1 major comments →

arxiv 2605.30452 v1 pith:LCZ76F5U submitted 2026-05-28 cs.LG cs.AImath.OC

A Unified Framework for Gradient Aggregation in Multi-Objective Optimization

Zeou Hu , Kelvin Ho , Yaoliang Yu This is my paper
classification cs.LG cs.AImath.OC
keywords multi-objective optimizationgradient aggregationPareto stationarityconvergence ratesMGDAdual cone projectionfederated learning
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a unifying framework for gradient aggregation across multiple objectives. It proves that directions staying inside the convex hull and free of mutual conflict form a sufficient condition for convergence to Pareto stationarity. From this single condition the authors obtain optimal convergence rates and show how a dual-cone projection step makes the condition feasible for a wider set of rules. The same lens recovers existing algorithms through a primal optimization view and produces a new capped MGDA variant based on CVaR. The result matters for any setting where several loss surfaces must be balanced simultaneously, because it replaces case-by-case proofs with one reusable argument.

Core claim

Non-conflicting directions, when chosen within the convex hull of gradients, form a fundamental sufficient condition for convergence to Pareto stationarity, from which optimal rates are derived. Feasibility is ensured through projection onto the dual cone, and a primal optimization perspective unifies established algorithms while enabling new variants such as capped MGDA.

What carries the argument

The sufficient alignment condition on non-conflicting directions chosen inside the convex hull of the component gradients, which directly yields convergence to Pareto stationarity.

Load-bearing premise

The sufficient alignment condition holds for the directions produced by the aggregation rule or can be enforced via dual-cone projection without invalidating the convergence argument.

What would settle it

Run a simple two-objective quadratic problem where the convex hull contains non-conflicting directions; an aggregation rule that systematically outputs conflicting directions should fail to reach Pareto stationarity while a projected rule succeeds.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Any aggregation rule whose output satisfies the alignment condition inherits the optimal convergence rates to Pareto stationarity.
  • Projection onto the dual cone extends convergence guarantees to a broader family of methods that would otherwise produce conflicting directions.
  • Capped MGDA, derived from the CVaR formulation, inherits the same rates and shows improved robustness in adversarial federated learning.
  • The primal optimization view recovers MGDA, linear scalarization, and other standard methods as special cases with explicit relationships among them.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Aggregation rules that occasionally violate alignment may still converge in practice but lose the paper's rate guarantees.
  • The framework suggests testing whether the dual-cone projection improves stability on non-convex or stochastic multi-objective problems.
  • Similar alignment ideas could be applied to federated or distributed settings beyond the adversarial case shown.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes a unified framework for gradient aggregation in multi-objective optimization. It introduces a sufficient alignment condition leading to a theorem that non-conflicting directions chosen within the convex hull of the gradients ensure convergence to Pareto stationarity, from which optimal rates are derived. The framework allows enforcing the condition via dual-cone projection, provides a primal-dual perspective unifying existing algorithms, introduces capped MGDA based on CVaR, and includes experimental validation on synthetic and practical tasks like adversarial federated learning.

Significance. If the central theorem applies after projection, the work offers a general analysis tool for MOO methods, potentially clarifying their relationships and enabling new convergent variants. The primal perspective and new algorithm are positive contributions.

major comments (1)
  1. [Central theorem (as stated in abstract and developed in the main analysis)] The central theorem requires a direction to be simultaneously non-conflicting and inside conv{g1,...,gm} to guarantee descent to Pareto stationarity. The paper uses dual-cone projection to enforce the non-conflicting property. Nothing shows that the projected vector remains inside the convex hull; if the projection can map a point in conv{G} to a point outside conv{G}, the hypothesis of the convergence theorem is no longer satisfied and the derived rates do not apply. This is load-bearing for all convergence claims.
minor comments (1)
  1. [Experiments] The abstract mentions experiments on synthetic problems and practical benchmarks, but additional details on statistical significance, hyperparameter sensitivity, and comparison metrics would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point regarding the central theorem. We address the comment below.

read point-by-point responses

  1. Referee: [Central theorem (as stated in abstract and developed in the main analysis)] The central theorem requires a direction to be simultaneously non-conflicting and inside conv{g1,...,gm} to guarantee descent to Pareto stationarity. The paper uses dual-cone projection to enforce the non-conflicting property. Nothing shows that the projected vector remains inside the convex hull; if the projection can map a point in conv{G} to a point outside conv{G}, the hypothesis of the convergence theorem is no longer satisfied and the derived rates do not apply. This is load-bearing for all convergence claims.

    Authors: We agree that the manuscript does not explicitly prove that the dual-cone projection of a vector from conv{G} necessarily remains inside conv{G}. This is a valid observation, and the current presentation leaves a gap in rigorously connecting the projection step to the hypothesis of the convergence theorem. In the revised manuscript we will add a lemma establishing the required invariance (either by showing that the particular projection operator employed maps conv{G} into itself, or by redefining the feasible set as the intersection of the dual cone with conv{G} and proving that the resulting projection satisfies both conditions simultaneously). This will be accompanied by the corresponding updates to the statement of the main theorem and the derived rates. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from independent sufficient condition

full rationale

The paper's central claim derives convergence to Pareto stationarity from a stated sufficient alignment condition on non-conflicting directions inside the convex hull of gradients, followed by a separate argument that dual-cone projection can enforce feasibility. No quoted step reduces the theorem to a fitted parameter, self-definition, or load-bearing self-citation chain; the alignment condition is presented as an external hypothesis whose satisfaction is argued separately. The analysis is therefore self-contained against its stated assumptions rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the sufficient alignment condition as a domain assumption in optimization theory; no free parameters or invented entities are identifiable from the provided text.

axioms (1)
  • domain assumption A sufficient alignment condition on gradient directions guarantees convergence to Pareto stationarity in multi-objective optimization.
    Described as central to the analysis and theorem in the abstract.

pith-pipeline@v0.9.1-grok · 5736 in / 1284 out tokens · 29270 ms · 2026-06-29T08:42:13.623940+00:00 · methodology

0 comments
read the original abstract

Many machine learning problems involve multiple inherent trade-offs that are best addressed by gradient-based multi-objective optimization (MOO) algorithms. Existing methods are often proposed with various motivations, analyzed case by case, and differ algorithmically in how the component gradients are aggregated at each step. In this work, we develop a unifying framework for gradient aggregation in MOO, establishing (optimal) rates of convergence to Pareto stationarity, the standard measure of performance in MOO. Central to our analysis is a sufficient alignment condition, from which we derive a theorem showing that non-conflicting directions, when chosen within the convex hull of gradients, form a fundamental sufficient condition for convergence. We further show that feasibility can be ensured through projection onto the dual cone, broadening the scope of methods that admit convergence guarantees. In parallel, we present a primal optimization perspective of gradient aggregation that encompasses established algorithms, clarifies their theoretical relationships, and enables the design of new variants. As an illustration, we introduce capped MGDA, derived from a CVaR-based formulation, and demonstrate its robustness in adversarial federated learning. Finally, we validate our theory through experiments on synthetic problems and practical benchmarks.

Figures

Figures reproduced from arXiv: 2605.30452 by Kelvin Ho, Yaoliang Yu, Zeou Hu.

Figure 1
Figure 1. Figure 1: An overview of the relationships between our key theoretical results. Our pivotal result, Theorem 1 and Corollary 1, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Dynamics of non-conflicting aggregators on VL [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Dynamics of mixed aggregator scheduling on [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Capped MGDA vs. MGDA in adversarial feder [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Non-conflicting aggregators on LibMOON fairness classification benchmark. Nash-MTL is unstable, while [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Mixed aggregator scheduling on fairness classification benchmark. [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Capped MGDA vs ‘MGDA + coefficient clipping’. We observe the later is not as effective, given the same [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Average client accuracy vs Epoch. We can see a clear improvement when [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

18 extracted references · 1 canonical work pages

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    Convergence guarantees of linear scalarization.The aggregation rule in (23) corresponds precisely to performing gradient descent on the scalarized objective F= P i fi

    Moreover, the scalarization weights need not be fixed at 1 m; any conic scalarization preserves this convergence property, and it is common to tune the weights for better performance. Convergence guarantees of linear scalarization.The aggregation rule in (23) corresponds precisely to performing gradient descent on the scalarized objective F= P i fi. Conse...

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    is argmin d max k − ⟨d,g k⟩(24) which corresponds to choosings(x) = max k(−xk),r(∥ · ∥) = 1 2 ∥ · ∥2 in (12). The resulting gradient aggregation rule (used in practice) is d=Jλ ∗,whereλ ∗ = argmin λ∈∆ ∥J ⊤λ∥2.(25) MGDA is automatically a non-conflicting aggregator, which is easy to see from the primal perspective e.g., (Fliege and Svaiter, 2000). Its upda...

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    Convergence guarantees of MGDA.To the best of our knowledge, the most complete complexity analysis of MGDA to date is provided by Fliege et al. (2019). Our framework extends this analysis to a broader class of gradient aggregation schemes. When specialized to MGDA, our general results (Theorems 2 and

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    apply without requiring additional assumptions and recover the same convergence rates as those established by Fliege et al. (2019) for both non-convex and convex settings. A.1.3 Nash Bargaining Multi-Task Learning (Nash-MTL) The primal optimization subproblem formulation of Nash-MTL (Navon et al.,

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    is argmin ∥d∥≤ϵ − X k log⟨d,g k⟩,(26) which corresponds to choosings(x) =− P k logx k,r(∥ · ∥) =ι Bϵ(·) := ( 0,∥d∥ ≤ϵ +∞,otherwise Note that this optimization formulation is equivalent to the −(Q k xk)1/m we presented in the main paper, by moving the negative sign out and take the log. Also, the hard ball constraint ∥d∥ ≤ϵ in (26) can be equivalently repl...

    2022

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    (2022) and our analysis (Theorem

    For theconvexcase, both Navon et al. (2022) and our analysis (Theorem

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    rely on the same standard assumptions. While Navon et al. (2022) proves convergence of wt to a weakly Pareto-optimal solution, our framework provides a rate of O(1/t) in terms of the function value gap to optimality. A.1.4 Fair Resource Allocation in MTL (FairGrad) To provide an α-fair framework for MTL (and MOO in general), FairGrad (Ban and Ji,

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    is argmin d − 1 m mX k=1 ⟨d,p k⟩+ 1 2 ∥d∥2,(30) wherep k := Pcone∗(J)(gk), Jα k :=p k.(31) which corresponds to choosings(x) =−( 1 m P k αk)⊤x,r(∥ · ∥) = 1 2 ∥ · ∥2 in (12). The resulting gradient aggregation rule (used in practice) is d= 1 m X k pk =J( 1 m mX k=1 αk).(32) which first projects each gradient onto the dual cone{d:J ⊤d≥0}and then averages th...

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    Within our framework, Corollary 2 establishes an O(1/ √ t) convergence rate for DualProj in terms of the Pareto stationarity measure γ(wt) for the non-convex setting

    Convergence guarantees of DualProj.The original work of Lopez-Paz and Ranzato (2017) does not appear to provide a formal convergence analysis. Within our framework, Corollary 2 establishes an O(1/ √ t) convergence rate for DualProj in terms of the Pareto stationarity measure γ(wt) for the non-convex setting. For the convex setting, we can apply Theorem 3 ...

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    • PCGrad can be modified to repeatedly project until gPC k lies in the dual cone, and we name this new variantPCGrad+ (see Algorithm 2). In this case, PCGrad+ again resembles UPGrad, except that UPGrad performs a one-step projection directly onto the dual cone C ∗, whereas PCGrad+ repeatedly performs alternating projections onto the half-spaces Hi ={z:⟨g ...

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    Proof.We omit the indextin the following to simplify the notation

    Theorem 2(Convergence of Non-Conflicting Directions).If the direction dt ∈conv(J f(wt)) and dt ∈cone ∗(Jf(wt)) (i.e., non-conflicting), then condition(A)and hence Corollary 1 holds withc t ≡1andF= P k fk. Proof.We omit the indextin the following to simplify the notation. LetF= P k fk andd=J f(w)λfor someλ∈∆. We directly verify (A): ⟨d,∇F(w)⟩= X k ⟨d,∇f k(...

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    as a reference. For Nash-MTL (Navon et al., 2022), we adopt the official implementation’s default, which always clips the aggregated update direction to satisfy∥dt∥= 1 . All examined methods are run in their deterministic, full-batch form, without momentum. For the normalized variants (e.g., Nash-MTL*, UPGrad*, DualProj*), we keep the original implementat...

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    for both dataset preprocessing and model architecture. Specifically, we use the function libmoon.util.mtl.get_dataset("adult") to generate the train, validation, and test splits. For the model, we adopt LibMOON’sM4 fair_model architecture: a fully connected neural network consisting of three hidden layers of dimension 256 each, with ReLU activations. The ...

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    We observe that the Pareto stationarity measure γ(wt) converges to 0 for all methods except Nash-MTL (without normaliza- tion), which suffers from overshooting because ∥d∥ is fixed at 1, leading to instability near Pareto stationarity. Applying convex-hull normalization to Nash-MTL yields smoother convergence, and a similar but less pronounced effect is a...

    2022