Geometry-Aware Multi-Objective Optimization: Norm-Constrained Oracles with Entropic Weight Updates

arXiv:2610.13773 · math.OC, cs.LG · Submitted 2026-10-09 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom.

Jane: And I'm Jane, here with Lu, Meng and our in-house large language model, Lalam.

Tom: Today's paper: "Geometry-Aware Multi-Objective Optimization".

Jane: The gist Multiple-gradient descent is typically formulated in Euclidean geometry, whereas matrix-aware optimizers use directions induced by other norms,

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So we were just looking at the title and authors for "Geometry-Aware Multi-Objective Optimization: Norm-Constrained Oracles with Entropic Weight Updates." It sounds very technical, which is typical when you’re diving into this kind of geometric optimization.

Jane: It really does sound like it, but what that means simply is they’re looking at how to optimize multiple objectives when the usual geometry—Euclidean space—doesn't capture the full picture of what we need.

Lu: They are dealing with directions induced by other norms instead of just sticking to the standard two geometry that most of us are used to in machine learning <ref:2610.13773#pg1>.

Meng: It sounds like they’re making a structural change to how we find the best step when we have several competing objectives, which is something I worry about when it comes to deploying models at scale.

Tom: Exactly, and they're using a linear minimization oracle combined with entropic updates of task weights to do this <ref:2610.13773#pg1>.

Jane: So the core idea is using these different norms to guide the optimization path while simultaneously adjusting how much we care about each objective based on entropy.

Lu: They define a primal norm that measures parameter displacements and a dual gradient norm that measures gradients, which sets up Holder’s inequality to link them together <ref:2610.13773#pg3>.

Meng: So the mathematical foundation is using these two different norms to constrain the movement while simultaneously steering the weight distribution <ref:2610.13773#pg3>.

Tom: And they show that this structure allows them to establish an O(T-one/two) ergodic stationarity rate deterministically for any finite-dimensional norm pair <ref:2610.13773#pg1,establish an O(T-1/2) ergodic stationarity rate>.

Jane: That means their method is fundamentally more flexible than standard gradient descent because it handles the geometry of the problem better <ref:2610.13773#pg2>.

The paper's summary: Tom: So what’s the actual mechanism they are using? They combine a norm-constrained linear minimization oracle with entropic updates of task weights to tackle multi-objective optimization.

Jane: It’s like they have a tool that selects the best move constrained by a certain norm, and at the same time, they are updating how important each objective is based on entropy <ref:2610.13773#pg1>.

Lu: The setup defines a primal norm measuring parameter displacements and a dual gradient norm measuring gradients, which sets up Holder’s inequality to link them together <ref:2610.13773#pg3>.

Meng: So the mathematical foundation is using these two norms to constrain the movement while simultaneously steering the weight distribution <ref:2610.13773#pg3>.

Lu: They also derive a required trajectory gradient bound from smoothness and lower-boundedness in the unregularized case, which is a key theoretical contribution they make here <ref:2610.13773#pg2>.

Meng: Deriving that bound directly from function properties without assuming one beforehand sounds like they are making their assumptions more concrete <ref:2610.13773#pg2>.

Tom: And for the stochastic case, they use one persistent blended-gradient momentum buffer to get an O(T-one/four) rate where the leading term has no explicit dependence on the number of tasks <ref:2610.13773#pg1,leading term has no explicit dependence on the number of tasks>.

Jane: That part is what I find most compelling—the convergence rate doesn't blow up as you add more objectives to your problem <ref:2610.13773#pg2>.

Lu: They also establish static regret for the inner weighting objective on arbitrary bounded gradient sequences, which means their method is robust against how those gradients behave over time <ref:2610.13773#pg2>.

Meng: Robustness against gradient behavior is exactly what we need when deploying models that might be noisy or unpredictable in production <ref:2610.13773#pg2>.

Tom: And finally, they get iteration-wise inner-gap control after burn-in with entropy regularization and a weight floor, which means they maintain stability while tracking the optimum <ref:2610.13773#pg1>.

Jane: So to sum up: it’s a method that is geometrically aware, handles multiple norms, and provides strong convergence guarantees in both deterministic and stochastic settings <ref:2610.13773#pg2>.

The paper's improvements: Tom: Now let’s talk about the specific improvements they highlight, because these aren't just general ideas; they are concrete ways this method beats existing approaches.

Jane: The most striking improvement is the continuous tracking of Pareto optimality with iteration-wise gap control. They achieve "iteration-wise inner-gap control after burn-in while retaining ergodic stationarity" <ref:2610.13773#pg1>.

Lu: They do this by setting specific parameters like tau = beta/two K, kappa = beta/4MK, and eta R = epsilon two beta/seven hundred sixty-eight M four K to ensure the gap is bounded by something like g apt at most ten-five for all time t after some burn-in <ref:2610.13773#pg1>.

Meng: Controlling the gap every single iteration, even after a burn-in, that’s a huge win for stability in production environments where you don't want the objective to jump around wildly <ref:2610.13773#pg1>.

Tom: They also show robust convergence rates in stochastic settings with one persistent buffer yielding an O(epsilon-four) rate under a trajectory gradient bound <ref:2610.13773#pg2>.

Jane: And as mentioned before, that leading term not depending on the number of tasks is a big win for scalability when you scale up to many objectives <ref:2610.13773#pg1>.

Lu: They also show they can achieve this general-norm optimization capability, establishing an O(T-one/two) ergodic stationarity rate in arbitrary norm pairs <ref:2610.13773#pg2,an O(T-1/2) ergodic stationarity rate in>.

Meng: So, they’re claiming that the method works well across different mathematical structures, which is a strong claim when you're dealing with diverse real-world datasets <ref:2610.13773#pg2>.

Tom: On the other hand, they also show selection-independent inner optimality through Theorem four point five, establishing static regret on arbitrary bounded gradient sequences <ref:2610.13773#pg2>.

Jane: That means the optimal blended gradient direction doesn't have to be unique in terms of spectral or sign geometry for this method to work correctly <ref:2610.13773#pg2>.

Lu: This is important because it shows the process is robust against those non-unique optimal directions, which motivates why they focus on that inner gap <ref:2610.13773#pg2>.

Conclusion: Tom: So to wrap up, the paper "Geometry-Aware Multi-Objective Optimization: Norm-Constrained Oracles with Entropic Weight Updates" establishes O(epsilon-two) deterministic stationarity and O(epsilon-four) stochastic stationarity under certain conditions <ref:2610.13773#pg1>.

Jane: It’s a method that provides iteration-wise inner-gap control after burn-in in any norm, which is a distinct conclusion compared to just average-gap convergence <ref:2610.13773#pg2>.

Lu: The main point is that they get strong guarantees: O(epsilon-two) deterministic, O(epsilon-four) stochastic, and static regret on arbitrary bounded gradient sequences <ref:2610.13773#pg2>.

Meng: So the key takeaway for deployment is that we have a method with strong performance guarantees conditional on having a trajectory gradient bound M <ref:2610.13773#pg2>.

Tom: It’s a very solid piece of work that shows how to leverage geometry more effectively in multi-objective settings without getting stuck only in the standard two world <ref:2610.13773#pg1>.

Jane: It really moves the discussion toward developing optimizers that are inherently more adaptable to complex, non-Euclidean optimization landscapes <ref:2610.13773#pg2>.

Lu: The paper sets a high bar for what we expect from these kinds of methods by providing such detailed analysis on how these components interact with each other <ref:2610.13773#pg1>.

Meng: I just hope we can see this level of rigor reflected in the next set of models we ship to production, conditional on having that trajectory gradient bound M <ref:2610.13773#pg2>.

Tom: Definitely. This work on "Geometry-Aware Multi-Objective Optimization: Norm-Constrained Oracles with Entropic Weight Updates" shows how to leverage geometry more effectively in multi-objective settings without getting stuck only in the standard two world <ref:2610.13773#pg1>.

El Mahdi Chayti, Alireza Mirrokni, Martin Jaggi

Machine Learning and Optimization Laboratory (MLO), EPFL

math.OC, cs.LG

Submitted: 2026-10-09

Updated: 2026-10-09

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: The gist Multiple-gradient descent is typically formulated in Euclidean geometry, whereas matrix-aware optimizers use directions induced by other norms, and this work analyzes a single-loop method

Key concepts

Norms and the Oracle
The method operates using a fixed pair of norms: one to measure parameter changes (primal norm) and another to measure gradients (dual norm). The linear minimization oracle selects a direction that minimizes a specific quantity while respecting constraints on both norms, allowing the optimization to proceed in any chosen norm geometry.
Entropic Updates of Task Weights
Instead of traditional methods, this approach uses entropic updates to adjust the weights assigned to different tasks. This mechanism is key for multi-objective optimization, guiding the search towards a Pareto optimal solution by managing trade-offs between conflicting objectives within the optimization loop.
Inner Gap Control
The inner gap refers to how close the current solution is to the true optimum along a specific direction. The method achieves iteration-wise control over this gap after an initial burn-in period, ensuring that even though it's not perfectly optimal every step, the error remains bounded and controllable.
Ergodic Stationarity Rate
This refers to the long-term convergence behavior of the optimization process. The analysis proves that under deterministic settings, the method converges to a state where it no longer significantly changes its direction over time, establishing a specific rate for this stable convergence.

Terminology

Summary

The gist Multiple-gradient descent is typically formulated in Euclidean geometry, whereas matrix-aware optimizers use directions induced by other norms, and this work analyzes a single-loop method combining a norm-constrained linear minimization oracle with entropic updates of task weights to achieve strong convergence rates in arbitrary norm pairs.

How it works

The proposed method combines a norm-constrained linear minimization oracle with entropic updates of task weights for multi-objective optimization, establishing an O(T−1/2) ergodic stationarity rate in the deterministic setting for any finite-dimensional norm pair. For stochastic gradients and a bounded-gradient assumption, one persistent blended-gradient momentum buffer yields an O(T−1/4) rate whose leading term has no explicit dependence on the number of tasks. The analysis establishes static regret for the inner weighting objective on arbitrary bounded gradient sequences. Furthermore, with entropy regularization and a weight floor, iteration-wise inner-gap control is achieved after burn-in while retaining ergodic stationarity.

Norms and the oracle

The method operates on a fixed norm pair where the primal norm measures parameter displacements, with dual g∗ measuring gradients. The linear minimization oracle is any selection lmo(g) in arg min v∈B ⟨g, v⟩, such that lmo(g) ≤ R and ⟨g, lmo(g)⟩ = −Rg∗. For a network with parameter blocks x = (xl)l the norm used in practice is x:= maxl xl(l)/ρl, with dual g∗ = P l ρ∥gl∥(l),∗, blockwise oracle lmo(g) = (ρ lmo l gl)l of per-block radius Rρell.

Convergence and Regret

The deterministic convergence is established by Theorem 4.4, which gives 1/T P t ∇F(xt)wt∗ ≤ (√2∆L + M p log K/2)/√T. With the step sizes of Lemma 4.3, this leads to T = O(L∆ log2 K/ϵ2) iterations for ϵ-accurate Pareto stationarity. Under a trajectory gradient bound, Theorem 4.9 gives O(ϵ−4) iterations with momentum on the blended gradient, and a weighted-average bound on noisy weight increments keeps explicit K dependence out of the leading T−1/4 term.

Inner Optimality and Tracking

The optimal blended gradient can be nonunique in spectral and sign geometries, motivating the inner gap. Theorem 4.5 gives static regret on arbitrary bounded gradient sequences. With regularization and clipping, Theorem 5.1 controls the inner gap at every iteration after a O˜(M4 log2 K/ϵ4) burn-in while retaining ergodic stationarity in any norm but with a non-competitive exponent.

Stochastic Analysis

The stochastic convergence with one persistent buffer yields O(ϵ−4) iterations under a trajectory gradient bound, and a weighted-average bound on noisy weight increments keeps explicit K dependence out of the leading T−1/4 term. The tracking error is bounded by Eλt∗ ≤ ζσ√α + (1 − α)t + LγR + Mω/α for t ≥ 0.

Iteration-wise Inner Gap

Theorem 5.1 provides a per-iterate guarantee, showing that for any primal norm, the gap(pt) is bounded by β. This tracking conclusion is achieved after burn-in and is conditional on the gradient bound M.

Experimental Results

Experiments compare the method with existing multi-task optimizers, examining its oracle, weighting, and momentum choices. The results on Cityscapes show that Entropic LMO-MGDA achieves the highest segmentation scores and the lowest absolute depth error among the reported methods. On NYU-v2, our method achieves the lowest reported aggregate degradation. The best and second-best values are highlighted in bold and underlined.

Conclusion

The analysis establishes O(ϵ−2) deterministic stationarity with a derived gradient bound, O(ϵ−4) stochastic stationarity with one persistent buffer, static regret of the weights on bounded gradient sequences, and iteration-wise inner-gap control after burn-in. The distinct tracking conclusion is the every-iterate guarantee after burn-in, not average-gap convergence alone. The content is an iteration-wise inner-gap guarantee with simultaneous ergodic progress and one tail-selected iterate carrying both, in any norm, conditional on the gradient bound M.

The paper is 2610.13773.

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The gist Multiple-gradient descent is typically formulated in Euclidean geometry, whereas matrix-aware optimizers use directions induced by other norms, and this work analyzes a single-loop method combining a norm-constrained linear minimization oracle with entropic updates of task weights to achieve strong convergence rates in arbitrary norm pairs.

Norms and the oracle

The method operates on a fixed norm pair where the primal norm measures parameter displacements, with dual g∗ measuring gradients. The linear minimization oracle is any selection lmo(g) in arg min v∈B ⟨g, v⟩, such that lmo(g) ≤ R and ⟨g, lmo(g)⟩ = −Rg∗. For a network with parameter blocks x = (xl)l the norm used in practice is x:= maxl xl(l)/ρl, with dual g∗ = P l ρ∥gl∥(l),∗, blockwise oracle lmo(g) = (ρ lmo l gl)l of per-block radius Rρell.

Inner Optimality and Tracking

The optimal blended gradient can be nonunique in spectral and sign geometries, motivating the inner gap. Theorem 4.5 gives static regret on arbitrary bounded gradient sequences. With regularization and clipping, Theorem 5.

Improvements for AI systems

  1. Continuous tracking of Pareto optimality with iteration-wise gap control: The method can achieve iteration-wise inner-gap control after burn-in while retaining ergodic stationarity, which means it can track a moving target by setting parameters like τ = β/2 ln K, κ = β/4MK, ηR = ϵ 2β/768 M 4 ln K to ensure the gap is bounded: gapt ≤ ϵ for all t ≥ T0.

  2. Robust convergence rates in stochastic settings: The system can achieve an O(T-1/4) rate whose leading term has no explicit dependence on the number of tasks under a bounded-gradient assumption, providing a strong convergence guarantee when using one persistent blended-gradient momentum buffer.

  3. General-norm optimization capability: The model can be optimized in arbitrary norm pairs, as the analysis establishes an O(T-1/2) ergodic stationarity rate and derives a required trajectory gradient bound from smoothness and lower-boundedness in the unregularized case, extending optimization beyond Euclidean geometry.

  4. Selection-independent inner optimality: The optimization process is robust against non-unique optimal directions, as Theorem 4.5 establishes a static regret on arbitrary bounded gradient sequences, meaning the optimal blended gradient can be nonunique in spectral and sign geometries, motivating a selection-independent inner gap.

  5. Improved performance under approximate oracle information: When the oracle is inexact (using assumption (34)), the system can still guarantee convergence, showing that even with an error of (1 + ϑ)R on the gradient direction, fixed relative oracle error satisfying (34) is already tolerated for deterministic stationarity and average inner optimality.

Abstract

Multiple-gradient descent is typically formulated in Euclidean geometry, whereas matrix-aware optimizers use directions induced by other norms. We analyze a single-loop multi-objective method combining a norm-constrained linear minimization oracle with entropic updates of task weights. For any finite-dimensional norm pair, we establish an O(T-1/2) ergodic stationarity rate in the deterministic setting, deriving the required trajectory gradient bound from smoothness and lower-boundedness in the unregularized case. With stochastic gradients and a bounded-gradient assumption, one persistent blended-gradient momentum buffer yields an O(T-1/4) rate whose leading term has no explicit dependence on the number of tasks. We also establish static regret for the inner weighting objective on arbitrary bounded gradient sequences and, with entropy regularization and a weight floor, iteration-wise inner-gap control after burn-in while retaining ergodic stationarity, conditional on a trajectory gradient bound. The optimal blended gradient can be nonunique in spectral and sign geometries, motivating a selection-independent inner gap. Our analysis extends the spectral construction studied by concurrent MOON to general norms. Experiments compare the method with existing multi-task optimizers and examine its oracle, weighting, and momentum choices.

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