MOON: Multi-Objective OrthoNormalized Updates for Multitask Learning

Kunlin Lyu; Lei Zhang; Ruodong Wang; Shiji Zhou; Yifan Sun

REVIEW 4 major objections 5 minor 300 references

MOON: Multi-Objective OrthoNormalized Updates for Multitask Learning

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read MOON claims that multi-task gradient manipulation should happen in the spectral-nuclear norm geometry of matrix parameters, not in flattened Euclidean space, and proves Pareto-stationarity convergence at O(T^{-1/2}) deterministic and…

desk verdict Genuinely new geometry for multi-task gradient manipulation, but the convergence theorem covers only a single matrix block, so the headline rates don't formally apply to the blockwise algorithm used in the experiments. read the letter →

arxiv 2608.11749 v1 pith:RDMAFZNE submitted 2026-08-12 cs.LG cs.AIstat.ML

Shiji Zhou , Kunlin Lyu , Lei Zhang , Ruodong Wang , Yifan Sun This is my paper · ORCID

classification cs.LGcs.AIstat.ML MSC 90C2990C2665K0568T07

keywords multi-tasklearningmulti-objectiveoptimizationgradientmanipulationspectralnormnuclearpolarfactororthonormalizedupdatesParetostationarity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that standard multi-objective optimization methods for multi-task learning flatten parameters into vectors and combine task gradients in Euclidean space, which is not the steepest descent direction when parameters are matrices. It proposes MOON, which solves a minimax problem over matrix updates under the spectral norm; the dual selects task weights by minimizing the nuclear norm of the weighted average gradient, and the update direction is the polar factor of that average. For smooth non-convex objectives, the averaged Pareto-stationarity measure converges at O($T^{{-1/2}}$) with exact gradients and O($T^{{-1/4}}$) with unbiased stochastic gradients. Experiments across five multi-task benchmarks show faster optimization and competitive or better final performance.

What carries the argument

The load-bearing object is the nuclear-norm Pareto-stationarity measure G(Theta) = min_{z in Delta_m} ||sum_i z_i grad_i(Theta)||_{S1}, the dual of the spectral-norm minimax subproblem min_W max_i [-<grad_i, W> + 1/2||W||^2_{S∞}]. Its exact solution is C * U V^T, where C is the nuclear norm and U V^T is the polar factor of the weighted aggregate gradient; the polar factor carries the argument because under spectral-norm smoothness it is the steepest descent direction, and the nuclear norm provides a matrix-geometric suboptimality certificate via duality. The proof works by telescoping weighted losses, bounding the momentum-vs-current-gradient error, and using softmax Lipschitzness to control weight drift; finite-step Newton-Schulz error is shown to alter only constants under a uniform approximation condition.

What would settle it

Run MOON under deterministic gradients on a network with at least two interacting weight matrices (for instance, a two-layer linear network) and measure (1/T) sum_t min_z ||sum_i z_i grad_i(Theta_t)||_{S1}; if the measure does not decay at the O($T^{{-1/2}}$) rate (or climbs, or plateaus far above zero) while a single-block toy case does decay, the blockwise transfer claimed is false. Alternatively, a two-block example where each block's polar direction is a descent direction but the coupled update raises a shared loss would falsify the blockwise application.

Watch

Extended reading notes

Core claim

MOON establishes that the correct geometry for multi-objective gradient manipulation on matrix-valued parameters is the spectral-nuclear norm pair: each objective is assumed L-smooth with respect to the spectral norm, the common descent direction is chosen by the minimax problem min_W max_i [-<grad_i, W> + 1/2||W||^2_{S∞}], whose dual minimizes the nuclear norm of the weighted aggregate gradient, and the primal solution is the polar factor times the nuclear norm. Because the polar factor discards singular-value magnitudes, the update is governed by singular subspaces rather than by the largest singular components, which the paper identifies as the reason Euclidean flattening is suboptimal under matrix geometry. The paper proves that Algorithm 1 (with momentum, Newton-Schulz polar approximation, and online softmax weight updates) drives the averaged nuclear-norm Pareto-stationarity measure to zero at the stated rates, and that at any fixed iterate the MOON weighting gives a nuclear-norm stationarity certificate no worse than Euclidean min-norm weighting.

Load-bearing premise

The convergence proof assumes a single matrix block and exact polar factors, but the paper applies the construction blockwise and uses Newton-Schulz approximations in practice, so the transfer to full networks rests on approximation conditions not proven for multi-block interaction.

Editorial extensions

If this is right

MOON attains the standard sublinear rate for smooth non-convex multi-objective optimization: O(T^{-1/2}) deterministic, O(T^{-1/4}) with unbiased bounded-variance stochastic gradients.
At any fixed iterate, the MOON dual weighting yields a nuclear-norm Pareto-stationarity certificate no larger than Euclidean min-norm (MGDA-type) weighting, so the matrix-geometric dual is never worse at that iterate.
The polar-factor update removes singular-value magnitude information, so updates are driven by singular subspaces, which the paper argues is why MOON avoids being dominated by large singular components and converges faster in practice.
The practical algorithm's finite Newton-Schulz approximation preserves the O(T^{-1/2}) rate whenever the approximation error is uniformly bounded below 1 in operator norm.
On the tested benchmarks (MultiMNIST, NYU-v2, CityScapes, QM9, CelebA), MOON reduces the average performance drop and reaches the same training loss faster in wall-clock time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper does not develop: replacing the spectral norm by Schatten p-norms would interpolate between polar-factor updates (p = infinity) and Euclidean updates, giving a tunable family whose rates should degrade continuously.
The nuclear-norm certificate suggests a practical diagnostic: monitor min_z ||sum_i z_i grad_i(Theta)||_{S1} during training as a geometric early-stopping or task-conflict signal, independent of validation metrics.
If the blockwise convergence gap is closed, the same spectral-nuclear geometry could be composed with block preconditioners to yield multi-objective optimizers for very large models.
The two-task toy example shows the benefit is not update scaling: under an equal spectral-norm budget, the MOON direction gives strictly larger common descent, implying the geometry itself, not step size, drives the gain.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things about this paper. First, the core idea is genuinely new: formulate multi-task gradient manipulation under spectral–nuclear norm geometry, derive the dual as nuclear-norm minimization, and update with the polar factor. That is a clean and principled departure from the usual Euclidean flatten-and-combine. Second, the convergence theorem only covers a single matrix block, while Algorithm 1 applies the update blockwise to real networks. The gap between theory and algorithm is real and load-bearing.

What the paper does well: the derivation is honest and the single-block analysis is internally consistent. The dual of the spectral-norm minimax is the nuclear-norm minimization, the primal recovery via the polar factor is right, and the descent lemmas in Appendix A.3 and A.4 support the stated O(T^{-1/2}) and O(T^{-1/4}) rates under the stated assumptions. The experiments are broad — MultiMNIST, NYU-v2, CityScapes, QM9, CelebA, plus a Qwen3 RL fine-tuning — and the public code is a plus. The ablations show that both the orthonormalized update and the momentum matter; the toy example in Appendix E nicely illustrates why Euclidean and matrix-geometric directions differ.

Now the soft spots, in proportion. The blockwise gap is the main one. The theory assumes each loss is a function of a single matrix Θ in R^{p×q}, and every lemma and theorem is stated for that setting. Algorithm 1, however, updates a full network with many matrix blocks. The phrase "the same construction is applied blockwise" is not a proof: there is no joint stationarity measure for a tuple of blocks, no descent lemma that controls cross-block terms, and no argument that blockwise iteration preserves the rates. This is a scope gap rather than an internal contradiction, but it directly affects the central claim that MOON converges for the networks used in the experiments. Second, Proposition 3 — the dual representation that motivates the entire method — is stated without proof. It deserves a proof in the appendix. Third, the Limitations section says the Newton–Schulz approximation error is omitted from the analysis, but Appendix D analyzes exactly that error under Assumption 4 and shows the O(T^{-1/2}) rate is preserved up to a factor 1/(1-ε_q). Those two statements contradict each other; a referee should ask which version the authors actually stand behind. Fourth, baseline error bars are missing. The appendix gives MOON's own standard deviations, which are small, but without the same for baselines the 0.2–0.3% MultiMNIST gains are hard to judge; the CityScapes gains are larger and more convincing.

Who this is for: people working on matrix-aware optimizers or multi-task gradient manipulation. The single-block theory is a useful building block even if the blockwise extension is unfinished. The citation pattern looks fine — self-citations are to directly relevant prior work.

Recommendation: this deserves a serious peer review, not a desk reject. A good referee should push the authors to either prove a joint blockwise convergence result or explicitly reframe the theory as single-block with the blockwise version framed as empirical, and also to reconcile the Limitations section with Appendix D and prove Proposition 3. Those are fixable in revision, and the core idea is worth engaging with.

Referee Report

4 major / 5 minor

Summary. The paper proposes MOON, a multi-task learning method that performs gradient manipulation under spectral--nuclear norm geometry for matrix-valued parameters. The main algorithmic update aggregates task gradients with learned simplex weights, applies momentum, and uses the polar factor of the momentum matrix as the update direction, with a finite Newton--Schulz approximation in practice. The theoretical contribution is a convergence analysis for smooth, possibly non-convex objectives: an averaged nuclear-norm Pareto-stationarity measure converges at rate O(T^{-1/2}) with exact gradients and O(T^{-1/4}) with unbiased stochastic gradients. The empirical section reports results on MultiMNIST, NYU-v2, CityScapes, QM9, and CelebA, plus toy, synthetic, and large-language-model fine-tuning experiments, claiming faster convergence and competitive or better final performance relative to Euclidean MOO baselines.

Significance. If the central claims hold, MOON is a principled bridge between matrix-aware optimizers such as Muon and multi-objective gradient manipulation, and the reported experiments suggest practical gains in wall-clock efficiency and final task quality. The paper has genuine strengths: the dual derivation via spectral--nuclear norm duality is elegant, the comparison inequality (Eq. 10) follows from feasibility rather than fitting, the descent lemmas in the appendix are explicit and internally consistent for the single-block setting, and the authors provide code, ablations, and a separate appendix addressing finite-step Newton--Schulz approximation. However, the convergence theory is developed only for a single matrix-valued parameter block, whereas the algorithm is applied blockwise to full networks; this is a load-bearing scope gap because the advertised rates are claimed for the network algorithm actually used in the experiments.

major comments (4)

[Section 'Multi-Objective Orthonormalized Updates' and Section 'Convergence Analysis'] The convergence analysis is stated for a single matrix block Θ in R^{p×q}, with every lemma and theorem assuming ℓ_i is L-smooth with respect to the spectral norm and that the update is Θ_{t+1} = Θ_t − α W_t for one matrix. Algorithm 1, however, is applied to full networks by the sentence 'the same construction is applied blockwise', and no joint norm, joint Pareto-stationarity measure, or descent lemma for a product of blocks is provided. The stationarity measure G(Θ) = min_z ||Σ z_i ∇ℓ_i(Θ)||_{S1} is not well-defined when ∇ℓ_i is a tuple of block gradients, and cross-block interactions are not controlled by the single-block proof. Since the O(T^{-1/2}) and O(T^{-1/4}) rates are the paper's central theoretical claim, this gap must be closed by either extending the analysis to blockwise parameterizations or sharply restricting the stated claims to the single-block setting.
[Proposition 3, Section 'Multi-Objective Orthonormalized Updates'] Proposition 3 states that the dual of the minimax problem (7) is min_{z∈Δ_m} (1/2)||Σ z_i ∇ℓ_i(Θ_t)||_{S1}^2 and that the exact primal solution is recovered as C U V^T with C equal to the nuclear norm of the weighted gradient. This is a central step in the derivation, yet no proof is given in the main text or the appendix. A standard Fenchel-duality argument can establish the result, but the manuscript should include it explicitly, and it should also address the rank-deficient case rather than assuming full rank.
[Section G 'Limitations' and Appendix D] The Limitations section states that the Newton--Schulz approximation error is omitted from the current analysis and 'leave[s] its precise theoretical characterization to future work', but Appendix D does precisely the opposite: it introduces Assumption 4, proves Lemmas 5--7, and derives Theorem 8, concluding that a fixed number q of Newton--Schulz iterations preserves the O(T^{-1/2}) deterministic rate up to a factor 1/(1−ε_q). These two passages directly contradict each other. Moreover, the stochastic setting is dismissed with 'the same argument' after conditioning on the history, without a formal statement or proof. The authors should reconcile the limitation statement with Appendix D and provide the full stochastic Newton--Schulz analysis if Theorem 2 is to be claimed for Algorithm 1 as implemented.
[Theorem 2 and Appendix A.4] Theorem 2 is stated for Algorithm 1, but Algorithm 1 computes W_t by Newton--Schulz iterations while the proof of Theorem 6 assumes the exact polar factor of the momentum matrix. The stochastic proof also uses δ_t = [⟨W_t, ∇ℓ_1(Θ_t)⟩, ..., ⟨W_t, ∇ℓ_m(Θ_t)⟩] while the stochastic model assumes access only to unbiased stochastic gradients; the manuscript does not specify how δ_t is computed from stochastic gradients in the analysis. These mismatches between the theorem statement, the algorithm, and the proof assumptions should be resolved or explicitly declared as idealized analysis.

minor comments (5)

[Introduction] The sentence 'creates a theoretical gap between their underlying assumptions, creates a mismatch between their underlying optimization principles' contains a duplicated and ungrammatical construction; please revise.
[Tables 1--4] The baseline results are reported without standard deviations, while Appendix F.9 reports error bars only for MOON. Given that the empirical claims rest on comparisons over three seeds, adding error bars for the baselines (or stating that they are omitted for space) would substantially strengthen the comparison.
[Table 4 and Appendix F] Several table entries contain formatting errors, for example the SI row in Table 4 reads '135.71.00' and '4.5155.3', and Appendix F.7 uses 'Euclidian' instead of 'Euclidean'. These should be corrected.
[Theorems 1 and 2] The notation min_{z*_t∈Δ_m} ||∇L(Θ_t) z*_t||_{S1} is ambiguous because ∇L(Θ_t) is not defined as a matrix for a tuple of task gradients; writing Σ_{i=1}^m z_i ∇ℓ_i(Θ_t) would be clearer.
[Appendix D] Assumption 4 is a uniform, trajectory-dependent bound on the Newton--Schulz approximation error, but no empirical estimate of ε_q is provided for the experiments; a measurement or a justification of why the bound is reasonable for the tested architectures would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: MOON's convergence theory and empirical benchmarks are self-contained; only non-circular scope gaps remain.

full rationale

MOON's central derivation, Proposition 2–3, starts from a spectral-norm quadratic upper bound and derives the dual nuclear-norm weighting subproblem and polar-factor primal solution by Lagrangian duality; nothing in this step presupposes the convergence measure G(Θ). The convergence proofs (Theorems 3–6) bound G(Θ_t) by the algorithm's own weighted aggregate ||Σ_i z_{i,t} ∇ℓ_i(Θ_t)||_{S1}, then control that quantity through the softmax/momentum lemmas and telescoping of the weighted losses—an inequality-chain argument, not an identity that reinstalls the target. Equation (10) is an optimality/feasibility inequality for the dual minimizer, so it is a genuine mathematical property rather than a fitted quantity relabeled as a prediction. The empirical sections compare MOON against external baselines on MultiMNIST, NYU-v2, CityScapes, QM9, and CelebA, and the ablations isolate the orthonormalized update and momentum; no fitted parameter is renamed a prediction. The self-citations (Zhou et al. 2022 and He et al. 2024) are contextual literature references and are not load-bearing. Two manuscript-level gaps are real but are not circularity: the single-block convergence proof is asserted to extend blockwise ('the same construction is applied blockwise') without a joint stationarity measure or descent lemma, and Section G says Newton–Schulz error is 'omitted' from the analysis while Appendix D analyzes exactly that error; both are correctness/scoping concerns, not cases of the derivation reducing to its inputs. Therefore no circular step is identified.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central convergence claim depends on smoothness and boundedness assumptions, on a uniform approximation condition for Newton-Schulz, and on an unproved blockwise extension. The method introduces no new physical entities; the hyperparameters alpha, beta, gamma, mu, and q are free but standard tuning knobs.

free parameters (5)

learning rate alpha = not stated; validation-tuned
The convergence theorems require alpha = sqrt(B/(LT)) (deterministic) or alpha = sqrt(mu B/(TL)) (stochastic), with unknown H, B, L, sigma; the practical algorithm tunes alpha per benchmark.
task-weight step size beta = not stated; validation-tuned (ablation in Table 8)
Theory sets beta = 1/(mHT) or beta = 1/((H+sqrt(m)rho sigma) mT), which depend on unknown H and sigma; experiments tune beta.
logit weight decay gamma = not stated; validation-tuned (ablation in Table 7)
Theory only constrains 0 < gamma < mHT (or similar), leaving gamma free.
momentum mu = not stated; presumably tuned
Theory allows any fixed mu in (0,1] in the deterministic setting and sets a specific mu in the stochastic setting; the algorithm requires selecting mu.
Newton-Schulz iterations q = not stated
The practical algorithm uses a finite number of NS iterations; Assumption 4 requires epsilon_q < 1 but no recommended q is given.

assumptions (5)

domain assumption Each objective ell_i is L-smooth with respect to the spectral norm: ||nabla ell_i(Theta) - nabla ell_i(Theta')||_{S1} <= L ||Theta - Theta'||_{Sinfty} (Assumption 1, Appendix A.1).
All quadratic upper bounds and convergence proofs rely on this condition, which is not verified for the benchmark losses.
domain assumption Gradients and losses are uniformly bounded: ||nabla ell_i(Theta_t)||_{S1} <= H and |ell_i(Theta_t)| <= B (Assumptions 2-3).
Used in Lemmas 1-4 and the final rate; requires compactness or boundedness of the trajectory.
ad hoc to paper The finite-step Newton-Schulz approximation errors satisfy the uniform bound epsilon_{t,q} <= epsilon_q < 1 (Assumption 4, Appendix D).
This condition is assumed to hold along the whole trajectory; the paper gives convergence of NS under standard conditions but does not prove the uniform bound for the particular matrices M_t arising in training.
standard math Proposition 1 from Bernstein and Newhouse (2024): steepest descent under the spectral norm is the polar factor of the gradient.
Imported as a building block for the multi-objective derivation; includes the quadratic spectral-norm upper bound (Eq. 3).
domain assumption For multiple matrix blocks, the same construction applies blockwise (Section 'Multi-Objective Orthonormalized Updates').
The theory is only for a single block; the extension to products of blocks is asserted, not proved.

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Pith. "Pith review of MOON: Multi-Objective OrthoNormalized Updates for Multitask Learning." pith.science (2026). https://pith.science/paper/RDMAFZNE

@misc{pith2026260811749,
  author       = {Pith},
  title        = {Pith review of: MOON: Multi-Objective OrthoNormalized Updates for Multitask Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RDMAFZNE}},
  note         = {Machine review of arXiv:2608.11749}
}

abstract

Multi-objective optimization (MOO) has demonstrated significant success in multi-task learning by mitigating task conflicts through gradient manipulation. However, most existing methods flatten model parameters into vectors and perform gradient manipulation under Euclidean geometry, thereby overlooking the matrix structure prevalent in modern architectures such as Transformers. In this paper, we show that gradient manipulation in Euclidean space does not generally yield the steepest descent direction under matrix geometry, potentially limiting optimization efficiency. Drawing from the theory of steepest descent for matrix-valued parameters, we propose MOON (Multi-Objective OrthoNormalized Updates), which performs gradient manipulation under spectral--nuclear norm geometry and uses the orthonormalized manipulated gradient for parameter updates. Theoretically, for smooth non-convex objectives, we establish convergence of the averaged Pareto-stationarity measure at rates of $\mathcal{O}(T^{-1/2})$ in the deterministic setting and $\mathcal{O}(T^{-1/4})$ under stochastic gradients. Empirical results across various benchmarks show that MOON consistently improves both optimization efficiency and final multi-task performance. Our code is available at https://github.com/KunlinLyu/MOON.

Figures

Figures reproduced from arXiv: 2608.11749 by the authors.

**Figure 1.** Average loss curve on the toy case. 0 20 40 60 80 100 Epoch 2 1.5 1 0.5 0.2 0.1 Training Loss FAMO MGDA Ours (a) ViT left task 0 20 40 60 80 100 Epoch 2 1.5 1 0.5 0.2 0.1 Training Loss FAMO MGDA Ours (b) ViT right task [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

**Figure 2.** Convergence of per-task training cross-entropy losses vs. training epochs on MultiMNIST. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Performance drop ∆m% relative to STL vs. training epochs on NYU-v2 and CityScapes benchmarks. encoder-decoder as well as the MTAN attention, which account for 99.57% of all parameters. We measure multi-objective optimization performance using the average performance drop ∆m% relative to the single-task learning (STL) baseline [Navon et al., 2022]. For each method m, the performance drop is calculated by ∆m% = 1 K P… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Average squared loss after 1000 training steps on the synthetic data multi-objective [PITH_FULL_IMAGE:figures/full_fig_p041_4.png]

**Figure 5.** Figure 5: Multi-objective reinforcement fine-tuning of Qwen3-1.7B-Base on MATH500 following [PITH_FULL_IMAGE:figures/full_fig_p042_5.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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