Simple Optimizers for Convex Aligned Multi-Objective Optimization

Ben Kretzu; Karen Ullrich; Yonathan Efroni

REVIEW 3 major objections 4 minor 48 references

Simple Optimizers for Convex Aligned Multi-Objective Optimization

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that for convex aligned multi-objective problems, two simple gradient methods—PAMOO and the meta-algorithm MG-AMOO—make the maximum gap across objectives shrink at rates independent of the number of objectives, while the na

desk verdict Solid reduction and m-free upper bounds for convex AMOO, but the equal-weights lower bound is false as stated. read the letter →

arxiv 2509.05811 v1 pith:6MXB3QFU submitted 2025-09-06 cs.LG

Ben Kretzu , Karen Ullrich , Yonathan Efroni This is my paper

classification cs.LG MSC 90C2590C29

keywords alignedmulti-objectiveoptimizationconvexmaximumgapPolyakstepsizeonlinelearninggradientdescentmulti-tasklowerbound

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

Many machine learning problems optimize several tasks at once, and in the aligned multi-objective setting those tasks share at least one common optimal point. Previous theory required strong convexity to guarantee convergence; this paper removes that assumption and works only with convex functions that are Lipschitz or smooth. Convergence is measured by the Maximum Gap, the largest shortfall among the objectives, and the paper proves that PAMOO and the new MG-AMOO meta-algorithm drive this gap to zero at rates that do not depend on how many objectives there are. It also proves a lower bound showing the naive equal-weight baseline can converge polynomially slower as the number of objectives grows. If correct, this gives practitioners a cheap, provable alternative to equal-weight averaging in multi-task optimization.

What carries the argument

The Maximum Gap metric, MG(x) = max_i (f_i(x) - f_i^*), replaces point-wise convergence since convex aligned objectives may have many shared optima. PAMOO selects a weight vector that maximizes the one-step decrease bound derived from convexity. MG-AMOO is the load-bearing reduction: at each iteration it selects the objective with the largest current gap and applies any single-objective optimizer to it; Lemma 2 then bounds the maximum gap of the averaged iterate by the average regret of the selected sequence. This reduction carries the m-independence of the rates.

What would settle it

Run PAMOO and MG-AMOO on a convex aligned family with known shared minima (for example f_i(x)=|x_i| with all functions minimized at zero), with the number of objectives exceeding the number of iterations K. If the equal-weight baseline's maximum gap falls substantially below the Omega(sqrt(m)/sqrt(K)) lower bound, Theorem 1 is false; if MG-AMOO's gap grows with m, the claimed m-independence is false. Alternatively, perturbing the supplied f_i^* values by a small epsilon and observing non-convergence would show the exact-optimal-values premise is load-bearing.

Watch

Extended reading notes

Core claim

The central claim is that for G-Lipschitz convex AMOO, both PAMOO and MG-AMOO instantiated with Polyak steps or gradient descent achieve MG(bar x) bounded by roughly G times the initial distance to a shared optimum divided by sqrt(K), and for beta-smooth objectives the bound is proportional to beta times the squared initial distance divided by K, with no factor depending on the number of objectives m. The counterpoint is a lower bound: the equal-weight baseline with the Polyak step size has worst-case MG in Omega(sqrt(m) G ||x1 - x*_EW|| / sqrt(K)) for K <= m, so its convergence degrades as objectives are added. The paper's contribution is a new analysis of PAMOO without strong convexity, pl

Load-bearing premise

Every proposed algorithm must know the exact minimum value of each objective in advance; if those values are only estimated, both the step-size choice and the selection of the worst objective can go wrong, and the paper does not analyze that situation.

Editorial extensions

If this is right

Equal-weight baselines are provably worse as the number of objectives grows, while the proposed methods keep a single-objective-like rate.
MG-AMOO costs only O(m) per iteration to pick the worst objective and can use standard optimizers such as GD or Adam, making the theory accessible to deep learning practice.
Both Lipschitz and smooth convex objectives converge at rates matching single-objective optimization up to constants, so aligned multi-task problems are not inherently harder than single-task ones.
The epsilon-approximate AMOO extension shows that small misalignment costs only an additive epsilon, keeping the algorithms usable when objectives are nearly aligned.
The online-to-AMOO reduction means any online learner with sublinear regret yields a converging optimizer without knowing smoothness or Lipschitz constants in advance.

Reading between the lines

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

The requirement to know every optimal value f_i^* exactly is likely the main practical hurdle; in deep learning these minima are unknown, and misestimates would break both the Polyak step-size rule and the max-gap selection rule, a case the paper leaves unanalyzed.
The lower bound suggests equal-weight averaging may be a poor default in many-task regimes with large m; a direct test would measure MG on synthetic convex AMOO instances with known shared minima across different m.
The reduction is built on choosing the worst objective adversarially, so replacing the online learner with a stochastic or bandit variant should yield corresponding stochastic regret guarantees, a testable extension.
For non-convex deep learning losses the convex theory does not apply directly, but the worst-objective selection rule is a natural heuristic; the experiments hint it transfers, which could be tested more broadly.

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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

Colleague,

The paper has two real contributions. The first is the MG-AMOO reduction: selecting the function with the largest current gap and feeding it to any single-objective optimizer, then averaging, turns AMOO into an online-learning problem. Lemma 2 and Corollary 1 are clean and correct, and they give m-independent rates for free. The second is a fresh analysis of PAMOO under plain convexity plus Lipschitz or smoothness, which removes the strong-convexity assumption from prior work. Those upper bounds are simple and, apart from transcription errors, sound.

The problem is Theorem 1, the headline lower bound for equal-weights. It is false as stated. For G-Lipschitz convex functions, any point y within distance R of the shared minimizer set has MG(y) ≤ G R by Lipschitz continuity. The equal-weight method with Polyak steps is distance-nonincreasing, so its average iterate satisfies MG ≤ G R. The claimed lower bound Ω(√m G R / √K) exceeds G R whenever K ≲ m, exactly the range K ≤ m in which the theorem is stated. In the paper's own construction with m=10, K=1, the claimed bound is √10/3 ≈ 1.054 times the actual gap. The proof's final inequality fails at that point. So the abstract's polynomial-in-m separation between EW and the proposed methods is unsupported. This is load-bearing, not a footnote.

The other soft spots are smaller. All algorithms need exact f_i^*, and the paper never discusses how to estimate them or what happens under misspecification; in machine-learning practice that is a real limitation. The smooth rates in Theorems 2 and 3 are missing the square on the distance (the proofs give 2β R^2 / K, not 2β R / K), and the Lipschitz GD rate has a misplaced square root. Mechanical, but they matter. The experiments are on synthetic teacher-student problems, no error bars, so they are illustrative only.

My take: this deserves a careful referee, not a desk reject. The reduction is a genuinely useful way to think about aligned multi-objective optimization, and the upper bounds are likely correct after minor fixes. But the central lower bound has to be repaired—restrict the iteration count, change the constant, or reformulate the claim—before the separation result can be believed. If the authors fix that, the paper is a solid contribution. As it stands, the headline claim is wrong.

I'd bring it to the reading group because the reduction is worth discussing, and I'd send it to review, but I wouldn't cite the lower bound until it's corrected.

Referee Report

3 major / 4 minor

Summary. The paper studies convex aligned multi-objective optimization (AMOO), where all objectives share a common minimizer. It introduces the Maximum Gap metric MG(x)=max_i(f_i(x)-f_i^*) and proposes two optimizer families: a new analysis of PAMOO and a reduction-based method MG-AMOO that wraps any single-objective optimizer. For G-Lipschitz objectives the claimed rates are O(G R / sqrt(K)), and for beta-smooth objectives O(beta R^2 / K), both independent of the number of objectives m. The paper also claims a lower bound showing that the equal-weights (EW) baseline with the Polyak step size has worst-case MG rate Omega(sqrt(m) G R / sqrt(K)) for K <= m, and extends the algorithms to an epsilon-approximate alignment setting.

Significance. The upper-bound part of the paper is a useful contribution: the MG-AMOO reduction in Lemma 2 is simple and elegant, and the PAMOO proof in Appendix E is self-contained and plausibly correct under the stated oracle assumptions. If the upper bounds are accepted, they genuinely extend AMOO from the strongly convex setting to convex Lipschitz/smooth settings and give an online-to-batch style conversion. However, the paper's advertised separation between the proposed methods and the EW baseline rests entirely on Theorem 1, and that theorem is false as stated. Since the abstract, Table 1, and the introduction all present the EW lower bound as a central novelty, the main claim of the paper is not supported. The requirement of exact knowledge of all f_i^* is also a substantial caveat that is not discussed in the limitations section.

major comments (3)

[Section 4, Theorem 1 proof] Theorem 1 is false as stated. The proof asserts ma/(m+2K) >= sqrt(m) a/(3 sqrt(K)) for every K <= m. This is equivalent to 3 sqrt(mK) >= m+2K, which fails for example when m=10, K=1. The failure is not a small constant: for K=1 the Polyak update returns x1, and the construction gives MG(x_bar)=a, while the claimed lower bound is sqrt(10) a/3 about 1.054a, which is larger than the true value. More generally, EW with the Polyak step is distance-nonincreasing with respect to x*_EW, and every f_i is G-Lipschitz, so MG(x_bar) <= G ||x1 - x*_EW||. Therefore a lower bound with factor sqrt(m/K) cannot hold when K << m, and the range K <= m includes this regime. The claimed polynomial-in-m separation is unsupported.
[Section 4, construction after Eq. (8)] Even if one restricted K to a range where the disputed inequality holds, the construction does not deliver the advertised dependence on m. The exact lower bound obtained in the proof is ma/(m+2K) = a/(1+2K/m), while R = ||x1 - x*_EW|| = a sqrt(m/(m-1)). Thus for a fixed ratio K/m the normalized lower bound is O(1) in m; at K=m it is exactly a/3. The factor sqrt(m/K) in Theorem 1 comes solely from the invalid inequality. Hence the abstract's claim that EW 'may converge with polynomial dependence on the number of functions' is not established by this example.
[Algorithms 1 and 2; Theorems 2 and 3] All proposed methods require exact knowledge of the optimal values f_i^*. Algorithm 1 uses {f_i^*} in Delta_k, Algorithm 2 uses {f_i^*} in the gap-selection step, and the Polyak step-size variants use f_i^* directly. The paper motivates the setting by deep-learning practice, where minimal losses are typically unknown, yet it does not discuss how f_i^* could be estimated or how errors in f_i^* affect the rates. The experiments, especially P2 and P3 with offset losses, do not state how f_i^* is obtained. As stated, the contribution is an oracle result, and the practical claims in the abstract are overstated without this caveat.

minor comments (4)

[Section 5.2, Theorem 3(1) and Appendix F.3, Theorem 8] The displayed bound in Theorem 3(1) is missing the division by sqrt(K): it should be (3/2) G ||x1 - x*|| / sqrt(K). Appendix Theorem 8 writes (3/2) G ||x1 - x*|| / K, which would give an O(1/K) rate for nonsmooth functions and is inconsistent with the O(G/sqrt(K)) claim in Table 1.
[Corollary 1 and Algorithm 2] The online-gradient-descent regret bound in Theorem 7 requires a bounded convex domain and a projection step. Algorithm 2 does not include a projection or specify the domain for the online learner. Please state the domain explicitly and add the projection step to the algorithm.
[Section 4 and Appendix C] The paper uses gradients throughout, but the lower-bound construction uses the nonsmooth functions f_i(x)=|x_i| and subgradients. A sentence clarifying the subgradient convention for the G-Lipschitz case would avoid ambiguity.
[Proof of Theorem 1, final display] The displayed chain contains a typo, 'sqrt(m)/sqrt(K) a / 3K', which should read 'sqrt(m) a / (3 sqrt(K))'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the convergence proofs are self-contained, and self-citations to the prior AMOO paper are attribution/reuse rather than load-bearing.

full rationale

The paper's central claims are derived in-line rather than imported. Theorem 2 (PAMOO) is proved from Lemma 6, which uses only convexity and the standard inequality f_w(x_k) - f_w(x*) <= grad f_w(x_k)^T (x_k - x*), and Lemma 5, which telescopes the squared-distance recursion and applies either Lipschitz or smoothness (Lemma 1, a standard Boyd-Vandenberghe result). No fitted constant or measured MG value enters the bound. Theorem 3 / MG-AMOO is established through Lemma 2, a Jensen/convexity inequality that upper-bounds MG of the averaged iterate by the average of the selected gaps, and then applies either the explicit Polyak/GD lemmas (Appendix F) or standard online regret bounds cited from Hazan et al.; the online learner is treated as a black box with its published regret guarantee, not as an author-specific theorem. Theorem 1 is a constructive example, not a reduction of the lower bound to its own statement. The only self-referential elements are attribution of the AMOO framework and the PAMOO algorithm to Efroni et al. (2025) and reuse of that implementation in experiments; the new convergence proofs do not rely on the earlier paper's convergence theorems, so this self-citation is not load-bearing. The reliance on exact f_i^* values is a genuine practical assumption/limitation, but it is not circular because f_i^* are inputs to the algorithms rather than outputs the proofs presuppose. The skeptical concern that Theorem 1's lower bound may fail for small K is a potential mathematical correctness issue, not a circularity issue, and is therefore not reflected in this score.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No data-fitted constants appear in the theory. The main burden is the assumption that the optimal values f_i^* are exactly available, plus the structural constants beta, G, and bounded-domain conditions. There are no newly invented particles, forces, or physical entities; the maximum-gap metric is a definition rather than an entity.

free parameters (2)

GD step size eta=1/(2 beta) = 1/(2 beta)
Used in Lemma 8 and Theorem 6 for the smooth MG-AMOO guarantee; it presumes the smoothness constant beta is known.
Online GD step size eta_k = ||x1-x*||/(G sqrt(k)) = R/(G sqrt(k))
Used in Theorem 7 via Hazan et al.; presumes knowledge of the Lipschitz constant G and the initial distance R.

assumptions (6)

domain assumption Each objective f_i is convex and differentiable on R^n
Stated in Section 2 and used throughout the descent inequalities, Jensen arguments, and gradient updates.
domain assumption The objectives are aligned: C* = {x : f_i(x)=f_i^* for all i} is nonempty
This is the AMOO assumption introduced in Section 2; it is the defining structural condition of the paper.
domain assumption Each f_i is either G-Lipschitz or beta-smooth
Stated in Section 2; the rates in Theorems 2 and 3 depend on these constants and the proofs use the corresponding inequalities.
ad hoc to paper Exact optimal values f_i^* are known to the algorithms
Algorithms 1 and 2 initialize with {f_i^*}, and Polyak step sizes plus gap selection use these exact values; this is a practical burden the paper does not address.
domain assumption For the online-learner instantiation, C* and the convex hull of the iterates are bounded
Corollary 1 explicitly assumes C* is bounded and defines X as a bounded convex hull; standard online regret guarantees require a bounded domain.
standard math Standard convex analysis results: Lemma 1, Jensen's inequality, and gradient descent inequalities
Used in Lemmas 5 through 13 as background without proof.

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Cite this review

Pith. "Pith review of Simple Optimizers for Convex Aligned Multi-Objective Optimization." pith.science (2026). https://pith.science/paper/6MXB3QFU

@misc{pith2026250905811,
  author       = {Pith},
  title        = {Pith review of: Simple Optimizers for Convex Aligned Multi-Objective Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MXB3QFU}},
  note         = {Machine review of arXiv:2509.05811}
}

read the original abstract

It is widely recognized in modern machine learning practice that access to a diverse set of tasks can enhance performance across those tasks. This observation suggests that, unlike in general multi-objective optimization, the objectives in many real-world settings may not be inherently conflicting. To address this, prior work introduced the Aligned Multi-Objective Optimization (AMOO) framework and proposed gradient-based algorithms with provable convergence guarantees. However, existing analysis relies on strong assumptions, particularly strong convexity, which implies the existence of a unique optimal solution. In this work, we relax this assumption and study gradient-descent algorithms for convex AMOO under standard smoothness or Lipschitz continuity conditions-assumptions more consistent with those used in deep learning practice. This generalization requires new analytical tools and metrics to characterize convergence in the convex AMOO setting. We develop such tools, propose scalable algorithms for convex AMOO, and establish their convergence guarantees. Additionally, we prove a novel lower bound that demonstrates the suboptimality of naive equal-weight approaches compared to our methods.

Figures

Figures reproduced from arXiv: 2509.05811 by the authors.

**Figure 2.** The behavior of the three weights (reds) in (P1) shown as a function of the loss (blue) and [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

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2003

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

@esa (Ref

\@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later remove the command n...

[2] [2]

\@lbibitem[] @bibitem@first@sw\@secondoftwo \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 @tmp #1 NAT@b@open@#2 NAT@b@shut@#2 \@ifnum @merge>\@ne @bibitem@first@sw \@firstoftwo \@ifundefined NAT@b*@#2 \@firstoftwo @num @NAT@ctr \@secondoft...

[3] [3]

@open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibsetup #1 @NAT@ctr @ @openbib .11em \@plus.33em \@minus.07em 4000 4000 `\.\@m @bibit...

arXiv

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Bayesian uncertainty for gradient aggregation in multi-task learning

Idan Achituve, Idit Diamant, Arnon Netzer, Gal Chechik, and Ethan Fetaya. Bayesian uncertainty for gradient aggregation in multi-task learning. arXiv preprint arXiv:2402.04005, 2024

arXiv 2024

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Hamsa Bastani. Predicting with proxies: Transfer learning in high dimension. Management Science, 67 0 (5): 0 2964--2984, 2021

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Aharon Ben-Tal and Arkadi Nemirovski. Robust optimization--methodology and applications. Mathematical programming, 92: 0 453--480, 2002

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Robust optimization

Aharon Ben-Tal, Arkadi Nemirovski, and Laurent El Ghaoui. Robust optimization. 2009

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Convex optimization

Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004

2004