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Objective-Function Free Multi-Objective Optimization: Rate of Convergence and Performance of an Adagrad-like algorithm

T0 review · 2 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper claims that multi-objective optimization can be done without ever evaluating objective functions, using an Adagrad-style adaptive step, and still converge to Pareto critical points at rate O(1/√k).

desk verdict A useful MO-Adagrad algorithm with a clean O(1/√k) rate proof, except the main theorem's proof has a threshold gap for the practical ς<1/2 range; worth refereeing but needs a fix. read the letter →

arxiv 2602.05893 v2 pith:QIZC3PAQ submitted 2026-02-05 math.OC

Marianna De Santis , Gabriele Eichfelder , Margherita Porcelli This is my paper · ORCID
classification math.OC MSC 90C2990C30
keywords multi-objectiveoptimizationobjective-function-freeAdagrad-likealgorithmcommondescentdirectionconvergencerateParetocriticalpointmulti-tasklearningnoisy
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 proposes MO-Adagrad, a first-order method for unconstrained multi-objective problems that never evaluates the objective functions. At each step it computes a common descent direction — the negative of the minimal-norm convex combination of the gradients — and scales it by an Adagrad-style weight built from past gradient norms. The paper proves that the average squared norm of this direction decays as θ/(k+1), so the best direction found in the first k iterations has norm at most O(1/√k). This matters because the method needs no line search, no Lipschitz constants, and no dominance test for accepting new points, making it attractive when objective evaluations are expensive, noisy, or comparable in cost to gradient evaluations. Numerical tests on bi-objective benchmarks and small multi-task learning problems support the theory and show improved robustness over a line-search descent method.

What carries the argument

The load-bearing object is the common descent direction -g^s_k, where g^s_k is the unique minimal-Euclidean-norm vector in the convex hull of the gradients at x_k, obtained by solving the small convex subproblem Ω(x_k); if this vector is zero, x_k is Pareto critical. The step is s_k = -g^s_k / w_k with w_k = √(ς + Σ_{ℓ≤k}∥g^s_ℓ∥²), an Adagrad-norm accumulator that makes the stepsize adaptive. The key identity ∥g^s_k∥² = -max_j ∇f_j(x_k)ᵀ(-g^s_k) shows that the same quantity measures both criticality and the local decrease of the max objective Φ, which lets the convergence proof proceed without evaluating Φ or any f_j. A descent lemma for Φ then yields the telescoping sum whose careful accoun

What would settle it

Run MO-Adagrad on a smooth bi-objective problem, solving the subproblem exactly or with a certified tolerance, and plot min_{ℓ≤k}∥g^s_ℓ∥ against √(θ/(k+1)) with θ from Theorem 15; a persistent violation of that bound would refute the theorem. A cheaper check is to evaluate the identity ∥g^s_k∥² = -max_j ∇f_j(x_k)ᵀ(-g^s_k) at every iterate: if it fails by more than numerical noise, the descent argument in Lemma 10 is invalid.

Watch

Extended reading notes

Core claim

The central result is Theorem 15: if MO-Adagrad is run on a smooth multi-objective problem whose gradients are Lipschitz and for which max_j f_j is bounded below, then the average over the first k+1 iterates of the squared norm of the common descent direction g^s_ℓ is at most θ/(k+1), where θ depends only on the initial gap in the max objective, the Lipschitz constants, and the parameter ς. Consequently the minimum of ∥g^s_ℓ∥ over ℓ ≤ k is at most √(θ/(k+1)), an O(1/√(k+1)) global rate for the norm of the common descent direction. The proof rests on the identity ∥g^s_k∥² = -max_j ∇f_j(x_k)ᵀ(-g^s_k), which connects the minimal-norm convex combination of gradients to the steepest common descen

Load-bearing premise

The proof assumes the convex subproblem that produces the common descent direction is solved exactly at every iteration; if it is only solved approximately, the key identity ∥g^s_k∥² = -max_j ∇f_j(x_k)ᵀ(-g^s_k) can fail, and the paper gives no bound on how this error affects the O(1/√k) rate.

Editorial extensions

If this is right

  • A multi-objective optimizer can drive all objectives toward a Pareto critical point using only gradient information; objective values are never needed, so the method applies when function evaluations are noisy, expensive, or unavailable.
  • The stepsize is fully automatic: past gradient norms set the learning rate, so no line search and no knowledge of Lipschitz constants is required, matching the simplicity of single-objective Adagrad.
  • The convergence guarantee is global and explicit: after k iterations the best common descent direction found has norm at most √(θ/(k+1)), so the worst-case iteration count to reach criticality ε is O(1/ε²).
  • In the paper's experiments, MO-Adagrad solved more of the 124 bi-objective test instances than the line-search descent method (89% versus 77%), and under added noise its identified critical points stayed closer to the noise-free solutions.
  • On the two small multi-task classification problems, MO-Adagrad reached comparable test accuracy roughly three times faster than the line-search method, because it never evaluates the losses.

Reading between the lines

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

  • A natural testable extension is to allow approximate solutions of the subproblem and quantify how the approximation error enters θ; the current proof requires the exact minimal-norm convex combination, while the experiments use a finite-precision solver.
  • The average-type rate implies that a uniformly random iterate among the first k is near-critical in the Cesàro sense; this could support cheap front exploration by multistart, an extension the paper flags but does not develop.
  • The robustness to noise seen in the preliminary experiments suggests that a stochastic version using gradient estimates could inherit the same adaptive-stepsize structure; providing high-probability rates in that setting is a natural next step not covered in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper proposes MO-Adagrad, an Adagrad-norm-type method for unconstrained multi-objective optimization. At each iteration it computes a common descent direction -g^s_k by solving the convex subproblem that minimizes the norm of a convex combination of gradients, then takes a step -g^s_k / w_k, where w_k accumulates past values of ||g^s_ℓ||. The method never evaluates objective functions and does not use line search or knowledge of Lipschitz constants. The main theoretical result, Theorem 15, gives an O(1/√(k+1)) bound on the average of ||g^s_ℓ||², hence on the criticality measure ω, under smoothness and boundedness of Φ = max_j f_j. Numerical experiments on CUTEst-based bi-objective instances, noisy variants, and two multi-task classification problems compare MO-Adagrad with a line-search steepest-descent method.

Significance. If the rate theorem is correct, the paper provides a clean function-free, line-search-free multi-objective method with essentially the same O(1/√k) worst-case criticality rate as standard first-order methods, extending objective-function-free Adagrad theory to the multi-objective setting. Strengths include self-contained proofs of the relation between the subproblem solution and the criticality measure (Lemmas 9–10), a concise descent analysis (Lemma 11), and broad numerical validation including a noisy setting. The main theorem, however, has a proof gap in the parameter regime actually used in the experiments; the gap is fixable and does not appear to threaten the rate itself. The analysis also assumes exact subproblem solutions while the implementation uses approximate ones, an issue that should be addressed.

major comments (2)
  1. [Theorem 15 / proof of Eqs. (15)–(16)] The proof assumes that S := Σ_{ℓ=0}^k ||g^s_ℓ||² ≥ max{ς, ς² e^{2Γ0/Lmax}} implies Γ0 ≤ (Lmax/2) log(2S/ς). This implication requires S ≥ (ς/2) e^{2Γ0/Lmax}. For ς < 1/2, ς² e^{2Γ0/Lmax} < (ς/2) e^{2Γ0/Lmax}, so (15) does not imply (16). Example: ς=0.01, Γ0=Lmax=1, S=0.02 satisfies (15) but (16) gives 1 ≤ 0.5 log(4) ≈ 0.693, which is false. Since Algorithm 1 is run with ς=10^{-2} in Section 4, this is the operative parameter regime, and the stated θ is not proven. The fix is to replace the threshold by max{ς, (ς/2)e^{2Γ0/Lmax}} and the second term of θ by (ς/2)e^{2Γ0/Lmax}; alternatively, one can apply Lemma 13 directly to (14) with b=Γ0, avoiding (16) entirely and yielding a smaller constant. The O(1/√k) rate is preserved, but Theorem 15 as written is not proven.
  2. [Algorithm 1, Step 2 and Theorem 15] The convergence analysis assumes that (Ω(x_k)) is solved exactly at every iteration: Lemma 10's identity ||g^s_k||² = -max_j ∇f_j(x_k)^T(-g^s_k) and Lemma 11's descent inequality use the exact minimizer g^s_k. In the experiments, (Ω) is solved with fmincon to finite tolerance, and no account is taken of the resulting inexactness. The numerical section therefore does not directly validate the theorem, and the algorithm as implemented is not covered by the rate guarantee. Please either analyze the inexact case (for example, with a subproblem tolerance criterion that preserves the descent bound) or clearly state and discuss the assumption and its practical implications.
minor comments (5)
  1. [Section 2, after Definition of weak efficiency] The sentence 'any efficient point for (MOP) is also efficient for (MOP)' should read 'also weakly efficient for (MOP)'.
  2. [Abstract] The phrase 'does not require neither knowledge of Lipschitz constants nor the use of line search procedures' contains a double negative; use 'requires neither ... nor' or 'does not require either ... or'.
  3. [Footnote 3] The definition of O uses a limit, but ratios of sequences need not have a limit; limsup is the standard notion and avoids excluding oscillating cases.
  4. [Theorem 15 proof, after Eq. (19)] The constant '64² Lmax⁴ / ς²' is easy to misread after OCR; please write it explicitly as '4096 Lmax⁴ / ς²' for clarity.
  5. [Section 4.2, Table 2] The text says both algorithms solve the 5 benchmark problems for all noise levels, but Table 2 includes CUTEst instances with failures; please clarify the distinction between the benchmark set and the CUTEst-based set.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the O(1/√k) bound is derived from the algorithm's own descent inequality and problem constants; the sole self-citation affects only a parameter choice.

full rationale

The central convergence claim, Theorem 15, is a self-contained descent-based complexity argument. It starts from the mean-value descent inequality in Lemma 11 (Eqs. (10)–(11)), uses the Adagrad-style weight recursion w_k^2 = w_{k-1}^2 + ||g^s_k||^2, and invokes Lemma 14 to bound the logarithmic sum. The constants in θ = max{ς, ς²e^{2Γ0/Lmax}, 2048Lmax^4/ς} depend only on the user parameter ς and problem data (Lmax and Γ0 = Φ(x0) − Φlow). Nothing in the proof is fitted to the quantity being bounded; the bound is on the algorithm's own criticality measure ω(xℓ) = ||g^s_ℓ||², which is legitimate because Pareto criticality is independently characterized by ω(x) = 0 (Lemma 4). The identity ||g^s_k||² = −max_j ∇f_j(x_k)ᵀ(−g^s_k) in Lemma 10 is proved via convex duality, not assumed as an input. The only self-citation, [33], appears in the numerical section for the choice ς = 10^{-2} — 'we set ς = 10^{-2} (see e.g. [21, 33])' — and is not load-bearing for the theoretical result. Concerns about exact subproblem solves versus the fmincon finite-tolerance implementation, and the threshold-gap concern in the proof of Theorem 15, are correctness/robustness issues rather than circularity: they do not reduce the claimed rate to the paper's own assumptions or fitted values.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central result is a self-contained rate proof using standard convex analysis and two lemmas borrowed from prior work (with the one from [2] proved in the appendix). The only hand-chosen parameter is the initial weight offset ς. No new physical/mathematical entities are postulated.

free parameters (1)
  • ς (initial weight parameter) = 10^-2 in numerical experiments; any value in (0,1) in theory
    User-chosen constant in Algorithm 1 (Step 1). It sets w_{-1}=√ς and enters the rate constant θ in Theorem 15. It is not fitted to data; the O(1/√(k+1)) rate holds for any ς∈(0,1). Listed because it is chosen by hand.
assumptions (6)
  • domain assumption Each f_j is continuously differentiable with L-Lipschitz gradients (for some Lj>0).
    Invoked in Lemma 11 (mean value theorem bound) and in the definition of Lmax in eq. (1).
  • domain assumption The MOP has at least one weakly efficient point, so Φ(x)=max_j f_j(x) is bounded below (Lemma 1).
    Defines Φlow and Γ0; without it the uniform O(1/(k+1)) bound in Theorem 15 does not follow.
  • domain assumption At each iteration the subproblem (Ω(x_k)) is solved to exact optimality.
    Lemmas 9-10 and the key identity (9) require that g^s_k is the exact minimum-norm convex combination; Algorithm 1 Step 2 assumes this.
  • standard math Lemma 13 (au ≤ b + c log u ⇒ u ≤ max{e^{b/c}, 4c^2/a^2}) holds.
    Attributed to [2] via private communication; proof reproduced in Appendix A. Used in Theorem 15.
  • standard math Lemma 14: Σ c_j/(ξ+Σ_{ℓ≤j} c_ℓ) ≤ log(1 + (1/ξ) Σ c_j).
    Taken from [20, Lemma 3.1]; used to control the quadratic term in the descent inequality.
  • standard math KKT conditions and Slater/strong duality hold for the convex subproblems (mgrad'(x)), (pgrad'(x)).
    Used in Lemmas 5 and 9 to relate Pareto criticality to (mgrad(x)) and to establish dual relation (7).

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Cite this review

Pith. "Pith review of Objective-Function Free Multi-Objective Optimization: Rate of Convergence and Performance of an Adagrad-like algorithm." pith.science (2026). https://pith.science/paper/QIZC3PAQ

@misc{pith2026260205893,
  author       = {Pith},
  title        = {Pith review of: Objective-Function Free Multi-Objective Optimization: Rate of Convergence and Performance of an Adagrad-like algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIZC3PAQ}},
  note         = {Machine review of arXiv:2602.05893}
}
abstract

We propose an Adagrad-like algorithm for multi-objective unconstrained optimization that relies on the computation of a common descent direction only. Unlike classical local algorithms for multi-objective optimization, our approach does not rely on the dominance property to accept new iterates, which allows for a flexible and function-free optimization framework. New points are obtained using an adaptive stepsize that does not require neither knowledge of Lipschitz constants nor the use of line search procedures. The rate of convergence is analyzed and is shown to be $\mathcal{O}(1 / \sqrt{ k+1})$ with respect to the norm of the common descent direction. The method is extensively validated on a broad class of unconstrained multi-objective problems and simple multi-task learning instances, and compared against a first-order line search algorithm. Additionally, we present a preliminary study of the behavior under noisy multi-objective settings, highlighting the robustness of the method.

Figures

Figures reproduced from arXiv: 2602.05893 by the authors.

Figure 1
Figure 1. Performance profiles (log10-scale) on the number of gradient (and function) evalua￾tions on instances derived from CUTEst problems by including ℓ2-norm regularizer as a second objective. 10 0 10 1 10 2 10 3 10 4 Logarithm of the performance ratio 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fraction of problems solved MO-Adagrad MO-Descent We report in [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Quadrants-Circle: Task 1 on the left, Task 2 on the right. Different colors denote different labels for the points. -2 -1 0 1 2 x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y Task 1: Quadrants (Predicted) Q1 Q2 Q3 Q4 -2 -1 0 1 2 x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y Task 2: Circle (Predicted) Outside Inside x 2 +y2 = 1 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Diagonals-Circle: Task 1 on the left, Task 2 on the right. Different colors denote different labels for the points. -2 -1 0 1 2 x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y Task 1: Diagonals Q2/Q4 (Class 0) Q1/Q3 (Class 1) -2 -1 0 1 2 x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y Task 2: Circle Outside Inside Boundary [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

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