Joint Gradient Balancing for Data Ordering in Finite-Sum Multi-Objective Optimization
Authors: Hansi Yang
James Kwok
Venue: ICLR 2025 Spotlight (2025)
Keywords: multi-objective optimization
Primary area: optimization

Abstract: In finite-sum optimization problems, the sample orders for parameter updates can significantly influence the convergence rate of optimization algorithms. While numerous sample ordering techniques have been proposed in the context of single-objective optimization, the problem of sample ordering in finite-sum multi-objective optimization has not been thoroughly explored. To address this gap, we propose a sample ordering method called JoGBa, which finds the sample orders for multiple objectives by jointly performing online vector balancing on the gradients of all objectives. Our theoretical analysis demonstrates that this approach outperforms the standard baseline of random ordering and accelerates the convergence rate for the MGDA algorithm. Empirical evaluation across various datasets with different multi-objective optimization algorithms further demonstrates that JoGBa can achieve faster convergence and superior final performance than other data ordering strategies.

Published as a conference paper at ICLR 2025
JOINT GRADIENT BALANCING FOR DATA ORDERING IN FINITE-SUM MULTI-OBIECTIVE OPTIMIZATION
Hansi Yang, James Kwok
Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Hong Kong, China (hyangbw, jamesk) (eessk)ssssees decce. ust.
ABSTRACT
In finite-sum optimization problems, the sample orders for parameter updates can significantly influence the convergence rate of optimization algorithms. While numerous sample ordering techniques have been proposed in the context of single-objective optimization, the problem sample ordering in finite-sum multi- objective optimization has not been thoroughly explored. To address this gap, we propose sample ordering method called JoGBa (Joint Gradient Balancing). which finds the sample orders for multiple objectives by jointly performing on- line vector balancing on the gradients of all objectives. Our theoretical analysis demonstrates that this approach outperforms the standard baseline of random or- dering and accelerates the convergence rate for the MGDA algorithm. Empiri- cal evaluation across various datasets with different multi-objective optimization algorithms further demonstrates that JoGBa can achieve faster convergence and superior final performance than other data ordering strategies.
INTRODUCTION
Many well-known machine learning problems involve jointly optimizing multiple objectives in model training. Examples include multi-task learning (Sener & Koltun 2018), meta-learning (Ye al. 202T), learning with fairness and safety constraints (Zaaar al., 2017) and multi-agent re- inforcement learning (Moffaert & Nowe, 2014). Mathematically, these problems share the same formulation of minimizing a vector-valued loss function and can be defined as:
min L(w)= = [w) Las (w)].
(1)
WER"
Here, each loss function (m))), I...., M corresponds to a training objective and can be expressed by (w)) = Σn (nww, E..) where each çn denotes a training sample and en is the per-sample loss. Solving problem (T) is fundamentally different from common single-objective optimization problems as different objectives may have conflicts with each other. A straight- forward baseline is to optimize weighted average of the multiple objectives, also known static or unitary weighting (Kurin et al. 2022] Xin et al.) (022). Its performance then largely depends on how to choose the weights balance different objectives, and may involve huge amount of tuning efforts. A popular alternative is thus to dynamically weight gradients from different objectives to avoid conflicts between them. Generally, these methods share the same procedure: First, compute all the gradients of each objective, then compute set of weights for different objectives based on their gradients. The model is updated by the weighted sum of all gradients, while the weights can dynamically change. The pioneering work of this approach is the multi-gradient descent algorithm (MGDA) (Désidéri) 2012) and its stochastic variants (Liuu Vicente 2021] Fernando et al.) [023]
Zhou al. 2022 Chen et al. 2024). Later works further improve upon MGDA by considering the worst improvement among different objectives (Liu et al.) 2021 Ban & Ji 2024), as well as constructing bargaining game between different objectives (Navon al. 2022).
While the aforementioned methods can be used to compute weights dynamically based on the loss gradients, less-investigater issue for the stochastic optimization of finite-sum multi-objective prob- lem is how we should order the samples for the computation of their stochastic gradients so that problem (1) be solved efficiently. For single-objective optimization in the finite-sum setting.

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many methods have been proposed for ordering the samples in stochastic optimization (Ying al.
2017 Gürbüzbalaban et al. 2019; Lu et al. 2021: Mohtashami et al. 2022 Lu et al. 2022). With multiple objectives, simple generalization of these methods to treat the weighted average of all loss gradients as sample "gradient" for update (Figure T(a). However, the gradient weights may change drastically during model update, making existing methods unstable and often do not outper- form the simple baseline of random ordering. Another simple extension is to run the singlc-objective sample ordering algorithm on each objective separately, but this can lead different orderings for different objectives (Figure l(b). Moreover, it overlooks possible conflicts between gradients from different samples, and thus may still yield limited improvement over random ordering.
Motivated by the above limitations, in this paper, we propose novel sample ordering framework for multi-objective optimization. As illustrated Figure [[c] the proposed method jointly provides sample orderings for different objectives by solving an online vector balancing problem with the gradients on each objective. The online vector balancing problem allows us to control the maximum norm of total model update in each epoch, leading to the theoretical guarantee of accelerated conver- gence. Specifically, our theoretical results demonstrate that the proposed method improves over the random ordering baseline for finite-sum multi-objectivv optimization, with smaller sample variance and faster convergence. Empirical results on a variety of data sets with multiple learning objectives demonstrate that the proposed method achieves faster convergence and better performance than the other data sampling methods.
Our contributions can be summarized as follows:
We propose a novel data ordering method that uses gradient balancing across different objectives to accelerate convergence.
We propose a novel theoretical framework analyze multi-objective optimization with different data ordering for each objective.
Empirical results on various multi-task learning data sets demonstrate effectiveness of the pro- posed method.
ordenig
Suit plu ordenng
onds Vng
Sample ordluing
(a) Same ordering for all objec- (b) Data ordering on each objective (c) Proposed method: joint data separately.
tives.
dering all objectives.
Figure 1: Visualization of different data ordering approaches for multi-objective optimization.
RELATED WORKS
2.1 PERMUTATION- BASED SGD FOR FINITE- SUM OPTIMIZATION
Stochastic optimization often assumes that training samples are independently from an underlying distribution in each iteration. However, this assumption does not match with practical implementa- tions that typically use finite training samples in certain order. Instead, permutation-basee SGD proposes to first sort all the training samples by order, and then use the samples following this order. An example is random reshuffling (Ying al.) 2017) and the related shuffle-once method (Bertsekas] 2 Gürbüzbalaban et al. 2019). which first generates random order of all training samples in each epoch, and then uses the training samples following this order in the subsequent iterations. Thcoretical analysis of random reshuffling dates back to Recht & RR (2012). Rajput ct al. (2021) introduces variant of random reshuffling that reverses the order every two epochs, and theoretically demonstrates that this variant achieves faster convergence for quadratic objectives.
Instead of using a random order, Lu et al. 2021 Mohtashami et al. (0022). and Lu et al. (0022) try to find better sample orders. These works mostly based on the herding problem (Welling 2009). which minimizes the consecutive errors of stochastic gradients. Theoretical analysis (Cha et al. 2023) demonstrates that the ordering based on the herding problem is asymptotically optimal. There
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different methods to solve the herding problem. Mohtashami et al. (2022) evaluates gradients on all samples first and then solves the herding problem to obtain the order for all samples before
starting an epoch. [u et al. (202T) uses stale gradients from the previous epoch to estimate the gradient on each sample. Lu et al. (2022) proposes to solve the herding problem via online vector
balancing, which removes the additional storage cost in (Mohtashami et al. 2022 Lu al. 2021).
Despite the aforementioned improvements, existing works on permutation- based SGD only focus on single-objective optimization problems. While some simple extensions exist for training with multiple objectives (e.g.. by using the weighted gradient or ordering samples for each objective sep- arately), these simple extensions do not always yield much improvements, as will be demonstrated in our empirical results.
2.2 GRADIENT- BASED MULTI-OBJECTIVE OPTIMIZATION
To balance the optimization on different objectives, most existing algorithms use the weighted aver- age of all objective gradients to update the model. There are different ways to compute the weights for different objectives. Some works perform the weighting based on some heuristics. Examples include using the prediction uncertainty (Kendall et al. 2017), gradient norms (Chen et al. 2018) or task difficulty (Guo et al. 2018). Another line of works propose to compute the objective weights from some sub-problems
the objective gradients. The pioneering work is MGDA (Désidéri 2012), which computes the weights by avoiding conflicts across any objective. Stochastic variants of MGDA with optimization convergence guarantees have been proposed in (Liu Vicente [2021]
Zhou et al. 2022, Fernando et al. 2023 Chen et al., 2024). PCGrad (Yu et al.) 2020) proposes to project the gradients of tasks to the normal plane of the other tasks with conflicting gradients. CA-
Grad (Liu et al. 2021) searches for an update direction in a neighborhood of the average gradient
that maximizes
worst improvement of any task. Nash-MTI. (Navon al. 2022) proposes to look for a fair gradient direction based on bargaining game between different objectives.
Convergence analysis for the deterministic MGDA algorithm dates back to (Fliege et al. 2019) Later on, stochastic variants of MGDA are introduced (Liu & Vicente 202 Zhou et al. 2022 Fer- nando et al. 2023; Chen et al., 2024). However, the vanilla stochastic MGDA introduces a biased estimate of the dynamic weight, which results in the biased estimate of update direction during op- timization. To address this issue, Liu & Vicente (202I) propose to increase the batch size during optimization, and prove the convergence of stochastic MGDA with the Lipschitz continuity assump- tion on the objective weights X" (ww) with respect to the loss gradients V VL(ww. Nevertheless, as first proved in (Zou et al.) 2022 Proposition 2), this assumption does not hold in general. To address this problem, momentum based bias reduction algorithms (Zhou et al. [002 Fernando et al. 2023) are proposed eliminate such unrealistic assumptions. The convergence of the MGDA algorithm without the unrealistic Lipschitzness assumption is first established in (Chen et al. 2024), which propose to mitigate the bias in update direction via double sampling. Most existing works focus on the convergence analysis under an online setting instead of the finite-sum setting, and ignores the impact of sample orders their theoretical analysis.
PROPOSED METHOD
3.1 MULTIPLE SAMPLE ORDERINGS FOR MULTIPLE OBJECTIVES
A simple extension of existing single-objective sample ordering methods to multi-objective opti- mization is to use the weighted average of all loss gradients as sample "gradient", and follow ex- isting data ordering methods on the weighted gradient. When the objective weights do not change with different samples, this extension can be regarded as using the weighted objective as the only objective in the existing methods. However, since the objective weights are constantly changing, using the same sample order cannot well tackle the possible conflicts between different objectives.
To alleviate this problem, we propose to use different sample orders for different objectives. Specif- ically, for a data set with K samples, we generate an order .,." {1...., K} > {1...., ,K} for the m-th objective. Some simple methods to generate the order π."" in each epoch t are listed below:
1. Random: In each epoch t, the data sets are randomly shuffled to generate an ordering ग."" for each objective.
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2. FlipFlop: For each objective, create new order mi+1 by reversing the previous mp", i..., Ti+1 (A) = -. +1 k).
3. Random FlipFlop, which performs Random on even epochs and FlipFlop on odd epochs.
3.2 SAMPLE ORDERING BY ONLINE VECTOR BALANCING
some recent works (Lu et al. 2021 Mohtashami et al. 2022 Lu et al. 2022 propose adaptively find good order for all training samples in each stochastic optimization with a single objective. An example is GraB (Lu al. 2022), which tries find sample ordering ι that minimizes the maximum norm of parameter update in each epoch, i.e., maxk (k') w With a single objective L, the model parameters updated by (x(++)) (b( Vl(w, (n)) at each iteration k in epoch. This problem is then transformed to the online vector balancing problem defined below:
Besides the simple ordering methods in Section 3.1
epoch for faster convergence
Definition 3.1 (Online Vector Balancing (Spencer 1977)). Given K vectors {=x) 1 k=1 E IRd, arriv-
ing one time, the goal of online vector balancing to assign a sign EK {-1,+1} to each vector upon receiving it so as to minimize maxmE (1.....K >> k= Cx
We propose to generalize this problem to multiple objectives by replacing the gradients on a single objective to those multiple objectives and jointly consider their influence to the model updates. The complete procedure, called JoGBa (Joint Gradient Balancing), is shown in [ Specif- ically, the k-th iteration of epoch
compute the gradients (Vlin (ur) (ung (R) (k))) )= for all
M objectives w.r.. the current model parameter wi (k) The sample order x" for each objective is then determined by solving the balancing problem on the gradients from different objectives, im- plemented by routine Balancinn ng in step 11. While there exists different ways to solve the online vector balancing problem and compute the gradient sign km.k.t here we follow GraB and use a greedy algorithm that works well in practice. As shown in Algorithm ? we compare two vector norms || + fm.k,elooo and || Am.k. so. where 8 + km,k.t corresponds to putting this sample at the beginning and km. corresponds to putting this sample the end. Since the online vector balancing problem in Definition 3.1 tries to minimize the vector sum's 's norm, we choose the sample order that can lead to the smallest norm, is indicated by the value of Em. The vector is shared among different objectives to enable joint balancing across their corresponding gradients. After the balancing routine is complete, we compute the objective weights by any multi-task learning algo- rithm (routine MTL), such as MGDA (Désidéri 2012) or Nash-MTL (Navon et al. (022). We then update the mean of all gradients and perform model update un())
3.3 THEORETICAL ANALYSIS
In this section, theoretically demonstrate how Algorithm improves upon simple extensions of sample ordering methods to multi-objective optimization. Since the convergence analysis of multi-objective optimization is different from optimizing single objective, we first introduce the definition of Pareto stationary. Denote the gradients for all M objectives as VL(w) IRdxM where [(w) is defined as in (1 and define A M as the following set:
MM XERM
> M
Am = 1, Am 2 vm
M
m=1
Analogous to the stationary and optimal solutions for a single objective, we define Pareto stationary and Pareto-optimal solutions for the multi-objective optimization problem minwer L(w).
Definition 3.2 (Pareto stationary and Pareto optimality (Momma et al. 2022 If there exists convex combination of the gradient vectors that equals to zero, i.e., there exists À E M such that (( O, then E Rd is Pareto stationary for L. If there is such that, for all Lm (w) defined in (1) with m 1,..., M, Lm (m)) < Lm(w") (w"), and for at least m' 1,..,, M, Lm (w) Lm (w J, then w" is Pareto-optimal for L.
Rdd and w + w"
By definition, Pareto stationary point, there is no common descent direction for all objectives. A necessary and sufficient condition for w being Pareto stationary for smooth objectives is that minseam [[W]()))|] 0 (Tanabe al. 2019), which corresponds to the stationary condition
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Algorithm JoGBa: Joint Gradient Balancing for Multi-Objective Optimization.
1: Input: number of epochs T' initialized order nı, initialized weight wa. stale mean vo 0. step size for t= .......2
{t is the number of epochs]
3: for m = 1.... M do fm is index on different objectives}
4: Initialize left index im + 1, right index Im + K
5: end for
6: Initialize running average <- stale w++1 0.
A I..... do [A is number iterations each epoch, of (k)...... mi (k) indicates the
sample index we select for each objective)
8:
Sample data (R) &e,M (k) from data set D
9:
for m 1,..., M do {Compute the gradient on the m-th objective and updates its sample order #/41 for next epoch + 1}
10:
Compute gradient Vl (w, (w, (k)" and centered gradient gm.,k. Vem(w, Enr (1)) ve
Compute sign for the current gradient: 6m.k,t + Ba Balancing gm.k.t) if km.k.e = +1 then
12:
13:
Update and left index bm: f- 9m mi (lm) T!" (k);; ) Lm +- mm + 1.
14:
else
Update and right index Ti: f- km,ket m (rmm) <- oi (k)); rin +- Tim end
16:
17:
end for
18:
Compute weights X from multi-task learning algorithms X MTL((LLLL(Mee)) (e)) |
19:
Update stale mean 01.411 <- 1141 + Σ (mng (()) Optimizer Step: w (k+1) +- wy
20:
(K) E Am Ve Vem(mu)) (w)) (m))))
21:
end for
22:
Use the model parameter from last itcration as the initialization for next epoch t 1:0041 w. (K+1)
23:
end for
Algorithm 2 Online greedy implementation of Ba Balancing(s, Im,k.))
1: Input: 9.
Cm,A,1 || Im,, ||s 9m.,k, else km,k,1 1.
3: Return km.k.t-
I (w)l = 0 for a specific objective Lm. Then, similar to the gradient norm (w) for
single-objective optimization, the quantity min sA M [V[(w) ) can be used measure of Pareto
stationarity (Désidéri 2012 Fliege et al. 2019 Liu & Vicente 2021 Fernando et al. 2023
Now we list several assumptions that are necessary for our theoretical results. These assumptions are all commonly used in previous theoretical analysis (Liu & Vicente [02T] Fernando et al. 2023
Zhou al. 2022: Chen et al., 2024) on the convergence of multi-objective optimization methods:
Assumption 3.3 (Lipschitzness of (m))sss and L(w)). For all m E (1....M, M), em (w. () is f-
Lipschitz continuous for all training samples ç. L(w) is then F-Lipschitz continuous in the Frobe- nius norm with F /M
Assumption 3.4 (Lipschitz smoothness of em (w)'s and L(w)). The gradient Ve(w, {) is fi- Lipschitz continuous for all m E M) for all g. VLm is then Lipsshhiit continuous in the Frobenius norm with Fi = f1.
Assumption 3.5 (Bounded gradient variance for each objective). For any w and sample ç, the mn-th loss function satisfies V E - V (w)ll < o for some given dm-
We then have the following convergence result if we use the MGDA algorithm (Désidéri, 2012 Sener & Koltun| 2018) to compute the objective weights X. Proof is in Appendix(A.1
Theorem 3.6 (Random Ordering). Suppose Assumptions 3.3 3.4 and 3.5) hold. Define = maxXeα" L(wr))> min M L(w)> the maximum difference belween objective val- ues at initialization and that at Pureto optimality. Consider the model parameters {w, gen- erated by the MGDA algorithm with random sample ordering. Set
Fi(F"to*)KT where
'Supersrript indicates the model parameters the beginning of each epoch.
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8 max om with on defined in Assumption 3.5 then,
Σ
min A
[VL(w, )211
2F FF F2 + +z²)
o +11+ lTg(T))
T
KT
To analyze the convergence rate of Algorithm I that uses online gradient balancing to determine sample ordering for different objectives, we first need an additional assumption on the Ba 1 aanct cing ng subroutine, which also used for gradient balaneing with single objective in (I.u al. 2022)
Assumption 3.7. (Balancing Bound) For the subroutine Ba Balanci ng in Algorithm I denote its
input vectors as .1,...; 2n E IRd which satisfy ||zi||2 < 1, = I....n Suppose the sub- routine assigns each vector 24 a sign 6 E f-1,+1}. There exists a constant A such that
|| o - A for all E (1..
From Definition 3.1 solving the online vector balancing problem corresponds to minimizing A in Assumption 3.7 We also have the following Proposition that controls the maximum norm of parameter updates in each epoch. Proof is Appendix A.2
Proposition 3.8. Under Assumption 3.3 and 3.7 Algorithm
satisfies: |!l w0 1bo S AF for
all k: E ( (1...., K} and t E {0,
Based on this Proposition, we can then prove the following convergence result. Theorem 3.9. Set
min
32K 32 26(K + A)(F Fi)
where o² Algorithm
maxm on with ot defined in Assumption 3.5
Under Assumptions
3.3
3.4
and
3.5
wields
T-1 E
(2 + t02)
d²
65 SAAF KT
[V YEAM
min
<I1
+ T
K2T2
T
Proof is Appendix A.3 Compared to random ordering in Theorem 3.6 the convergence rate of Algorithm 1 has a different term O (KKT) on the right hand side, which better than the O(KT))1//3 1/2) term in Theorem 3.6 As such, Algorithm | can achieve faster convergence than random ordering as is implemented in existing multi-objective optimization methods. We also note that a smaller A leads to faster convergence, which demonstrates that solving the online vector balancing problem (minimizing A) is indeed useful to find better orders on the training samples.
The naive extension of GraB (Lu al.) 2022) that performs online vector balaneing gradients of each objective separately can also be analyzed under the same framework as follows:
suppose that the sample order ग.h in Algorithm 1 is separately generated for each objective. We have ||w, MAF for all k E (1. K} and {O...., 1}.
Proposition 3.10 (Separate Ordering). Under Assumption 3.3 and
(k) w ( S
Proof is in Appendix A.4. Compared to the results in Proposition 3.8 the bound here is M times larger if we apply gradient balancing separately each objective. Recall that M is the total number of objectives. Thus, the convergence can be much slower than that in Theorem 3.9
4 EXPERIMENTS
In this section, we demonstrate the effectiveness of the proposed method for multi-objective opti- mization. We consider the following baselines: (i) Random reshuffling (Random), which is used in most existing implementations to randomly shuffle the whole data set in each epoch t, (ii) FlipFlop, which creates new order Ā,+1 by reversing the previous order, i.c., T (k) = n. (K' +1 k). (ii) Random FlipFlop, the combination of random reshuffling and FlipFlop, and (iv) GraB (Lu et al.

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(d) Nash-MTL.
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Figure 2: Training losses (objective values) of different tasks on NYUv2 data with different data ordering methods. Top: Loss on semantic segmentation task (semantic loss). Middle: Loss on depth
estimation task (depth loss). Bottom: Loss on surface normal prediction task (normal loss).
2022]. which performs gradient balancing on the weighted gradient of all objectives, and the weight computed using the combined dynamic weighting algorithm.
The proposed method is independent of the dynamic weighting algorithms. In the following, we
combine it with variety of dynamic weighting algorithms, including: (i) MGDA (Désidéri 2012
Liu & Vicente| 2021 Zhou et al. 2022; |eernando et al.] (023), (ii) PCGrad (Yu et al.] 2020). (ii)
CAGrad (Liu et al., 2021), and (iv) Nash-MTL (Navon et al. 2022). We select these methods as
they generally have good empirical performance. The proposed method can also be easily combined
with other dynamic weighting algorithms. Moreover, we include the single-task learning (Liu al.
2021 Navon et al. 2022) baseline (STL), which trains on one task only and serves performance upper bound.
We consider two data sets that are commonly used for multi-objective optimization in machine learn- ing: (i) NYUv2 (Silberman et al.] 2012). an indoor scene data set that involves three different tasks: semantic segmentation, depth estimation, and surface normal prediction. (ii) QM9 (Ramakrishnan jet al. 2014), which a widely used benchmark for graph neural networks predicting 11 properties of molecules. More details on the setup can be found in Appendix B
4.1 NYUv2
Figure 2 compares the convergence curves of different ordering methods with the proposed method. We can see that the impact of sample order on the convergence can be different for different objec- tives. Both depth estimation and surface normal prediction tasks are more influenced by different sample ordering methods, while such influence becomes less significant for the semantic segmenta- tion task. FlipFlop and GraB generally achieve worse performance than the other methods, while the proposed JoGBa is the only method that consistently outperforms existing baselines with random ordering.
Table I compares the testing performance of different data ordering combined with different multi- objective optimization methods. FlipFlop generally performs worse than the other methods as it only reverses the sample ordering after each epoch. Random FlipFlop slightly improves upon the standard random baseline. While GraB does not yield faster convergence rate in Figure 2 its testing
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Table 1: Test performance (averaged over random seeds) for the three tasks on NYUv2 data: semantic segmentation, depth estimation, and surface normal prediction.
Segmentation
Depth
Surface Normal
Angle Distance
Within 1" f
A %
mloU
Pix Acc
Abs For
Rel Er
Mcan
Modian
11.25
22.5
STL
38.30
63.76
0.6754
0.2780
25.01
19.21
30.14
57.20
69.15
MGDA (Random)
30.48
50.77
0.0020
0.2535
19:22
29.51
57.11
1.31
MGDA+ MiplooFippllpp
29.47
$7.90
0.6270
0.2755
24.98
19.45
29.18
55.88
68.36
1.58
MGDDARandom FlipFlop
30.52
0.0018
0.2536
19.16
1.28
MGDA+GraB
30.74
59.92
0.6011
0.2524
24.12
19.11
29.54
37.35
69.76
1.25
MGDA+JoGBa
31.02
00:21
0.0008
0.2508
24.08
19.08
29.55
57.47
70.03
1.19
PCGrad (Random)
38.06
64.64
0.5550
0. 0.23225
27.41
22.80
23, 86
49.53
63.14
3.97
PCGrad-FFipPlog
$7.71
61.63
0.5590
0.2285
26.52 81
22.
23.96
19. 19.30
62.91
3.89
PCGrud-Runšom Flippllpp
38.12
0.5570
2329
3.86
POGrad+GraB
38 38.31
61.66
0.5352
0.2311
26.79
22. 87
23.66
19.76
63,22
3.78
PCGrad+JoGBa
38.59
64.67
0.5545
53
3.56
CAGrud (Random)
39.79
65.49
0.5486
0.2250
26.31
21.58
25.61
52.36
65.58
0.20
C
39.42
63.55
0.5437
0.22119
25.79
21.75
52.17
65.34
0.27
CAGrad CAGral-andomm FlipFlop
39.85
63.73
0.5467
0.22226
26. 14
21.46
25. 62
52.24
65.62
0.17
CAGmddGmmB
30.91
66.09
0.54228
0.22144
25.79
21.44
25.64
82.26
65.44
0.18
CoGrad-JoGBa
0.5410
21.50
Nash-MML (Random)
40.13
65,93
0.5261
0.2171
25. 26
20.08
28. 40
36.47
68.
-1.04
Boab-MTL=FFippoopp
0.5813
20.99
Nash-MTL RRndoo HlipHlp
40.67
66.32
0,5184
0,2009
25. 34
19.73
28.54
55.35
68.07
-4.16
40.84
66.51
0.5156
0.2087
26
19.45
62
55.37
68.11
-4.19
Nast-MTL+JoOBa
41.13
66.71
0.5112
0,2009
25.11
19.19
28.77
55.28
68.18
-4.27
performance is comparable Random FlipFlop. The proposed method JoGBa achieves the best overall performance across different performance metrics for all three tasks.
4.2
QM9
Due to the large number of objectives in the QM9 data, here only plot the average of all training objectives. The convergence curves are shown in Figure B for different sample ordering methods. Compared to the NYUv2 data set, the effect of sample ordering becomes less significant for the QM9 data. Only GraB and JoGBa achieve slight improvements than the other ordering methods.
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k15
$10
5.5
84
kpare
35c
130
100
110
200
(a) MGDA.
(b) CAGrad.
(c) PCGrad.
(d) Nash-MTL.
Figure Training losses on QM9 data with different data ordering methods.
Table 2 compares the testing performance of different data ordering methods. Similar to the results for NYUv2, FlipFlop generally performs worse as it only reverses the sample ordering after each epoch. Random FlipFlop achieves comparable performance with the random ordering baseline, and GraB slightly improves upon it. The proposed method JoGBa achieves the best overall performance.
4.3 COMPARISON ON TIME COSTS
The proposed JoGBa has two key steps in each iteration: (i) sample ordering, in which we determine the order of a sample based on its gradients, and (ii) model updating, in which we compute the objective weights and update the model with the weighted gradients. Table 3 compares the time costs of these two steps in each iteration for different multi-objective optimization algorithms on NYUv2 and QM9 data. As can be seen, the time cost of sample ordering is almost negligible compared to that of model update, and is generally the same for the same data set across different oblti-obeective optimization algorithms. This is intuitive sample ordering is not related any specific multi-objective optimization algorithm, and demonstrates that the proposed method does not introduce much additional time cost.
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Published as a conference paper at ICLR 2025
Table 2: Test performance (averaged over random seeds) all property prediction tasks in QM9.
FHOMO FLUMO (R²)
ZPVE
Up
U
Co
MAE
Am %
STL
0.067 0.181
60.57
53.91
0.502
4.53
58.8
64.2
63.8
66.2
0.072
MGDA (Random)
0.217 0.368
126.8
104.6
3.22
5.69
88.37
89.40
89.32
88.01
0.120
120.5
MGpA+PlipFlop
0,221 0.371
130.9
104.5
3.32
5.62
88.31
89.45
89.71
88.84
0.124
121.4
RGDAoRandom FlipPlop
0.216 0.365
126.7
103.2
3.19
89.27
88.74
87.34
0.115
118.9
MGDA+GraB
0.206 0.343
120.8
101.4
3.16
5.44
87.68
88.63
88.87
87.26
0.119
118.4
MGDA+JoGBa
0.202 0.332
117.3
99.2
3.12
5.37
87.48
88.37
88.80
87.04
0.116
116.7
PCGrad (Random)
0,106 0.293
75.85
88.33
3.94
9.15
116.36
116.8
117.2
114.5
0.110
125.7
PCGrad+PlipFlop
0.106 0.306
75.15
88.29
3.87
9.17
120.17
117.4
117.8
114.1
0.113
126.3
PCGrad-Random FlipFlop
0.104 0.293
75.05
88.25
3.83
14.88
116.4
116.9
114,1 0.106
125.2
PCGrad+GraB
0.098 0.281
74.91
86.98
3.75
115.66
114.4
117.1
113.6
0.102
124.2
PCGrad+JoGBa
0.098 0.271
74.43
84.30
3.56
8.78
113.15
113.2
117.1
0.096 113.5
123.5
CAGrad (Random)
0.118 0.321
83.51
94.81
3.21
6.93
113.99
114.3
114.5
112.3
0.116
112.8
CAirad=FlipFlop
0.115 0.325
85.13
94.94
3.24
114.32
115.2
114.9
113.1 0.117
113.1
CAGrad-Random FlipFlop
0.113 0.322
83.19
94.87
3.15
6.92
114.18
113.8
113.8
111.6 0.113
112.8
CAGrad+CiraB
0.111 0.312
82.49
94.71
2.96
6.77
113.89
113.7
110.4
111.8 0.108
112.1
CAGrad-JoGBa
0.110 0.304
82.38
94.49
2.92
6.49
113.22
113.5
110.2
111. 0.104
111.9
Nash-MTL (Random)
0.102
82.95
81.89
2.42
74.50
75.02
75.10
74.16 0.093
62.0
Nash-MTLL+FlipFlog
0.106 0.255
82.79
82.01
2.45
5.42
74.52
75.07
75.13 74.27 0.096
62.2
Mash-MTI=+Random FlipFlop
0.097 0.254
82.53
81.47
2.42
5.29
74.41
75.08
75.07
74.22 0.094
61.6
Nash-MTL+Gnah
0.099 0.252 82,64
$1.68
2.38
5.31
74.43
74.94
75.05
74.13 0.091
61.7
Nash-MTL+JoCBa
0.094 0.231 82.24
80.73
2.29
5.24
74.37
74.84
75.03
74.05 0.087
59.2
Table 3: Per-iteration CPU time cost (in seconds) of the two key steps JoGBa combined with different multi-objective optimization algorithms.
NYUv2
QM9
MGDA
PCGrad
CAGrad
Nash-MTL
MGDA
PCGrad
CAGrad
Nash-MTL.
Sample ordering 0.02
0.03
0.03
0.03
0.06
0.05
0.04
0.05
Model update
1.04
0.91
0.99
1.06
2.97
1.37
1.17
1.62
Table Test performance (averaged over random seeds) for three tasks on NYUv2 with different sample ordering methods for the proposed multi-ordering framework.
Segmentation
Depth
Surface Normal
Angle Distance
Within (
Am %
mloU
Pix
Abs Eir
Rel Em
Mean
Median
11.25
MGDA+Random
30.18
59.77
0.6020
0.2555
21.13
19.22
29.51
57.11
69.58
1.31
MGDA+FlipFlop
29.17
57.90
0.6270
0.2755
21.88
19:45
29.18
35.58
08.36
1.58
MGDA+Random Flipflop
30.52
59.81
0.6018
0.2556
24.11
19.16
29.52
57.23
69.56
1.28
MGDA-GraB
19.11
MGDA+JoGBB
31.02
60.21
0.6008
0.2508
24.08
19:08
29.55
37.47
70.03
1.19
PCGrud-Random
38.06
80.
3.97
PCGrnd Flipllo
37.74
64.63
0.5590
0.2285
26.94 94.94
22.
23.06
49.30
62.94
3.89
PCGrad-Ranšom Flipplop
38.1
64.64
0.5570
0.2329
26.99 99
22.67
23.56
49.65
63.18
3.86
POGrad&GraR
38.31
0.5552
0.2311
3.78
PCGrud-JoGBa
38.50
64.67
0.5545
0.2270
26.53
22.40
23.87
49.95
63.87
3.56
CAGrad-Random
39.79
65.49
0.5486
0.2250
26.31
21.58
25.61
32.36
65.58
0.20
CAGrad+FlipFop
39.12
63.55
0.5437
0.22119
25.79 79
21.75
25.97
52.17
65.31
(1.27
CAGrad-Random Flipillpp
CAGmd+GraB
30.91
66.09
0.5428
0.22144
25. 70
21.44
25.66
52.26
65.44
0.18
CAGrad-JoGBa
40.42
66.08
0,5410
0.2205
25.52
21.50
26.04
52.43
65.73
0.03
Nush -an--
40.13
65.93
0.5261
0.2171
25. 26
20.08
28.40
55.47
68.15
4.04
Nash-MLL Fabl-MML=FFipFopp
39.16
63.82
0.5313
0.2190
26.12
20.99
28.05
54.64
67.77
-3888
MTL.MMMLLRaaaao FlipFlop
19.73
Nush-MLL-GrraB
40.84
66.51
0.5156
0.2087
25.26
19.45
28.62
$5.37
68.11
-4.19
Nash-MILL+ooiB
0.5112
19.19
-4.27
4.4 ABLATION STUDY
To solve the underlying online vector balancing balancing problem. besides using Algorithm ! other data ordering methods (such those mentioned in Section 3.1) may also be used to obtain

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Published as a conference paper at ICLR 2025
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(i) CAGrad.
(k) PCGrad.
(I) Nash-MTL
Figure 4: Training loss (objective values) of different tasks on NYUv2 data with different data order- ing methods for the proposed multi-ordering framework. Top: Loss on the semantic segmentation task (semantic loss). Middle: Loss on the depth estimation task (depth loss). Bottom: Loss on the surface normal prediction task (normal loss).
sample orders for different objectives. In this experiment, we consider the following sample ordering
methods for comparison: (i) Random reshuffling (Random), (ii) FlipFlop, which creates the new order 1 by reversing the previous order for each objective, i.e., T (k) =="(K + 1+ k). (iii) Random FlipFlop (Random FF), the combination of random reshuffling and FlipFlopp and (iv) GraB (Lu et al. (022). which applies GraB to all objectives separately. Experiment is performed on the same NYUv2 data set and training setup as in Section 4.1
Figure 4 compares the convergence curves of different sample ordering methods. Similar to Figure the influence of sample orders on the convergence rate is generally different for different objectives. The surface normal prediction task is more influenced by different sample ordering methods than the other two tasks. FlipFlop and GraB generally achieves worse performance than the other methods, while JoGBa is the only method that consistently outperforms existing baselines.
Table 4 compares the testing performance of different data ordering combined with different multi- objective optimization methods. FlipFlop generally performs worse than other methods it only reverses the sample ordering after each epoch. Both Random FlipFlop and GraB improve upon the standard random baseline. but their performance is still worse than the proposed method JoGBa, which demonstrate the effectiveness of joint sample ordering in multi-objective optimization.
5 CONCLUSION
In this paper, we propose novel sample ordering framework for multi-objective optimization. The proposed framework determines sample orders for each objective by performing online vector bal- ancing with the gradients on different objectives. It can be seamlessly combined with existing multi- objective optimization methods. Theoretical results demonstrate that the proposed method improves upon the baseline of random ordering with faster convergence. Empirical results different multi- objective optimization problems demonstrate that the proposed method achieves faster convergence and better performance than the other data ordering methods.
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Published as a conference paper at ICLR 2025
ACKNOWLEDGMENTS
This research is supported in part by the Research Grants Council of the Hong Kong Special Ad- ministrative Region (Grants 16200021, 16202523 and C7004-22G-I).
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Published as a conference paper at ICLR 2025
A PROOFS
A.1 PROOF OF THEOREM 3.6
Theorem 3.6
By the Fr-smoothness of [(w)) for all À AM have
((w++1)4 - [(w;)) = (VV wi+1 wi) 2
W1+1 w9 1
(3)
For notation simplicity, we define Qi V.(w),, and XQ. = arg M [L w, A/| Then we have:
where Wi+11 wy 04 (ww)) s.t. X E arg min V VL (w,, All
[(w,+1) ((w;) v az (V Q, 2. + _ az 1012,,
(4)
The inner product term can be bounded as - -(VL(w;)), Qe1s,) = (VL(w,)), VL(W:)) Q:12,) (L((,)), VC((x))(((w)))
VL(m)) VL( - Q2, - [V((mn
<F (17 (w) - Q11 | ((w;))
(6)
2 V IVL (w,)
where (a) follows from (3 in Lemma A.3 (b) follows from Lemma
A.4
Plugging
5
into
taking expectations on both sides and rearranging yields
(u) s Ea(C(w:) ((w + 2F BBllllQ [[(w))]
+ (F² + {F
Fi
2
For all t E plugging in 0g = a, and taking the telescope sum on both sides of the last inequality yield
T
E ExllQ - vl((u))]] Fi (r² + o
Ba(C(ur) [(w++ 20
2
EAIIQ: - VL(W=))11) + -i ( o
at EA(C(w)) [(w++ 1)) 20
By increasing the batch size during optimization with a batch size of Ott), it holds that
0² + log(T))
EallQQ VL(w.)
T
Plugging 7 back into (6. its optimization error given by:
T EAll (x]lV(((())
EA
min [E[T].EAA
vL((w))111
1=1
2 (T)
Ex(C(w:) - L(Wr+1)l
Fi [12 1 +02)a
+
+
T
aKT
2
0 2 log (T)
A
F (r2 2 +o²)a,
(8)
T
KTT
2
where the last inequality the definition of A maXAEAM L(wa)) )-
WEE L(w)
Then setting
Fo have:
2F + +
a² + log(T))
E min ME(T)
[VL(W;
<
KT
T
which concludes our proof.
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A.2 PROOF OF PROPOSITION 3.8
Proof. The proof follows directly by using Assumption 3.3 in Assumption [V]]( F.
3.7 for each
A.3 PROOF OF THEOREM 3.9
Proof. From Lemma A. Appendix A.5
we have
Σ
min VL
1012
24
247
max
w;
w; +
(1) a2F1 (1122000
kKT
T
1=)
1=0
On the other hand, from Lemma
A.2
we obtain
T-1
T-1
Σ A < 120a
A 48a2
Σ max [V )1l3.
Combining them together gives:
T
min
VL
)A12
T
AM
t=0
2A
F7
120a K2l + 64a 2 A 48a 2²
Σ
max V ((w ((w())))))) )0
aKT
T
t=0
a² Fi (FF +2²
2A
120a FFT
48a K2 F2
T-1
+ 640 2AFO222F00
max | (((
aKT
T
T
a F (F2 + +
2
Note that for any a E Rd, ||0/|/0 S |2||| so the last term can by bounded by its nomm Moving it the left side of the inequality gives:
480 2 2 F2
T-1
2A
120o2FRK22
min XEAM
aKT +
+ A F2FO²
T
T
2 FFF +
Finally, set the value of as:
A
1
a min
32K 2 FTT'
KF' 26(K + A)F
and we obtain
T-1
A A T02) 02
65 F.)
SAAR
min M
K²T2
T
KT
which concludes the proof.
A.4 PROOF FOR PROPOSITION 3.10
Proof. Similar to Proposition 3.8 the proof follows directly by using Assumption 3.3 in Assump- tion 3.7 for each (V((w) lo I F, and repeated for M objectives.
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Published as a conference paper at ICLR 2025
A.5
TECHNICAL LEMMAS
Lemma A.I. In Algorithm 7 ifaKF < 1 holds and Assumption
3.3 and 3.4
hold, then
Σ min XEAM
IVC(W,))lI S
2A
2FF
T-1
Σ
max
w
(k)
w. (1)
+
a FFF +
T
aKT
T
Proof. Note that the update can be written
K
M
(1) = a
wi
Akm Ve (mm (e)
(K)
k=
m=1
By Taylor's Theorem, for all f = 0,.... ,T 11
(1 $ C(w, ( ) + (VL(I))), w - w (2)) +
Fi
wi+11 - wi w
2
K M
I ((w,)) - akE
V[(we)),
d (e, (8); (arr
K
k= m=1
F2K2F E
M
(K), (mntt)) A (ew
K
k= m
((w()))
ak
| |VL(w, (2) )a1
aK 1
d (l (oo)))
+ 2
aK
VL(Ww
) (1) (2)))
1 K M
d (lw (2) Sox
K
k= m=
a
FF
K
X Ve, (2)
; Eomco)
2
k=1 1=1
K M
((w())
aK
[V[(w |211
aK
/ (1)
Vl (8), :Ea :oo(t)))
K
k=
m=1
(F2 + +0²)
In the second step, we apply - (a, b) = - =1aa|| + _ la b11 [1|-
Va. b. In the third step, we use the condition that anL Expanding the last term using Assumption 3.4 we get
K
M
(2) a^
1
(k) Am,der Ve ( (x))
K
k= m=1
K
K
M
1
VL(W
11) )
Am, Vmm (k), (opr())))))))))))))))
K
m=
1
1) V (1)) 2
VL(mn
K
K
(1) 2 un
F?
m (1)
K
In the second step we apply Jensen's Inequality. Putting it back, we obtain
aK
ak r
02F1(12+03
((w < ((w
V vL(w!
2
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Published as a conference paper at ICLR 2025
Finally, summing from t 0 to T 1, and using the definition A
maX L(wa)d
win NEAM ((w),, we have:
min AEA
= akT
2A
+
2F
T-1
max &
wi
(k)
w
(0)2
+
a
FFF2 +
T
T
2
That completes our proof.
Lemma A.2. In Algorithm ! if the learning rate satisfies
1
1
S min
32n
16H
then the following holds:
D S 20Hs + (8amts + 4aHH2))4--1 + 20m 2 V Ak 2 2.
and
A < 8annn' n
Finally,
K A S 16a + 2 + 2 Σ v [(ww))l
K
1=1
Proof. Without loss generality, for all m E (2. n 13 and all k (,. K}.
(m) w
=
WR
W : (t)
(t)
t=1
(o (ox (t)))
=
#-11
(()
t=1
W (t)
:oo((t))
-Vf
Wk-1
(0--1(02(1) (ELL11))
Now add and subtract
a(m 1)
v
(8) Wk-1
v
w (oo
T1
which gives
W
g
(m)
Σ
4.
Wx-1
(02210=(11)) (oo
1 Σ
vs
(d)
wk (o) (8)
(e)
1=1
a(m 1)
Vʃ
1 100-1:20001100
(1) (1)
1=1
m-1
(o)))) ( ))
(t)
Vf W
wx-1
1=1
We further add and subtract
a(m
K
V VL(Wi (o, a1mmammm1m)) 1)V((w))
K
16

---

Published as a conference paper at ICLR 2025
to arrive at
(m) k
= 17 Σ
10 ((xx(())
:For(e) ) - - E vs :on
k=1
t=1
a(m 1)V((w))) + a(m- 1)
Vf (wki coo- - Vf (w Fo.
vi w :
-V/ -V
We now rearrange, take norms on both sides and apply the triangle inequality,
(m)
20x
a
m-1
Σ
Vf
w
k-1
(o
(ox()))))))
xEL(E) )
-
1
vs Σ
n2
w ioo
71
1=1
+a(mm
1) (ww=))l
a(m- 1)
Σ Vf (fxiiian 1(1) - Vf (w (w Ton 1(t)
7
m-1
(0: (oo())) Vf ( (o (
+o
Σ
Vf
W
(t)
(9)
There are four terms on the right hand side. We apply Assumption 3.7 on the first term, and As- sumption 3.4 on the last two terms. First, for the first term,
(ox(t)))
1 Σ
w k-
Foo(1)
vs
W wk-
f
Wk--
n0 (ox()))
For(t))
1
Σ
Vf
W k-1
(o1-)(01(1)),
x zor-
8=1
ʃ
k-1
(o (ox
x Fo)-
Σ
vs
Assumption 3.4 and 3.5
T.2
Σ n
(a (on(t))
(x)
stoit t +
W 1-
20
s=1
L2
(o.
(x)
1A IA
max mm m
W
w+1
wk-1
w
w
7
00
max Fm -222AA-11
Denote
Ni-, (tt) (t)
1
U1 = Ve
w k-1
Foc 1(9))
n
We can use Assumption 3.7 to obtain a bound on the prefix sum
U
SA
S + 2L A-1
that is,
Σ ff
wx-1
(o, (0,_,0x (ox(t)),
(te(t))
1 vf
(c)
+, -2L2Ax-1).
<
n1 s=1
17

---

Published as a conference paper at ICLR 2025
Now have bound for the first term in Equation proceed to bound the last two terms into
where we apply Assumption 3.4 We can then rewrite Equation
(m) w
wx
aA(s t 2L2 + a(m 1)IlV[(w>)ll
alloo (m - 1) n W/ w
Σ
12
+ abox
m
w
k-1
(o (ox2(()x())))
n
Furthermore, applying the triangle inequality the norms in the last two terms, we obtain
w
(0)
wk
+ wk-1 wx= (n-1)
I 2A
and similarly,
(2)
(o (oxx())))
w (o 1 (ox
w
un
w wk + Ww wk-1 + wk-1
4 2-
This gives
w
(m)
w/
$ aA(s + 2L24 + a(m 1)]lV((w))]>> +2abo (m 1)
+a4 (m 1)(24 : 1 + AA)
aA(c -o + 2L A 11 + amm L)
(m 1m k-1 Ax).
(10)
Note that Equation 110 only holds with A: (....., K] and m (2..... 1}. We now discuss the boundary cases. Note that the bound of Equation (T)) trivially holds with m for any k since the left hand side becomes zero. On the other hand, when k- = 1, we have
1
w (m)
w (1)
; x00
W1 - a
t=1
1
= wi
a
W1); 16 (o))
+ a Σ
Vʃ
; mo.(0)
[=]
t=1
Vf
(i (w1:T((())
Vf (wi: -
Σ 1=1
Σ
Vf (1i: (o))))
1=1
Take norms and apply the triangle inequality, we obtain
w (m)
w1
<
(w Tor (s))
n
m 1
ta
Vf w
(t) :oa
Vf [wi I To(()))
+a
V (1i; (o)(0)))))))))))
Σ vs (wi (w1:2o(1o))
2=1
a(m V[ + + - 1) A + a(m 1)(s +0;)
+ An + an(s + a.).
(11)
Now that we have the bounds for A,. we next will sum them up. Taking a max over on both sides Equation (10), this implies for all the & > 2,
AS < alf(s a, + 1-) + co
A&) + am VLww)\l
as m < n. Considering the fact that aLoon < 1/2, we get
A < 2aHH + a, (8anLzo 4a A4-1 2am VL(ww)lles
18

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Published as a conference paper at ICLR 2025
This completes the proof of the first inequality in the lemma. Applying this recursively from any 2 to 2, this gives
A <88amLoc + 44HHL))
(8ante + 4aHL2) (saH(s + +,,) + + (cmm))(())))))))
Applying the learning rate conditions that 32am S and L6 S 1, we obtain
k-1
A + dall(s o;) 4an V[(ws)]l>
2
Square on both sides,
8-1
A 33
A + 48a 2 o,) + 48a V
We can apply similar trick Equation
11 and get
A S 80 n
V (w Il + 8a2 (s
This completes the proof of the second inequality in the lemma. Summing from l: = to K, we get
A
K
R-1
K
A
3
+ 48a +48a2 + o,) - + 28
Σ
V
vlww))ll
k=2
k=
K
<
A +334
48a 22 + 2:)2((K K - 1) + 48
V ((
k=1
k=
s 1602n ( 1130 + 16a 0²)
K
+48a2 + 0,) , 1) + 2 >
k=
< 16a2n22s + 0;,2 48a² 2 + o;)² + 48a 2n2
k=1
That completes the third inequality, and we have finished proving all three inequalities. Lemma A.3 (CChen et al. 2024 ). Given Q E RdxM, rreaall X2. with 2 0 is defined as
Xa.0 E arg min [|QAI] + p|lAl1? AEAM
(12)
Then, for any E A M holds that
(Q Q1) 2 11Q2 - P.
(13)
and [|QA - 12 1|QA112 11Q26, 2p.
(14)
Proof. By the first-order optimality condition for equation
12). for any AM have
(QT QQ2 A X²,o) 2 -p.
By rearranging the above inequality, we obtain
(Q>2 QA) 2 1Q p. which is precisely the first inequality in the claim. Furthermore, we also have
I|QA - 0!2,p11? = || + || - 2(Q22 Q2)
< ||QA||2 + ||@ - 211@> 2p ||QAi1 -1Qx 2p,
which is the desired second inequality in the claim. Hence, the proof is complete.
19

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Published as a conference paper at ICLR 2025
Lemma A.4 (Hölder continuity of do w.r.t Q Chen al. 2024)). For all Q,Q' RAXM define X E arg m ||Qi|? and X E arg min QQ') 2 and de = QX", d'q = Q'X", then
1Q22N11
|lde de < 4mxx
sup XEAM
||Qi|, sup
sup IIQ QQ)AII (15)
AEAM
AEAM
Proof. We first rewrite l|q do 1IQ 2'1"1120 as
IQQ - Q'X"'|2 =1QQ>1| + 2 - "QQA"Q''""
=1|QA*|| - -"'""]==2('''"""" 2(Q''",''""""""""""""""""""""""" Q''" -
=1|Q^-|2 1 +2 Q'X"
- -QXx) Q'X-) +2(0'x" '''' QX"),
where (Q'A", Q'X- - Q'x2) - 0 by 13 in Lemma A.3
Then it can be further bounded by
(2)
Q'X-11
min NEAM
||1112
- Q)1-II min IQ + YEAM
max
-1QQA112 + max
- -QQ' "10""""1I'I""""0''0000'''"'"'''" Q)
AEA
XEAM
(1QA12--199) +
maX
AEAM
- 1'0--2AAI(IIAA))) + +QQ' + 211 - QIXT
max AEAM M
<4 max
sup
|QQIlI sup
1Q'
sup AEAM
IIQ - QQ)All
AEAM
N
where (a) follows from the Cauchy-Schwartz inequality; (b) follows from subadditivity of maxi- mum operator; (c) follows from the triangle inequality. The proof is complete.
A.6 PROOF ON THE CONVERGENCE RATE OF ALGORITHM I WITH RANDOM ORDERING
The following Theorem studies the convergence rate of Algorithm with random ordering.
Theorem A.5. Set a min
21A KIT SK
with random yields:
R of
V2KL' AL-KET173
24LA
K
4 T K
B FLVL(w)))
<
of
KT
ΣK
kk=1 Ok
To prove Theorem A.5 first need the following Lemma:
Lemma A.6. Suppose that Assumption 3.4 holds. Then for iterates w, generated by Algorithm 1 with stepsize S Lm we have
aK +
aL?
aLL
K
f(w++1) S L(w))
[L(w:))11
V;
+
o
(16)
2
K
2
k=1
where V2 = Ek w. - w
(k))
1..0
20

---

Published as a conference paper at ICLR 2025
Proof. Recall that w++1 wt agt, where 9t Ei=o V,n((w)) Using smsoothsssss of f. we get
K
BL(w++1) < E.((w) - akE
V[(wx)
1
Ve Ve(ukk ;Ex()) :ox(1))
K
==1
a²K=L
E
1
Ve(wg
(t)
(ox(
2
n1
2m1
K
= El(ws)
ak
VL((ws)]
am
V
(w0 (1)
2
2
K
k=1
aK
[[(w=)) - K
Σ
K
(
(1))
n22 2 I
K Σ
Vi(w
(1) Son(t))
2
2
K
k=1
[L(w))
ak
+ am
VL(w))
Σ
K
V (20k)) (1))
+
a² I
K
o
2
2
K
2
k=1
Then note that
K
1 K
1
VC(we)
1
(1)
1
V (20
(1))
K
K
K
k=1
k=1
K
(une - (v0 (x)) 1|2
K
k=1
K
13
wi
w (2)
K
k=1
L
V1
which completes the proof.
Lemma A.7. Suppose that Assumption 3.4 holds and that Algorithm D is used with stepsize S
Then
2LK
E[V
S <2² +a²k?s?
(17)
where V1 =K
ww, w (k) 12
Proof.
Let fix any l: E [1, K.
1] and find an upper bound for Eell well2. First, note that
w w. a A-1
Therefore, by Young's inequality, Jensen's inequality and gradient Lipschitzness,
Egllw, -w,ll = o²E,
Σ Ve(wi {t)
2
2
<
TT,
Ve(wi () - Ve(w,,
+ ++2 2
Ve(w, 5t)
1=0
k-1
< 20 Σ
Eel ()( ve(1),,)))1'- + 20
Ve(we, Et
1=
k-1 Σ
k
< 2a k2 2z
Eg l
w1l12 + 2a²F.
Σ
Vi(wi, 5
21

---

Published as a conference paper at ICLR 2025
Let us bound the second term. For any i we have E, (Veee [i) V((ww) So using (with vectors V (x)), Vz (re)...., fm (2t)), we obtain
E2
Vilwi Et)
A [( + E.
Ve(wi: Ei- vl(ww))
1=0
1=0
S (] + k(K k)
K-1 (s + maxo
k
Combining the produced bounds yields
En || - - 2a 2 V [22 2a ki(K - k)
E,llu// -well2 < 2a L2k
K K-1 (s + max oxx
1=0
+ 2a² k(K - h:)
S L + 2a²
K - (5 + max 0z
whence
K
B[V]
E, 2 wrl1
1
S 1)E| 7 - +
+ K(K + + + max oR)
3
3
k
Since E appears in both sides of the equation, we rearrange and use that a S which leads to
1 2LK by assumption,
[[V 1 (1- a² 1))
4
$ nn 1)m(2n (m))) 1| + 4 goān(n + 1)0,
4
9
g
< <nnn"||VL(u)) n + max maxox)
Now we are ready to prove theorem A.5
Proof. Taking expectation in Lemma A.6 6 and then using A.7 we have that for any t E (0, 1,..., T-
1},
16
aK
2 L
[.[[(we+1)]
[(w:))
V VL(um))1
aT.
+
+
2
1=1
((w,) aK
a² k2 +maxo=)
L²L
K
2
2
k=1
-(ww) (1 a2 - L2 maxox)² + a²L K
aK
Σ o
2
Let de = [(w,) - L. Adding -L" to both sides gives:
8, ak (1 a²
+ aLL Σ o
E,(8,41]
ak
< (1+ 22
2 (1
3 k22 + max
21 K
2
2
k=1
22

---

Published as a conference paper at ICLR 2025
Taking unconditional expectations in the last inequality and using that by assumption have 1 L²K 2 1. we get the estimate
2'
1 (1 + 0 AL aK B(IV
a²L
+ max +
Σ
o
2
k=1
Then have:
4(1 + A
[[)))( <
(L(wo) - -(")
akT
at. K
+2a + maxox)2
Σ0 oz
2 k=1
Using that + S exp(x) and that the stepsize a satisfies a S (AL2 K 22T) 1/3, we have
(1 a AL?
AL.22TT exp(1) 3.
Using this in the previous bound, we finally obtain
12(((wo)) - L*))
aL
min t=.... {
+ 22 1.2 K(s + max max
aKT
& 2
k
B DETAILS ON EXPERIMENTAL SETUP
All experiments are conducted on a server with an Intel Xeon Gold 6342 CPU and an NVIDIA RTX A6000 GPU. We use the PyTorch version 1.10.1 with CUDA version 11.7. For experiments on the NYUv2 data set, we train Multi- Task Attention Network (MTAN) (Liu et al.) 2019) following previous works on multi-task learning (Yu et al.] 2020 Navon
al. 2022). We also follow the
training procedure from (Liu et al. 2019: Yu et al. [2020] Navon et al. 2022). Each method is
trained for 200 epochs with the Adam optimizer (Kingma & Ba 2015). We set the learning rate
= x 10 at the beginning of training, and reduce to
after 100 epochs. The batch
size is set for all methods.
For experiments on QM9 data set, we use the MPNN model proposed in (Gilmer et al. (017). Each method is trained for 300 epochs with the Adam optimizer (Kingma & Ba 2015) and we set the learning rate a = 1 x 10-4 throughout the whole training process. The batch size is set to 120 for all methods.
23


Reviews:
[{"review_id": "zJxAFeYc9U", "reviewer": "Reviewer_M4jx", "summary": "The paper proposes a novel framework JoGBa for sample ordering in finite-sum multi-objective optimization problem, and demonstrate its empirical effectiveness on various MOO methods.", "questions": "See the question in previous section.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "Data ordering for better optimization performance (that can especially outperform random shuffling) is important, and I think bring this concept to multi-objective optimization is an interesting and important contribution.\n\nThe paper theoretically analyze the convergence of MGDA for finite-sum MOO problem, with both random shuffling and JoGBa ordering.\n\nEmpirical results look promising.", "weaknesses": "For audience like me that is kind of familiar with MOO but not sample ordering, I think the presentation of algorithm 1 and figure 1 can be improved. Specifically, \n(1) I cannot directly see from Figure 1, how the JoGBa approach (Fig 1 Right) is different from data ordering on each objective separately (Fig 1 Middle). In other words, Figure might not be illustrative enough.\n(2) Algorithm 1 contains numerous superscripts and subscripts, making it challenging to interpret (which is acceptable if it prioritizes rigor). However, a clearer description might help—for example, one or two sentences explaining how JoGBa differs from ordering data for each objective separately. From the current description in lines 176 to 178, \"The sample ordering is then determined based on the results of solving the balancing problem (routine Balancing) with the gradient on each objective,\" it’s still unclear (at least for me) how this approach differs from the one illustrated in the middle of Figure 1.\n\nI believe the paper will be a clear accept for me, if the authors can clarify my concern or improve the presentation of the main methodology as mentioned above.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "asIa0BRkVE", "reviewer": "Reviewer_wbXj", "summary": "The paper proposes a novel sample order approach for multi-objective optimization problems. The method is inspired by the online vector balancing problem which tries to make the average of the centered gradient as close as to zero. Compared with different existing sample order approaches, the author shows that the proposed algorithm can achieve faster convergence. The experimental results on the NYUv2 and QM9 data sets also support the theory because the proposed algorithm outperforms different baseline methods.", "questions": "1.\tIn line 189, the stale mean should be $\\nu_{t+1}$. You write it as $m_{t+1}$. Please double check the notation.\n\n2.\tIn Line 226, a necessary and sufficient condition for $\\lambda$ not $x$.\n\n3.\tIn Line 654, “beginning” instead of “begining”.\n\n4.\tIn Definition 3.1, the authors mentioned online vector balancing which is the core of how to balance the gradient. If possible, please try to explain more clearly about the intuition (e.g., online vector balancing problem tries to make the average of the centered gradient as close as to “zero”). This will make the reader have a smooth reading experience.\n\n5.\tIn Theorem 3.6 and Theorem 3.7, it seems the convergence rate of random sample ordering and the proposed algorithm are the same. Though in Theorem 3.6, the convergence rate is “=” while in Theorem 3.7, it is “<=”, one cannot conclude that the proposed algorithm converges faster than random sample ordering. The experimental results also show that the proposed algorithm converges just a little bit faster than other methods. Please try to explain clearly about the advantage of the proposed algorithm (maybe add some remarks/comments to give comparisons of different algorithms both theoretically and experimentally).  \n\n6.\tIf possible, let the reader know that the paper has a clear set up of the experiments in the appendix. This could help practitioners better understand the method’s sensitivity and optimize it for specific tasks.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Novel Approach to Multi-Objective Optimization: Inspired by the online vector balancing problem, The JoGBa method introduces a unique framework for ordering samples in multi-objective optimization by jointly balancing gradients across objectives . This is a new contribution to the multi-objective optimization field.\n\nTheoretical Convergence Rate: The paper provides a thorough theoretical analysis. The convergence proofs are grounded in established assumptions, lending credibility to the claims.\n\nExtensive Empirical Validation: JoGBa is validated on multiple datasets (e.g., NYUv2 and QM9) across diverse multi-objective optimization algorithms, such as MGDA, PCGrad, and Nash-MTL. This empirical diversity strengthens the generalizability of the findings.", "weaknesses": "Improved Convergence and Performance not clear: The authors claims that the method consistently outperforms existing data ordering strategies and dynamic weighting approaches. However, it seems the convergence rate for different algorithms are similar based on Theorem 3.6 and Theorem 3.7.\n\nPresentation and Clarity: Certain sections, especially the theoretical parts and Algorithm 1, could be more reader-friendly. A more structured breakdown of steps and implications of theoretical results would enhance readability, making the technical details accessible to a broader audience.\n\nIntuitions not sufficient: the online vector balancing problem is the core of balancing the gradient, but the authors fail to give enough intuition and explanation about how to implement the method across different objectives.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zfp2dChfgZ", "reviewer": "Reviewer_ntx7", "summary": "The authors in this paper modify the classical method in finite-sum multi-objective optimization such as MGDA algorithm by developing a sample ordering rule. JoGBa jointly balances gradients from multiple objectives during optimization. It is demonstrated for both theoretical analysis and empirical results, the proposed sample ordering outperforms random ordering.", "questions": "See weakness", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The main idea of this paper is clearly presented through the incorporation of the MGDA algorithm with a novel sample ordering method. Figure 1 offers a great visualization and improves the audience's understanding. The numerical results demonstrate the practical superiority of the proposed algorithm.", "weaknesses": "1. line 65 typo \"oerdering\"\n4. for better presentation, the balancing routine could be written as an algorithm instead of a definition\n5. The authors need to better explain the balancing routine\n6. The algorithm is complicated, the authors should provide with more motivation, remarks, and discussion about the design of algorithm. \n7. Please provide a detailed analysis of the time and space complexity of solving the balancing problem, or to compare its complexity to existing methods.\n8. typo at the end of line 19 of Algorithm 1 \n9. typo in Assumption 3.3 \n10. In the statement of Theorem 3.6 and 3.9, the definitions for $w$ and $\\sigma$ are missing. \n11. As a remark of Theorem 3.9, the authors might want to do discuss about the step size (depending on T). As T going large, the step size is diminishing. But why the converge rate is similar compare to Theorem 3.6 where a fixed step size is used? Please provide a detailed explanation of this apparent discrepancy, possibly including a comparison of the convergence behaviors under different step size regimes.\n12. Please provide interpretations of the third and fourth terms in Theorem 3.9 in the context of the algorithm's behavior or performance. You could also suggest they discuss how these terms relate to key aspects of the method. \n13. There is a gap between (2) and (13), so as Theorem 3.9 and its proof. I'm convinced with the proof and the theorem statement. \n14. The authors need to discuss about the computation cost for the balance problem in the numerical section to make a fair comparison.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Spotlight)