Gradient Deconfliction via Orthogonal Projections onto Subspaces For Multi-task Learning
Authors: Shijie Zhu
Hui Zhao
Pengjie Wang
Hongbo Deng
Jian Xu
Bo Zheng
Venue: Submitted to ICLR 2023 (2023)
Keywords: multi-task learning
deep learning

Abstract: Although multi-task learning (MTL) has been a preferred approach and successfully applied in many real-world scenarios, MTL models are not guaranteed to outperform single-task models on all tasks mainly due to the negative effects of conflicting gradients among the tasks. In this paper, we fully examine the influence of conflicting gradients and further emphasize the importance and advantages of achieving non-conflicting gradients which allows simple but effective trade-off strategies among the tasks with stable performance. Base on our findings, we propose the Gradient Deconfliction via Orthogonal Projections onto Subspaces (GradOPS) spanned by other task-specific gradients. Our method not only solves all conflicts among the tasks, but can also effectively search for diverse solutions towards different trade-off preferences among the tasks. Theoretical analysis on convergence is provided, and performance of our algorithm is fully testified on multiple benchmarks in various domains. Results demonstrate that our method can effectively find multiple state-of-the-art solutions with different trade-off strategies among the tasks on multiple datasets.

Under review as a conference paper at ICLR 2023
GRADIENT DECONELICTION VIA ORTHOGONAL PROJECTIONS ONTO SUBSPACES FOR MULTI-TASK LEARNING
Anonymous authors Paper under double-blind review
ABSTRACT
Although multi-task learning (MTL) has been preferred approach and success- fully applied in many real-world scenarios, MTL models are not guaranteed outperform single-task models on all tasks mainly due to the negative effects of conflicting gradients among the tasks. this paper, we fully examine the influence of conflicting gradients and further emphasize the importance and ad- vantages achieving non-confficting gradients which allows simple but effective trade-off strategies among the tasks with stable performance. Based on our find- ings, we propose the Gradient Deconfliction via Orthogonal Projections onto Sub- spaces (GradOPS) spanned by other task-specific gradients. Our method not only solves all conflicts among the tasks, but can also effectively search for diverse solutions towards different trade-off preferences among the tasks. Theoretical analysis on convergence is provided, and performance of our algorithm fully testified on multiple benchmarks in various domains. Results demonstrate that our method can effectively find multiple state-of-the-arr solutions with different trade-off strategies among the tasks on multiple datasets.
INTRODUCTION
Multi-task Learning (MTL) aims at jointly training one model to master different tasks via shared
representations and bottom structures to achieve better and more generalized results. Such positive
knowledge transfer is the prominent advantage of MTL and is key to the successful applications of
MTL various domains, like computer visions (Misra al.] 2016 Kokkinos
[207]] Zamir et al.
2018
Liu et al. 2019)., natural language processing (Dong et al. 2015 McCann et al.) 2018 Wang
al.
2020; Radford et al.) 2019), and recommender systems (Ma et al., 20188 bj Tang et al. 2020
Wen al. 2020). Many works also make further improvements via task-relationship modelling
(Misra et al. 2016 Ma et al. 2018a Xia et al.] 2018 Liu et al.) 2019 Tang et al.) (0200 Xi et al,
2021) to fully exploit the benefit from shared structures. However, while such design allows pos- itive transfer among the tasks, it also introduces the major challenges in MTL, of which the most dominating one the conflicting gradients problem.
Gradients of two tasks, Di and gj. are considered conflicting if their dot product Gj < O. With conflicting gradients, improvements
some tasks may be achieved the expense of undermining other tasks. The most recent and representative works that seek to directly solve conflicts among the tasks are PCGrad (Yu et al., (220)) and Grad (Wang et al., 2020). For each task T,, PCGrad iteratively projects its gradient 9; onto gradient directions of other tasks and abandons the conflicting part; GradVac argues that each task pair (Ti, T,) should have unique gradient similarity, and chooses to modify 9i towards such similarity goals. However, these methods only discussed convergence guarantee and gradient deconffiction under two-task settings (Liu et al., (021a). For MTLs with 3 or more tasks, the properties of these algorithms may not hold because the sequential gradient modifications are not guaranteed to produce non conn--ooffiiinnggg gradients among any two task pairs. Specifically, demonstrated in Figure [(b) and [(c) the aggregated update direction G' of these methods (e.g. PCGrad) still randomly conflicts with different original 9; because of the randomly selected processing orders of task pairs, leading to decreasing performance corresponding tasks.

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2164667,28
01-6422,35
W
092 -59.25)
Trich
0
1
2-2
2/22
(a) GD
(b) PCGrad
(c) PCGrad
(d) GradOPS
Figure 1: Illustrative example of gradient conflicts in three-task learning problem using gradient descent (GD), PCGrad and GradOPS. Task-specific gradients are labeled 91, 92 and Øs- The aggre- gated gradient G or G' in (b). and (c) conflicts with the original gradient 93, 92 and 9s. respec- tively, resulting in decreasing performance of corresponding tasks. Note that different processing orders in PCGrad (11,2,3) for (b), [3,2,1] for (c)) lead to conflicts of G with different original Øt- In contrast, the GradfeS-modified 9i is orthogonal to S = span (92, 93) with the conflicting part on S removed, similarily for 2/2 and g/s (omitted). Thus, neither each g nor G' conflicts with any of {g. ).
Another stream of work (Sener & Koltun 2018 Lin et al., 2019b a Liu al., 2021a) avoids di- rectly dealing with gradient conflicts but casting the MTL problems as multi nulti-objecttiv optimization (MOO) problems. Though practically effective, applications of most MOO methods are greatly lim- ited since these methods have to delicately design complex algorithms for certain trade-off among the tasks based on conflicting gradients. However, solutions toward certain trade-off may not always produce expected performance, especially when the convex loss assumption fails and thus conver- gence Pareto optimal fails. In addition, different scenarios may require divergent preferences over the tasks, whereas most MOO methods can only search for one Pareto optimal toward certain trade-off because their algorithms are designed to be binded with certain trade-off strategy. There- fore, it's difficult to provide flexible and different trade-off's given conflicting gradients. Even for Liu et al. (2021a) [iu et al. (2021b) and Navon et al. (2022) which claim to seek Pareto points with balanced trade-ofis among the tasks, their solutions might not necessarily satisfy the MTL practi- tioners' needs, since it's always better to provide the ability of reaching different Pareto optimals and leave the decision to users (Lin et al. 2019a).
In this paper, we propose a simple yet effective MTL. algorithm: Gradient Deconfliction via Orthog- onal Projections onto Subspaces spanned by other task-specific gradients (GradOPS). Compared with existing projection-based methods (Yu et al.) [2200] [Wang et al.) 2020). our method not only completely solves all conflicts among the tasks with stable performance invariant to the random pro- cessing orders during gradient modifications, but is also guaranteed to converge to Pareto stationary points regardless of the number of tasks be optimized. Moreover, with gradient conflicts among the tasks completely solved, our method can effectively search for diverse solutions toward differ- ent trade-off preferences simply via different non-negative linear combinations of the deconfficted gradients, which is controlled by a single hyperparameter (see Figure
The main contributions of this paper can be summarized as follows:
Focused on the conflicting gradients challenge, we propose orthogonal projection based gradient modification method that not only completely solves all conflicts among the tasks with stable and invariant results regardless of the number of tasks and their processing orders, the aggregated final update direction also non-conflicting with all tasks.
With non-conflicing gradients obtained, simple reweighting strategy designed to of- fer the ability of searching Pareto stationary points toward different trade-offs. We also empirically testified that, with gradient-conflicts completely solved, such a simple strat- egy is already effective and flexible enough to achieve similar trade-offs with some MOO methods and even outperform them.
Theoretical analysis on convergence is provided and comprehensive experiments are pre- sented to demonstrate that our algorithm can effectively find multiple state-of- tfate-of-the-as art solu- tions with different trade-offs among the tasks on MTL benchmarks various domains.
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Intial Paint
Pareto
MCDD
CAGaad
IMTL-G
GndarSie-0j
GmdO#Sia
G
12
Figure 2: Visualization of trade-off's in a 2D multi-task optimization problem. Shown are trajectories of each method with different initial points (labeled with black +) using Adam optimizer (Kingma & Ba 2014). See Appendix B for details. GD is unable to traverse the deep valley on two of the initial points because there are conflicting gradients and the gradient magnitude of one task much larger than the other. For MGDA (Sener & Koltun 2018), CAGrad (Liu al. 202 1a), and IMTL-G (Liu et al. 2021b), the final convergence point is fixed for each initial point. In contrast, GradOPS could converge to multiple points in the Pareto set by setting different
PRELIMINARIES
In this section, we first clarify the formal definition of MTL. Then discuss the major problems in MTL and state the importance of achieving non-confiicing gradients among the tasks.
Problem Definition. For a multi-takk learning (MTL) problem with T > tasks [T,,... Tr). each task is associated with loss function (0) for shared set of parameters 6. Normally, standard objective for MTL is to minimize the summed loss over all tasks: 8* arg ming E, (0).
Pareto Stationary Points and Optimals. solution 9 dominates another 0' (0) (0') for all T; and (i) < C,()') holds for at least one T. A solution 0" is called Pareto optimal if no solution dominates 0* A solution is called Pareto stationary there exists w IR such that wy 2 0, Σ mi = 1 and E (ig:(() = 0, where gi = VoL; (0) denotes the gradient T; All Pareto stationary, the reverse holds when L: is convex for all Ti (Drummond & lusem 2004 Cruz et al.) 20.I). Note that MTL methods can only ensure reaching Pareto stationary points, convergence to Pareto optimals may holds without the convex loss assumption.
Pareto optimals
Conflicting Gradients.
Two gradients 91 and 92 confficting if the dot product 91 92 0. Let G be the aggregated update gradient, i.e. G - Gi. we define types of confticting gradients:
conflicts among the tasks if there exists any two task T, and T, with 9i 9, < 0, confficts of the final update direction with the tasks if any task T, satisfies that G gi < O.
For clarity, we introduce the concept of strong non-conflicting if 9j 0, Vi,j holds, and weak non-conflicting if the aggregated gradient G dose not conflict with all original task gradients De- Note that strong non-conflicing gradients are always guaranteed to be weak non-conflicting, whereas weak non-conflicing gradients are not necessarily strong non-conflicting Most existing MOO methods focus seeking weak non-confficting G toward certain trade-offss which is directly responsible for task performance. Current projection based methods (Yu et al. [200 WWan et al,
2020) can only ensure strong non-conflicting gradients with T = 2, and when T 2 even weak non-confliction is not guaranteed.
Advantages of Strong Non- Con-Conflicting Gradients. Without strong non-conflicting gradients, MOO methods are limited to find only one solution toward certain trade-off by delicately balancing the conflicting gradients, which empirically may lead to unsatisfying results if such trade-off less effective when convex assumption of L; fails. In contrast, with strong non-conflicting gradients, we instantly acquire the advantage that will be much easier to apply different trade-offs on G simply via non non-negative linear combinations of the deconfficted gradients, all guaranteed to be non- conflicting with each original g,, With such ability, stable results with various trade-offs are also ensured since all tasks are always updated toward directions with non-decreasing performance.
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8
9
91
91
(a) GD
(b) MGDA
(c) IMTL-G
(d) PCGrad
(c) GradOPS
Figure 3: Visualization of the update direction (in yellow) obtained by various methods on a two- task learning problem. We rescaled the update vector to half for better visibility. 91 and 92 represent the two task-specific gradients. MGDA proposes to minimize the minimum possible convex combi- nation of task gradients, and the update vector is perpendicular to the dashed line. IMTL-G proposes to make the projections of the update vector onto (91. 92 } to be equal. PCGrad and GradOPS project each gradient onto the normal plane of the other obtain gi and 92 For PCGrad, the final update vector is the average of {.h 32 GradOPS further reweights (91. 923 to make trade-offs between two tasks. As result, the final update direction of GradOPS is flexible between gí and 2 covering the directions of MGDA and IMTL-G instead of been fixed as other methods, and always doesn't t conflict with each task lask-specific gradient.
Thus, our GradOPS aims solve MTL problems with the following goals: (1) to obtain stable and better performance on all the tasks by achieving strong non-conflicting gradients, (2) to provide simple yet effective strategies capable of performing different trade-offs.
METHOD
3.1
STRONG NON-CONFLICTING GRADIENTS
Though recent works Yu et al., 2020 Wang et al. 2020) have already sought to directly deconflict each task specific gradient gi with all other tasks iteratively, none of theses methods manage solve all conflicts among the tasks as stated in Section I let alone to further examine the benefits of strong non-oonflicting gradients. As shown in Figure ((b) and [(c] both the modified g and the aggregated gradient G' of existing methods may still confiict with some of the original task gradients {g:
To address the limits of existing methods and to ensure strong non-conflicting gradients, GradOPS deconflicts gradients by projecting each Ji onto the subspace orthogonal to the span of the other task gradients. Formally, GradOPS proceeds as follows: (i) For each g. any permutation of the original task gradients, GradOPS first identifies whether g: conflicts with any of the other tasks by checking the signs of 9i for all {9; }
(ii) If no conflict exists, g, remains unmodified: gf 9i- Otherwise, GradOPS projects 9i onto the subspace orthogonal to S = span Jizs- The orthogonal basis for as {u,) } is computed using the Gram-Schmin procedure:
U1 91, if o 1 else u2 92.
Σ projur (gg),} 1.j +
(1)
"
9j
where (v)
w. |
Given the orthogonal basis (uz), ), the modified of is orthogonal to S:
g: = 9i Σ proj., (9:).
jzi
Eventually, we have -9; 2 O. Vi,j and thus each of does not conflict with every original gj- (iii) With strong non-conflicting gradients {g(). the vanilla aggregated update direction G' Σ.
guaranteed be non-conflicting with any of the original 9j since G' 9j (E, g)) 9j
E,(g 9)) 0, Vj. the same holds for any non-negative lincar combinations of the {
With this procedure, GradOPS ensures that both each modified task gradient gf and the final update gradient G do not conflict with each original task gradient gü. thus completely solves the conflicting gradients problem.
addition, unlike (Yu et al., 2020 Wang et al., 2020), our result is stable and invariant to the permutations of tasks during the procedure. Theoretical analysis convergence of GradOPS is provided as Theorem in Appendix A.I under mild assumptions neural network and small update step size.
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3.2
OFAS AMONG TASKS
Algorithm GradOPS Update Rule
Require: task number T, constant a, initial model parameters 0
1: while not converged do
2: gi < (0), vi
3: for 1; do
4:
3k * i, O < then
5:
compute orthogonal basis {u,} according Eq.
6:
g/- - 9i - Ejzs proja, (gi)
7:
else
8:
g, 91
9:
end if
end for
11:
G'= E
compute scaling factors {w;} according to Eq.
13:
update& 9 with gradient Gnew = >, wig.
14:
end while
As shown in Figure most existing methods could only converge one point in the Pareto set, without the ability of flexibly performing trade-off's among the tasks. Figure 3(a) to 3(d) give more detailed illustration about the fixed update strategies of different methods, different solutions are achieved only through different initial points. However, note that MTL practitioners may need varying trade-offs among the tasks by demand in real-workh applications (Lin et al., 2019a Navon et al., 2021). In order to converge to points with different trade-offs, the most straightforward idea is to assign tasks different loss weights, i.e. to scale task gradients. However, this often unsatisfying for two reasons: (1) update strategies of some methods like IMTL-G (Liu et al. (021b) invariant to gradients scales, and (2) difficult to assign appropriate weights without prior knowledge. Therefore, we further propose dynamically scale the modified task gradients with strong conflicting guarantees obtained by GradOPS to adjust the update direction.
To this end, we first define R; as the scalar projection of G' = = onto the original 9::
R = =||| cos cos(óig..))) G'
||9:||
GradOPS always ensures 9i G' 0. Given the same G', each R₃ can be regarded as a measure of the angle between g, and G", or measure of the real update magnitude on direction of g,, indicating the dominance and relative update speed of task T; among the tasks. A new update direction Gheew Σ3 w.g is thus obtained by summing over the reweighted gf with:
Note that R2 2
R
r1
ET
(4)
w; EdT/T"
(5)
where u, functions as the scale factor calculated from fr:) and determines trade-offs among the tasks, ri reveals the relative update magnitude on direction g; and always splits the tasks into domi- nating ones (Tilr; > }} and the dominated ones with r; < 1. Thus, a single hyperparameter a E IR on rf will allow effective trade-offs among the dominating and dominated tasks via non-nee -egative we in Eq. 5 Details on convergence of GradOPS with different w are analysed in Theorem
Discussion about a. With a > 0, performance of the dominating tasks are expected to be im- proved since higher value of will enforce Guee toward {g[} of these tasks and yield greater {w:}. While a < 0, the dominated tasks are emphasized since lower value a will ensure greater {wx} for tasks with smaller {r,). Specifically. a proper a will pay more attention to the dominated tasks, and thus obtain more balanced update gradient, which similar to IMTL-G. Keep decreas- ing will further focus tasks with smaller gradient magnitudes, which coincides with the idea
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of MGDA. An example is provided in Figure 2 where the top right points in GradOPS(a= and GradOPS(a=-5) converge to the similar points to IMTL-G and MGDA, respectively. Note that nnw = G' when 0. A pictorial description of this idea is shown in Figure 3(e) Different will redirect Gnew between 9 and 9/-
The complete GradOPS algorithm is summarized in Algorithm case of our method when 2 and a O.
Note that PCGrad is special
3.3 ADVANTAGES OF GRADOPS
Now discuss the advantages of GradOPS. Both GradOPS and existing MOO methods verge to Pareto stationary points, which are also Pareto optimals under convex loss assumptions. Thus ideally, there exists no dominating methods but only methods whose trade-off strategy best suited for certain MTL applications. Realizing that the assumptions may fail in practice and con- vergence to Pareto optimals may not hold, we are interested in achieving robust performance via simplest strategies instead of designing delicate trade-off strategies, which may fail to reach ex- pected performance with violations of the assumptions.
Our Goal of securing robust performance is achieved via ensuring strong non-conflicting gradients. Though (Yu et al. 2020; Wang et al. 2020) are already aware of the importance solving gradient conflicts (see Section 1}, none of them manage guarantee strong non-confficting gradients, which achieved by GradOPS, In addition, we further conclude empirically the benefit of achieving strong non-confficting gradients, which is all fully exploited in GradOPS:
With all conflicts solved, effective trade-offs among the tasks are much easier. With conflicting gradients, existing methods have to conceive very delicate algorithms to achieve certain trade-off preferences. The trade-off strategy in Liu et al. (2021a) requires solving the dual problem of max- imizing the minimum local improvements, IMTL-G (Liu et al., (021b) applies complex matrix multiplications only to find G' with equal projection onto each gi- Unlike these methods, GradOPS can provide solutions with different trade-off's simply via non-negative linear combinations of the deconflicted gradients, and achieves similar trade-offs with different MOO methods, like MGDA and IMTL-G, or even outperforms them. Moreover, GradOPS, which only requires tuning a single hyperparameter, also outperforms the very expensive and exhaustive grid search procedures. See Section 4 and Appendix C.3 for details on effects of tuning a. Moreover, experiment results Sec- tion-iime that certain trade-off strategy may not always produce promising results on different datasets. Unlike most MOO methods which are binded with certain fixed trade-offs. GradOPS is also less affected and more stable since it flexibly supports different trade-off's and always ensures that all tasks are updated toward directions with non non-decreasing performance.
Lastly, it's worth mentioning that GradNorm (Chen al. 2018) also dynamically adjust gradient norms. However, its target of balancing the training rates of different tasks can be biased since relationship of the gradient directions are ignored, which can be improved by taking real update magnitude on each task into account as we do in Eq.
EXPERIMENTS
In this section, we evaluate our method on diverse multi-task learning datasets including two public benchmarks and one industrial large-scale recommendation dataset. For the two public benchmarks, instead of using the frameworks implemented by different MTL methods with various experimental details, we present fair and reprodueible comparisons under the unified training framework (Lin) & Zhang 2022). GradOPS inherits the hyperparameters of the respective baseline method in all experiments, except for one additional hyperparameter a. We include the additional experimental details and results Appendix B and rrespectivll.. Source codes will be publicly available.
Compared methods. (1) Uniform scaling: minimizing Σ, Ci: (2) Single-task: solving tasks inde-
pendently: (3) existing loss reweighting methods (Chen et al. 2018 projection methods (Yu et al.) 2020 Wang et al. 2020) and MMO methods (Sener & Koltun 2018 Liu et al. 2021a b).
Evaluation. In addition to common evaluation metrics in each experiment, we follow (Maninis
et al. 2019 Liu et al. 2021a Navon al. 2022) and report two metrics to capture the overall

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Under review as a conference paper at ICLR 2023
Method
Income
Marital
Education
Average
Am%
MR
Single-task
0.9454
0,9817
0.8875
0,9382
0.00
4.33
Uniform scaling
0.9407
0.9763
0.8829
0,9333
0.52
13.00
GradNorm
0.9435
0,9829
0.8873
0,9379
0.03
6.00
PCGrad
0,9428
0,9812
0.8856
0,9365
0.18
11.00
GradVac
0,9436
0.9798
0.8856
0,9363
0.20
10.33
MGDA
0.9460
0,9833
0.8870
0,9388
-0.06
3.00
CAGrad
0.9444
0.9820
0.8863
0.9376
0.07
7.00
IMTL-G
0,9443
0,9826
0.8864
0,9378
0.05
6.33
GradOPS(a=0)
0.9436
0.9817
0.8861
0,9371
0.12
9.33
GradOPS(a=2)
0,9432
0,9769
0.8864
0,9355
0.28
10.33
GradOPS(a=-I)
0,9443
0,9832
0,8866
0,9380
0.02
5.67
GradOPS(a=-2)
0,9453
0.9837
0.8871
0,9387
-0.05
3.33
GradOPS(a=-3)
0,9459
0.9854
0.8876
0.9396
-0.15
1.33
Table I: Experiment results on UCI Census-income dataset. The best scores are shown in bold and the second-best scores are underlined.
Segmentation
Depth
Surface Normal
Method
Am% 4.
MR
Angle Distance
Within 6
mloU 1
Pix Acc
Abs Err J
Rel Ext J
Mcan
Msdian L
11.25 f
22.5 1
30
Single-task
30.39
23.68
0.00
4.67
Uniform scaling
27.50
53.31
0.6061
0.2627
32,80
27.61
19.85
41.53
53.68
6.50
11.33
GrudNorm
26.54
52.69
0.5902
0.2500
31.40
26.00
21.62
44,02
56.29
3.10
7.78
PCGrad
27.53
53.51
0.5980
0,2606
32.61
27.32
20.10
42,00
54.11
5.72
9.67
GradVae
28.33
54.66
0,5908
0.2530
32.74
5.16
9.00
MGDA
18.57
46.99
0.7015
0.2913
29.83
23.27
24.84
48.63
60.58
5.64
6.78
CAGrud
27.07
54.14
0.5999
0,2598
30.93
25.35
22.30
45.09
57.34
1.93
7.67
IMTL-G
30.66
57.76
GraiOPS(a=t))
27.08
54,40
0.5862
0.2545
32.12
26.86
20.69
42.69
54,89
4.32
8.11
GradOPS(a=1)
26.56
53.26
0.6055
0,2600
34,30
29.39
18.79
39.79
$1.67
9.40
12.22
31.75
GraDOSS(a=-I)
28.45
55.67
0.5895
0.2550
30.46
24.85
22.73
45.91
58.20
-0.23
4.44
GraOOP((o=-L3))
27.34
34.71
0.6077
0,2509
30.34
24.80
22.75
45.94
58.34
0.43
5.11
GradOPS(om-3)
20.75
29.87
59.94
Table 2: Experimental results on NYUv2 dataset. Arrows indicate the values are the higher the better (f) or the lower the better (J). Best performance for each task bold, with second-bess underlined.
performance: (1) Mean Rank (MR): The average rank of each method across the different tasks (lower is better). A method receives the best value, MR = 1, it ranks first in all tasks. (2) Am: The average per-task performance drop of method m with respect to the single-tasking baseline b: Am D (-1)²(M,,.. M,, where 2, = 1 if higher value is better for criterion
M; on task i and otherwise.
4.1 MULTI-TASK CLASSIFICATION
The UCI Census-income dataset Duua & Grafj 2017) is a commonly used benchmark for multi-task learning, which contains 299,285 samples and 40 features extracted from the 1994 census database. Referring to the experimental settings in Ma et al. (0188)). we construct three multi-task learning problems from this dataset by setting some of the features as prediction targets. In detail, task Income aims to predict whether the income exceeds 50K, task Marital aims predict whether this person's 's marital status never married, and task Education aims predict whether the education level is at least college. Since all tasks are binary classification problems, we use the Area Under Curve (AUC) scores as the evaluation metrics.
We summarize the experimental results for all methods in Table I The reported results are the average performance 10 times experiments with random parameter initialization. As shown, GradOPS(a=0) outperforms the projection-based method PCGrad in all tasks, indicating the ben- efit of our proposed strategy which projects each task-specify gradient onto the subspace orthog- onal the span of the other gradients deconflict gradients. Then, compare the perfor-

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Method
Purchase
Cart
Wish
Average
Am%
MR
Single-task
0.8149
0.7616
0.8095
0.7953
0.00
7.33
Uniform scaling
0.8178
0.7606
0.8087
0.7957
-0.04
9.00
GradNorm
0.8188
0.7609
0.8084
0.7960
-0.08
6.67
PCGrad
0.8175
0.7609
0,8089
0.7958
-0.05
8.67
GradVac
0.8182
0.7609
0.8087
0.7959
-0.07
8.33
IMTL-G
0.8181
0.7602
0.8097
0.7960
-0.08
8.00
GradOPS(a=0)
0.8190
0.7620
0.8095
0.7968
-0.19
3.33
GradOPS(o=1)
0.8186
0.7620
0.8067
0.7958
-0.05
6.67
GraOOPS(a=2)
0.8199
0.7625
0.8073
0.7966
-0.15
4.33
GradOPS(a=0.5)
0.8185
0.7618
0.8101
0.7968
-0.18
4.00
GradOPS(a=-I)
0,8187
0.7616
0,8099
0.7967
-0.17
4.33
GradOPS(u=-2)
0.8178
0.7604
0.8117
0.7966
-0.16
7.33
Table 3: Experiment Results on the real large-scale recommendation system. The best and runner up results in each column are bold and underlined, respectively.
mance of GradOPS with different a and other MOO methods. We note that GradOPS(a=-1) covers CAGrad and IMTL-G, and GradOPS(o=-2) approximates MGDA. This results show that GradOPS with proper a < can roughly recover the final performance of CAGrad, IMTL-G and MGDA. GradOPS((==3) achieves the best performance in terms of Marital, Education and Av- erage. Although GradOPS(a=2) doesn't achieve as good experimental results GradOPS with = -- 33} do, it outperforms Uniform scaling baseline. And we find that for this dataset, almost any value of -00 S S 5 will improve Average performance over Uniform scaling base- line (see Appendix C for details). It suggests that the way to deconflict gradients proposed by us beneficial for MTI..
4.2 SCENE UNDERSTANDING
NYUv2 (Silberman al. 2012) is indoor dataset which contains 3 tasks: 13 class semantic segmentation, depth estimation, and surface normal prediction. We follow the experiment setup from (Liu et al. 2019 Yu al. 2020), and fix SegNet (Badrinarayanan al. 2017) as the network structure for all methods.
The results are presented in Table Each experiment is repeated times over different random parameter initialization and the mean performance is reported. Our methods GradOPS(a=-1.5) achieves comparable performance with IMTL--G. (0) and GradOPS(o=-I) achieve the best A., and MR respectively. As surface normal estimation owns the smallest gradient magni- tude, MGDA focuses on learning to predict surface normals and achieves poor performance on the other two tasks, which is similar to GradOPS with lower -3 It is worth noting that although MGDA has overall good results on UCI Census-income dataset, yet has not as good An on this dataset, which suggesting that it is hard to generalize different domains.
4.3 LARGE-SCALE RECOMMENDATION
In this subsection, we conduct offline experiments on large-scale recommendation system in Taobao Inc. to evaluate the performance proposed methods. We collect an industrial dataset through sampling user logs from the recommendation system during consecutive 7 days. There are 46 features, more than 300 million users, 10 millions items and about 1.7 billions samples in the dataset. We consider predicting three post-click actions: purchase, add shopping cart (Cart) and add to wish list (Wish). We implement the network as a feedforward neural network with several fully-connected layers with ReLU activation, and adopt AUC scores the evaluation metries.
Results are shown Table 3 As wish prediction is the dominated task in this dataset, GradOPS with negative {-0.5. -1 22 show better performance than GradOPS(a=0) in wish task, and GradOPS with a positive a = {1, 2} achieve better performance in the other two tasks. Our proposed GradOPS can successfully find of well-distributed solutions with different trade-offs,
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and it outperforms the single-task baseline where each task is trained with separate model. This experiment also shows that our method remains effective in real world large-scale applications.
5 RELATED WORK
Ever since the successful applications of multi-task learning in various domains, many methods have been released in pursuit of fully exploiting the benefits of MTL.. Early branches of popular approaches heuristically explore shared experts and task-relationship modelling via gating networks
(Misra et al.] 2016 Ma et al., [018a| Xia et al.] 20188 Liu et al.) [2019] Tang et al. 2020 Xi et al.)
2021), loss reweighting methods to ensure equal gradient norm for each tasks (Chen et al. 2018) 8) or
to uplift losses toward harder tasks (Guo et al. 2018, Kendall et al. 2018)) 8). GradOPS is agnostic to model architectures and loss manipulatioos, thus can be combined with these methods by demands.
Recent trends in MTL focus on gradient manipulations toward certain objectives. Chen et al. (2020) simply drops out task gradients randomly according to conflicting ratios among tasks. [Javaloy & Valera (202T) focuses on reducing negative transfer by scaling and rotating task gradients. Existing projection based methods explicitly study the confficting gradients problem and relationships among task gradients. Some methods more precisely alter each task gradient toward non-negative inner product (Yu et al. 2020) or certain values of gradient similarity (Wang et al. (020) with other tasks, iteratively. GradOPS makes further improvements over these methods, stated in Section
Originated from (Désidéri 2012), some methods seek to find Pareto optimals of MTL as solutions of multi-objectivee optimization (MOO) problems. (Sener& & Koltun (2018) apply the Frank-Wolft algo- rithm to solve the MOO problem defined in Désidéri (2012) Lin et al. (2019b) choose to project the closed-form solutions of a relaxed quadratic programming to the feasible space, hoping to achieve irvetion-efficiency More recent methods seek to find Pareto optimals with certain trade-off among the tasks instead of finding an arbitrary one. [Lu et al.] (2021a) seeks to maximize the worst local improvement of individual tasks to achieve Pareto points with minimum average loss. IMTL-G (Liu jet al., 2021b) seeks solutions with balanced performance among all tasks by searching for an update vector that has equal projections each task gradient. Note that some methods have already been proposed to provide Pareto solutions with different tradc-offs. Lin et al. (2019) explicitly splits the loss space into independent cones and applies constrained MMO methods to search single solution for each cone; MMahaparra & Rajan (2020) provides the ability of searching solutions toward certain desired directions determined by the input preference ray r: Navon et al. (202I) trains hypernet- work, with preference vector r as input, to directly predict weights of a certain MTL model with losses in desired ray r. Comparison of GradOPS and MOO methods are discussed in Section 333
Finally, as discussed in Yu et al. (2020), we also state the difference of GradOPS with gradient pro- jection methods (Lopez-Paz & Ranzato 2017 Chaudhuy et al.) | | Farajtabar et al.) (2200) applied in continual learning, which mainly solve the catastrophic forgetting problem in sequential lifelong learning. GradOPS is distinct from these methods two aspects: (1) instead of concentrate only on conflicts of current task with historical ones, GradOPS focus gradient deconfliction among all tasks within the same batch simultaneously to allow mutually positive transfer among tasks, (2) GradOPS aims to provide effective trade-off's among tasks with the non-conflictngg gradients.
CONCLUSION
In this work, focusing on the conflicting gradients challenge in MTL, we introduce the idea of strong non-conflicting gradients, and further emphasize the advantages of acquiring such gradients: it will be much casier to apply varying trade-offs among the tasks simply via different non non-negative linear combinations of the deconflicted gradients, which are all guaranteed to be non-conflicting with each original tasks. Thus, solutions with stable performance toward different trade-offs can also be achieved since all tasks are updated toward directions with non-decreasing performance in each training step for all trade-off preferences. To fully exploit such advantages, we propose a sim- ple algorithm (GradOPS) that ensures strong non-con conflicting gradients, based on which a simple reweighting strategy is also implemented to provides effective trade-offs among the tasks. Compre- hensive experiments show that GradOPS achieves state-of- state-of-thh-are the-art results and is also flexible enough to achieve similar trade-offs with some of the existing MOO methods and even outperform them.
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A DETAILED DERIVATION
A.1 PROOF OF THEOREM 1
Theorem 1, Assume individual loss functions L1, C2.... Ly are differentiable. Suppose the gradi- ent of C is L-Lipschitz with L 0. Then with the update step size < TL GradOPS in Section 3.1
will converge to Pareto stationary point.
Proof. As differential and L-smooth, we can obtain the following inequality:
C(0') < (()) + VL(O) 0) V² 011 $ C()) + VL(0) (0')))00000)) - -0) -0++ + 1+22/1 110 - 01
(6)
Plugging in the GradOPS update by letting 0' IG', we can conclude the following:
L(0') s c(0) - G' 1
(Expanding, using the identity G= Σ Ji, G'= == s
=(()) -1 Σ 9h Σdl
Lt
Σ g/ll?
2
1
[())
(g. 9/)
Lt
(gf
9/r)
2
(Using the inequality gs زوو 2 0, Vi,j from Section 3.1
(()
Σ (g. 9/) 1/2 L1 Σ (gb gb))
1
(Using the inequality d. J s |19hll
|loa|l| - 1/2 2(116611 + (10;(12)
<(()) -t
Σ (gs 96) 2/2
1
Lt
Σ
(118/11 + (110}17)
2
in
[()) - Σ (a. 9f)) 5 TLt Σ
11sfl1
(Using the identity of = |1 Vi)
C( - Σ |1
2 TLt Σ 1|
C(0) - t(1 2 TLt Σ 11
Note that ((1- Σ, |1/ > 0 when S - Further, when 2 (11-(111 Σ, || if and only if ||// 0, vi, i.. of Vi.
Hence repeatedly applying GradOPS process with < 2 reach some point 0" in the optimiza- tion landscape where gf = 0, Vi. According to Section 3.1 we have 9; = or gi belongs to subspace span(9,) hyyt This means that there exists convex combination of the gradients {g.} at this point 0* that equals zero, and therefore 0* is Pareto stationary under the definitions in Section 2
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108
0.8
-08
000
0.8
-16
146
1.6
Fap
0g
(a) Multi-Tass Task Objective
(b) Taskl Objective
(c) Task2 Objective
Figure 4: Visualization of the loss surfaces of Figure
10
62
30
-24
(a) Multi-Task Objective
(b) Taskl Objective
(c) Task2 Objective
Two-dimensional contour graphs of the three-dimensional surfaces in Figure the point with lowest single-task loss.
Figure 5:
*
denotes
A.2 PROOF OF THEOREM 2
Theorem 2. Assume individual loss fanctions L1, L2y..., Lr differentiable. Suppose the gradi-
ent of C is L-Liscchitz with L > 0. Then with the update step size t < min; TLu? los 11? GradOPS in Section 3.2 will converge to a Pareto stationary point.
2wr [|
Proof. Similar Section A.I plugging in Inequality by letting 0' bewn we can conclude the following:
C(0') C()) (IG. G Ghew
(Expanding, using the identity G
Σgi Dis G
Σ wigh)
L()) Σ9: g:
Σ wigh
T.ll Σ
wigfll
1 TL 2
(()) willo'/1?
Σ
i |1
(8)
If gf 0, vi, GradOPS reaches the Pareto stationary point. Otherwise, denotes T as the tasks
set where || > 0, E T Therefore, w; > 0, Vi + according to Section 3.2 In this case, with min 2w. 1| J 1| GradOPS can reach some point 6* where g' 0, Vi, i.e. the Pareto
stationary point.
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B IMPLEMENTATION DETAILS
Illustrative Example. We provide here the details for the illustrative example of Figure 2 We modify the illustrative example in (Liu al. 2021a) and consider + 02) R2 with the following individual loss functions:
(0) cifi (2) and = c2f2(0), where
[1( = (og(max((((0111 0.7) tanh(-3 02)], 0.0005))
(,0000) +
[2(0) log(max((6(-11 + 0.7) tanhi-33 02)], 0.0005)
(0) (1.5 tanh (-01 [[.))++7))))+ (0f + + (-02 0.8)3) 2.5
92( = (1.5 tanh(2 (-01 - 0.7) (0f + (-02 0.8)3) 2.5
Ci (0) mann(an 02), 0) and c2( max((anh(( 02), 0).
The multi-task objective is (()) = (1 + (2) The three- direeddimessonall loss surfaces and the corresponding wwodiimensional contour graphs are shown in Tigure4and5]] respectively. We pick initial parameter vectors A E {(-0.85, 0.75). (-0.55, -0.3), (0.9, 0.9) and performed 20,000 gra- dient updates minimize & using the Adam optimizer with learning rate 0.001. The corresponding optimization trajectories with different methods is shown in Figure
Multi- Task Classification. the UCI Census-income income dataset, there are 199,523 training exam- ples and 99,762 test examples. Following Ma et al. (2018a), we further randomly split test examples into a validation dataset and a test dataset by the fraction of 1:1. We apply a 2-layer fully-connected ReLU-activated neural network with 192 hidden units of each layer for all methods. We also place one dropout layer (Hinton et al. 2012) 2) with p = 0.3 for regularization. For GradNorm (Chen al. 2018) and CAGrad (Liu al. 202 baselines which have additional hyperparametens, the original papers and search a E {0.5, 2.0} for GradNorm and с E {0.1, 0.2..... 0.9} for CAGrad with the best average performance of the three tasks validation dataset (a 1.5 for GradNorm and e = 0.5 for CAGrad). We train each method for 200 epochs with batch-size of 1024, and the Adam optimizer with learning rate of le 4.
follow
Scene Understanding. We follow the training and evaluation procedure used in previous work on MTL (Liu et al., 2019: Yu al. 2020) and apply SegNet (Badrinarayanan et al. 2017 as the network structure for all methods. For GradNorm (Chen et al. 2018) and CAGrad (Liu et al. 2021a) baselines, we follow the hyperparameters settings in the original papers with a 1.5 for GradNorm and с = 0.4 for CAGrad. Each method is trained for 200 epochs with a batch size of 2, using the Adam optimizer with a learning rate of 4. The learning-rate is halved - after 100 epochs.
Large-scale Recommendation. We collect an industrial dataset through sampling user logs from the recommendation system Taobao during consecutive 7 days. There are 46 features, more than 300 million users, 10 millions items and about 1.7 billions samples in the dataset. We implement the network as a feedforward neural network with 4 fully-connected layers with ReLU activation. The hidden units are fixed for all models with hidden size (1024.512,256,12g) All methods are trained with Adam optimizer with a learning rate of le 3.
C ADDITIONAL EXPERIMENTS
C. EFFECTS TUNING a.
To better understand how a affects the {w;}. we show the traces of w, during training for different values of UCI Census-income dataset in Figure 6 The positive and negative have opposite effects on wi of tasks Income and Marital compared to the zero a. And the a with larger absolute values further widen the gap.
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Income
Marital
Education
3.0
2-3
8.5-
am
2.2-
2.0
1.3
-08
1.04
0.5
0.0
8.0-
2.0-
1.5
-0
3.0-
03
0.3
0.5-
Epach
Epach
Epoch
Figure 6: Traces of how wa change during training for different values of a on UCI Census-income dataset. On the top row, we show the comparison between E {1, 0, (1) and on the bottom row, we show the comparison between E {3, 0, -3]. For 0, w; remains at 1 for all three tasks. The positive a 1.0 assigns dominated task Income a greater w; Marital lower w, < 1. And larger value of pushes weights farther apart. While, negative has the opposite effect.
and dominating task
Training
Test
1.00
Income
forrencce (8)
Mar ital
0.
Educat ion
Averazee
0.50
0.5
0.00
0..5
-0.5
-0.75
-10.0
-7.5
-5.0
-2.5
0.0
2.5
5.0
7.5
10.0
-10.0
-7.5
-5.0
-2.5
0.0
2.5
5.0
7.5
10.0
Figure/ 7: Performance gains with different a on UCI Census-income dataset. We show both the training and test performance gains compared to Uniform scaling baseline across all three tasks and Average performance metrics. We enumerate a E [-10, 10] steps of 0.5.
C.2 TRAINING AND TEST PERFORMANCE GAINS WITH DIFFERENT
To test whether the performance of GradOPS are robust against the hyperparameter a changes, show both the training and test performance gains with different a on UCI Census-incoma dataset Figure 7 We note that the training performance gain of dominating task Marital rises as the value of a gets larger, while the gain of the dominated task Income goes down. This again proves the ability of GradOPS to obtain solutions with different trade-offs. It should be mentioned that higher training performance does not necessarily guarantee higher test performance. And the test performance of task Marital with a > is worse than the performance with a S 0, which is inconsistent with the observation of corresponding training performance. The reason may be that the model overfits on task Marital, which suggesting a better regularization scheme for this domain. Note that achieve Average performance gains on both training and test for almost all values of
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Average Task Per formance
0.9
statio
0. 936
939
932
0.
0.0
2.5 L2 distance to GradOPS we ights
Figure 8: Performance comparison between GradOPS(a=-3) and GradOPS-static with grid search weights watatic The x-axis denotes the L2 distance between w and the average weights of GradOPS(a==3) over training steps.
-10 < S 5, which indicating that GradOPS is numerically stable. Moreover, the consistently positive performance gains across all these values of suggest that the way to deconflict gradients introduced by GradOPS improve MTL performance.
C.3 COMPARISON BETWEEN GRADOPS AND GRADOPS WITH GRID-SEARCH WEIGHTS.
To further verify the effectiveness of the strategy introduced in Section 3.2 for obtaining wi, we con- duct comparative experiment where we train GradOPS with static w catatic (referred to as GradOPS- static) on UCI Census-income dataset. {wştatic } are considered as hyperparameters, and are sam- pled from candidates generated by grid search. The sum of {witatic] is guaranteed to equal T = 3, and G Stattic = E; watatic gf is used the final update gradient. Then, compare the performance of BradOPS-static to our GradOPS(a==3).
The results shown in Figure 8 We train GradOPS-static 100 times with different combina- tions of (tatttic). Even after this exhaustive grid search, GradOPS-static still falls short our GradOPS(a=-3), Therefore, GradOPS find the optimal grid search weights in single train- ing run.
17


Reviews:
[{"review_id": "hWQj3u2tG30", "reviewer": "Reviewer_tv3z", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper proposes a simple and effective gradient deconfliction algorithm, called GradOPS, for multi-task learning (MTL). Concretely, GradOPS projects the gradient associated with one task onto the subspace orthogonal to the span of the other task-specific gradients, achieving non-conflicting gradients for different tasks. Theoretical analysis of the convergence of GradOPS is provided, followed by extensive experiments on several multi-task learning datasets. Overall, this paper demonstrates the effectiveness of the proposed novel algorithms, both from theoretically and practically.", "summary_of_the_review": "Thank authors for their detailed responses. In terms of others’ review and authors’ feedback, I remain my score as “weak accept.”", "strength_and_weaknesses": "Strengths:\n1. The proposed algorithm is simple and effective. I am very happy to see that such simple algorithm can achieve non-conflicting gradients and improved performance for multi-task learning problems.\n2. Figure 1 is a good example to illustrate the differences between the proposed GradOPS and existing MTL algorithms. \n3. Theoretical analysis of the convergence of GradOPS is provided, ensuring that the output of GradOPS can converge to a Pareto stationary point.\n4. Authors also give a full discussion of the advantages of the proposed GradOPS in Section 3.3.\n5. I find almost no typos in the paper.\n\nWeaknesses and Suggestions:\n1. More clarity on Figure 1. I appreciate the illustration of Figure 1. However, I believe that it can be improved in terms of the clarity and beauty. For instance, the caption of Figure 1 in Page 2 should include the calculation of the aggregated gradient G = \\sum_{i=1}^{3}g_{i}, instead of letting readers seek the detailed aggregation rule of G in Subsection 3.1 in Page 4. Besides, different arrows with different colors in Figure1 represents different gradients, and it will be better to clarify them in the caption. Finally, the notation of GradOPS-modified g_{1}^{‘} looks not good enough. It should be g_{1}^{‘} instead of g^{‘}_{1} in Figure 1(b)(c)(d), to keep notation consistence with g_{1} in Figure 1(a).\n2. The theoretical results should be placed in the main paper, and in particular, the assumptions of the convergence of the proposed algorithm in Theorem 1 & 2 should be expressed explicitly in the main paper (e.g. in the bottom of page 4). For example, the convergence of GradOPS in Theorem 1 requires the Lipschitz properties of the gradient and the small step size of GD. Such assumptions are quite essential to derive the final theoretical results, and should be clarified in the main paper (e.g. at least with the claim “the convergence of GradOPS is provided in Theorem 1, under mild assumptions of neural network and the step size of GD”). A counterexample is that, if we use ReLU activation function in deep neural network, the smooth assumption of neural network could not be satisfied and the convergence of GradOPS in Theorem 1 could not be guaranteed.\n3. As far as I can tell, the proof of Theorem 1 and Theorem 2 is easy (and hence is correct), so the derived results should not be stated as Theorem. It will be better to call them as Proposition.\n4. More explanations on the architecture of the deep neural networks (DNN) in experiment Part. As far as I can see, this paper uses different deep neural networks as backbones for different datasets, and authors also point out the name of these networks in the main paper. However, the components of DNN is very important in our experiments, and so I will suggest the authors to give more explanations on the components of DNN in the Appendix C (e.g. the kernel size of convolutional network, and the activation function in DNN).\n5. Other minor suggestions: (1) Eqs.4 and 5 can be merged and put in one line for concise. (2) Avoid orphan line before Section 3.1. in Page 4 and the orphan line at the top of Page 9. (3) Typos: in caption of Figure 3, “the final updated direction”.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "Xn9Fc7pqVQ", "reviewer": "Reviewer_wEFB", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper proposes a new method for resolving the gradient conflicts encountered in multi-task learning problems. Gradient conflict means when two gradients have a negative cosine similarity. The idea is based on [1] to project gradients of tasks to be orthogonal to all the other tasks if there is any gradient conflict between the two tasks. The difference is that in [1] gradient is projected iteratively based on pairs of tasks. In this work, the gradient is projected to the subspace that is orthogonal to the span of all other gradients. The paper also explores tuning the weight of gradient aggregation after resolving all gradient conflicts, so that it allows controlling the balance over different tasks. The experiments show that the proposed method is able to optimize for different Pareto optimal solutions and the results are competitive.\n\n[1] Tianhe Yu, Saurabh Kumar, Abhishek Gupta, Sergey Levine, Karol Hausman, and Chelsea Finn. Gradient surgery for multi-task learning. Advances in Neural Information Processing Systems, 33:5824–5836, 2020.", "summary_of_the_review": "Overall, I think the paper presents an interesting perspective on how to do flexible re-weighting of gradients based on the cosine similarity of the independent task gradient and the aggregated gradient. The idea of gradient projection is not completely new but inherited from Yu et al. I have some reservations about whether it makes the most sense to project gradient to be orthogonal to all other gradients to resolve the conflict, as it can lose a lot of the gradient information. ", "strength_and_weaknesses": "Strength: \n1. The method proposed is guaranteed to resolve all gradient conflicts. \n2. The method allows an easy way to control the weights of gradients from different tasks, which is nice to have to explore different Pareto optimal solutions to the problem. \n\nWeaknesses:\n1. The idea of projection is not new but based on Yu et al. 2022. \n2. Not sure if strictly enforcing no gradient conflicts by projecting to the orthogonal subspace loses too much information about the gradient. In the worst-case scenario where there are a lot of gradient conflicts, this method will destroy a lot of the original gradient information by iterative projecting the gradients to a new orthogonal subspace. I would be more convinced if there is some evidence on how this procedure destroys or keeps most of the gradient information as compared to Yu et al. \n3. I would love to see how Yu et al. performs with a similar kind of gradient re-weighting, though it would be non-trivial since there could be gradient conflicts in Yu et al.’s method (i.e. not all cosine similarity between g_i and G are positive, maybe use an exponential of the cosine similarity). As it seems like when alpha = 0, (i.e. gradient averaging after resolving the gradient conflicts), the proposed method does not have a clear advantage over Yu et al. It is the flexible re-weighting that makes it interestingly competitive. I wonder if that is also true by applying a similar re-weighting idea to Yu et al.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "VVUVWpjFeq", "reviewer": "Reviewer_FaLj", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This manuscript proposes a novel projection-based method for tackling the issues of gradient conflict in the multi-task learning problem, GradOPS. The work improves previous projection-based methods like PCGrad by projecting conflict gradients onto an orthogonal subspace. In this way, it can not only resolve all conflicts in more than two tasks but also effectively searches for diverse solutions with different trade-off preferences among different tasks.", "summary_of_the_review": "This paper is well-organized and somewhat novel; however, there are some weaknesses required to address.", "strength_and_weaknesses": "Strengths:\n\n1. The underlying idea of this work is well-motivated. It hypothesizes a reasonable theory about the “conflicting gradients” issue in multi-task learning and follow-up with a simple algorithm.\n2. The concept definition of “strong non-conflicting” and “weak non-conflicting” provide a clear description of the core issue, which in turn strongly serves as the basis for method design.\n3. The proposed method is simple and general, it can be easily applied to various gradient-based approaches. Furthermore, the paper writing is professional and rigorous.\n\nWeaknesses:\n\n1. Although this work provides a novel projection-based approach, the core idea does not improve too prominently from previous work (Yu et al., 2020; Wang et al., 2020), in fact, PCGrad can also be generalized to the case of T > 2 as well (and the experimental results did not significantly exceed PCGrad, shown in Table 1, 2). Moreover, compared to PCGrad, the theoretical analysis of this paper is relatively weak.\n2. The experimental results are somewhat disappointing, which makes the proposed method less convincing. For instance, the results in Table 1 and Table 2 are not exactly better than the baseline method (MGDA). In addition, the performance of GradOPS shown in Table 3 is highly dependent on a good choice of α values, which may mean that the method is not so general for different types of datasets.\n3. There is no comparison with the state-of-the-art algorithm, in fact, the NashMTL (Navon et al., 2022) has surpassed the previous method, suggesting that the authors supplement the corresponding comparison experiments.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "NV4nW94usf", "reviewer": "Reviewer_JvTK", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "To address the task interference in multi-task learning, this paper presents an innovative method to obtain deconflicted gradients among tasks. This is done through projecting the gradient of a task to the subspace that is orthogonal to that spanned by the gradients of all other tasks. The authors provide the convergence analysis of the method and tested the method using two benchmark datasets and one large-scale recommendation dataset. ", "summary_of_the_review": "The idea is interesting and the method is technically sound. However, several weaknesses as listed, especially, the weak empirical results dampened my enthusiasm of this paper. ", "strength_and_weaknesses": "Strength\n+ The proposed method produces deconflicted gradients that are not in confliction with any original gradients. \n+ The deconflicted gradients make it much easy to define an overall updating gradient that does not conflict with any individual task specific gradient while weighting tasks during learning. \n+ Code will be provided\n+ Presentation of the manuscript is clear\n\nWeakness\n-  Regarding the discussion underneath Eq. (2), since g_i’ is located in the subspace that is orthogonal to that spanned by all g_j’s, should be g_i’• g_j = 0? \n- The underlying assumption of existence of all g_i’ is that all g_i’s should be linearly independent among them. A discussion of such assumption in reality would strength the approach.\n-  Resulting from using different \\alpha values on test set are provided. However, no discussion is made how \\alpha would be tuned in practice. Use results from the \\alpha value leading to the best test performance to compare with other methods is not a fair comparison. \n- Recent related work should be discussed and compared, for example:\nJavaloy, A., & Valera, I. (2022). RotoGrad: Gradient Homogenization in Multitask Learning. ICLR, 1–24. http://arxiv.org/abs/2103.02631\n- My biggest concern of this paper is the weak empirical results, with minimal difference in the results among compared the methods (especially, table 1 and 3), which does not support the advantage of the proposed method claimed by the authors.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "1Py3I3AdYo", "reviewer": "Reviewer_CAmc", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "Training multitask learning models can be difficult at times due to the existence of conflicting gradients. That is, when computing the gradient of the model's parameters, different tasks present different gradients which can differ in magnitude and direction, counteracting each other when added up, and leading to parameter updates that obtain unsatisfactory results. \n\nIn this paper, the authors argue the necessity and importance of the absence of gradient conflict during training, and provide a method to explicity ensure this setting. The method, GradOPS, generates a new set of gradients by first transforming the original gradients into an orthogonal set of vectors, and then subtracting their contribution (i.e. projection) from the original gradients, making sure that the resulting gradients do not conflict the original ones. Finally, these gradients are added up, weighted by their contribution to the final update direction (and a hyperparameter).\n\nThe authors then conduct three experiments on a multitask binary-classification, NYUv2, and a recommendation system, proving their method effective compared with other MTL approaches.\n", "summary_of_the_review": "Overall, I find the idea of the paper (robustifying, in some sense, projection-based methods to improve conflicting gradients) appealing, and it could be a nice direction to pursue. The paper also have some cool reflections regarding conflicting gradients.\n\nHowever, the proposed method has some questionable design choices and technical flaws that need further investigation/clarification, including the motivation of why the proposed procedure leads to sensible directions to follow during optimization.\n\nThis, added to some concerns on the experimental part, makes me believe that the manuscript needs a bit more of work, and hence leans me towards rejection.\n\n", "strength_and_weaknesses": "**Strengths**\n- S1: I like the distinction between strong and weak non-conflicting gradients, although I find the explanations a bit too hand wavy for my taste.\n- S2: The proposed method does not depend on the task ordering, which I find quite important.\n- S3: Using scalar projections to measure the \"dominance\" of one task in the training process is quite interesting.\n- S4: I really like the use of Mean Rank to compare methods in the experiments.\n\n**Weaknesses**\n\nProposed method\n- W1: I have serious concerns understanding the proposed method and the intuition behind it:\n  - As of Eq. 1, $u_i$ is not defined for $i \\neq 1$.\n  - For $i \\neq 1$, $u_2 = 0$ always.\n  - It is not clear to me why this method should work. For example, if there are more tasks than parameters, then the basis generated by GS would span the entire parameter space. Then Eq. (2) would produce $g_i = 0$ for all tasks, unless for some subtle reason in the algorithm, $u_i$ is always different from 0. \n   - I fail to understand the reason behind the double normalization in Eq. 4 and 5, rather than weighting $R_i$ directly in Eq. 4.\n- W2: One of the advantages of GradOPS is that \"it solves all conflicts among tasks,\" meaning that the resulting vectors $g'_i$ hold $\\langle g'_i, g_j \\rangle \\geq 0$ for all $i$ and $j$. I fail to see this as a milestone by itself, since this could very well be achieved by multiplying all gradients by 0. Why is this important? And why are the new update directions meaningful?\n- W3: While the introduction of the parameter $\\alpha$ is introduced with the advantage of exploring more solutions, there is no control or intuition for the practitioner on which values of $\\alpha$ set to get to a desired trade-off.\n\nExperiments\n- W4: While the experiments are repeated ten times, there are no standard deviations and no best averaged results were marked in bold, rather than running any statistical test to check for significant changes.\n- W5: Results on the NYUv2 dataset are significantly worse than those reported in the literature, for example, in RotoGrad [1] and RLW [2].\n\nLiterature review: \n- L1: Literature review is missing some conflicting gradient methods that do not fit in the MOO vs. projection discussion: e.g. GradDrop [3] randomly drops elements of the task gradients, and RotoGrad [1] applies rotation matrices in the heads to align task gradients.\n- L2: Some comments feel a bit off. For example, saying that IMTL-G is a state-of-the-art method is rather questionable.\n\n**Questions:**\n- Q1. How were the MR and $\\Delta m$ computed? Individually for each run, or over the averaged results?\n- Q2: Which values of $\\alpha$ were selected for CAGrad and GradNorm?\n\n[1] - Spotlight, ICLR 2022 - [RotoGrad: Gradient Homogenization in Multitask Learning](http://arxiv.org/abs/2103.02631)\n\n[2] - ArXiv - [A Closer Look at Loss Weighting in Multi-Task Learning](http://arxiv.org/abs/2111.10603) \n\n[3] -  NeurIPS 2021 - [Just Pick a Sign: Optimizing Deep Multitask Models with Gradient Sign Dropout](https://proceedings.neurips.cc//paper_files/paper/2020/hash/16002f7a455a94aa4e91cc34ebdb9f2d-Abstract.html)", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject