| Type | Model | N-M | ddk | | | | H MACs Param | BLEU | COMET-22 |
| Multi-Branch Architecture | Weighted Transformer (Ahmed et al.,2017) | 6-6 | 1024 | | | | 211M | 28.90 | |
| Multi-Unit Transformer (Yan et al.,2020) | 6-6 | | | | | 130M | 29.30 | |
| MAT (Fan et al., 2020) | 6-6 | | | | | 206M | 29.90 | |
| Multi-Path Transformer (Lin et al., 2022) | 6-6 | | | | | 193M | 29.68 | |
| Lightweight Architecture | Evolved Transformer (So et al., 2019) Delight (Mehta et al., 2021) | | | | | | 64M | 28.20 | |
| Weight Sharing | | | 640 | | | | 54M | 28.00 | |
| Universal Transformer (Dehghani et al.,2019) | | 1024 | | = | | 65M | 28.90 | |
| SubFormer (Reid et al., 2021) | | - | | = | | 63M | 28.50 | |
| SubFormer-big (Reid et al., 2021) | | | | | | 197M | 29.30 | |
| ODE Transformer (RK4) (Li et al., 2022) ODE Transformer (RK4) (Li et al., 2022) | 6-6 24-6 | 512 512 | | = | | 62M 118M | 29.03 29.80 | |
| Other Comparisons | | | | 64 | | | | | |
| RealFormer (He et al., 2021) DMAN (Fan et al., 2021) | 18-18 | 512 512 | | 8 8 | | 151M | 29.35 | |
| Mega-Softmax (Ma et al.,2022) | 6-6 6-6 | 512 | | | | 63M 67M | 29.10 29.01 | |
| Our System | Transformer | | | 64 | 1 | | | | |
| PartialFormer (w/o Head Scaling) | 24-6 | 512 | 8-8 | | 11.1B | 118M | 29.05 | 83.60 |
| PartialFormer | 24-6 | 512 | 64 24-16 | 8-8 | 8.8B | 66M | 28.86 30.09 | 83.35 84.17 |
| 24-6 | 512 | 64 | | 12.2B | 115M | | |
| Transformer | 6-6 | 512 | 64 45 | 8-8 | 9.9B | 62M | 27.43 | 82.19 |
| Transformer PartialFormer (w/o Head Scaling) | 24-6 24-6 | 360 360 | 45 | 8-8 8-8 | 6.3B 5.2B | 62M 36M | 28.00 27.88 | 82.72 82.49 |
| | 360 | 45 | 24-16 | 6.8B | 61M | 29.23 | |
| PartialFormer PartialFormer | 24-6 24-6 | 360 | 45 30-16 | | 6.9B | 68M | 29.56 | 83.74 83.94 |
+
+Table 1: Results on the WMT’14 En-De task. MACs denote the multiplication-addition operations. We compute them via 20 source and target tokens following Mehta et al. (2021).
+
+# 4 Experimental Setups
+
+In our evaluation, we assess the performance of PartialFormer across 9 machine translation tasks2. More details are given in Appendix A
+
+Dataset. We evaluate our approach on three widely-used datasets: WMT’14 English-German (En-De), WMT’14 English-French (En-Fr), and WMT’16 English-Romanian (En-Ro). Besides, to further validate the effectiveness of PartialFormer, we also evaluate PartialFormer on six translation tasks from WMT’17 benchmark. We preprocess the raw data following the standard strategy.
+
+Architectures and Selected Baselines. We use a 24-6 encoder-decoder PartialFormer architecture for its strong performance, on all 9 machine translation tasks. Detailed configurations are provided in the results tables. We compare our approach with various baselines, including vanilla Transformer models, multi-branch architecture, lightweight architecture, weight-sharing methods, and other strong baselines.
+
+Training & Evaluation. We train all the models on GeForce RTX 3090 cards via Fairseq (Ott et al., 2019) toolkit. For evaluation, we utilized multi-BLEU (Papineni et al., 2002) and COMET22 (Rei et al., 2022) scores. Beam sizes were 4, 4, and 5 for En-De, En-Fr, and En-Ro tasks respectively. Length_penalty of 0.6, 0.8, and 1.3 were applied to En-De, En-Fr, and En-Ro tasks respectively. For the WMT’17 benchmark, beam size and Length_penalty were set to 4 and 1, respectively. We used an ensemble of the last ten checkpoints.
+
+# 5 Experiments
+
+Results of WMT’14 En-De Table 1 presents the results for the WMT’14 En-De task. Note that we also provide a “strong” baseline which also benefits from deep model stacking. Even though the performance of PartialFormer (w/o Head Scaling) is slightly inferior to that of the Transformer model (27.88 vs. 28.00 and 28.86 vs. 29.05), it outshines the latter in terms of parameter efficiency, consuming significantly fewer parameters (36M vs. 62M, 66M vs. 118M). We attribute this phenomenon to our PG-FFN, which leverages a group of compact FFNs. This approach enables PG-FFN to maintain high hidden dimension, while drastically reducing parameter consumption.
+
+Upon utilizing our head scaling technique to amplify the capacity, our Partialformer delivers a BLEU score 29.56 and 30.09 on two configurations, respectively. This surpasses the standard Transformer by 1.56 BLEU points (29.56 vs. 28.00) and
+
+Table 2: Results on the WMT’14 En-Fr task.
+
+| Type | Model | N-M | ddk | | | | H MACs Param | BLEU | COMET-22 |
| Multi-Branch Architecture | Weighted Transformer (Ahmed et al.,2017) | 6-6 | 1024 | | | | 211M | 28.90 | |
| Multi-Unit Transformer (Yan et al.,2020) | 6-6 | | | | | 130M | 29.30 | |
| MAT (Fan et al., 2020) | 6-6 | | | | | 206M | 29.90 | |
| Multi-Path Transformer (Lin et al., 2022) | 6-6 | | | | | 193M | 29.68 | |
| Lightweight Architecture | Evolved Transformer (So et al., 2019) Delight (Mehta et al., 2021) | | | | | | 64M | 28.20 | |
| Weight Sharing | | | 640 | | | | 54M | 28.00 | |
| Universal Transformer (Dehghani et al.,2019) | | 1024 | | = | | 65M | 28.90 | |
| SubFormer (Reid et al., 2021) | | - | | = | | 63M | 28.50 | |
| SubFormer-big (Reid et al., 2021) | | | | | | 197M | 29.30 | |
| ODE Transformer (RK4) (Li et al., 2022) ODE Transformer (RK4) (Li et al., 2022) | 6-6 24-6 | 512 512 | | = | | 62M 118M | 29.03 29.80 | |
| Other Comparisons | | | | 64 | | | | | |
| RealFormer (He et al., 2021) DMAN (Fan et al., 2021) | 18-18 | 512 512 | | 8 8 | | 151M | 29.35 | |
| Mega-Softmax (Ma et al.,2022) | 6-6 6-6 | 512 | | | | 63M 67M | 29.10 29.01 | |
| Our System | Transformer | | | 64 | 1 | | | | |
| PartialFormer (w/o Head Scaling) | 24-6 | 512 | 8-8 | | 11.1B | 118M | 29.05 | 83.60 |
| PartialFormer | 24-6 | 512 | 64 24-16 | 8-8 | 8.8B | 66M | 28.86 30.09 | 83.35 84.17 |
| 24-6 | 512 | 64 | | 12.2B | 115M | | |
| Transformer | 6-6 | 512 | 64 45 | 8-8 | 9.9B | 62M | 27.43 | 82.19 |
| Transformer PartialFormer (w/o Head Scaling) | 24-6 24-6 | 360 360 | 45 | 8-8 8-8 | 6.3B 5.2B | 62M 36M | 28.00 27.88 | 82.72 82.49 |
| | 360 | 45 | 24-16 | 6.8B | 61M | 29.23 | |
| PartialFormer PartialFormer | 24-6 24-6 | 360 | 45 30-16 | | 6.9B | 68M | 29.56 | 83.74 83.94 |
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| Multi-Branch Architecture | Weighted Transformer (Ahmed et al.,2017) | 6-6 | 1024 | | | | 211M | 28.90 | |
| Multi-Unit Transformer (Yan et al.,2020) | 6-6 | | | | | 130M | 29.30 | |
| MAT (Fan et al., 2020) | 6-6 | | | | | 206M | 29.90 | |
| Multi-Path Transformer (Lin et al., 2022) | 6-6 | | | | | 193M | 29.68 | |
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| SubFormer-big (Reid et al., 2021) | | | | | | 197M | 29.30 | |
| ODE Transformer (RK4) (Li et al., 2022) ODE Transformer (RK4) (Li et al., 2022) | 6-6 24-6 | 512 512 | | = | | 62M 118M | 29.03 29.80 | |
| Other Comparisons | | | | 64 | | | | | |
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| Mega-Softmax (Ma et al.,2022) | 6-6 6-6 | 512 | | | | 63M 67M | 29.10 29.01 | |
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| 24-6 | 512 | 64 | | 12.2B | 115M | | |
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| Transformer PartialFormer (w/o Head Scaling) | 24-6 24-6 | 360 360 | 45 | 8-8 8-8 | 6.3B 5.2B | 62M 36M | 28.00 27.88 | 82.72 82.49 |
| | 360 | 45 | 24-16 | 6.8B | 61M | 29.23 | |
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| Multi-Branch Architecture | Weighted Transformer (Ahmed et al.,2017) | 6-6 | 1024 | | | | 211M | 28.90 | |
| Multi-Unit Transformer (Yan et al.,2020) | 6-6 | | | | | 130M | 29.30 | |
| MAT (Fan et al., 2020) | 6-6 | | | | | 206M | 29.90 | |
| Multi-Path Transformer (Lin et al., 2022) | 6-6 | | | | | 193M | 29.68 | |
| Lightweight Architecture | Evolved Transformer (So et al., 2019) Delight (Mehta et al., 2021) | | | | | | 64M | 28.20 | |
| Weight Sharing | | | 640 | | | | 54M | 28.00 | |
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| SubFormer (Reid et al., 2021) | | - | | = | | 63M | 28.50 | |
| SubFormer-big (Reid et al., 2021) | | | | | | 197M | 29.30 | |
| ODE Transformer (RK4) (Li et al., 2022) ODE Transformer (RK4) (Li et al., 2022) | 6-6 24-6 | 512 512 | | = | | 62M 118M | 29.03 29.80 | |
| Other Comparisons | | | | 64 | | | | | |
| RealFormer (He et al., 2021) DMAN (Fan et al., 2021) | 18-18 | 512 512 | | 8 8 | | 151M | 29.35 | |
| Mega-Softmax (Ma et al.,2022) | 6-6 6-6 | 512 | | | | 63M 67M | 29.10 29.01 | |
| Our System | Transformer | | | 64 | 1 | | | | |
| PartialFormer (w/o Head Scaling) | 24-6 | 512 | 8-8 | | 11.1B | 118M | 29.05 | 83.60 |
| PartialFormer | 24-6 | 512 | 64 24-16 | 8-8 | 8.8B | 66M | 28.86 30.09 | 83.35 84.17 |
| 24-6 | 512 | 64 | | 12.2B | 115M | | |
| Transformer | 6-6 | 512 | 64 45 | 8-8 | 9.9B | 62M | 27.43 | 82.19 |
| Transformer PartialFormer (w/o Head Scaling) | 24-6 24-6 | 360 360 | 45 | 8-8 8-8 | 6.3B 5.2B | 62M 36M | 28.00 27.88 | 82.72 82.49 |
| | 360 | 45 | 24-16 | 6.8B | 61M | 29.23 | |
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+ "html": "| Algorithm1Training stepwith RotoGrad. |
| Input input samples X,task labels {Yk}, network's (RotoGrad's) learning rate n (mroto) Output backbone (heads) parameters 0 ({Φk}), RotoGrad's parameters {Rk} |
| 1: compute shared feature Z = f(X;0) |
| 2: for k=1,2,...,K do |
| 3: compute task-specific loss Lk = ∑nLk(hk(Rk Zn; Φk), yn,k) |
| 4: compute gradient of shared feature Gk = VzLk |
| 5: compute gradient of task-specific feature Gk = RkGk Treated as constant w.r.t. Rk . |
| 6: compute unitary gradients Uk = Gk/llGkll |
| 7: compute relative task convergence αk = ||Gkll/IlG | |
| 8: end for |
| 9: make {αk} sum up to one [α1,α2,...,αk] = [α1,α2,...,αk]/∑k αk |
| 10: compute shared magnitude C = ∑k αkllGkll |
| 11: update backbone parameters 0 =0"- nVθz : C∑κUk compute target vector V = k ∑k Uk |
+
+Memory efficiency and time complexity. As we need one rotation matrix per task, we have to store $\mathcal { O } \hat { (} K d ^ { 2 } \hat { ) }$ additional parameters. In practice, we only need $K d ( d - 1 ) / 2$ parameters due to the aforementioned parametrization and, in most cases, this amounts to a small part of the total number of parameters. Moreover, as described by Casado & Martínez-Rubio (2019), parametrizing $\scriptstyle { R _ { k } }$ enables efficient computations compared with traditional methods, with a time complexity of $O ( d ^ { 3 } )$ independently of the batch size. In our case, the time complexity is of $O ( K d ^ { 3 } )$ , which scales better with respect to the number of tasks than existing methods (for example, $O ( K ^ { 2 } d )$ for PCGrad ( $\mathrm { T u }$ et al., 2020)). Forward-pass caching and GPU parallelization can further reduce training time.
+
+Scaling-up RotoGrad. Despite being able to efficiently compute and optimize the rotation matrix $\scriptstyle R _ { k }$ , in domains like computer vision, where the size $d$ of the shared representation $_ z$ is large, the time complexity for updating the rotation matrix may become comparable to the one of the network updates. In those cases, we propose to only rotate a subspace of the feature space, that is, rotate only $m < < d$ dimensions of $_ z$ . Then, we can simply apply a transformation of the form $r _ { k } = [ R _ { k } z _ { 1 : m } , z _ { m + 1 : d } ]$ , where $z _ { a : b }$ denotes the elements of $_ z$ with indexes $a , a + 1 , \ldots , b$ . While there exist other possible solutions, such as using block-diagonal rotation matrices $\scriptstyle { \cal R } _ { k }$ , we defer them to future work.
+
+# 4 ILLUSTRATIVE EXAMPLES
+
+In this section, we illustrate the behavior of RotoGrad in two synthetic scenarios, providing clean qualitative results about its effect on the optimization process. Appendix C.1 provides a detailed description of the experimental setups.
+
+To this end, we propose two different multitask regression problems of the form
+
+$$
+L ( \pmb { x } ) = L _ { 1 } ( \pmb { x } ) + L _ { 2 } ( \pmb { x } ) = \varphi ( R _ { 1 } f ( \pmb { x ; \theta } ) , 0 ) + \varphi ( R _ { 2 } f ( \pmb { x ; \theta } ) , 1 ) ,
+$$
+
+where $\varphi$ is a test function with a single global optimum whose position is parametrized by the second argument, that is, both tasks are identical (and thus related) up to a translation. We use a single input $\pmb { x } \in \mathbb { R } ^ { 2 }$ and drop task-specific network parameters. As backbone, we take a simple network of the form $z = W _ { 2 } \bar { \operatorname* { m a x } } ( \bar { W _ { 1 } x } + b _ { 1 } , 0 ) + b _ { 2 }$ with $b _ { 1 } \in \mathbb { R } ^ { 1 0 } , b _ { 2 } \in \mathbb { R } ^ { 2 }$ , and $W _ { 1 } , W _ { 2 } ^ { \top } \in \mathbb { R } ^ { 1 0 \times 2 }$ .
+
+For the first experiment we choose a simple (avocado-shaped) convex objective function and, for the second one, we opt for a non-convex function with several local optima and a single global optimum. Figure 1 shows the training trajectories in the presence (and absence) of RotoGrad in both experiments, depicted as level plots in the space of $_ { z }$ and $\mathbf { \nabla } r _ { k }$ , respectively. We can observe that in the first experiment (Figure 1a), RotoGrad finds both optima by rotating the feature space and matching the (unique) local optima of the tasks. Similarly, the second experiment (Figure 1b) shows that, as we have two symmetric tasks and a non-equidistant starting point, in the vanilla case the optimization is dominated by the task with an optimum closest to the starting point. RotoGrad avoids this behavior by equalizing gradients and, by aligning gradients, is able to find the optima of both functions.
+
+# 5 RELATED WORK
+
+Understanding and improving the interaction between tasks is one of the most fundamental problems of MTL, since any improvement in this regard would translate to all MTL systems. Consequently, several approaches to address this problem have been adopted in the literature. Among the different lines of work, the one most related to the present work is gradient homogenization.
+
+Gradient homogenization. Since the problem is two-fold, there are two main lines of work. On the one hand, we have task-weighting approaches that focus on alleviating magnitude differences. Similar to us, GradNorm (Chen et al., 2018) attempts to learn all tasks at a similar rate, yet they propose to learn these weights as parameters. Instead, we provide a closed-form solution in Equation 1, and so does IMTL-G Liu et al. (2021b). However, IMTL-G scales all task gradients such that all projections of $G$ onto $G _ { k }$ are equal. MGDA-UB (Sener & Koltun, 2018), instead, adopts an iterative method based on the Frank-Wolfe algorithm in order to find the set of weights $\left\{ \omega _ { k } \right\}$ (with $\textstyle \sum _ { k } \omega _ { k } = 1 )$ such that $\sum _ { k } \omega _ { k } G _ { k }$ has minimum norm. On the other hand, recent works have started to put attention on the conflicting direction problem. Maninis et al. (2019) and Sinha et al. (2018) proposed to make task gradients statistically indistinguishable via adversarial training. More recently, PCGrad (Yu et al., 2020) proposed to drop the projection of one task gradient onto another if they are in conflict, whereas GradDrop (Chen et al., 2020) randomly drops elements of the task gradients based on a sign-purity score. Contemporaneously to this work, improved versions of MGDA (Désidéri, 2012) and PCGrad have been proposed by Liu et al. (2021a) and Wang et al. (2021), respectively.
+
+The literature also includes approaches which, while orthogonal to the gradient homogenization, are complementary to our work and thus could be used along with RotoGrad. Next, we provide a brief overview of them. A prominent approach for MTL is task clustering, that is, selecting which tasks should be learned together. This approach dates back to the original task-clustering algorithm (Thrun & O’Sullivan, 1996), but new work in this direction keeps coming out (Standley et al., 2020; Zamir et al., 2018; Shen et al., 2021; Fifty et al., 2021). Alternative approaches, for example, scale the loss of each task differently based on different criteria such as task uncertainty (Kendall et al., 2018), task prioritization (Guo et al., 2018), or similar loss magnitudes (Liu et al., 2021b). Moreover, while most models fall into the hard-parameter sharing umbrella, there exists other architectures in the literature. Soft-parameter sharing architectures (Ruder, 2017), for example, do not have shared parameters but instead impose some kind of shared restrictions to the entire set of parameters. An interesting approach consists in letting the model itself learn which parts of the architecture should be used for each of the tasks (Guo et al., 2020; Misra et al., 2016; Sun et al., 2020; Vandenhende et al., 2020). Other architectures, such as MTAN (Liu et al., 2019b), make use of task-specific attention to select relevant features for each task. Finally, similar issues have also been studied in other domains like meta-learning (Flennerhag et al., 2019) and continual learning (Lopez-Paz & Ranzato, 2017).
+
+# 6 EXPERIMENTS
+
+In this section we assess the performance of RotoGrad on a wide range of datasets and MTL architectures. First, we check the effect of the learning rates of the rotation and network updates on the stability of RotoGrad. Then, with the goal of applying RotoGrad to scenarios with large sizes of $_ { z }$ , we explore the effect of rotating only a subspace of $_ z$ . Finally, we compare our approach with competing MTL solutions in the literature, showing that RotoGrad consistently outperforms all existing methods. Refer to Appendix C for a more details on the experiments and additional results.
+
+Relative task improvement. Since MTL uses different metrics for different tasks, throughout this section we group results by means of the relative task improvement, first introduced by Maninis et al. (2019). Given a task $k$ , and the metrics obtained during test time by a model, $M _ { k }$ , and by a baseline model, $S _ { k }$ , which consists of $K$ networks trained on each task individually, the relative task improvement for the $k$ -th task is defined as
+
+$$
+\Delta _ { k } : = 1 0 0 \cdot ( - 1 ) ^ { l _ { k } } \frac { M _ { k } - S _ { k } } { S _ { k } } ,
+$$
+
+where $l _ { k } = 1$ if $M _ { k } < S _ { k }$ means that our model performs better than the baseline in the $k \mathrm { . }$ -th task, and $l _ { k } = 0$ otherwise. We depict our results using different statistics of $\Delta _ { k }$ such as its mean $( \mathrm { a v g } _ { k } \Delta _ { k } )$ , maximum $( \operatorname* { m a x } _ { k } \Delta _ { k } )$ , and median $( { \mathrm { m e d } } _ { k } \Delta _ { k } )$ across tasks.
+
+Statistical significance. We highlight significant improvements according to a one-sided paired t-test $\langle \alpha = 0 . 0 5 \rangle$ ), with respect to MTL with vanilla optimization (marked with $\dagger$ in each table).
+
+# 6.1 TRAINING STABILITY
+
+At the end of $\ S$ we discussed that, by casting problem 4 as a Stackelberg game, we have convergence guarantees when the rotation optimizer is the slow-learner. Next, we empirically show this necessary condition.
+
+Experimental setup. Similar to (Sener & Koltun, 2018), we use a multitask version of MNIST (LeCun et al., 2010) where each image is composed of a left and right digit, and use as backbone a reduced version of LeNet (LeCun et al., 1998) with light-weight heads. Besides the leftand right-digit classification proposed in (Sener & Koltun, 2018), we consider three other quantities to predict: i) sum of digits; ii) parity of the digit product; and iii) number of active pixels. The idea here is to enforce all digit-related tasks to cooperate (positive transfer), while the (orthogonal) image-related task should not disrupt these learning dynamics.
+
+
+Figure 3: Test error on the sum of digits task for different values of RotoGrad’s learning rate on multi-MNIST.
+
+Results. Figure 3 shows the effect—averaged over ten independent runs—of changing the rotations’ learning rate in terms of test error in the sum task, while the rest of tasks are shown in Appendix C.2. We observe that, the bigger the learning rate is, in comparison to that of the network’s parameters (1e−3), the higher and noisier the test error becomes. MSE keeps decreasing as we lower the learning rate, reaching a sweet-spot at half the network’s learning rate (5e−4). For smaller values, the rotations’ learning is too slow and results start to resemble those of the vanilla case, in which no rotations are applied (leftmost box in Figure 3).
+
+# 6.2 ROTOGRAD BUILDING BLOCKS
+
+In this section, we empirically evaluate to which extent each of the RotoGrad building blocks, that we denote Scale Only (§3.1) and Rotate Only (§3.2), contribute to the performance gain in MTL.
+
+Experimental setup. We test all methods on three different tasks of NYUv2 (Couprie et al., 2013): 13-class semantic segmentation; depth estimation; and normal surfaces estimation. To speed up training, all images were resized to $2 8 8 \times 3 8 4$ resolution; and data augmentation was applied to alleviate overfitting. As MTL architecture, we use SegNet (Badrinarayanan et al., 2017) where the decoder is split into three convolutional heads. This is the same setup as that of Liu et al. (2019b).
+
+Results. The three top rows of Table 1 show the performance of RotoGrad and its both components in isolation. All the methods with the same number of parameters. Compared to Vanilla optimization (4th row), Rotate Only improves all metrics by homogenizing gradient directions. Scale Only avoids overlooking the normal estimation task and improves on segmentation by homogenizing gradient magnitudes, at the expense of higher depth estimation error. Remarkably, RotoGrad exploits its scaling and rotation components to obtain the best results in semantic segmentation and depth estimation, while still achieving comparable performance in the normal estimation task.
+
+# 6.3 SUBSPACE ROTATIONS
+
+We now evaluate the effect of subspace rotations as described at the end of $\ S$ , assessing the trade-off between avoiding negative transfer and size of the subspace considered by RotoGrad.
+
+Experimental setup. We test RotoGrad on a 10-task classification problem on CIFAR10 (Krizhevsky et al., 2009), using binary cross-entropy and f1-score as loss and metric, respectively, for all tasks. We use ResNet18 (He et al., 2016) without pre-training as backbone $\acute { a } = 5 1 2$ ), and linear layers with sigmoid activation functions as task-specific heads.
+
+Table 1: Median (over five runs) on the NYUv2 dataset. RotoGrad obtains great performance in segmentation and depth tasks, and significantly improves the results on normal surfaces. $\Delta _ { S }$ , $\Delta _ { D }$ , and $\Delta _ { N }$ denote the relative task improvement for each task.
+
+| Method | Left digit Right digit Product parity Sum digitsAcc.个 Acc. ↑ f1个 MSE↓ avg△k ↑ | Act. Pix.MSE↓ |
| SingleVanillat | 95.70± 0.20 94.05 ± 0.16 92.09 ± 0.76 1.90 ± 0.10 0.00±0.00 | 0.01 ± 0.01 |
| 94.94 ± 0.20 93.26±0.27 93.07 ± 0.48 2.10 ± 0.17 -3.26 ± 3.12 | 0.11 ± 0.01 |
| GradDrop | 95.33 ± 0.39 93.55 ± 0.29 | 93.32 ± 0.54 | 2.14 ± 0.07 -2.52 ± 1.63 | 0.13 ±0.02 |
| Whou PCGradMGDA-UBGradNormLSINN IMTL-GVanillaGradDrop | 95.07 ± 0.39 93.28 ± 0.18 | 93.34 ± 0.51 | 2.14 ± 0.19 -3.36 ±3.86 | 0.12 ±0.02 |
| 94.46 ± 1.04 92.23 ± 1.54 83.89 ± 1.84 2.50± 0.60-10.80±10.45 | 0.06± 0.02 |
| 95.19 ± 0.37 93.70± 0.31 93.31± 0.39 2.06 ± 28.71-1.81± 37.99 | 0.09 ± 7.46 |
| 95.28 ± 0.38 93.84 ± 0.21 93.24 ± 0.49 1.91 ± 6.61 -0.01 ± 82.48 | 0.07 ± 2.05 |
| 95.13 ± 0.20 93.41 ± 0.17 93.54± 0.50 1.99 ± 0.17 -1.43 ± 3.24 | 0.14 ±0.05 |
| 95.14 ± 0.16 93.47 ± 0.12 93.59 ± 0.32 | 2.00 ±0.06 -1.30 ±1.82 | 0.16 ± 0.04 |
| p PCGradMGDA-UBGradNormIMTL-GRotoGrad | 95.04 ± 0.26 93.36 ± 0.30 93.49 ± 0.30 | 1.98 ± 0.13 -1.22 ± 2.81 | 0.13 ± 0.01 |
| 89.99 ± 2.21 86.76 ± 1.18 79.24 ± 2.83 | 3.65± 0.42-29.14 ±9.23 | 0.06 ± 0.00 |
| 95.28 ± 0.18 93.56 ± 0.25 93.56± 0.57 | 1.86 ± 0.07 0.86 ± 1.93 | 0.09 ± 0.04 |
| 95.47 ± 0.27 93.79 ± 0.31 93.56± 0.5795.45 ± 0.19 93.83 ± 0.19 93.22 ± 0.35 | 1.73 ± 0.09 2.12 ± 1.461.85 ± 0.13 1.55 ± 2.22 | 0.07 ± 0.020.08 ± 0.03 |
| Single | 85.05 ± 0.45 84.58 ± 0.24 77.47 ± 1.13 5.84±0.14 0.00 ±0.00 | 0.17 ± 0.06 |
| Vanillat | 84.18 ± 0.30 84.18 ±0.38 80.11 ± 0.85 | 4.81 ± 0.06 5.14 ± 0.83 | 2.75 ± 3.17 |
| GradDrop | 84.38 ± 0.29 | 84.48 ± 0.41 80.11 ± 0.69 | 4.69 ± 0.12 5.68 ± 1.05 | 1.91 ± 0.86 |
| PCGrad | 84.22 ± 0.31 | 84.23 ± 0.21 79.92 ± 0.79 | 4.69 ± 0.09 5.50 ± 0.75 | 2.26±0.85 |
| MGDA-UBGradNormNHAS IMTL-GVanillaGradDrop | MGDA-UB | MGDA-UBGradNormIMTL-G | 84.61 ± 0.75 | 84.38 ± 0.45 77.44 ± 1.44 | 4.47 ± 0.18 5.99 ± 1.48 |
| 84.23 ± 0.33 | 84.13 ± 0.30 79.40 ± 0.87 | 4.92 ± 0.07 4.60 ± 1.01 | 4.30 ± 2.18 |
| 84.60 ± 0.45 | 84.39 ± 0.37 79.63 ± 1.10 | 4.57 ± 0.13 5.81 ± 0.85 | 2.47 ± 1.65 |
| 84.11 ±0.48 84.11 ±0.40 79.83 ± 0.79 | 4.84 ±0.10 4.78±0.88 | 3.04±2.53 |
| 84.23 ± 0.35 84.33 ±0.40 80.10 ± 0.83 | 4.73 ± 0.09 5.34 ± 0.92 | 2.99 ± 2.59 |
| PCGrad | 84.21 ± 0.21 84.26 ±0.38 79.64 ± 0.48 | 4.84 ± 0.06 5.01 ± 0.65 | 2.70± 2.25 |
| pm MGDA-UBGradNormIMTL-GRotoGrad | 77.05± 5.44 | 78.00±5.04 71.76± 4.32 | 5.27 ± 0.56 -4.36± 6.72 | 1.00 ± 0.57 |
| 84.37 ± 0.34 | 84.30 ±0.46 79.97 ± 0.75 | 4.72 ± 0.13 5.24± 0.89 | 4.12 ± 9.46 |
| 84.23 ± 0.34 | 84.23 ± 0.39 79.77 ± 1.04 | 4.51 ± 0.12 5.94 ± 0.99 | 1.70 ± 1.051.61 ± 2.72 |
| 84.60± 0.50 | 84.44 ± 0.45 79.14 ± 0.96 | 4.45 ± 0.10 6.08 ± 0.48 |
+
+# C.2.2 ILLUSTRATIVE EXAMPLES
+
+We complement the illustrative figures shown in Figure 1 by providing, for each example, an illustration of the effect of RotoGrad shown as an active and passive transformation. In an active transformation (Figure 6 left), points in the space are the ones modified. In our case, this means that we rotate feature $_ z$ , obtaining $\mathbf { \Delta } _ { \mathbf { r } _ { 1 } }$ and $\mathbf { \boldsymbol { r } } _ { 2 }$ while the loss functions remain the same. In other words, for each $_ z$ we obtain a taskspecific feature $\mathbfit { r } _ { k }$ that optimizes its loss function. In contrast, a passive transformation (Figure 6 right) keeps the points unaltered while applying the transformation to the space itself. In our case, this translates to rotating the optimization landscape of each loss function (now we have $L _ { k } \circ R _ { k }$ instead of $L _ { K }$ ), so that our single feature $_ z$ has an easier job at optimizing both tasks. In the case of RotoGrad, we can observe in both right figures that both optima lie in the same point, as we are aligning task gradients.
+
+Besides the two regression experiments shown in $\ S 4$ , we include here an additional experiment where we test RotoGrad in the worst-case scenario of gradient conflict, that is, one in which task gradients are opposite to each other. To this end, we solve a 2-task binary classification problem where, as dataset, we take 1000 samples from a 2D Gaussian mixture model with two clusters; $y _ { n , k } = 1$ if ${ \mathbf { \mathcal { x } } } _ { n }$ was sampled from cluster $k$ ; and $\scriptstyle y _ { n , k } = 0$ otherwise. We use as model a logistic regression model of the form ${ \pmb y } _ { k } = { \pmb W } _ { 2 } \operatorname* { m a x } ( { \pmb W } _ { 1 } { \pmb x } + { \pmb b } _ { 1 } , 0 ) + { \pmb b } _ { 2 }$ with $\dot { b } _ { 1 } \in \mathbb { R } ^ { 2 } , b _ { 2 } \in \dot { \mathbb { R } }$ , $W _ { 1 } \in \mathbb { R } ^ { 2 \times 2 }$ , and $W _ { 2 } \in \mathbb { R } ^ { 1 \times 2 }$ . Because rotations in 1D are ill-posed (there is a unique proper rotation), here we add task parameters to increase the dimensionality of $_ z$ and make all parameters shared, so that there is still no
+
+
+Figure 6: Level plots showing the illustrative examples of Figure 1 for RotoGrad. Top: Convex case. Bottom: Non-convex case. Left: Active transformation (trajectories of $\mathbf { \nabla } r _ { k }$ and the level plot of $L _ { 1 } + L _ { 2 }$ . Right: Passive transformation (trajectory of $_ { z }$ and level plot of $( L _ { 1 } \circ R _ { 1 } ) + ( L _ { 2 } \circ R _ { 2 } ) )$ .
+
+
+Figure 7: Logistic regression for opposite classification tasks. Test data is plotted scattered as gray dots. RotoGrad learns both opposite rotations $R _ { 1 } \stackrel { \mathbf { \Delta } } { = } \dot { R } _ { 2 } ^ { \top }$ .
+
+task-specific parameters. To avoid a complete conflict where $\nabla _ { z } L _ { 1 } + \nabla _ { z } L _ { 2 } = 0$ , we randomly flip the labels for the second tasks with $5 \%$ probability. Figure 7 shows that, in this extreme scenario, RotoGrad is able to learn both tasks by aligning gradients, that is, by learning that one rotation is the inverse of the other $R _ { 1 } = R _ { 2 } ^ { \top }$ .
+
+# C.2.3 TRAINING STABILITY
+
+While we showed in $\ S$ only the results for the sum-of-digits task as they were nice and clear, here we show in Figure 8 the results of those same experiments in $\ S 6 . 1$ for all the different tasks. The same discussion from the main manuscript can be carried out for all metrics. Additionally, we can observe that the vanilla case (learning rate zero) completely overlooks the image-related task (Active pixels) while performing the best in the parity task.
+
+Additionally, let us clarify what we mean here with stability, as in the main manuscript we mainly talked about convergence guarantees. In these experiments we measure the convergence guarantees of the experiments in terms of ‘training stability’, meaning the variance of the obtained results across different runs. The intuition here is that, since the model does not converge, we should expect some wriggling learning curves during training and, as we take the model with the best validation error, the individual task metrics should have bigger variance (that is, less stability) across runs.
+
+
+Figure 8: RotoGrad’s performance on all tasks for the experiments in $\ S 6 . 1$ for all metrics. We can observe training instabilities/stiffness on all tasks as we highly increase/decrease RotoGrad’s learning rate, as discussed in the main manuscript.
+
+# C.2.4 CIFAR10 AND CELEBA
+
+For the sake of completeness, we present in Table 6 and Table 7 the same tables as in $\ S 6 .$ , but with more statistics of the results. For CIFAR10, we now included in Table 6 the minimum task improvement across tasks and, while noisier, we can still observe that RotoGrad also improve this statistic. The standard deviation of the task improvement across tasks is, however, not too informative. We also include in Figure 9 the cosine similarity plots for the different subspace rotations from $\ S$ , showing a clear positive correlation between the cosine similarity and the size of the considered subspace. While the cosine similarities look low, we want to remark that we are computing the cosine similarity of a huge space, and we only align a subspace of it. If we, instead, showed the cosine similarity with respect to each specific subspace, the cosine similarity should look similar to that of RotoGrad 512. In the case of CelebA, we added in Table 7 the maximum f1-score across tasks and, similar to the last case, it is not too informative, as all methods achieve almost perfect f1-score in one of the classes. We also include the training times for some baselines, showing that RotoGrad stays on par with them.
+
+# C.2.5 NYUV2
+
+Complementing the results shown in $\ S 6$ , we show in Table 8 the results obtained combining RotoGrad with all other existing methods (rows within RotoGrad $+$ ), for gradient scaling methods we only apply the rotation part of RotoGrad. Results show that RotoGrad helps improve/balance all other methods, which is specially true for those methods that heavily overlook some tasks. Specifically, MGDA-UB stops overlooking the semantic segmentation and depth estimation tasks, while PCGrad and GradDrop stop completely overlooking the surface normal loss. Note that we also show in Table 8 the training times of each method, and RotoGrad stays on par with non-extended methods in training time. As mentioned in Appendix C.1, due to cluster overload, some times were deceivingly high (specifically those baselines with $\scriptstyle { \mathbf { } } _ { R _ { k } }$ ) as we had to run them on different machines, and were omitted to avoid confusion.
+
+Table 6: Complete task-improvement statistics in CIFAR10 for all competing methods and RotoGrad with different dimensionality for $\scriptstyle { \mathbf { } } _ { R _ { k } }$ , see $\ S 6$ .
+
+| Method d | mink △k ↑ med△k ↑ avgk△k ↑ std△k↓ maxk△k ↑ |
| Vanillat 0 | -0.81± 0.37 1.90 ± 0.53 2.58 ± 0.54 3.38 ± 0.94 11.14 ± 3.35 |
| RotoGrad 64 | -1.70 ± 0.81 1.79 ± 0.57 2.90 ± 0.49 3.98 ± 0.62 13.16 ± 2.40 |
| RotoGrad 128 | -1.12 ± 0.36 2.25 ± 1.07 2.97 ± 1.08 3.84 ± 0.87 | 12.64 ± 3.56 |
| RotoGrad 256 | 0.17 ± 1.01 2.16 ± 0.72 3.68 ± 0.68 3.83 ± 0.74 | 14.01 ± 3.22 |
| RotoGrad 512 | -0.43 ± 0.76 3.67 ± 1.40 4.48 ± 0.99 | 4.23 ± 0.82 | 15.57 ± 3.99 |
| Vanillat | -0.81± 0.37 1.90 ± 0.53 2.58 ± 0.54 | 3.38 ± 0.94 | 11.14 ± 3.35 |
| Wrehtntt GradDropPCGradMGDA-UBGradNormIMTL-G | -0.73 ± 0.33 2.80 ± 0.20 | 3.41 ± 0.45 | 4.08 ± 0.34 | 13.58 ± 1.50 |
| -1.52 ± 0.98 1.95 ± 0.87 | 2.86 ± 0.81 | 3.74 ± 0.69 | 12.01 ± 3.19 |
| -7.27 ± 1.36 -1.21 ± 0.74 | -1.75 ± 0.43 | 3.24 ± 0.55 | 3.67 ± 0.98 |
| -0.35 ± 0.59 2.45 ± 0.66 | 3.23 ± 0.35 | 4.02 ± 0.33 | 14.25 ± 1.35 |
| -0.39 ±0.82 1.97 ± 0.29 2.73 ± 0.27 | 3.25 ± 0.75 | 10.20 ± 2.98 |
| Vanilla | -0.85 ± 0.58 3.10 ± 1.29 3.12 ± 0.79 | 4.05 ± 0.56 | 14.23 ± 2.86 |
| GradDropHmA(1= p) PCGradMGDA-UBGradNormIMTL-G | -1.49 ± 0.78 3.27 ± 1.61 | 3.54 ±1.10 | 4.11 ± 0.56 | 13.88 ± 2.95 |
| -1.44 ± 0.58 2.67 ± 0.88 | 3.29 ± 0.46 | 3.90 ± 0.37 | 13.44 ± 1.86 |
| -3.59 ± 1.48 | 0.57 ± 0.62 | 0.21 ± 0.67 | 2.44 ± 0.52 | 4.78 ± 2.15 |
| -0.79 ±1.28 | 3.10 ± 1.01 | 3.21 ± 1.04 | 3.41 ± 0.86 | 10.88 ± 4.73 |
| -1.29 ± 0.52 | 1.81 ± 0.87 | 3.02 ± 0.69 | 3.81± 0.21 | 12.76 ± 1.77 |
| RotoGrad | -0.43 ± 0.76 3.67 ± 1.40 | 4.48 ± 0.99 | 4.23 ± 0.82 | 15.57 ± 3.99 |
+
+Table 7: Complete f1-score statistics and training hours in CelebA for all competing methods and two different architectures/settings (median over five runs), see $\ S$ . For the convolutional network we use $m = 2 5 6$ , and $m = 1 5 3 6$ for the residual network.
+
+| Algorithm1Training stepwith RotoGrad. |
| Input input samples X,task labels {Yk}, network's (RotoGrad's) learning rate n (mroto) Output backbone (heads) parameters 0 ({Φk}), RotoGrad's parameters {Rk} |
| 1: compute shared feature Z = f(X;0) |
| 2: for k=1,2,...,K do |
| 3: compute task-specific loss Lk = ∑nLk(hk(Rk Zn; Φk), yn,k) |
| 4: compute gradient of shared feature Gk = VzLk |
| 5: compute gradient of task-specific feature Gk = RkGk Treated as constant w.r.t. Rk . |
| 6: compute unitary gradients Uk = Gk/llGkll |
| 7: compute relative task convergence αk = ||Gkll/IlG | |
| 8: end for |
| 9: make {αk} sum up to one [α1,α2,...,αk] = [α1,α2,...,αk]/∑k αk |
| 10: compute shared magnitude C = ∑k αkllGkll |
| 11: update backbone parameters 0 =0"- nVθz : C∑κUk compute target vector V = k ∑k Uk |
",
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+ "type": "text",
+ "text": "Memory efficiency and time complexity. As we need one rotation matrix per task, we have to store $\\mathcal { O } \\hat { (} K d ^ { 2 } \\hat { ) }$ additional parameters. In practice, we only need $K d ( d - 1 ) / 2$ parameters due to the aforementioned parametrization and, in most cases, this amounts to a small part of the total number of parameters. Moreover, as described by Casado & Martínez-Rubio (2019), parametrizing $\\scriptstyle { R _ { k } }$ enables efficient computations compared with traditional methods, with a time complexity of $O ( d ^ { 3 } )$ independently of the batch size. In our case, the time complexity is of $O ( K d ^ { 3 } )$ , which scales better with respect to the number of tasks than existing methods (for example, $O ( K ^ { 2 } d )$ for PCGrad ( $\\mathrm { T u }$ et al., 2020)). Forward-pass caching and GPU parallelization can further reduce training time. ",
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+ "type": "text",
+ "text": "Scaling-up RotoGrad. Despite being able to efficiently compute and optimize the rotation matrix $\\scriptstyle R _ { k }$ , in domains like computer vision, where the size $d$ of the shared representation $_ z$ is large, the time complexity for updating the rotation matrix may become comparable to the one of the network updates. In those cases, we propose to only rotate a subspace of the feature space, that is, rotate only $m < < d$ dimensions of $_ z$ . Then, we can simply apply a transformation of the form $r _ { k } = [ R _ { k } z _ { 1 : m } , z _ { m + 1 : d } ]$ , where $z _ { a : b }$ denotes the elements of $_ z$ with indexes $a , a + 1 , \\ldots , b$ . While there exist other possible solutions, such as using block-diagonal rotation matrices $\\scriptstyle { \\cal R } _ { k }$ , we defer them to future work. ",
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+ "text": "4 ILLUSTRATIVE EXAMPLES ",
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+ "text": "In this section, we illustrate the behavior of RotoGrad in two synthetic scenarios, providing clean qualitative results about its effect on the optimization process. Appendix C.1 provides a detailed description of the experimental setups. ",
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+ "text": "To this end, we propose two different multitask regression problems of the form ",
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+ "img_path": "images/b5fb521f7c225c8f1fa958c7aa38c3d31e726a1340ece64d8b6989e97952956e.jpg",
+ "text": "$$\nL ( \\pmb { x } ) = L _ { 1 } ( \\pmb { x } ) + L _ { 2 } ( \\pmb { x } ) = \\varphi ( R _ { 1 } f ( \\pmb { x ; \\theta } ) , 0 ) + \\varphi ( R _ { 2 } f ( \\pmb { x ; \\theta } ) , 1 ) ,\n$$",
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+ "text": "where $\\varphi$ is a test function with a single global optimum whose position is parametrized by the second argument, that is, both tasks are identical (and thus related) up to a translation. We use a single input $\\pmb { x } \\in \\mathbb { R } ^ { 2 }$ and drop task-specific network parameters. As backbone, we take a simple network of the form $z = W _ { 2 } \\bar { \\operatorname* { m a x } } ( \\bar { W _ { 1 } x } + b _ { 1 } , 0 ) + b _ { 2 }$ with $b _ { 1 } \\in \\mathbb { R } ^ { 1 0 } , b _ { 2 } \\in \\mathbb { R } ^ { 2 }$ , and $W _ { 1 } , W _ { 2 } ^ { \\top } \\in \\mathbb { R } ^ { 1 0 \\times 2 }$ . ",
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+ "type": "text",
+ "text": "For the first experiment we choose a simple (avocado-shaped) convex objective function and, for the second one, we opt for a non-convex function with several local optima and a single global optimum. Figure 1 shows the training trajectories in the presence (and absence) of RotoGrad in both experiments, depicted as level plots in the space of $_ { z }$ and $\\mathbf { \\nabla } r _ { k }$ , respectively. We can observe that in the first experiment (Figure 1a), RotoGrad finds both optima by rotating the feature space and matching the (unique) local optima of the tasks. Similarly, the second experiment (Figure 1b) shows that, as we have two symmetric tasks and a non-equidistant starting point, in the vanilla case the optimization is dominated by the task with an optimum closest to the starting point. RotoGrad avoids this behavior by equalizing gradients and, by aligning gradients, is able to find the optima of both functions. ",
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+ "text": "5 RELATED WORK ",
+ "text_level": 1,
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+ "text": "Understanding and improving the interaction between tasks is one of the most fundamental problems of MTL, since any improvement in this regard would translate to all MTL systems. Consequently, several approaches to address this problem have been adopted in the literature. Among the different lines of work, the one most related to the present work is gradient homogenization. ",
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+ "text": "Gradient homogenization. Since the problem is two-fold, there are two main lines of work. On the one hand, we have task-weighting approaches that focus on alleviating magnitude differences. Similar to us, GradNorm (Chen et al., 2018) attempts to learn all tasks at a similar rate, yet they propose to learn these weights as parameters. Instead, we provide a closed-form solution in Equation 1, and so does IMTL-G Liu et al. (2021b). However, IMTL-G scales all task gradients such that all projections of $G$ onto $G _ { k }$ are equal. MGDA-UB (Sener & Koltun, 2018), instead, adopts an iterative method based on the Frank-Wolfe algorithm in order to find the set of weights $\\left\\{ \\omega _ { k } \\right\\}$ (with $\\textstyle \\sum _ { k } \\omega _ { k } = 1 )$ such that $\\sum _ { k } \\omega _ { k } G _ { k }$ has minimum norm. On the other hand, recent works have started to put attention on the conflicting direction problem. Maninis et al. (2019) and Sinha et al. (2018) proposed to make task gradients statistically indistinguishable via adversarial training. More recently, PCGrad (Yu et al., 2020) proposed to drop the projection of one task gradient onto another if they are in conflict, whereas GradDrop (Chen et al., 2020) randomly drops elements of the task gradients based on a sign-purity score. Contemporaneously to this work, improved versions of MGDA (Désidéri, 2012) and PCGrad have been proposed by Liu et al. (2021a) and Wang et al. (2021), respectively. ",
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+ "text": "The literature also includes approaches which, while orthogonal to the gradient homogenization, are complementary to our work and thus could be used along with RotoGrad. Next, we provide a brief overview of them. A prominent approach for MTL is task clustering, that is, selecting which tasks should be learned together. This approach dates back to the original task-clustering algorithm (Thrun & O’Sullivan, 1996), but new work in this direction keeps coming out (Standley et al., 2020; Zamir et al., 2018; Shen et al., 2021; Fifty et al., 2021). Alternative approaches, for example, scale the loss of each task differently based on different criteria such as task uncertainty (Kendall et al., 2018), task prioritization (Guo et al., 2018), or similar loss magnitudes (Liu et al., 2021b). Moreover, while most models fall into the hard-parameter sharing umbrella, there exists other architectures in the literature. Soft-parameter sharing architectures (Ruder, 2017), for example, do not have shared parameters but instead impose some kind of shared restrictions to the entire set of parameters. An interesting approach consists in letting the model itself learn which parts of the architecture should be used for each of the tasks (Guo et al., 2020; Misra et al., 2016; Sun et al., 2020; Vandenhende et al., 2020). Other architectures, such as MTAN (Liu et al., 2019b), make use of task-specific attention to select relevant features for each task. Finally, similar issues have also been studied in other domains like meta-learning (Flennerhag et al., 2019) and continual learning (Lopez-Paz & Ranzato, 2017). ",
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+ "text": "6 EXPERIMENTS ",
+ "text_level": 1,
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+ "text": "In this section we assess the performance of RotoGrad on a wide range of datasets and MTL architectures. First, we check the effect of the learning rates of the rotation and network updates on the stability of RotoGrad. Then, with the goal of applying RotoGrad to scenarios with large sizes of $_ { z }$ , we explore the effect of rotating only a subspace of $_ z$ . Finally, we compare our approach with competing MTL solutions in the literature, showing that RotoGrad consistently outperforms all existing methods. Refer to Appendix C for a more details on the experiments and additional results. ",
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+ "text": "Relative task improvement. Since MTL uses different metrics for different tasks, throughout this section we group results by means of the relative task improvement, first introduced by Maninis et al. (2019). Given a task $k$ , and the metrics obtained during test time by a model, $M _ { k }$ , and by a baseline model, $S _ { k }$ , which consists of $K$ networks trained on each task individually, the relative task improvement for the $k$ -th task is defined as ",
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+ "text": "$$\n\\Delta _ { k } : = 1 0 0 \\cdot ( - 1 ) ^ { l _ { k } } \\frac { M _ { k } - S _ { k } } { S _ { k } } ,\n$$",
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+ "text": "where $l _ { k } = 1$ if $M _ { k } < S _ { k }$ means that our model performs better than the baseline in the $k \\mathrm { . }$ -th task, and $l _ { k } = 0$ otherwise. We depict our results using different statistics of $\\Delta _ { k }$ such as its mean $( \\mathrm { a v g } _ { k } \\Delta _ { k } )$ , maximum $( \\operatorname* { m a x } _ { k } \\Delta _ { k } )$ , and median $( { \\mathrm { m e d } } _ { k } \\Delta _ { k } )$ across tasks. ",
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+ "text": "Statistical significance. We highlight significant improvements according to a one-sided paired t-test $\\langle \\alpha = 0 . 0 5 \\rangle$ ), with respect to MTL with vanilla optimization (marked with $\\dagger$ in each table). ",
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+ "text": "6.1 TRAINING STABILITY ",
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+ "text": "At the end of $\\ S$ we discussed that, by casting problem 4 as a Stackelberg game, we have convergence guarantees when the rotation optimizer is the slow-learner. Next, we empirically show this necessary condition. ",
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+ "text": "Experimental setup. Similar to (Sener & Koltun, 2018), we use a multitask version of MNIST (LeCun et al., 2010) where each image is composed of a left and right digit, and use as backbone a reduced version of LeNet (LeCun et al., 1998) with light-weight heads. Besides the leftand right-digit classification proposed in (Sener & Koltun, 2018), we consider three other quantities to predict: i) sum of digits; ii) parity of the digit product; and iii) number of active pixels. The idea here is to enforce all digit-related tasks to cooperate (positive transfer), while the (orthogonal) image-related task should not disrupt these learning dynamics. ",
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+ "Figure 3: Test error on the sum of digits task for different values of RotoGrad’s learning rate on multi-MNIST. "
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+ "text": "Results. Figure 3 shows the effect—averaged over ten independent runs—of changing the rotations’ learning rate in terms of test error in the sum task, while the rest of tasks are shown in Appendix C.2. We observe that, the bigger the learning rate is, in comparison to that of the network’s parameters (1e−3), the higher and noisier the test error becomes. MSE keeps decreasing as we lower the learning rate, reaching a sweet-spot at half the network’s learning rate (5e−4). For smaller values, the rotations’ learning is too slow and results start to resemble those of the vanilla case, in which no rotations are applied (leftmost box in Figure 3). ",
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+ "text": "6.2 ROTOGRAD BUILDING BLOCKS ",
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+ "type": "text",
+ "text": "In this section, we empirically evaluate to which extent each of the RotoGrad building blocks, that we denote Scale Only (§3.1) and Rotate Only (§3.2), contribute to the performance gain in MTL. ",
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+ "text": "Experimental setup. We test all methods on three different tasks of NYUv2 (Couprie et al., 2013): 13-class semantic segmentation; depth estimation; and normal surfaces estimation. To speed up training, all images were resized to $2 8 8 \\times 3 8 4$ resolution; and data augmentation was applied to alleviate overfitting. As MTL architecture, we use SegNet (Badrinarayanan et al., 2017) where the decoder is split into three convolutional heads. This is the same setup as that of Liu et al. (2019b). ",
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+ "text": "Results. The three top rows of Table 1 show the performance of RotoGrad and its both components in isolation. All the methods with the same number of parameters. Compared to Vanilla optimization (4th row), Rotate Only improves all metrics by homogenizing gradient directions. Scale Only avoids overlooking the normal estimation task and improves on segmentation by homogenizing gradient magnitudes, at the expense of higher depth estimation error. Remarkably, RotoGrad exploits its scaling and rotation components to obtain the best results in semantic segmentation and depth estimation, while still achieving comparable performance in the normal estimation task. ",
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+ "text": "6.3 SUBSPACE ROTATIONS ",
+ "text_level": 1,
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+ "text": "We now evaluate the effect of subspace rotations as described at the end of $\\ S$ , assessing the trade-off between avoiding negative transfer and size of the subspace considered by RotoGrad. ",
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+ "text": "Experimental setup. We test RotoGrad on a 10-task classification problem on CIFAR10 (Krizhevsky et al., 2009), using binary cross-entropy and f1-score as loss and metric, respectively, for all tasks. We use ResNet18 (He et al., 2016) without pre-training as backbone $\\acute { a } = 5 1 2$ ), and linear layers with sigmoid activation functions as task-specific heads. ",
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+ "Table 1: Median (over five runs) on the NYUv2 dataset. RotoGrad obtains great performance in segmentation and depth tasks, and significantly improves the results on normal surfaces. $\\Delta _ { S }$ , $\\Delta _ { D }$ , and $\\Delta _ { N }$ denote the relative task improvement for each task. "
+ ],
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+ "table_body": "| Method | Left digit Right digit Product parity Sum digitsAcc.个 Acc. ↑ f1个 MSE↓ avg△k ↑ | Act. Pix.MSE↓ |
| SingleVanillat | 95.70± 0.20 94.05 ± 0.16 92.09 ± 0.76 1.90 ± 0.10 0.00±0.00 | 0.01 ± 0.01 |
| 94.94 ± 0.20 93.26±0.27 93.07 ± 0.48 2.10 ± 0.17 -3.26 ± 3.12 | 0.11 ± 0.01 |
| GradDrop | 95.33 ± 0.39 93.55 ± 0.29 | 93.32 ± 0.54 | 2.14 ± 0.07 -2.52 ± 1.63 | 0.13 ±0.02 |
| Whou PCGradMGDA-UBGradNormLSINN IMTL-GVanillaGradDrop | 95.07 ± 0.39 93.28 ± 0.18 | 93.34 ± 0.51 | 2.14 ± 0.19 -3.36 ±3.86 | 0.12 ±0.02 |
| 94.46 ± 1.04 92.23 ± 1.54 83.89 ± 1.84 2.50± 0.60-10.80±10.45 | 0.06± 0.02 |
| 95.19 ± 0.37 93.70± 0.31 93.31± 0.39 2.06 ± 28.71-1.81± 37.99 | 0.09 ± 7.46 |
| 95.28 ± 0.38 93.84 ± 0.21 93.24 ± 0.49 1.91 ± 6.61 -0.01 ± 82.48 | 0.07 ± 2.05 |
| 95.13 ± 0.20 93.41 ± 0.17 93.54± 0.50 1.99 ± 0.17 -1.43 ± 3.24 | 0.14 ±0.05 |
| 95.14 ± 0.16 93.47 ± 0.12 93.59 ± 0.32 | 2.00 ±0.06 -1.30 ±1.82 | 0.16 ± 0.04 |
| p PCGradMGDA-UBGradNormIMTL-GRotoGrad | 95.04 ± 0.26 93.36 ± 0.30 93.49 ± 0.30 | 1.98 ± 0.13 -1.22 ± 2.81 | 0.13 ± 0.01 |
| 89.99 ± 2.21 86.76 ± 1.18 79.24 ± 2.83 | 3.65± 0.42-29.14 ±9.23 | 0.06 ± 0.00 |
| 95.28 ± 0.18 93.56 ± 0.25 93.56± 0.57 | 1.86 ± 0.07 0.86 ± 1.93 | 0.09 ± 0.04 |
| 95.47 ± 0.27 93.79 ± 0.31 93.56± 0.5795.45 ± 0.19 93.83 ± 0.19 93.22 ± 0.35 | 1.73 ± 0.09 2.12 ± 1.461.85 ± 0.13 1.55 ± 2.22 | 0.07 ± 0.020.08 ± 0.03 |
| Single | 85.05 ± 0.45 84.58 ± 0.24 77.47 ± 1.13 5.84±0.14 0.00 ±0.00 | 0.17 ± 0.06 |
| Vanillat | 84.18 ± 0.30 84.18 ±0.38 80.11 ± 0.85 | 4.81 ± 0.06 5.14 ± 0.83 | 2.75 ± 3.17 |
| GradDrop | 84.38 ± 0.29 | 84.48 ± 0.41 80.11 ± 0.69 | 4.69 ± 0.12 5.68 ± 1.05 | 1.91 ± 0.86 |
| PCGrad | 84.22 ± 0.31 | 84.23 ± 0.21 79.92 ± 0.79 | 4.69 ± 0.09 5.50 ± 0.75 | 2.26±0.85 |
| MGDA-UBGradNormNHAS IMTL-GVanillaGradDrop | MGDA-UB | MGDA-UBGradNormIMTL-G | 84.61 ± 0.75 | 84.38 ± 0.45 77.44 ± 1.44 | 4.47 ± 0.18 5.99 ± 1.48 |
| 84.23 ± 0.33 | 84.13 ± 0.30 79.40 ± 0.87 | 4.92 ± 0.07 4.60 ± 1.01 | 4.30 ± 2.18 |
| 84.60 ± 0.45 | 84.39 ± 0.37 79.63 ± 1.10 | 4.57 ± 0.13 5.81 ± 0.85 | 2.47 ± 1.65 |
| 84.11 ±0.48 84.11 ±0.40 79.83 ± 0.79 | 4.84 ±0.10 4.78±0.88 | 3.04±2.53 |
| 84.23 ± 0.35 84.33 ±0.40 80.10 ± 0.83 | 4.73 ± 0.09 5.34 ± 0.92 | 2.99 ± 2.59 |
| PCGrad | 84.21 ± 0.21 84.26 ±0.38 79.64 ± 0.48 | 4.84 ± 0.06 5.01 ± 0.65 | 2.70± 2.25 |
| pm MGDA-UBGradNormIMTL-GRotoGrad | 77.05± 5.44 | 78.00±5.04 71.76± 4.32 | 5.27 ± 0.56 -4.36± 6.72 | 1.00 ± 0.57 |
| 84.37 ± 0.34 | 84.30 ±0.46 79.97 ± 0.75 | 4.72 ± 0.13 5.24± 0.89 | 4.12 ± 9.46 |
| 84.23 ± 0.34 | 84.23 ± 0.39 79.77 ± 1.04 | 4.51 ± 0.12 5.94 ± 0.99 | 1.70 ± 1.051.61 ± 2.72 |
| 84.60± 0.50 | 84.44 ± 0.45 79.14 ± 0.96 | 4.45 ± 0.10 6.08 ± 0.48 |
",
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+ "text": "C.2.2 ILLUSTRATIVE EXAMPLES ",
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+ "text": "We complement the illustrative figures shown in Figure 1 by providing, for each example, an illustration of the effect of RotoGrad shown as an active and passive transformation. In an active transformation (Figure 6 left), points in the space are the ones modified. In our case, this means that we rotate feature $_ z$ , obtaining $\\mathbf { \\Delta } _ { \\mathbf { r } _ { 1 } }$ and $\\mathbf { \\boldsymbol { r } } _ { 2 }$ while the loss functions remain the same. In other words, for each $_ z$ we obtain a taskspecific feature $\\mathbfit { r } _ { k }$ that optimizes its loss function. In contrast, a passive transformation (Figure 6 right) keeps the points unaltered while applying the transformation to the space itself. In our case, this translates to rotating the optimization landscape of each loss function (now we have $L _ { k } \\circ R _ { k }$ instead of $L _ { K }$ ), so that our single feature $_ z$ has an easier job at optimizing both tasks. In the case of RotoGrad, we can observe in both right figures that both optima lie in the same point, as we are aligning task gradients. ",
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+ "text": "Besides the two regression experiments shown in $\\ S 4$ , we include here an additional experiment where we test RotoGrad in the worst-case scenario of gradient conflict, that is, one in which task gradients are opposite to each other. To this end, we solve a 2-task binary classification problem where, as dataset, we take 1000 samples from a 2D Gaussian mixture model with two clusters; $y _ { n , k } = 1$ if ${ \\mathbf { \\mathcal { x } } } _ { n }$ was sampled from cluster $k$ ; and $\\scriptstyle y _ { n , k } = 0$ otherwise. We use as model a logistic regression model of the form ${ \\pmb y } _ { k } = { \\pmb W } _ { 2 } \\operatorname* { m a x } ( { \\pmb W } _ { 1 } { \\pmb x } + { \\pmb b } _ { 1 } , 0 ) + { \\pmb b } _ { 2 }$ with $\\dot { b } _ { 1 } \\in \\mathbb { R } ^ { 2 } , b _ { 2 } \\in \\dot { \\mathbb { R } }$ , $W _ { 1 } \\in \\mathbb { R } ^ { 2 \\times 2 }$ , and $W _ { 2 } \\in \\mathbb { R } ^ { 1 \\times 2 }$ . Because rotations in 1D are ill-posed (there is a unique proper rotation), here we add task parameters to increase the dimensionality of $_ z$ and make all parameters shared, so that there is still no ",
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+ "img_path": "images/78ffefbfae88244f23f8e3836db9f3cab3ef7efeb2bc0b9cd2fc7f4bc36204b3.jpg",
+ "image_caption": [
+ "Figure 6: Level plots showing the illustrative examples of Figure 1 for RotoGrad. Top: Convex case. Bottom: Non-convex case. Left: Active transformation (trajectories of $\\mathbf { \\nabla } r _ { k }$ and the level plot of $L _ { 1 } + L _ { 2 }$ . Right: Passive transformation (trajectory of $_ { z }$ and level plot of $( L _ { 1 } \\circ R _ { 1 } ) + ( L _ { 2 } \\circ R _ { 2 } ) )$ . "
+ ],
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+ "img_path": "images/de287210430ce69c1ea927f9d84c2ef8529527ad5ea281127801030478ee6306.jpg",
+ "image_caption": [
+ "Figure 7: Logistic regression for opposite classification tasks. Test data is plotted scattered as gray dots. RotoGrad learns both opposite rotations $R _ { 1 } \\stackrel { \\mathbf { \\Delta } } { = } \\dot { R } _ { 2 } ^ { \\top }$ . "
+ ],
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+ {
+ "type": "text",
+ "text": "task-specific parameters. To avoid a complete conflict where $\\nabla _ { z } L _ { 1 } + \\nabla _ { z } L _ { 2 } = 0$ , we randomly flip the labels for the second tasks with $5 \\%$ probability. Figure 7 shows that, in this extreme scenario, RotoGrad is able to learn both tasks by aligning gradients, that is, by learning that one rotation is the inverse of the other $R _ { 1 } = R _ { 2 } ^ { \\top }$ . ",
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+ "type": "text",
+ "text": "C.2.3 TRAINING STABILITY ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "While we showed in $\\ S$ only the results for the sum-of-digits task as they were nice and clear, here we show in Figure 8 the results of those same experiments in $\\ S 6 . 1$ for all the different tasks. The same discussion from the main manuscript can be carried out for all metrics. Additionally, we can observe that the vanilla case (learning rate zero) completely overlooks the image-related task (Active pixels) while performing the best in the parity task. ",
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+ "text": "Additionally, let us clarify what we mean here with stability, as in the main manuscript we mainly talked about convergence guarantees. In these experiments we measure the convergence guarantees of the experiments in terms of ‘training stability’, meaning the variance of the obtained results across different runs. The intuition here is that, since the model does not converge, we should expect some wriggling learning curves during training and, as we take the model with the best validation error, the individual task metrics should have bigger variance (that is, less stability) across runs. ",
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+ "Figure 8: RotoGrad’s performance on all tasks for the experiments in $\\ S 6 . 1$ for all metrics. We can observe training instabilities/stiffness on all tasks as we highly increase/decrease RotoGrad’s learning rate, as discussed in the main manuscript. "
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+ "text": "C.2.4 CIFAR10 AND CELEBA ",
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+ "text": "For the sake of completeness, we present in Table 6 and Table 7 the same tables as in $\\ S 6 .$ , but with more statistics of the results. For CIFAR10, we now included in Table 6 the minimum task improvement across tasks and, while noisier, we can still observe that RotoGrad also improve this statistic. The standard deviation of the task improvement across tasks is, however, not too informative. We also include in Figure 9 the cosine similarity plots for the different subspace rotations from $\\ S$ , showing a clear positive correlation between the cosine similarity and the size of the considered subspace. While the cosine similarities look low, we want to remark that we are computing the cosine similarity of a huge space, and we only align a subspace of it. If we, instead, showed the cosine similarity with respect to each specific subspace, the cosine similarity should look similar to that of RotoGrad 512. In the case of CelebA, we added in Table 7 the maximum f1-score across tasks and, similar to the last case, it is not too informative, as all methods achieve almost perfect f1-score in one of the classes. We also include the training times for some baselines, showing that RotoGrad stays on par with them. ",
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+ "text": "C.2.5 NYUV2",
+ "text_level": 1,
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+ "text": "Complementing the results shown in $\\ S 6$ , we show in Table 8 the results obtained combining RotoGrad with all other existing methods (rows within RotoGrad $+$ ), for gradient scaling methods we only apply the rotation part of RotoGrad. Results show that RotoGrad helps improve/balance all other methods, which is specially true for those methods that heavily overlook some tasks. Specifically, MGDA-UB stops overlooking the semantic segmentation and depth estimation tasks, while PCGrad and GradDrop stop completely overlooking the surface normal loss. Note that we also show in Table 8 the training times of each method, and RotoGrad stays on par with non-extended methods in training time. As mentioned in Appendix C.1, due to cluster overload, some times were deceivingly high (specifically those baselines with $\\scriptstyle { \\mathbf { } } _ { R _ { k } }$ ) as we had to run them on different machines, and were omitted to avoid confusion. ",
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+ "Table 6: Complete task-improvement statistics in CIFAR10 for all competing methods and RotoGrad with different dimensionality for $\\scriptstyle { \\mathbf { } } _ { R _ { k } }$ , see $\\ S 6$ . "
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+ "table_body": "| Method d | mink △k ↑ med△k ↑ avgk△k ↑ std△k↓ maxk△k ↑ |
| Vanillat 0 | -0.81± 0.37 1.90 ± 0.53 2.58 ± 0.54 3.38 ± 0.94 11.14 ± 3.35 |
| RotoGrad 64 | -1.70 ± 0.81 1.79 ± 0.57 2.90 ± 0.49 3.98 ± 0.62 13.16 ± 2.40 |
| RotoGrad 128 | -1.12 ± 0.36 2.25 ± 1.07 2.97 ± 1.08 3.84 ± 0.87 | 12.64 ± 3.56 |
| RotoGrad 256 | 0.17 ± 1.01 2.16 ± 0.72 3.68 ± 0.68 3.83 ± 0.74 | 14.01 ± 3.22 |
| RotoGrad 512 | -0.43 ± 0.76 3.67 ± 1.40 4.48 ± 0.99 | 4.23 ± 0.82 | 15.57 ± 3.99 |
| Vanillat | -0.81± 0.37 1.90 ± 0.53 2.58 ± 0.54 | 3.38 ± 0.94 | 11.14 ± 3.35 |
| Wrehtntt GradDropPCGradMGDA-UBGradNormIMTL-G | -0.73 ± 0.33 2.80 ± 0.20 | 3.41 ± 0.45 | 4.08 ± 0.34 | 13.58 ± 1.50 |
| -1.52 ± 0.98 1.95 ± 0.87 | 2.86 ± 0.81 | 3.74 ± 0.69 | 12.01 ± 3.19 |
| -7.27 ± 1.36 -1.21 ± 0.74 | -1.75 ± 0.43 | 3.24 ± 0.55 | 3.67 ± 0.98 |
| -0.35 ± 0.59 2.45 ± 0.66 | 3.23 ± 0.35 | 4.02 ± 0.33 | 14.25 ± 1.35 |
| -0.39 ±0.82 1.97 ± 0.29 2.73 ± 0.27 | 3.25 ± 0.75 | 10.20 ± 2.98 |
| Vanilla | -0.85 ± 0.58 3.10 ± 1.29 3.12 ± 0.79 | 4.05 ± 0.56 | 14.23 ± 2.86 |
| GradDropHmA(1= p) PCGradMGDA-UBGradNormIMTL-G | -1.49 ± 0.78 3.27 ± 1.61 | 3.54 ±1.10 | 4.11 ± 0.56 | 13.88 ± 2.95 |
| -1.44 ± 0.58 2.67 ± 0.88 | 3.29 ± 0.46 | 3.90 ± 0.37 | 13.44 ± 1.86 |
| -3.59 ± 1.48 | 0.57 ± 0.62 | 0.21 ± 0.67 | 2.44 ± 0.52 | 4.78 ± 2.15 |
| -0.79 ±1.28 | 3.10 ± 1.01 | 3.21 ± 1.04 | 3.41 ± 0.86 | 10.88 ± 4.73 |
| -1.29 ± 0.52 | 1.81 ± 0.87 | 3.02 ± 0.69 | 3.81± 0.21 | 12.76 ± 1.77 |
| RotoGrad | -0.43 ± 0.76 3.67 ± 1.40 | 4.48 ± 0.99 | 4.23 ± 0.82 | 15.57 ± 3.99 |
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+ "Table 7: Complete f1-score statistics and training hours in CelebA for all competing methods and two different architectures/settings (median over five runs), see $\\ S$ . For the convolutional network we use $m = 2 5 6$ , and $m = 1 5 3 6$ for the residual network. "
+ ],
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+ "table_body": "| Method R@1 R@10 R@100 MRR |
| Sparse retrieval BM25 [32] 39.1 69.1 86.2 48.6 |
| DocT5Query [24] 46.7 76.5 90.4 56.2 |
| Dense retrieval ANCE [42] 45.6 75.7 89.6 55.6 |
| Sentence-T5t [22] 41.8 75.4 91.2 52.8 |
| Generative retrieval |
| GENRE+ [5] 35.6 57.6 79.1 42.3 40.0 |
| Ultron-URL [46] 29.6 67.8 - |
| Ultron-PQ [46] 31.6 73.1 45.4 - |
| Ultron-Atomic [46] 32.8 74.1 46.9 = |
| Ours 47.9 79.8 91.6 58.1 |
+
+Table 3: nDCG $@ 1 0$ results on BEIR. The results of the methods marked with † are from our own re-implementation, other results are cited from the original paper or implemented using official code. ST5 GPL denotes Sentence-T5 trained on GPL datasets [40].
+
+| Method R@1 R@10 R@100 MRR |
| Sparse retrieval BM25 [32] 39.1 69.1 86.2 48.6 |
| DocT5Query [24] 46.7 76.5 90.4 56.2 |
| Dense retrieval ANCE [42] 45.6 75.7 89.6 55.6 |
| Sentence-T5t [22] 41.8 75.4 91.2 52.8 |
| Generative retrieval |
| GENRE+ [5] 35.6 57.6 79.1 42.3 40.0 |
| Ultron-URL [46] 29.6 67.8 - |
| Ultron-PQ [46] 31.6 73.1 45.4 - |
| Ultron-Atomic [46] 32.8 74.1 46.9 = |
| Ours 47.9 79.8 91.6 58.1 |
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