Title: FAMO: Fast Adaptive Multitask Optimization
Venue: NeurIPS 2023
Accepted: yes
Keywords: multitask learning

multitask optimization

conflicting gradients

knowledge transfer
Abstract: One of the grand enduring goals of AI is to create generalist agents that can learn multiple different tasks from diverse data via multitask learning (MTL). However, in practice, applying gradient descent (GD) on the average loss across all tasks may yield poor multitask performance due to severe under-optimization of certain tasks. Previous approaches that manipulate task gradients for a more balanced loss decrease require storing and computing all task gradients ($\mathcal{O}(k)$ space and time where $k$ is the number of tasks), limiting their use in large-scale scenarios. In this work, we introduce Fast Adaptive Multitask Optimization (FAMO), a dynamic weighting method that decreases task losses in a balanced way using $\mathcal{O}(1)$ space and time. We conduct an extensive set of experiments covering multi-task supervised and reinforcement learning problems. Our results indicate that FAMO achieves comparable or superior performance to state-of-the-art gradient manipulation techniques while offering significant improvements in space and computational efficiency. Code is available at \url{https://github.com/Cranial-XIX/FAMO}.

FAMO: Fast Adaptive Multitask Optimization

Abstract

One of the grand enduring goals of AI is to create generalist agents that can learn multiple different tasks from diverse data via multitask learning (MTL). However, in practice, applying gradient descent (GD) on the average loss across all tasks may yield poor multitask performance due to severe under-optimization of certain tasks. Previous approaches that manipulate task gradients for a more balanced loss decrease require storing and computing all task gradients ( O( k ) space and time where k is the number of tasks), limiting their use in large-scale scenarios. In this work, we introduce Fast Adaptive Multitask Optimization (FAMO), a dynamic weighting method that decreases task losses in a balanced way using O( 1 ) space and time. We conduct an extensive set of experiments covering multi-task supervised and reinforcement learning problems. Our results indicate that FAMO achieves comparable or superior performance to state-of-the-art gradient manipulation techniques while offering significant improvements in space and computational efficiency. Code is available at https://github.com/Cranial-XIX/FAMO .

1 Introduction

Large models trained on diverse data have advanced both computer vision [20] and natural language processing [4], paving the way for generalist agents capable of multitask learning (MTL) [5]. Given the substantial size of these models, it is crucial to design MTL methods that are effective in terms of task performance and efficient in terms of space and time complexities for managing training costs and environmental impacts. This work explores such methods through the lens of optimization.

Perhaps the most intuitive way of solving an MTL problem is to optimize the average loss across all tasks. However, in practice, doing so can lead to models with poor multitask performance: a subset of tasks are severely under-optimized . A major reason behind such optimization failure is that a subset of tasks are under-optimized because the average gradient constantly results in small (or even negative) progress on these tasks (see details in Section 2).

To mitigate this problem, gradient manipulation methods [43, 25, 7, 24] compute a new update vector in place of the gradient to the average loss, such that all task losses decrease in a more balanced way. The new update vector is often determined by solving an additional optimization problem that involves all task gradients. While these approaches exhibit improved performance, they become computationally expensive when the number of tasks and the model size are large [41]. This is because they require computing and storing all task gradients at each iteration, thus demanding O( k ) space and time complexities, not to mention the overhead introduced by solving the additional optimization problem. In contrast, the average gradient can be efficiently computed in O( 1 ) space and time per iteration because one can first average the task losses and then take the gradient of the average loss. 1 To this end, we ask the following question:

AvgLossLandscape

L1 contour

L² contour

TimeComsumption(sec.)

10

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995

0

Q

89

60

54

33

39

-10

-10

-10

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Adam

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PCGrad

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Adam

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PCGrad

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FAMO

[L1

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-5

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L2

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-2.0-1.5-1.0-0.50.00.5

-2.01.5-1.00.5 0.0 0.5-2.0-1.5-1.0-0.5 0.0 0.5-2.0—1.5-1.0—0.5 0.0 0.5-2.01.5-1.00.5 0.0 0.5-2.0-1.5-1.0-0.5 0.0 0.5

( Q ) Is it possible to design a multi-task learning optimizer that ensures a balanced reduction in losses across all tasks while utilizing O( 1 ) space and time per iteration?

In this work, we present Fast Adaptive Multitask Optimization (FAMO), a simple yet effective adaptive task weighting method to address the above question. On the one hand, FAMO is designed to ensure that all tasks are optimized with approximately similar progress. On the other hand, FAMO leverages the loss history to update the task weighting, hence bypassing the necessity of computing all task gradients. To summarize, our contributions are:

We introduce FAMO, an MTL optimizer that decreases task losses approximately at equal rates while using only O( 1 ) space and time per iteration.

We demonstrate that FAMO performs comparably to or better than existing gradient manipulation methods on a wide range of standard MTL benchmarks, in terms of standard MTL metrics, while being significantly computationally cheaper.

2 Background

In this section, we provide the formal definition of multitask learning, then discuss its optimization challenge, and provide a brief overview of the gradient manipulation methods.

Multitask Learning (MTL) MTL considers optimizing a single model with parameter θ ∈ R m that can perform k ≥ 2 tasks well, where each task is associated with a loss function ℓ i ( θ ) ∶ R m → R ≥ 0 . 2 Then, it is common to optimize the average loss across all tasks:

Optimization Challenge Directly optimizing (1) can result in severe under-optimization of a subset of tasks. A major reason behind this optimization challenge is the 'generalized" conflicting gradient phenomenon, which we explain in the following. At any time step t , assume one updates the model

Algorithm 1 Fast Adaptive Multitask Optimization (FAMO)

1: Input : Initial parameter θ 0 , task losses { ℓ i } k i = 1 (ensure that ℓ i ≥ ϵ > 0 , for instance, by ℓ i ← ℓ i -ℓ ∗ i + ϵ , ℓ ∗ i = inf θ ℓ i ( θ ) ), learning rate α and β , and decay γ ( = 0 . 001 by default).

2: ξ 1 ← 0 . // initialize the task logits to all zeros

3: for t = 1 ∶ T do

4: Compute z t = Softmax ( ξ t ) , e.g.,

5: Update the model parameters:

6: Update the logits for task weighting:

7: end for

parameter using a gradient descent style iterative update: θ t + 1 = θ t -αd t where α is the step size and d t is the update at time t . Then, we say that conflicting gradients (CG) [24, 43] happens if

In other words, certain task's loss is increasing. CG often occurs during optimization and is not inherently detrimental. However, it becomes undesirable when a subset of tasks persistently undergoes under-optimization due to CG. In a more general sense, it is not desirable if a subset of tasks has much slower learning progress compared to the rest of the tasks (even if all task losses are decreasing). This very phenomenon, which we call the 'generalized" conflicting gradient, has spurred previous research to mitigate it at each optimization stage [43].

Gradient Manipulation Methods Gradient manipulation methods aim to decrease all task losses in a more balanced way by finding a new update d t at each step. d t is usually a convex combination of task gradients, and therefore the name gradient manipulation (denote ∇ ℓ i,t = ∇ θ ℓ i ( θ t ) for short):

Here, S k = { w ∈ R k ≥ 0 ∣ w ⊺ 1 = 1 } is the probabilistic simplex, and w t is the task weighting across all tasks. Please refer to Appendix A for details of five state-of-the-art gradient manipulation methods (MGDA, PCGRAD, CAGRAD, IMTL-G, NASHMTL) and their corresponding f . Note that existing gradient manipulation methods require computing and storing k task gradients before applying f to compute d t , which often involves solving an additional optimization problem. As a result, we say these methods require at least O( k ) space and time complexity, which makes them slow and memory inefficient when k and model size m are large.

3 Fast Adaptive Multitask Optimization (FAMO)

In this section, we introduce FAMO that addresses question Q , which involves two main ideas:

At each step, decrease all task losses at an equal rate as much as possible (Section 3.1).

Amortize the computation in 1. over time (Section 3.2).

3.1 Balanced Rate of Loss Improvement

At time t , assume we perform the update θ t + 1 = θ t -αd t , we define the rate of improvement for task i as

FAMO then seeks an update d t that results in the largest worst-case improvement rate across all tasks ( 1 2 ∥ d t ∥ is subtracted to prevent an under-specified optimization problem where the objective can be infinitely large):

When the step size α is small, using Taylor approximation, the problem (4) can be approximated by

Instead of solving the primal problem in (5) where d ∈ R m ( m can be millions if θ is the parameter of a neural network), we consider its dual problem:

Proposition 3.1. The dual objective of (5) is

where z ∗ t = [ z ∗ t,i ] is the optimal combination weights of the gradients, and the optimal update direction is d ∗ t = J t z ∗ t .

Proof.

Write g ( d, z ) = ( ∑ k i = 1 z i ∇ log ℓ i,t ) ⊺ d -1 2 ∥ d ∥ 2 , then by setting

Plugging in d ∗ back, we have

At the optimum, we have d ∗ t = J t z ∗ t .

The dual problem in (6) can be viewed as optimizing the log objective of the multiple gradient descent algorithm (MGDA) [9, 35]. Similar to MGDA, (6) only involves a decision variable of dimension k ≪ m . Furthermore, if the optimal combination weights z ∗ t is an interior point of S k , then the improvement rates r i ( α,d ∗ t ) of the different tasks i equal, as we show in the following result.

Proposition 3.2. Assume { ℓ i } k i = 1 are smooth and the optimal weights z ∗ t in (6) is an interior point of S k , then

where r ∗ i ( d ∗ t ) = lim α → 0 1 α r i ( α,d ∗ t ) .

Proof. Consider the Lagrangian form of (6)

When z ∗ reaches the optimum, we have ∂ L( z, λ, µ )/ ∂z = 0 , recall that d ∗ t = J t z ∗ t , then

When z ∗ t is an interior point of S k , we know that µ = 0 . Hence J ⊺ t d ∗ t = -λ . This means,

3.2 Fast Approximation by Amortizing over Time

Instead of fully solving (6) at each optimization step, FAMO performs a single-step gradient descent on z , which amortizes the computation over the optimization trajectory:

But then, note that

so we can use the change in log losses to approximate the gradient.

In practice, to ensure that z always stays in S k , we re-parameterize z by ξ and let z t = Softmax ( ξ t ) , where ξ t ∈ R K are the unconstrained softmax logits. Consequently, we have the following approximate update on ξ from (8):

Remark: While it is possible to perform gradient descent on z for other gradient manipulation methods in principle, we will demonstrate in Appendix B that not all such updates can be easily approximated using the change in losses.

3.3 Practical Implementation

To facilitate practical implementation, we present two modifications to the update in (10).

Re-normalization The suggested update above is a convex combination of the gradients of the log loss, e.g.,

When ℓ i,t is small, the multiplicative coefficient z i,t ℓ i,t can be quite large and result in unstable optimization. Therefore, we propose to multiply d ∗ by a constant c t , such that c t d ∗ can be written as a convex combination of the task gradients just as in other gradient manipulation algorithms (see (2) and we provide the corresponding definition of w in the following):

Regularization As we are amortizing the computation over time and the loss objective { ℓ i (⋅)} s are changing dynamically, it makes sense to focus more on the recent updates of ξ [46]. To this end, we put a decay term on w such that the resulting ξ t is an exponential moving average of its gradient updates:

We provide the complete FAMO algorithm in Algorithm 1 and its pseudocode in Appendix C.

3.4 The Continuous Limit of FAMO

One way to characterize FAMO's behavior is to understand the stationary points of the continuoustime limit of FAMO (i.e. when step sizes ( α,β ) shrink to zero). From Algorithm 1, one can derive the following non-autonomous dynamical system (assuming { ℓ i } are all smooth):

(13) reaches its stationary points (or fixed points) when (note that c t > 0 )

Therefore, the minimum points of ∑ k i = 1 log ℓ i ( θ ) are all stationary points of (13).

4 Related Work

In this section, we summarize existing methods that tackle learning challenges in multitask learning (MTL). The general idea of most existing works is to encourage positive knowledge transfer by sharing parameters while decreasing any potential negative knowledge transfer (a.k.a, interference) during learning. There are three major ways of doing so: task grouping, designing network architectures specifically for MTL, and designing multitask optimization methods.

Task Grouping Task grouping refers to grouping K tasks into N < K clusters and learning N models for each cluster. The key is estimating the amount of positive knowledge transfer incurred by grouping certain tasks together and then identifying which tasks should be grouped [39, 45, 38, 36, 11].

Multitask Architecture Novel neural architectures for MTL include hard-parameter-sharing methods, which decompose a neural network into task-specific modules and a shared feature extractor using manually designed heuristics [21, 29, 2], and soft-parameter-sharing methods, which learn which parameters to share [30, 34, 12, 27]. Recent studies extend neural architecture search for MTL by learning where to branch a network to have task-specific modules [14, 3].

Multitask Optimization The most relevant approach to our method is MTL optimization via task balancing. These methods dynamically re-weight all task losses to mitigate the conflicting gradient issue [40, 43]. The simplest form of gradient manipulation is to re-weight the task losses based on manually designed criteria [6, 13, 18], but these methods are often heuristic and lack theoretical support. Gradient manipulation methods [35, 43, 25, 7, 16, 24, 32, 26, 47] propose to form a new update vector at each optimization by linearly combining task gradients. The local improvements across all tasks using the new update can often be explicitly analyzed, making these methods better understood in terms of convergence. However, it has been observed that gradient manipulation methods are often slow in practice, which may outweigh their performance benefits [22]. By contrast, FAMO is designed to match the performance of these methods while remaining efficient in terms of memory and computation. Another recent work proposes to sample random task weights at each optimization step for MTL [23], which is also computationally efficient. However, we will demonstrate empirically that FAMO performs better than this method.

5 Empirical Results

We conduct experiments to answer the following question:

How does FAMO perform in terms of space/time complexities and standard MTL metrics against prior MTL optimizers on standard benchmarks (e.g., supervised and reinforcement MTL problems)?

In the following, we first use a toy 2-task problem to demonstrate how FAMO mitigates CG while being efficient. Then we show that FAMO performs comparably or even better than state-of-theart gradient manipulation methods on standard multitask supervised and reinforcement learning benchmarks. In addition, FAMO requires significantly lower computation time when K is large compared to other methods. Lastly, we conduct an ablation study on how robust FAMO is to γ . Each subsection first details the experimental setup and then analyzes the results.

5.1 A Toy 2-Task Example

oT

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To better understand the optimization trajectory of FAMO, we adopt the same 2D multitask optimization problem from NASHMTL [32] to visualize how FAMO balances different loss objectives. The model parameter θ = ( θ 1 , θ 2 ) ∈ R 2 . The two tasks' objectives and their surface plots are provided in Appendix D and Figure 2. We compare FAMO against ADAM [19], MGDA [35], PCGRAD [43], CAGRAD [24], and NASHMTL [32]. We then pick 5 initial points θ init ∈ {(-8 . 5 , 7 . 5 ) , (-8 . 5 , 5 ) , ( 0 , 0 ) , ( 9 , 9 ) , ( 10 , -8 )} and plot the corresponding optimization trajectories with different methods in Figure 1. Note that the toy example is constructed such that naively applying ADAM on the average loss can cause the failure of optimization for task 1.

Findings: From Figure 1, we observe that FAMO, like all other gradient manipulation methods, mitigates the CG and reaches the Pareto front for all five runs. In the meantime, FAMO performs similarly to NASHMTL and achieves a balanced loss decrease even when the two task losses are improperly scaled. Finally, as shown in the top-right of the plot, FAMO behaves similarly to ADAM in terms of the training time, which is 25 × faster than NASHMTL.

5.2 MTLPerformance

Multitask Supervised Learning. We consider four supervised benchmarks commonly used in prior MTL research [24, 27, 32, 33]: NYU-v2 [31] (3 tasks), CityScapes [8] (2 tasks), QM-9 [1] (11 tasks), and CelebA [28] (40 tasks). Specifically, NYU-v2 is an indoor scene dataset consisting of 1449 RGBD images and dense per-pixel labeling with 13 classes. The learning objectives include image segmentation, depth prediction, and surface normal prediction based on any scene image. CityScapes dataset is similar to NYU-v2 but contains 5000 street-view RGBD images with per-pixel annotations. QM-9 dataset is a widely used benchmark in graph neural network learning. It consists of > 130K molecules represented as graphs annotated with node and edge features. We follow the same experimental setting used in NASHMTL [32], where the learning objective is to predict 11 properties of molecules. We use 110K molecules from the QM9 example in PyTorch Geometric [10], 10K molecules for validation, and the rest of 10K molecules for testing. The characteristic of this dataset is that the 11 properties are at different scales, posing a challenge for task balancing in MTL. Lastly, CelebA dataset contains 200K face images of 10K different celebrities, and each face image is provided with 40 facial binary attributes. Therefore, CelebA can be viewed as a 40-task MTL problem. Different from NYU-v2, CityScapes, and QM-9, the number of tasks ( K ) in CelebA is much larger, hence posing a challenge to learning efficiency.

We compare FAMO against 11 MTL optimization methods and a single-task learning baseline: (1) Single task learning (STL), training an independent model ( θ for each task; (2) Linear scalarization (LS) baseline that minimizes L 0 ; (3) Scale-invariant (SI) baseline that minimizes ∑ k log L k ( θ ) , as SI is invariant to any scalar multiplication of task losses; (4) Dynamic Weight Average (DWA) [27], a heuristic for adjusting task weights based on rates of loss changes; (5) Uncertainty Weighting (UW) [18] uses task uncertainty as a proxy to adjust task weights; (6) Random Loss Weighting (RLW) [23] that samples task weighting whose log-probabilities follow the normal distribution; (7) MGDA [35] that finds the equal descent direction for each task; (8) PCGRAD [43] proposes to project each task gradient to the normal plan of that of other tasks and combining them together in the end; (9) CAGRAD [24] optimizes the average loss while explicitly controls the minimum decrease across tasks; (10) IMTL-G [25] finds the update direction with equal projections on task gradients; (11) GRADDROP [7] that randomly dropout certain dimensions of the task gradients based on how much they conflict; (12) NASHMTL [32] formulates MTL as a bargaining game and finds the solution to the game that benefits all tasks. For FAMO, we choose the best hyperparameter γ ∈ { 0 . 0001 , 0 . 001 , 0 . 01 } based on the validation loss. Specifically, we choose γ equals 0 . 01 for the CityScapes dataset and 0 . 001 for the rest of the datasets. See Appendix E for results with error bars.

Evaluations: We consider two metrics [32] for MTL: 1) ∆ m % , the average per-task performance drop of a method m relative to the STL baseline denoted as b : ∆ m % = 1 K ∑ K k = 1 (-1 ) δ k ( M m,k -M b,k )/ M b,k × 100 , where M b,k and M m,k are the STL and m 's value for metric M k . δ k = 1 (or 0 ) if the M k is higher (or lower) the better. 2) Mean Rank (MR) : the average rank of each method across tasks. For instance, if a method ranks first for every task, MR will be 1.

Findings: Results on the four benchmark datasets are provided in Table 1, 2 and 3. We observe that FAMO performs consistently well across different supervised learning MTL benchmarks compared

to other gradient manipulation methods. In particular, it achieves state-of-the-art results in terms of ∆ m % on the NYU-v2 and QM-9 datasets.

Multitask Reinforcement Learning. We further apply FAMO to multitask reinforcement learning (MTRL) problems as MTRL often suffers more from conflicting gradients due to the stochastic nature of reinforcement learning [43]. Following CAGRAD [24], we apply FAMO on the MetaWorld [44] MT10 benchmark, which consists of 10 robot manipulation tasks with different reward functions. Following [37], we use Soft Actor-Critic (SAC) [15] as the underlying RL algorithm, and compare against baseline methods including LS (SAC with a shared model) [44], Soft Modularization [42] (an MTL network that routes different modules in a shared model to form different policies), PCGRAD [43], CAGRAD and NASHMTL [32]. The experimental setting and hyperparameters all match exactly with those in CAGRAD. For NASHMTL, we report the results of applying the NASHMTL update once per { 1 , 50 , 100 } iterations. 4 The results for all methods are provided in Table 5.2.

Method

Success

↑

(mean

±

stderr)

LS (lower bound)

0.49

±

0.07

STL (proxy for upper bound)

0.90

±

0.03

PCGRAD [43]

0.72

±

0.02

SOFT MODULARIZATION [42]

0.73

±

0.04

CAGRAD

0.83

±

0.05

NASHMTL [32] (every 1)

0.91

±

0.03

NASHMTL [32] (every 50)

0.85

±

0.02

NASHMTL [32] (every 100)

0.87

±

0.03

NASHMTL (ours) (every 1)

0.80

±

0.13

NASHMTL (ours) (every 50)

0.76

±

0.10

NASHMTL (ours) (every 100)

0.80

±

0.12

UW[18]

0.77

±

0.05

FAMO (ours)

0.83

±

0.05

LS

NashMTL (1)

(50)

1(001)

FAMO(=0.0001)

(0.001)

(0.01)

1.0

Rate

0.75

R

Success

0.5

S

0.25

1M

2M

0

Training Steps

11.0

ime

2.0

1.5

1.0

1.0

Findings: From Table 5.2, we observe that FAMO performs comparably to CAGRAD and outperforms PCGRAD and the average gradient descent baselines by a large margin. FAMO also outperforms NASHMTL based on our implementation. Moreover, FAMO is significantly faster than NASHMTL, even when it is applied once every 100 steps.

5.3 MTLEfficiency (Training Time Comparison)

Figure 4 provides the FAMO's average training time per epoch against that of the baseline methods.

LS

MGDA

PCGrad

CAGrad

IMTL-G

NashMTL

FAMO

1.91

1.71

1.83

3

8

6.58

7.30

250

268.84

2.37

1.00

1.38

1.17

1.07

2

2.10 2.15

1.94

2.00

6

4

4.68

3.99

4.24

200

150

135.40

1.00

1.02

100

89.63

67.26

2

53.77

1.00

1.17

50

1.00

3.43

U

CityScapes (2 tasks)

NYU-v2 (3 tasks)

QM-9 (9 tasks)

Q

CelebA (40 tasks)

Findings: From the figure, we observe that FAMO introduces negligible overhead across all benchmark datasets compared to the LS method, which is, in theory, the lower bound for computation time. In contrast, methods like NASHMTL have much longer training time compared to FAMO. More importantly, the computation cost of these methods scales with the number of tasks. In addition, note that these methods also take at least O( K ) space to store the task gradients, which is implausible for large models in the many-task setting (i.e., when m = ∣ θ ∣ and K are large).

5.4 Ablation on γ

In this section, we provide the ablation study on the regularization coefficient γ in Figure 5.

=0.0001

=0.001

=0.01

40

36.08

0

91.03

86.86

3

3.02

△m%

30

-1

80

20

19.62

-2

-2.73

60

40

58.50

²

1.93

1.21

10

8.13

-3

-3.59

20

-4.10

0

CityScapes

NYU-v2

QM-9

CelebA

Findings: From Figure 5, we can observe that choosing the right regularization coefficient can be crucial. But except for CityScapes, FAMO performs reasonably well using all different γ s. The problem with CityScapes is that one of the task losses is close to 0 at the very beginning, hence small changes in task weighting can result in very different loss improvement. Therefore we conjecture that using a larger γ , in this case, can help stabilize MTL.

6 Conclusion and Limitations

In this work, we introduce FAMO, a fast optimization method for multitask learning (MTL) that mitigates the conflicting gradients using O( 1 ) space and time. As multitasking large models gain more attention, we believe designing efficient but effective optimizers like FAMO for MTL is crucial. FAMO balances task losses by ensuring each task's loss decreases approximately at an equal rate. Empirically, we observe that FAMO can achieve competitive performance against the state-of-the-art MTL gradient manipulation methods. One limitation of FAMO is its dependency on the regularization parameter γ , which is introduced due to the stochastic update of the task weighting logits w . Future work can investigate a more principled way of determining γ .

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A Gradient Manipulation Methods

In this section, we provide a brief overview of representative gradient manipulation methods in multitask/multiobjective optimization. Specifically, we will also discuss the connections among these methods.

Multiple Gradient Descent Algorithm (MGDA) [9, 35] The MGDA algorithm is one of the earliest gradient manipulation methods for multitask learning. In MGDA, the per step update d t is found by solving

As a result, the solution d ∗ of MGDA optimizes the 'worst improvement" across all tasks or equivalently seeks an equal descent across all task losses as much as possible. But in practice, MGDA suffers from slow convergence since the update d ∗ can be very small. For instance, if one task has a very small loss scale, the progress of all other tasks will be bounded by the progress on this task. Note that the original objective in (6) is similar to the MGDA objective in the sense that we can view optimizing (6) as optimizing the log of the task losses. Hence, when we compare FAMO against MGDA, one can regard FAMO as balancing the rate of loss improvement while MGDA balances the absolute improvement across task losses.

Projecting Gradient Descent (PCGRAD) [43] PCGRAD initializes v i PC = ∇ ℓ i,t , then for each task i , PCGRAD loops over all task j ≠ i (in a random order, which is crucial as mentioned in [43]) and removes the 'conflict"

In the end, PCGRAD produces d t = 1 k ∑ k i = 1 v i PC . Due to the construction, PCGRAD will also help improve the 'worst improvement" across all tasks since the 'conflicts" have been removed. However, due to the stochastic iterative procedural of this algorithm, it is hard to understand PCGRAD from a first principle approach.

Conflict-averse Gradient Descent (CAGRAD) [24] d t is found by solving

Here, ℓ 0 ,t = 1 k ∑ k i = 1 ℓ i,t . CAGRAD seeks an update d t that optimizes the 'worst improvement" as much as possible, conditioned on that the update still decreases the average loss. By controlling the hyperparameter c , CAGRAD can recover MGDA ( c →∞ ) and the vanilla averaged gradient descent ( c → 0 ). Due to the extra constraint, CAGRAD provably converges to the stationary points of ℓ 0 when 0 ≤ c < 1 .

Impartial Multi-Task Learning (IMTL-G) [25] IMTL-G finds d t such that it shares the same cosine similarity with any task gradients:

The constraint that d t = ∑ k i = 1 w i,t ∇ ℓ i,t is for preventing the problem from being under-determined. From the above equation, we can see that IMTL-G ignores the 'size" of each task gradient and only cares about the 'direction". As a result, one can think of IMTL-G as a variant of MGDA that applies to the normalized gradients . By doing so, IMTL-G does not suffer from the straggler effect due to slow objectives. Furthermore, one can view IMTL-G as the equal angle descent , which is also proposed in Katrutsa et al. [17], where the objective is to find d such that

NASHMTL[32] NASHMTL finds d t by solving a bargaining game treating the local improvement of each task loss as the utility for each task:

Note that the objective of NASHMTL implicitly assumes that there exists d such that ∀ i, ∇ ℓ ⊺ i,t d > 0 (otherwise we reach the Pareto front). It is easy to see that

Therefore, due to the log , NASHMTL also ignores the 'size" of task gradients and only cares about their 'directions". Moreover, denote u i = ∇ ℓ i,t ∥∇ ℓ i,t ∥ . Then, according to the KKT condition, we know:

Consider when k = 2 , if we take the equal angle descent direction: d ∠ = ( u 1 + u 2 )/ 2 (note that as u 1 and u 2 are normalized, their bisector is just their average). Then it is easy to check that

As a result, we can see that when k = 2 , NASHMTL is equivalent to IMTL-G (or the equal angle descent) . However, when k > 2 , this is not in general true.

Remark Note that all of these gradient manipulation methods require computing and storing K task gradients before applying f to compute d t , which often involves solving an additional optimization problem. Hence, these methods can be slow for large K and large model sizes.

B Amortizing other Gradient Manipulation Methods

Although FAMO uses iterative update on w , it is not immediately clear whether we can apply the same amortization easily on other existing gradient manipulation methods. In this section, we discuss such possibilities and point out the challenges.

Amortizing MGDA This is almost the same as in FAMO, except that MGDA acts on the original task losses while FAMO acts on the log of task losses.

Amortizing PCGRAD For PCGRAD, finding the final update vector requires iteratively projecting one task gradient to the other, so there is no straightforward way of bypassing the computation of task gradients.

Amortizing IMTL-G The task weighting in IMTL-G is computed by a series of matrix-matrix and matrix-vector products using task gradients [25]. Hence, it is also hard to amortize its computation over time.

Therefore, we focus on deriving the amortization for CAGRAD and NASHMTL.

Amortizing CAGRAD For CAGRAD, the dual objective is

where g 0 = ∇ ℓ 0 ,t and g w = ∑ k i = 1 w i ∇ ℓ i . Denote

Now, if we take the gradient with respect to w in (15), we have:

As a result, in order to approximate this gradient, one can separately estimate:

Once all these are estimated, one can combine them together to perform a single update on w . But note that this will require 3 forward and backward passes through the model, making it harder to implement in practice.

Amortizing NASHMTL Per derivation from NASHMTL [32], the objective is to solve for w :

One can therefore form an objective:

Taking the derivative of F with respect to w , we have

Therefore, to approximate the gradient of w , one needs to first estimate

Then we estimate

Again, this results in 3 forward and backward passes through the model, let alone the overhead of resetting the model back to θ (requires a copy of the original weights).

In short, though it is possible to derive fast approximation algorithm to approximate the gradient update on w for some of the existing gradient manipulation methods, it often involves much more complicated computation compared to that of FAMO.

C FAMO Pseudocode in PyTorch

We provide the pseudocode for FAMO in Algorithm 2. To use FAMO, one just first compute the task losses, call get_weighted_loss to get the weighted loss, and do the normal backpropagation through the weighted loss. After that, one call update to update the task weighting.

D Toy Example

We provide the task objectives for the toy example in the following. The model parameter θ = ( θ 1 , θ 2 ) ∈ R 2 and the task objectives are L 1 and L 2 :

Algorithm 2 Implementation of FAMO in PyTorch-like Pseudocode

E Experimental Results with Error Bars

We followed the exact experimental setup from NASHMTL [32]. Therefore, the numbers for baseline methods are taken from their original paper. In the following, we provide FAMO's result with error bars.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses an important issue with (most) prior MTL optimization techniques which requires the computation of all per-task gradients, during training, for obtaining the update direction. This results in a $\\mathcal{O}(K)$ requirement in space and time where $K$ is the number of tasks. The authors propose a novel approach for dynamically weighting tasks using $\\mathcal{O}(1)$ space and time.\n\nSTRENGTHS:\n1. The paper proposes a novel approach for addressing a known pain point with (most) SoTA multi-task optimization methods.\n2. The paper provides strong empirical evidence which indicates the method performs on par with SoTA MTL methods (e.g., NashMTL, IMTL-G, CAGrad) while significantly reducing the optimization time.\n3. The paper is well-written and well-structured. Connection to prior works in MTL literature is well established (Appendix A-B).\n\nWEAKNESSES:\n1. While the division of the proposed method follows a different trajectory, the proposed algorithm is fairly similar to MGDA. Hence, more discussion is needed on the connection between the proposed method and MGDA.\n2. Please provide additional details on the experimental sections (data split sizes, HPs, etc.).\n3. Hyperparameter (HP) optimization:\n    - Previous papers on MTL (e.g., CAGrad) did not include a validation split for NYU/Cityscapes. If the authors follow the same settings, how did you tune FAMO’s specific HPs? \n    - Missing details on the tuning of the lr for the task weights.\n4. From section 5.4 (and as stated by the authors), it appears the method is highly sensitive to the choice of HPs.\n5. It would be beneficial to include figures of the losses throughout the optimization process and some metric to verify that the losses decrease in an (approximately) equal rate.\n6. It would be beneficial if you could provide a figure of the task weights throughout the optimization process.\n\nMinor: \nTable 2, MR column missing indication for best-performing method.\n\nQUESTIONS:\n1. Could you provide some intuition as to why the proposed method alleviates the CG problem in MTL, only by ensuring each task’s loss decreases at an equal rate?\n2. Have you tried performing several optimization steps for approximating the solution in Eq. 7 at each training iteration?\n\nLIMITATIONS:\nLimitations addressed.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new multi-task balancing method with claimed O(1) efficiency, which is much more efficient than previous gradient manipulation methods.\n\nThe core idea is to let the multiple losses decrease at roughly the same speed.\n\nSTRENGTHS:\nThe method is more efficient than previous gradient manipulation methods.\nAuthors conducted lots of experiments on several datasets and show that the method is performing well on them.\n\nWEAKNESSES:\n1. I think the contribution is little bit over-stated. When I read the title, abstract and introduction, I thought authors will present a novel method to manipulate task gradients with O(1) time and space complexity, which is awesome. Why I thought that? For example, in Figure 1, they compare FAMO with other gradient balancing methods like PCGrad and CAGrad, which balance task gradients with O(K) time and space complexity.\n\nHowever, authors actually do not manipulate gradients but weight losses. Losses weighting are not novel path with previous methods DWA and UW. If you balance losses rather gradients, of course you can have O(1) time and space complexity since you do not touch parameter gradients w.r.t to K losses. \nIn Question [highlight in the introduction], authors want to design an MTL optimizer, which should handle the gradients of parameters. I am not sure if the algorithm proposed in the paper can be called a new optimizer?\nMy suggestion is that authors can re-factory the story [title, abs, intro] as an efficient loss balancing methods, which seems more suitable to their contributions.\n\n2.  The method part is not presented very clearly, see the questions.\n\nQUESTIONS:\n1. Equation 6 is transformed from Equation 5 using L'Hopital's rule, based on the step size \\alpha is reaching to 0, however, learning rate \\alpha is normally something like 0.001, which is much larger than 0. So, Equation 6 is an approximation here and should be made clear.\n\n2. Could authors elaborate a bit on why Equation 7 is the dual form for (6)?\n\n3. In Section 3.2,\"But then, assume [author's equation but hard to write for me here], one can approximate (8) by:\", may I ask why we can assume this equation? what is the intuition for this assumption?\n\n4. What is w? the final weighs losses I guess?. In the paper, authors have used v_{t} and z_{t} as task weight. And they did not clearly explain w.\n\nLIMITATIONS:\nNo social impact", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a dynamic convex combination multi-task learning losses reweighting so that it can decrease different task losses more balancedly while having little computational overhead than simple task loss average. Specifically, they formulate an optimization problem to find a new “gradient” direction that can achieve sufficient relative-to-magnitude decrease for the worst task loss. Instead of solving this optimization exactly every update, the authors amortize the optimization by keeping dual variables on the probability simplex across updates and update them through one-step gradient descent after every primal variable (learnable parameter) update. Experimentally, the authors show that their proposed method FAMO can achieve comparable/superior on 4 multitask supervised learning benchmarks and 1 multi-task RL benchmark, while being more time-saving compared to other gradient manipulation methods.\n\nSTRENGTHS:\n1. The paper is in general well-written, with a nice discussion about different multi-task gradient manipulation technique in Appendix A.\n2. The proposed method FAMO empirically is competitive with other methods on multiple benchmarks.\n\nWEAKNESSES:\n**[Lack of proof for significant contribution]**\n\nIn terms of methodology, as the authors mention themselves, FAMO’s motivation formulation is basically applying MGDA to the logarithm of all task losses. Thus the formulation is practical but not very novel. What distinguishes FAMO from MGDA fundamentally is FAMO’s amortized optimization of the dual variable. I will raise additional question later in this section about solidifying this contribution. In terms of experiment results, the authors claim FAMO can perform better/equal to other baseline methods in terms of (1) final performance, and (2) wall-clock time. For final performance, it is not convincing enough that FAMO can always match/outperform other MTL methods. On CityScapes and CelebA, it seems to fall behind some other methods. On QM-9, it’s also not clear to me what’s the value of multi-task learning — it seems single task learning (STL) is better (with lower MAE values) than almost all MTL methods  (including FAMO) (please do correct me if I’m misunderstanding this). I would appreciate authors to provide further clarifications about these points. In terms of wall-clock time, the authors only show the wall clock time for each epoch for different methods. However, this neglects the fact that different methods might require different number of epochs to converge. It is conceivable that a non-optimization-amortized method could take longer per epoch but converge in fewer epochs than FAMO and thus could use less total time. I think the authors should also report the total time as it should be final metric to judge the wall-clock savings.\n\n**[Understanding the impact of amortizing the optimization]** A primary contribution of the paper is to propose to amortize the optimization to find the update direction instead of solving it exactly each update. Hence I believe it is important and useful that the authors provide experiment results on the non-amortized version of FAMO, where the optimization in equation (5) is solved exactly in each iteration. With this non-amortized result, it would provide the context of whether the amortization deteriorates the performance, allowing the readers to understand the room of improvement with amortization. Besides, it would also be interesting to see the amortized vs non-amortized comparison for MGDA to further understand this point.\n\n**[Potentially incorrect proof for proposition 3.1]** In the proof for proposition 3.1, the authors use the Lagrangian of the dual by adding $cz^\\top \\mathbf{1}$ to the objective to get rid of the simplex constraint. However, the simplex constraint is not only about sum of the variables being 1, but also about having nonnegative individual values. Thus, I believe these nonnegative constraint should also be reflected in the Lagrangian. Otherwise it needs to be specifically argued that the optimal solution cannot occur on the edges of the simplex.\n\nQUESTIONS:\n**[Why is FAMO $O(1)$ in space?]** For performing backprop over the simple case of the average task loss (LS), either all the $K$ tasks’ computation graphs are computed simultaneously and stored in the memory (needed for backprop) at the same time, or the tasks’ losses are computed one after another. In the first case, the computation graph would already have space complexity of O(K), while in the second case, the time to iterate over all tasks would have time complexity of O(K). FAMO is basically calling backprop on a reweighting of the logarithm of the task losses so it should at least have the same space and time complexity as LS. So why do the authors claim FAMO is $O(1)$ in space? Besides, storing the K weights are already technically O(K) in space. I can understand that FAMO would take less time than the other multi task gradient manipulation methods which require non-amortized optimization, but I would appreciate the authors be more rigorous about their claims with more explanation.\n\n**[Why is STL so good?]** On QM-9 and MTRL results, it seems STL performs (much) better than other methods. Can the authors provide explainations on why that is the case?\n\n**[Mean Rank]** In Table 3, why is IMTL-G’s mean rank (0.84) smaller than 1? On page 7 the paper says for a method that ranks first for every task, its MR is still 1.\n\n**[Obeying the simplex constraint]** This question is relatively minor but I would suggest the authors consider alternative ways to  obey the simplex constraint other than using a softmax which couldn’t really exactly represent the corners of the simplex. Instead, one could apply $L^2$-projected gradient descent instead of gradient descent on $z$ by projecting the updated $z$ back onto the simplex. There exist efficient algorithms for simplex projection which don’t require solving the projection optimization problem iteratively: https://arxiv.org/pdf/1309.1541.pdf.\n\nLIMITATIONS:\nThe authors mention their approach depending an extra weight decay hyperparameter as an limitation of their work. I don’t believe this paper needs to discuss negative societal impacts. I currently rate the paper as borderline accept because of the weaknesses described above. However, I’m open to increase my score given satisfactory author rebuttal responses.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposed a novel multi-task optimization method aimed at mitigating conflict between task gradients without inducing the substantial time slowdown that typically comes with specialized multi-task optimizers.\nThe proposed method, called FAMO, aims at improving the worst-case rate of improvement across all tasks at each step.\nAt a high level, it does this by setting the weights of each task to minimize the gradient norm of the log task loss (similar to how MGDA sets weights by minimizing the gradient norm of the task loss). They show that finding these weights is the dual of finding the direction that maximizes the minimum rate of improvement across tasks. In practice however, these weights are learned over time via SGD updates after each optimization step of the main model, rather than optimized in full at each model update step.\n\nThe authors then demonstrate their method on a toy 2D loss landscape (used in prior work), showing that their method reaches the Pareto frontier of this problem, like other optimization methods (and unliked naive Adam), but has the closest overall compute time to Adam. Next, the authors compare their method to 11 other optimization methods for MTL on 4 supervised MTL and 1 RL setting finding that FAMO compares competitively (it is consistently in the top 2 or 3 performing methods) in all settings.  Finally, the authors demonstrate the performance efficiency of their method in the supervised learning settings showing that FAMO is slightly faster than comparable methods on settings with 2 or 3 tasks, and orders of magnitude faster as the number of tasks increases to 10 and 60.\n\nSTRENGTHS:\n- The method being proposed has clear benefits in terms of speed / efficiency, especially as the number of tasks scales, which is very important in many MTL settings.\n- The method is compared to a broad and representative set of optimization methods, and performs competitively on all benchmarks proposed even outperforming slower methods consistently.\n- Moreover, the experimental methodology seems sound; all experiments consider multiple random seeds and 2 sensible metrics to compare methods are considered.\n- The paper is well written, with good figures, clear descriptions, and the motivation is straightforward. Moreover, I find the proposed method to be explained well.\n\nWEAKNESSES:\n- The primary contribution of FAMO (in my view) is the decision to maximize minimum task rate of improvement, by focusing on the gradient of the log loss, rather than maximizing the minimum total improvement (i.e. MGDA). However, this is not the “fast” part. The fast part of the method comes from optimizing the task-weights slowly, via gradient updates, throughout the optimization process. The authors do well to point this out in the appendix, but the effects of these contributions should be analyzed separately. In a deep neural network it is possible that the generalization benefits come from the amortized optimization of task weights, rather than the specific weight objectives. This could easily be addressed by, for instance, comparing amortized MGDA to FAMO, which is perhaps the most comparable setting.\n- I feel as though GradNorm is also natural comparison to this method, given that it also learns task weights during training (in an amortized fashion), with the goal of balancing the average rate of improvement across tasks.\n- [21] is referenced as suggesting that current optimization methods aren’t worth the tradeoff in computation time, but iirc their conclusion was that optimization methods actually don’t help performance over the baseline, which appears to not be true in this papers experiments. This discrepancy should likely be commented on.\n\nQUESTIONS:\n- Is there a typo in the psuedo code of algo 2? z is computed via softmax, but then the raw weights w are used for the rest of the get_weighted_loss method.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-12"}