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Title: Mint: Matrix-Interleaving for Multi-Task Learning

Paper content:
Deep learning enables training of large and flexible function approximators from scratch at the cost of large amounts of data.
Applications of neural networks often consider learning in the context of a single task.
However, in many scenarios what we hope to learn is not just a single task, but a model that can be used to solve multiple different tasks.
Such multi-task learning settings have the potential to improve data efficiency and generalization by sharing data and representations across tasks.
However, in some challenging multi-task learning settings, particularly in reinforcement learning, it is very difficult to learn a single model that can solve all the tasks while realizing data efficiency and performance benefits.
Learning each of the tasks independently from scratch can actually perform better in such settings, but it does not benefit from the representation sharing that multi-task learning can potentially provide.
In this work, we develop an approach that endows a single model with the ability to represent both extremes: joint training and independent training.
To this end, we introduce matrix-interleaving (Mint), a modification to standard neural network models that projects the activations for each task into a different learned subspace, represented by a per-task and per-layer matrix.
By learning these matrices jointly with the other model parameters, the optimizer itself can decide how much to share representations between tasks.
On three challenging multi-task supervised learning and reinforcement learning problems with varying degrees of shared task structure, we find that this model consistently matches or outperforms joint training and independent training, combining the best elements of both.
While deep learning has enabled remarkable levels of generalization through the use of function approximators, this comes at the cost of large amounts of data, which remains a critical challenge in deploying deep learning to a number of domains.
When combined with deep networks, multitask learning offers the promise of building more powerful representations using less data per task, leading to greater performance and data efficiency.
However, multi-task deep learning has also posed considerable challenges.
Numerous works have observed that joint training on multiple tasks can actually decrease task performance due to the negative influence of other tasks (Parisotto et al., 2015; Rusu et al., 2016a) .
Indeed, training networks entirely independently on each task has remained a strong approach, to the point that multiple multi-task methods have first trained models independently before using them to train a multi-tasking model (Parisotto et al., 2015; Rusu et al., 2016a; Ghosh et al., 2017; Teh et al., 2017; .
Moreover, our experiments in Section 6 indicate that three recently proposed methods for multi-task learning are all surpassed by training models independently per task.
However, training independent models will only work well when provided enough data per task, and precludes potential positive data-efficiency gains from multi-task learning, only providing protection against negative transfer.
Further, while a number of works have successfully shared parameters, finding an architecture with the appropriate level of parameter sharing for a given problem domain can require a considerable amount of manual engineering.
In this work, we aim to develop a multi-task learning method that can perform well both when tasks share very little and when they share a large amount of structure.
To address this problem, we consider how a single neural network model can represent two extremes: independent models, when optimization challenges prevail, or a single model with shared weights, when sharing is beneficial.
Further, we would like such a model to be able to represent intermediate levels of model sharing, when appliable.
One option for performing independent training within a single model is to put separate networks with independent weights into a single model, using the task ID to select which network prediction to output.
However, this prevents any sharing.
An alternative approach is to condition the model on the task ID, through various conditioning approaches, including additive and multiplicative approaches such as FiLM (Perez et al., 2018) .
In fact, point-wise multiplicative conditioning, as proposed in FiLM, can indeed represent separate networks by selecting which parts of the network to be used for different tasks, as can a number of other approaches in multi-task learning (Rosenbaum et al., 2017; 2019; Fernando et al., 2017 ).
Yet, these approaches still require an optimization over shared parameters in order to select which parameters are used for each task.
These shared parameters can introduce significant optimization challenges.
We instead consider how to allow a model to perform optimization on only shared parameters, only disjoint parameters, or any combination thereof.
We can achieve this by simply interleaving learned per-task matrices at each layer of a jointly-trained neural network.
When optimization over shared parameters is ineffective, the model can still represent a full neural network per task using only the per-task matrices, resulting in independent training; while using identical per-task matrices results in standard joint training.
Intermediately, a mix of shared and per-task parameters may be used.
In effect, by incorporating these matrices into the network, the optimizer itself can automatically and dynamically modulate the degree to which a representation is shared between tasks, depending on the problem domain and the optimization progress, and can do so without having to optimize shared parameters.
The primary contribution of this paper is a simple yet effective approach for multi-task learning that can represent and smoothly interpolate between independent training and joint training, via matrix interleaving (Mint).
We describe how we can implement Mint in deep multi-task models and show its effectiveness in improving data efficiency and generalization in multi-task settings while providing intuition about the reasons why this architecture performs so well.
Further, we show that the model can be extended to goal-conditioned reinforcement learning in a straightforward manner by allowing the model to generate the interleaved matrices conditioned on task information such as the goal.
We evaluate Mint on sets of tasks with both high and low levels of shared structure and find that it performs well in both settings, performing comparably to or outperforming both joint training and independent training, effectively combining the best elements of both.
Further, in comparison to previous methods that use multiplicative interactions for continual learning (Cheung et al., 2019) and for general conditioning (Perez et al., 2018) , Mint is better able to separate tasks by avoiding the need to optimize over shared parameters and can empirically produce substantially better performance on a range of challenging multi-task problems.
Finally, Mint also outperforms state-of-the-art approaches for multi-task learning while being significantly simpler to implement.
Simultaneous optimization of multiple, potentially unrelated tasks can prove challenging for deep neural networks.
Recent multi-task learning architectures attempt to mitigate this issue by providing alternative pathways for information to flow through a neural network for each task.
In this paper, we introduce a new multi-task learning module, Mint, which provides theoretical guarantees of universal approximation even for multi-task settings with no shared structure.
We conjecture that this property, not shared by similar multi-task architectures, enables Mint to outperform other multi-task approaches on a variety of reinforcement learning benchmarks.
We also observe that Mint is able to match or improve upon the performance of independent training.
While Mint exhibits strong performance gains over previous methods, one potential limitation is that the task matrices may introduce a significant number of parameters, particularly as the number of tasks increases.
As discussed, this can be alleviated for problem domains with many tasks, by learning a single neural network that produces the matrices and biases conditioned on the task descriptor.
Further, in our experiments, we find that Mint-based networks can outperform prior methods while using comparable or fewer parameters.
In summary, Mint is a simple, yet effective approach for deep multi-task learning.
Its implementation requires minimal modifications over standard deep networks.
As a result, we expect it to be straightforward for future work to build upon or use Mint for more effective multi-task learning in deep networks.
A PROOF OF THEOREM 1 Lemma 1.
For a given α i , applying Mint to y (l−1) can express an arbitrary affine transformation at layer l for each task.

assistant: text: We propose an approach that endows a single model with the ability to represent both extremes: joint training and independent training, which leads to effective multi-task learning.