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Title: Sharing Knowledge in Multi-Task Deep Reinforcement Learning

Paper content:
We study the benefit of sharing representations among tasks to enable the effective use of deep neural networks in Multi-Task Reinforcement Learning.
We leverage the assumption that learning from different tasks, sharing common properties, is helpful to generalize the knowledge of them resulting in a more effective feature extraction compared to learning a single task.
Intuitively, the resulting set of features offers performance benefits when used by Reinforcement Learning algorithms.
We prove this by providing theoretical guarantees that highlight the conditions for which is convenient to share representations among tasks, extending the well-known finite-time bounds of Approximate Value-Iteration to the multi-task setting.
In addition, we complement our analysis by proposing multi-task extensions of three Reinforcement Learning algorithms that we empirically evaluate on widely used Reinforcement Learning benchmarks showing significant improvements over the single-task counterparts in terms of sample efficiency and performance.
Multi-Task Learning (MTL) ambitiously aims to learn multiple tasks jointly instead of learning them separately, leveraging the assumption that the considered tasks have common properties which can be exploited by Machine Learning (ML) models to generalize the learning of each of them.
For instance, the features extracted in the hidden layers of a neural network trained on multiple tasks have the advantage of being a general representation of structures common to each other.
This translates into an effective way of learning multiple tasks at the same time, but it can also improve the learning of each individual task compared to learning them separately (Caruana, 1997) .
Furthermore, the learned representation can be used to perform Transfer Learning (TL), i.e. using it as a preliminary knowledge to learn a new similar task resulting in a more effective and faster learning than learning the new task from scratch (Baxter, 2000; Thrun & Pratt, 2012) .
The same benefits of extraction and exploitation of common features among the tasks achieved in MTL, can be obtained in Multi-Task Reinforcement Learning (MTRL) when training a single agent on multiple Reinforcement Learning (RL) problems with common structures (Taylor & Stone, 2009; Lazaric, 2012) .
In particular, in MTRL an agent can be trained on multiple tasks in the same domain, e.g. riding a bicycle or cycling while going towards a goal, or on different but similar domains, e.g. balancing a pendulum or balancing a double pendulum 1 .
Considering recent advances in Deep Reinforcement Learning (DRL) and the resulting increase in the complexity of experimental benchmarks, the use of Deep Learning (DL) models, e.g. deep neural networks, has become a popular and effective way to extract common features among tasks in MTRL algorithms (Rusu et al., 2015; Liu et al., 2016; Higgins et al., 2017) .
However, despite the high representational capacity of DL models, the extraction of good features remains challenging.
For instance, the performance of the learning process can degrade when unrelated tasks are used together (Caruana, 1997; Baxter, 2000) ; another detrimental issue may occur when the training of a single model is not balanced properly among multiple tasks (Hessel et al., 2018) .
Recent developments in MTRL achieve significant results in feature extraction by means of algorithms specifically developed to address these issues.
While some of these works rely on a single deep neural network to model the multi-task agent (Liu et al., 2016; Yang et al., 2017; Hessel et al., 2018; Wulfmeier et al., 2019) , others use multiple deep neural networks, e.g. one for each task and another for the multi-task agent (Rusu et al., 2015; Parisotto et al., 2015; Higgins et al., 2017; Teh et al., 2017) .
Intuitively, achieving good results in MTRL with a single deep neural network is more desirable than using many of them, since the training time is likely much less and the whole architecture is easier to implement.
In this paper we study the benefits of shared representations among tasks.
We theoretically motivate the intuitive effectiveness of our method, deriving theoretical guarantees that exploit the theoretical framework provided by Maurer et al. (2016) , in which the authors present upper bounds on the quality of learning in MTL when extracting features for multiple tasks in a single shared representation.
The significancy of this result is that the cost of learning the shared representation decreases with a factor O( 1 / √ T ), where T is the number of tasks for many function approximator hypothesis classes.
The main contribution of this work is twofold.
As stated in the remarks of Equation (9), the benefit of MTRL is evinced by the second component of the bound, i.e. the cost of learning h, which vanishes with the increase of the number of tasks.
Obviously, adding more tasks require the shared representation to be large enough to include all of them, undesirably causing the term sup h,w h(w(X)) in the fourth component of the bound to increase.
This introduces a tradeoff between the number of features and number of tasks; however, for Figure 1 : (a) The architecture of the neural network we propose to learn T tasks simultaneously.
The w t block maps each input x t from task µ t to a shared set of layers h which extracts a common representation of the tasks.
Eventually, the shared representation is specialized in block f t and the output y t of the network is computed.
Note that each block can be composed of arbitrarily many layers.
a reasonable number of tasks the number of features used in the single-task case is enough to handle them, as we show in some experiments in Section 5.
Notably, since the AVI/API framework provided by Farahmand (2011) provides an easy way to include the approximation error of a generic function approximator, it is easy to show the benefit in MTRL of the bound in Equation (9).
Despite being just multi-task extensions of previous works, our results are the first one to theoretically show the benefit of sharing representation in MTRL.
Moreover, they serve as a significant theoretical motivation, besides to the intuitive ones, of the practical algorithms that we describe in the following sections.
We have theoretically proved the advantage in RL of using a shared representation to learn multiple tasks w.r.t. learning a single task.
We have derived our results extending the AVI/API bounds (Farahmand, 2011) to MTRL, leveraging the upper bounds on the approximation error in MTL provided in Maurer et al. (2016) .
The results of this analysis show that the error propagation during the AVI/API iterations is reduced according to the number of tasks.
Then, we proposed a practical way of exploiting this theoretical benefit which consists in an effective way of extracting shared representations of multiple tasks by means of deep neural networks.
To empirically show the advantages of our method, we carried out experiments on challenging RL problems with the introduction of multi-task extensions of FQI, DQN, and DDPG based on the neural network structure we proposed.
As desired, the favorable empirical results confirm the theoretical benefit we described.
A PROOFS

assistant: text: A study on the benefit of sharing representation in Multi-Task Reinforcement Learning.