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Title: Discovery of Predictive Representations With a Network of General Value Functions

Paper content:
The ability of an agent to {\em discover} its own learning objectives has long been considered a key ingredient for artificial general intelligence.
Breakthroughs in autonomous decision making and reinforcement learning have primarily been in domains where the agent's goal is outlined and clear: such as playing a game to win, or driving safely.
Several studies have demonstrated that learning extramural sub-tasks and auxiliary predictions can improve (1) single human-specified task learning, (2) transfer of learning, (3) and the agent's learned representation of the world.
In all these examples, the agent was instructed what to learn about.
We investigate a framework for discovery: curating a large collection of predictions, which are used to construct the agent's representation of the world.
Specifically, our system maintains a large collection of predictions, continually pruning and replacing predictions.
We highlight the importance of considering stability rather than convergence for such a system, and develop an adaptive, regularized algorithm towards that aim.
We provide several experiments in computational micro-worlds demonstrating that this simple approach can be effective for discovering useful predictions autonomously.
The idea that an agent's knowledge might be represented as predictions has a long history in machine learning.
The first references to such a predictive approach can be found in the work of BID4 , BID2 BID6 who hypothesized that agents would construct their understanding of the world from interaction, rather than human engineering.
These ideas inspired work on predictive state representations (PSRs), as an approach to modeling dynamical systems.
Simply put, a PSR can predict all possible interactions between an agent and it's environment by reweighting a minimal collection of core test (sequence of actions and observations) and their predictions, as an alternative to keeping a finite history or learning the one step latent dynamics of the world, as in a POMDP.
Extensions to high-dimensional continuous tasks have demonstrated that the predictive approach to dynamical system modeling is competitive with state-of-the-art system identification methods BID9 .
One important limitation of the PSR formalism is that the agent's internal representation of state must be composed exclusively of predictions.Recently, BID23 introduced a formalism for specifying and learning large collections of predictions using value functions from reinforcement learning.
These General Value Functions (GVFs), can represent a wide array of multi-step state contingent predictions BID19 , while the predictions made by a collection of GVFs can be used to construct the agent's state representation (Schaul and Ring, 2013; BID29 . State representation's constructed from predictions have been shown to be useful for reward maximization tasks BID21 Schaul and Ring, 2013) , and transfer learning (Schaul et al., 2015) .
One of the great innovation of GVFs is that we can clearly separate (1) the desire to learn and make use of predictions in various ways, from (2) the construction of the agent's internal state representation.
For example, the UNREAL learning system BID10 , learns many auxiliary tasks (formalized as GVFs) while using an actor-critic algorithm to maximize a single external reward signal-the score in an Atari game.
The auxiliary GVFs and the primary task learner share the same deep convolutional network structure.
Learning the auxiliary tasks results in a better state representation than simply learning the main task alone.
GVFs allow an agent to make use of both increased representational power of predictive representations, and the flexibility of state-of-the-art deep learning systems.
In all the works described above, the GVFs were manually specified by the designer; an autonomous agent, however, must discover these GVFs.
Most work on discovery has been on the related topics of temporal difference (TD) networks BID15 and options BID11 BID17 BID16 BID28 BID16 BID1 .
Discovery for options is more related than TD networks, because similarly to a GVF, an option (Sutton et al., 1999) specifies small sub-tasks within an environment.
Option discovery, however, has been largely directed towards providing temporally abstract actions, towards solving the larger task, rather than providing a predictive representation.
For example, BID1 formulated a gradient descent update on option parameters-policies and termination functions-using policy gradient objectives.
The difficulty in extending such gradient-based approaches is in specifying a suitable objective for prediction accuracy, which is difficult to measure online.We take inspiration from ideas from representation search methods developed for neural networks, to tackle the daunting challenge of GVF discovery for predictive representations.
Our approach is inspired by algorithms that search the topology space of neural networks.
One of the first such approaches, called the cascade correlation network learning typifies this approach BID7 ).
The idea is to continually propose new hidden units over time, incrementally growing the network to multiple levels of abstraction.
To avoid the computation and memory required to pose units whose activation is de-correlated with the network activations, BID24 empirically demonstrated that simply generating large numbers of hidden units outperformed equal sized fixed networks in online supervised learning problems.
Related approaches demonstrated that massive random representations can be highly effective (Rahimi and Recht, 2009; BID0 BID8 .
This randomized feature search can be improved with the addition of periodic pruning, particularly for an incremental learning setting.In this paper, we demonstrate such a curation framework for GVF discovery, with simple algorithms to propose and prune GVFs.
To parallel representation search, we need both a basic functional form-a GVF primitive-for each unit in the network and an update to adjust the weights on these units.
We propose a simple set of GVF primitives, from which to randomly generate candidate GVFs.
We develop a regularized updating algorithm, to facilitate pruning less useful GVFs, with a stepsize adaptation approach that maintains stability in the representation.
We demonstrate both the ability for the regularizer to prune less useful GVFs-and the corresponding predictive features-as well as utility of the GVF primitives as predictive features in several partially observable domains.
Our approach provides a first investigation into a framework for curation of GVFs for predictive representations, with the aim to facilitate further development.
In this paper, we proposed a discovery methodology for GVF networks, to learn of predictive representations.
The strategy involves iteratively generating and pruning GVF primitives for the GVF network, with a new algorithm called AdaGain to promote stability and facilitate pruning.
The results demonstrate utility of this curation strategy for discovering GVF networks.
There are many aspects of our system that could have been designed differently, namely in terms of the learning algorithm, generation approach and pruning approach; here, our goal was to provide a first such demonstration, with the aim to facilitate further development.
We discuss the two key aspects in our system below, and potential avenues to expand along these dimensions.In the development of our learning strategy, we underline the importance of treating the predictive representation as a dynamical system.
For a standard supervised learning setting, the representation is static: the network can be queried at any time with inputs.
The predictive representations considered here cannot be turned off and on, because they progressively build up accurate predictions.
This dynamic nature necessitates a different view of learning.
We proposed a focus on stability of the predictive system, deriving an algorithm to learn a stepsize.
The stepsize can be seen as a control input, to stabilize the system, and was obtained with a relatively straightforward descent algorithm.
More complex control inputs, however, could be considered.
For example, a control function outputting a stepsize based on the current agent state could be much more reactive.
Such an extension would the necessitate a more complex stability analysis from control theory.Our discovery experiments reflect a life-long learning setting, where the predictive representation is slowly built up over millions of steps.
This was slower than strictly necessary, because we wanted to enable convergence for each GVF network before culling.
Further, the pruning strategy was simplistic, using a threshold of 10%; more compact GVF networks could likely be learned-and learned more quickly-with a more informed pruning approach.
Nonetheless, even when designing learning with less conservative learning times, building such representations should be a long-term endeavour.
A natural next step is to more explicitly explore scaffolding.
For example, without compositional GVFs, myopic discounts were less frequently kept; this suggests initially preferring horizon and termination discounts, and increasing preference on myopic discounts once compositional GVFs are added.
Further, to keep the system as simple as possible, we did not treat compositional GVFs differently when pruning.
For example, there is a sudden rise in prediction error at about 80 million steps in FIG2 (b); this was likely caused by pruning a GVF whose prediction was the cumulant for a critical compositional GVF.
Finally, we only considered a simple set of GVF primitives; though this simple set was quite effective, there is an opportunity to design other GVF primitives, and particularly those that might be amenable to composition.

assistant: text: We investigate a framework for discovery: curating a large collection of predictions, which are used to construct the agent’s representation in partially observable domains.