The challenge of selecting the most relevant features of a given dataset arises ubiquitously in data analysis and dimensionality reduction. However, features found to be of high importance for the entire dataset may not be relevant to subsets of interest, and vice versa. Given a feature selector and a fixed decomposition of the data into subsets, we describe a method for identifying selected features which are compatible with the decomposition into subsets. We achieve this by re-framing the problem of finding compatible features to one of finding sections of a suitable quiver representation. In order to approximate such sections, we then introduce a Laplacian operator for quiver representations valued in Hilbert spaces. We provide explicit bounds on how the spectrum of a quiver Laplacian changes when the representation and the underlying quiver are modified in certain natural ways. Finally, we apply this machinery to the study of peak-calling algorithms which measure chromatin accessibility in single-cell data. We demonstrate that eigenvectors of the associated quiver Laplacian yield locally and globally compatible features.

Learning optimal behavior from existing data is one of the most important problems in Reinforcement Learning (RL). This is known as "off-policy control" in RL where an agent's objective is to compute an optimal policy based on the data obtained from the given policy (known as the behavior policy). As the optimal policy can be very different from the behavior policy, learning optimal behavior is very hard in the "off-policy" setting compared to the "on-policy" setting where new data from the policy updates will be utilized in learning. This work proposes an off-policy natural actorcritic algorithm that utilizes state-action distribution correction for handling the off-policy behavior and the natural policy gradient for sample efficiency. The existing natural gradient-based actor-critic algorithms with convergence guarantees require fixed features for approximating both policy and value functions. This often leads to sub-optimal learning in many RL applications. On the other hand, our proposed algorithm utilizes compatible features that enable one to use arbitrary neural networks to approximate the policy and the value function and guarantee convergence to a locally optimal policy. We illustrate the benefit of the proposed off-policy natural gradient algorithm by comparing it with the vanilla gradient actor-critic algorithm on benchmark RL tasks. Reinforcement Learning (RL) [2], [24] deals with learning algorithms for an agent to learn optimal behavior based on the feedback received from an unknown environment.

This paper makes a first step towards compatible and hence reusable network components. Rather than training networks for different tasks independently, we adapt the training process to produce network components that are compatible across tasks. We propose and compare several different approaches to accomplish compatibility. Our experiments on CIFAR-10 show that: (i) we can train networks to produce compatible features, without degrading task accuracy compared to training networks independently; (ii) the degree of compatibility is highly dependent on where we split the network into a feature extractor and a classification head; (iii) random initialization has a large effect on compatibility; (iv) we can train incrementally: given previously trained components, we can train new ones which are also compatible with them. This work is part of a larger goal to increase network reusability: we envision that compatibility will enable solving new tasks by mixing and matching suitable components.

Recent policy optimization approaches have achieved substantial empirical success by constructing surrogate optimization objectives. The Approximate Policy Iteration objective (Schulman et al., 2015a; Kakade and Langford, 2002) has become a standard optimization target for reinforcement learning problems. Using this objective in practice requires an estimator of the advantage function. Policy optimization methods such as those proposed in Schulman et al. (2015b) estimate the advantages using a parametric critic. In this work we establish conditions under which the parametric approximation of the critic does not introduce bias to the updates of surrogate objective. These results hold for a general class of parametric policies, including deep neural networks. We obtain a result analogous to the compatible features derived for the original Policy Gradient Theorem (Sutton et al., 1999). As a result, we also identify a previously unknown bias that current state-of-the-art policy optimization algorithms (Schulman et al., 2015a, 2017) have introduced by not employing these compatible features.