A central problem in the domain of reinforcement learning is how to incentivize exploration and diversity of experience, since RL agents can typically only learn from states they go to, and it can often be the case that states with high reward don't have an obvious trail of high-reward states leading to them, meaning that algorithms that are naively optimizing for reward will be relatively unlikely to discover them. One potential way to promote exploration is to train an ensemble of agents, and have part of your loss that incentivizes diversity in the behavior of those agents. Intuitively, an incentive for diversity will push policies away from one another, which will force them to behave differently, and thus reach a wider range of different states. 

Current work in diversity tends to penalize, for each agent, the average pairwise distance between it and every other agent. The authors of this paper have two critiques with this approach:

1. Pure pairwise distance doesn't fully capture the notion of diversity we might want, since if you have policies that are clustered together into multiple clusters, average pairwise distance can be increased by moving the clusters farther apart, without having to break up the clusters themselves
2. Having the diversity term be calculated for each policy independently can lead to "cycling" behavior, where policies move in ways that increase distance at each step, but don't do so when each agent's independent steps are taken into account 

As an alternative, they propose calculating the pairwise Kernel function similarity between all policies (where each policy is represented as the average action probabilities it returns across a sample of states), and calculating the determinate of that matrix. The authors claim that this represents a better measure of full population diversity. I can't fully connect the dots on this intuitively, but what I think they're saying is: the determinant of the kernel matrix tells you  something about the effective dimensionality spanned by the different policies. In the same way that, if a matrix is low-rank, it tells you that some vectors within it can be nearly represented by linear combinations of others, a low value of the Kernel determinant means that some policies can be sufficiently represented by combinations of other policies, such that it isn't really adding diversity value to the ensemble. 

https://i.imgur.com/CmlGsNP.png

Another contribution of this paper is to propose an interesting bandit-based way of determining when to incentivize diversity vs focus on pure reward. The diversity term in the loss is governed by a lambda parameter, and the paper's model sets up Thompson sampling to determine what the value of the parameter should be at each training iteration. This bandit setup works by starting out uncertain over whether to include diversity in the loss, and building a model of whether reward tends to increase during steps where diversity is used. Over time, if diversity consistently doesn't produce benefits, the sampler will tend more towards excluding it from the loss. This is just a really odd, different idea, that I've never heard of before in the context of hyperparameter scheduling during training. I will say that I'm somewhat confused about the window of time that the bandit uses for calculating rewards. Is it a set of trajectories used in a single training batch, or longer than that? Questions aside, they do so fairly convincingly that the adaptive parameter schedule outperforms over a fixed schedule, though they don't test it against simpler forms of annealing, so I'm less clear on whether it would outperform those. 

Overall, I still have some confusions about the method proposed by this paper, but it seems to be approaching the question of exploration from an interesting direction, and I'd enjoy trying to understand it further in future.