In the natural world, life has found innumerable ways to survive and often thrive. Between and even within species, each individual is in some manner unique, and this diversity lends adaptability and robustness to life. In this work, we aim to learn a space of diverse and high-reward policies in a given environment. To this end, we introduce a generative model of policies for reinforcement learning, which maps a low-dimensional latent space to an agent policy space. Our method enables learning an entire population of agent policies, without requiring the use of separate policy parameters. Just as real world populations can adapt and evolve via natural selection, our method is able to adapt to changes in our environment solely by selecting for policies in latent space. We test our generative model's capabilities in a variety of environments, including an open-ended grid-world and a two-player soccer environment. Code, visualizations, and additional experiments can be found at https://kennyderek.github.io/adap/.

To gain intuition for how p-locality functions, we will introduce another notion of locality, called Markov locality, which will use the language of Markov blankets. We will prove that under relatively relaxed conditions p-locality and Markov locality are equivalent. This will allow us to relate the notion of locality to various graph structures commonly used to represent probability distributions, and will be a key step in proving Properties 2.1 and 2.2. We start by defining the Markov boundary, M(X,S), of a random variable X contained in a set of random variables S, as a minimal set such that p(X|S) = p(X|M(X,S)). The Markov boundary defines a minimal set of variables such that, conditioned on these variables, conditioning on no additional random variables in S changes the probability of X [39]. Similarly, we define the Markov blanket, M(X,S) for X in S as any set of variables such that conditioning on M(X,S), makes X conditionally independent from all other variables [39]. In this way, the Markov boundary is a Markov blanket but not all blankets are boundaries. Markov locality: Given probability distribution p(Z) and function f: RNX+NΘ RNΘ, the update function f(Z) is Markov-local with respect to the distribution p over Z if and only if k: Z Ωs.t. AMarkov boundary can be thought of as the set of variables that'locally' communicate with the parameter Θk, thus providing a natural measure of locality. Importantly, for Markov-locality to be of use, we would like the Markov boundaries of random variables in the model of interest to be unique.

In statistical learning and analysis from shared data, which is increasingly widely adopted in platforms such as federated learning and meta-learning, there are two major concerns: privacy and robustness. Each participating individual should be able to contribute without the fear of leaking one's sensitive information. At the same time, the system should be robust in the presence of malicious participants inserting corrupted data. Recent algorithmic advances in learning from shared data focus on either one of these threats, leaving the system vulnerable to the other.

In recent centralized nonconvex distributed learning and federated learning, local methods are one of the promising approaches to reduce communication time. However, existing work has mainly focused on studying first-order optimality guarantees. On the other side, second-order optimality guaranteed algorithms, i.e., algorithms escaping saddle points, have been extensively studied in the nondistributed optimization literature. In this paper, we study a new local algorithm called Bias-Variance Reduced Local Perturbed SGD (BVR-L-PSGD), that combines the existing bias-variance reduced gradient estimator with parameter perturbation to find second-order optimal points in centralized nonconvex distributed optimization. BVR-L-PSGD enjoys second-order optimality with nearly the same communication complexity as the best known one of BVR-L-SGD to find first-order optimality. Particularly, the communication complexity is better than non-local methods when the local datasets heterogeneity is smaller than the smoothness of the local loss. In an extreme case, the communication complexity approaches to eΘ(1)when the local datasets heterogeneity goes to zero.