A.1 Datasets Details of the datasets we introduce are presented in this section. Specific details about generation as well as statistics from the resulting datasets are delineated for each one below. A.1.1 Prefix sum data Binary string inputs of length nare generated by selecting a random integer in [0,2n)and expressing its binary representation with n digits. Datasets are produced by repeating this random process 10,000 times without replacement. Because the number of possible points increases exponentially as a function of n and the size of the generated dataset is fixed, it is important to note that the dataset becomes sparser in its ambient hypercube as nincreases.

Deep neural networks are powerful machines for visual pattern recognition, but reasoning tasks that are easy for humans may still be difficult for neural models. Humans possess the ability to extrapolate reasoning strategies learned on simple problems to solve harder examples, often by thinking for longer. For example, a person who has learned to solve small mazes can easily extend the very same search techniques to solve much larger mazes by spending more time. In computers, this behavior is often achieved through the use of algorithms, which scale to arbitrarily hard problem instances at the cost of more computation. In contrast, the sequential computing budget of feed-forward neural networks is limited by their depth, and networks trained on simple problems have no way of extending their reasoning to accommodate harder problems. In this work, we show that recurrent networks trained to solve simple problems with few recurrent steps can indeed solve much more complex problems simply by performing additional recurrences during inference. We demonstrate this algorithmic behavior of recurrent networks on prefix sum computation, mazes, and chess. In all three domains, networks trained on simple problem instances are able to extend their reasoning abilities at test time simply by "thinking for longer."

In this section we prove the theoretical results around the dual curriculum game and use these results to show approximation bounds for our methods, given that they have reached a Nash equilibrium (NE). The first theorem is the main result that allows us to analyze dual curriculum games. The high-level result says that the NE of a dual curriculum game are approximate NE of the base game from the perspective of any of the individual players, or from the perspective of the joint strategy. Let Bbe the maximum difference between U1t and U2t, and let (π,θ1,θ2) be a NE for G. Then (π,pθ1 + (1 p)θ2) is an approximate NE for the base game with either teacher or for a teacher optimizing their joint objective. More precisely, it is a 2Bp(1 p)-approximate NE when Ut = pU1t + (1 p)U2t, a 2B(1 p)-approximate NE when Ut = U1t, and a 2Bp-approximate NE when Ut = U2t. At a high level, this is true because, for low values of p, the best-response strategies for the individual players can be thought of as approximate-best response strategies for the joint-player, and vis-versa. Since the Nash Equilibrium consists of each of the players playing their own best response, they must be playing an approximate best response for the joint-player. We provide a formal proof below: Proof. Let B be the maximum difference between U1t and U2t, and let (π,θ1,θ2) be a Nash Equilibrium for G. Then consider pθ1 + (1 p)θ2 as a strategy in the base game for the joint player pU1t + (1 p)U2t.

Recent developments in reinforcement learning (RL) led to algorithms that surpass human experts in a broad range of tasks [Mnih et al., 2015, Vinyals et al., 2019, Schrittwieser et al., 2020, Wurman et al.,

However, we have identified that the effectiveness of these rewards declines as the environmental complexity rises. Therefore, we present a novel USD algorithm, skill disco very with gui dance ( DISCO-DANCE), which (1) selects the guide skill that possesses the highest potential to reach unexplored states, (2) guides other skills to follow guide skill, then (3) the guided skills are dispersed to maximize their discriminability in unexplored states. Empirical evaluation demonstrates that DISCO-DANCE outperforms other USD baselines in challenging environments, including two navigation benchmarks and a continuous control benchmark.

A.1 Datasets Details of the datasets we introduce are presented in this section. Specific details about generation as well as statistics from the resulting datasets are delineated for each one below. A.1.1 Prefix sum data Binary string inputs of length n are generated by selecting a random integer in [0, 2 Datasets are produced by repeating this random process 10,000 times without replacement. Because the number of possible points increases exponentially as a function of n and the size of the generated dataset is fixed, it is important to note that the dataset becomes sparser in its ambient hypercube as n increases. Moreover, we are limited to datasets with binary strings of length n>13 to avoid duplicate data points.