Hierarchical inductive biases are hypothesized to promote generalizable policies in reinforcement learning, as demonstrated by explicit hyperbolic latent representations and architectures. Therefore, a more flexible approach is to have these biases emerge naturally from the algorithm. We introduce Free Random Projection, an input mapping grounded in free probability theory that constructs random orthogonal matrices where hierarchical structure arises inherently. The free random projection integrates seamlessly into existing in-context reinforcement learning frameworks by encoding hierarchical organization within the input space without requiring explicit architectural modifications. Empirical results on multi-environment benchmarks show that free random projection consistently outperforms the standard random projection, leading to improvements in generalization. Furthermore, analyses within linearly solvable Markov decision processes and investigations of the spectrum of kernel random matrices reveal the theoretical underpinnings of free random projection's enhanced performance, highlighting its capacity for effective adaptation in hierarchically structured state spaces.

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