FracTrace: Long-horizon credit assignment in deep reinforcement learning via fractional eligibility traces

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Temporal credit assignment becomes intractable when reward signals are displaced hundreds of steps from causal actions, limiting learning in robotic manipulation, autonomous navigation, and long-horizon planning. Conventional exponentially-decaying eligibility traces allocate negligible credit beyond moderate horizons, as their geometric forgetting renders causal attribution effectively impossible at extended lags. This paper introduces FracTrace, a Fractional Trace Actor-Critic framework that replaces exponential decay with power-law memory kernels derived from fractional calculus. Variable-order Mittag-Leffler functions are incorporated at two architectural levels: hierarchical traces maintain multi-timescale state representations across fast, medium, and slow dynamics, while fractional eligibility traces induce algebraic decay that sustains meaningful credit assignment on tasks with reward delays of 500–800 steps. A key design principle retains the standard distributional Bellman operator for the critic, preserving -contraction in Wasserstein metric, while fractionalization is applied exclusively to policy-side advantages. For an idealized policy-gradient variant of the algorithm, almost-sure convergence of the critic to the true value function and of the policy to a stationary point neighborhood of radius , a truncation-dependent bias term, are established under two-timescale stochastic approximation, where vanishes as the truncation horizon ; the practical algorithm uses the PPO clipped surrogate and is evaluated empirically. The critic achieves sample complexity , matching classical actor-critic rates while capturing substantially longer dependencies. Empirical evaluation across six long-horizon tasks demonstrates substantial improvements over PPO, PPO-LSTM, R2D2, and GTrXL, with FracTrace uniquely solving extreme-horizon tasks (800 steps) at per-step complexity. The kernel-swap ablation, which isolates kernel choice within an otherwise identical architecture, supports the conclusion that these gains stem from power-law memory structure; the PPO-LSTM comparison provides converging evidence at the level of a full baseline architecture rather than a single-factor control.