Correcting Stochastic Update Bias in Preconditioned Language Model Optimizers

Preconditioned optimizers are central to language model training, but their stochastic update rules are usually treated as direct approximations to population preconditioned descent. We show that this view misses two finite-sample biases. First, the gradient and preconditioner are typically estimated from the same minibatch, introducing gradient--preconditioner coupling bias. Second, even when the preconditioner estimate is unbiased, its inverse or inverse-root is generally biased because inversion is nonlinear. We propose a single-batch bias-correction framework that addresses both effects: cross-fitted preconditioning estimates the numerator and preconditioner from independent microbatch groups, while variance-corrected inversion uses microbatch variability to subtract the leading delta-method bias term. The framework applies to diagonal moment, diagonal curvature, and matrix preconditioning methods, instantiated in AdamW, Sophia, and Shampoo. Bias correction reduces held-out pretraining loss on Qwen2.5-0.5B by $0.15$, $0.07$, and $0.11$ nats, respectively; the effects on mixed-quality pretraining and downstream instruction tuning are consistently neutral-to-positive. Together, these results establish bias correction as a practical mechanism for reducing finite-sample update bias and improving the performance of preconditioned optimizers.

Reinforcement Learning in Agent-Based Market Simulation: Unveiling Realistic Stylized Facts and Behavior

Investors and regulators can greatly benefit from a realistic market simulator that enables them to anticipate the consequences of their decisions in real markets. However, traditional rule-based market simulators often fall short in accurately capturing the dynamic behavior of market participants, particularly in response to external market impact events or changes in the behavior of other participants. In this study, we explore an agent-based simulation framework employing reinforcement learning (RL) agents. We present the implementation details of these RL agents and demonstrate that the simulated market exhibits realistic stylized facts observed in real-world markets. Furthermore, we investigate the behavior of RL agents when confronted with external market impacts, such as a flash crash. Our findings shed light on the effectiveness and adaptability of RL-based agents within the simulation, offering insights into their response to significant market events.