Backpropagation is the default learning rule for artificial neural networks and is often treated as the settled approach whenever differentiability is available. In this work, we revisit this convention through a theoretical lens of sample efficiency. We introduce a unified vectorized feedback framework for loss-based and reward-based learning on computational graphs, in which synthetic gradients emerge as a natural alternative to backpropagation. We characterize the conditions under which synthetic gradients can achieve a lower gradient-estimation mean squared error than backpropagation. We construct examples illustrating that this sample efficiency advantage can be arbitrarily large. Experiments on contextual bandits and reinforcement learning tasks demonstrate the potential of our theoretical findings.
We consider the problem of Cost-Aware Learning, where sampling different components of a finite-sum objective incurs different costs. The objective is to reach a target error while minimizing the total cost. We propose Cost-Aware SGD, which uses a distribution based on gradient norms and costs to sample components. We provide a thorough analysis of this algorithm, including cost-improvement bounds over baselines, a characterization of distribution proxy sub-optimality, and a lower bound. We apply our theoretical insights to reinforcement learning with language models, where the computational cost of sequence-level policy gradients varies with length. We find that the advantage magnitude serves as a high-fidelity proxy for gradient norms, and use this to introduce Cost-Aware GRPO. Empirical results on 1.5B, 4B, and 8B LLMs demonstrate that this algorithm significantly reduces the tokens used in policy optimization while matching or exceeding baseline accuracy.
Synthetic data becomes crucial for large language model training, but its effectiveness is highly inconsistent. We provide an information-theoretic account of this inconsistency: synthetic data improves a model only when the generation-training loop is information-open, i.e., shaped by external signals (verifiers, environments, or rubrics) that inject task-relevant information beyond the model's current distribution. When the loop is information-closed (relying on the model's own outputs without such signals), the data processing inequality ensures that task-relevant information can only decrease, making collapse a predicted outcome. Among information-open pipelines, both efficiency and generalization hinge on the meta-level of supervision: a coarser signal such as binary correctness treats all acceptable outputs as equivalent, so the behavior it teaches is not tied to any particular domain or surface form and generalizes naturally across tasks and domains. These observations lead to a guiding thesis: learning preferentially converges to the most information-efficient signal component available, which accelerates learning when that component is the intended one, but causes reward hacking when a spurious pattern happens to be simpler.
Model-free reinforcement learning algorithms such as Proximal Policy Optimization (PPO) treat the environment as a black box, estimating policy gradients from sampled rewards; this process demands millions of interactions and relies on high-variance advantage estimates. When environment dynamics are differentiable, the return is an end-to-end differentiable function of the policy parameters, enabling exact gradient computation via backpropagation through simulation. We term this approach Analytic Policy Gradients (APG) and evaluate it against PPO on four continuous control tasks of increasing dynamical complexity: a one-dimensional point-mass target-reaching task, a 2D point-mass navigation task with obstacle avoidance, a 2D rigid-body T-block pushing task, and a 7-DOF Franka FR3 end-effector reaching task. Both algorithms share identical model architectures, observation normalization, and optimizer settings. To decouple sample efficiency from compute efficiency, we design a multi-axis evaluation protocol that records performance against environment steps and gradient steps. We report a segmented backpropagation scheme with MC and critic-based bootstrap modes that mitigates gradient degradation on long-horizon tasks, and present ablations over segment length and bootstrap strategy.