Thomas T. Zhang

Many modern applications of deep learning involve training a neural network via a one-step prediction loss (e.g., L^2 regression, cross-entropy), but deploy the network by rolling out along its own predictions. Key examples include autoregressive language modeling, flow-based generative modeling, and robot policy learning. It is well-documented that these settings induce a phenomenon we call test-time feedback (TTF): the mismatch between the training/validation loss and downstream metrics of interest, such as task success rate and generation quality, which grows with task length. While data curation, architecture, and objective design have been proposed to combat train-test shift in TTF settings, this paper proposes optimization as a new design axis to mitigate error accumulation. Specifically, we introduce a new optimization paradigm called double-preconditioning (DoPr) uniquely tailored to the challenges of TTF. DoPr combines gradient-wise preconditioning, as in Adam and Muon, with activation-wise preconditioning (AP), such as in KFAC. We show that the addition of AP yields a drop-in intervention for increasing downstream model performance across a range of TTF settings. Interestingly, these gains in test-time performance do not consistently accompany improvements in validation loss, opening new questions about how to properly evaluate models trained with one-step supervised objectives.

This paper presents a theoretical analysis of two of the most impactful interventions in modern learning from demonstration in robotics and continuous control: the practice of action-chunking (predicting sequences of actions in open-loop) and exploratory augmentation of expert demonstrations. Though recent results show that learning from demonstration, also known as imitation learning (IL), can suffer errors that compound exponentially with task horizon in continuous settings, we demonstrate that action chunking and exploratory data collection circumvent exponential compounding errors in different regimes. Our results identify control-theoretic stability as the key mechanism underlying the benefits of these interventions. On the empirical side, we validate our predictions and the role of control-theoretic stability through experimentation on popular robot learning benchmarks. On the theoretical side, we demonstrate that the control-theoretic lens provides fine-grained insights into how compounding error arises, leading to tighter statistical guarantees on imitation learning error when these interventions are applied than previous techniques based on information-theoretic considerations alone.

Layer-wise preconditioning methods are a family of memory-efficient optimization algorithms that introduce preconditioners per axis of each layer’s weight tensors. These methods have seen a recent resurgence, demonstrating impressive performance relative to entry-wise ("diagonal") preconditioning methods such as Adam(W) on a wide range of neural network optimization tasks. Complementary to their practical performance, we demonstrate that layer-wise preconditioning methods are provably necessary from a statistical perspective. To showcase this, we consider two prototypical models, linear representation learning and single-index learning, which are widely used to study how typical algorithms efficiently learn useful features to enable generalization. In these problems, we show SGD is a suboptimal feature learner when extending beyond ideal isotropic inputs \mathbf{x} ∼\mathsf{N}(\mathbf{0}, \mathbf{I}) and well-conditioned settings typically assumed in prior work. We demonstrate theoretically and numerically that this suboptimality is fundamental, and that layer-wise preconditioning emerges naturally as the solution. We further show that standard tools like Adam preconditioning and batch-norm only mildly mitigate these issues, supporting the unique benefits of layer-wise preconditioning.

We propose Taylor Series Imitation Learning (TaSIL), a simple augmentation to standard behavior cloning losses in the context of continuous control. TaSIL penalizes deviations in the higher-order Taylor series terms between the learned and expert policies. We show that experts satisfying a notion of \textit{incremental input-to-state stability} are easy to learn, in the sense that a small TaSIL-augmented imitation loss over expert trajectories guarantees a small imitation loss over trajectories generated by the learned policy. We provide sample-complexity bounds for TaSIL that scale as \tilde{\mathcal{O}}(1/n) in the realizable setting, for n the number of expert demonstrations. Finally, we demonstrate experimentally the relationship between the robustness of the expert policy and the order of Taylor expansion required in TaSIL, and compare standard Behavior Cloning, DART, and DAgger with TaSIL-loss-augmented variants. In all cases, we show significant improvement over baselines across a variety of MuJoCo tasks.