From Cursed to Competitive: Closing the ZO-FO Gap via Input-to-State Stability

Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across a range of deep learning training problems. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue.

Zeroth-Order (ZO) optimization is pivotal for scenarios where backpropagation is unavailable, such as memory-constrained on-device learning and black-box optimization. However, existing methods face a stark trade-off: they are either sample-inefficient (e.g., standard finite differences) or suffer from high variance due to randomized estimation (e.g., random subspace methods). In this work, we propose Coherent Coordinate Descent (CoCD), a deterministic, sample-efficient, and budget-aware ZO optimizer. Theoretically, we formalize the notion of gradient coherence and demonstrate that CoCD is equivalent to Block Cyclic Coordinate Descent (BCCD) with ``warm starts,'' effectively converting historical (stale) gradients from a liability into a computational asset. This mechanism enables $O(1)$ query complexity per step while maintaining global descent directions. Furthermore, we derive error bounds revealing a counter-intuitive insight: larger finite-difference step sizes can induce an implicit smoothing effect on the optimization landscape by reducing the effective smoothness constant, thereby improving convergence stability. Experiments on MLP, CNN, and ResNet architectures (up to 270k parameters) demonstrate that CoCD significantly outperforms BCCD in terms of sample efficiency and convergence loss/accuracy, and exhibits superior stability over randomized ZO methods. Our results suggest that deterministic, structure-aware updates offer a superior alternative to randomization for lightweight ZO optimization.

The z-transform of a sequence is a classical tool used within signal processing, control theory, computer science, and electrical engineering. It allows for studying sequences from their generating functions, with many operations that can be equivalently defined on the original sequence and its $z$-transform. In particular, the z-transform method focuses on asymptotic behaviors and allows the use of Taylor expansions. We present a sequence of results of increasing significance and difficulty for linear models and optimization algorithms, demonstrating the effectiveness and versatility of the z-transform method in deriving new asymptotic results. Starting from the simplest gradient descent iterations in an infinite-dimensional Hilbert space, we show how the spectral dimension characterizes the convergence behavior. We then extend the analysis to Nesterov acceleration, averaging techniques, and stochastic gradient descent.