Research Report on Noise-Shaped One-Bit Coefficients in Discrete Polynomial Fourier Extension

We develop a second-order theory of quantization noise in matrix multiplication in which the quantization format is characterized by the variance it assigns to each element. The constant variance profile of integer quantization recovers existing integer-noise theory, while the multiplicative profile of floating-point rounding reduces the data dependence to a scalar, the participation factor $κ$, yielding a closed-form signal-to-noise-ratio law. The resulting functional also admits a closed-form upper bound $κ^{*}$ that no function-preserving linear transform can exceed and that is attained by a recent state-of-the-art method. Building on this analysis, we introduce KBBQ (\textbf{K}appa-\textbf{B}raked \textbf{B}lockwise \textbf{Q}uantization), which parameterizes the extent to which a transform approaches this ceiling. At W4A4, across four base models and two FP4 formats, KBBQ outperforms the prior state of the art without additional deployment-time computation.

Motivated by the optimization of bounded binary black-box functions, we study the problem of learning polynomial surrogates over the Boolean hypercube. To ensure that optimizing the surrogate yields good solutions for the underlying objective, we require uniform $L_\infty$-error guarantees rather than the usual $L_2$-type guarantees. We characterize the minimax sample complexity of uniform estimation under subgaussian noise for two classes of bounded polynomials. First, for polynomials of degree at most $d$ on $n$ variables, the sample complexity scales as $n^{d+1}$. Second, for $s$-sparse Fourier-Walsh polynomials with $s \leq n$, it scales as $ns^2$. These rates differ structurally from the noiseless setting, where uniform exact recovery scales as $n^d$ and $ns$, respectively. Our lower bounds hold even for arbitrary adaptive learners, showing that the additional factors are intrinsic to the noisy cases. Standard Fourier-analysis tools for the $L_2$-norm do not naturally extend to the $L_\infty$-setting in a way that yields uniform guarantees. Our proofs overcome this difficulty by relying on suitably chosen auxiliary norms that serve as proxies for controlling the $L_\infty$-error. Together, our results provide a tight characterization of the sample complexity of learning optimization-safe polynomial surrogates.

Low-bit forward evaluation is an attractive route to memory-efficient zeroth-order (ZO) adaptation: the optimizer needs only scalar losses, and the model can be queried near deployment precision. The obstacle is that a quantized ZO query is not a continuous finite difference followed by harmless storage rounding. The query chooses endpoints, the low-precision engine rounds them, and the loss difference is measured along the rounded chord. For nonuniform companding quantizers, this makes the codebook insufficient to predict ZO behavior: a fixed weight-space radius can collapse in dense cells, over-span sparse cells, or assign a rounded chord to an unrounded update direction. We identify the missing object as query geometry and model scalar nonuniform quantization as $Q = φ^{-1} \circ U \circ φ$. CAQ-ZO (Compander-Aligned Queries for Zeroth-Order Optimization) forms one-grid-step Rademacher stencils $z \pm Δr$ in $z = φ(x)$, maps endpoints back through $φ^{-1}$, and updates in $z$. Our theory proves the grid-span mismatch, decomposes endpoint-rounding estimator residuals, and gives stationarity bounds in which generic off-grid queries retain a $Δ^2/μ^2$ residual channel while CAQ-ZO makes the query-time residual exactly zero. Synthetic experiments isolate this channel, and matched NF4 Qwen/Llama fine-tuning shows that CAQ-ZO improves the trained NF4 baseline under the same quantizer and evaluation budget.