Uniform Approximation of Functions with Asymmetric Growth and Decay by Deep Weighted Polynomials

Technical Summary: Uniform Approximation of Functions with Asymmetric Growth and Decay by Deep Weighted Polynomials

Problem Statement
The paper addresses the challenge of uniformly approximating functions on unbounded domains that exhibit asymmetric behavior: growing without bound on one side of the real line while decaying to zero on the other. Classical polynomial approximation fails in this setting because non-constant polynomials diverge at infinity, making uniform approximation impossible on unbounded domains. While rational functions or exponential sums can handle such targets, they often require divisions or suffer from ill-conditioning and non-density issues (e.g., pure exponential sums are not dense in $C[a,b]$ without confluent families). The authors specifically target functions with "kinks" or sharp transitions (such as option pricing payoffs) combined with one-sided decay, where standard minimax polynomials are forced to distribute error uniformly, including on the decaying tail where the function is negligible.

Methodology
The authors propose a class of one-sided weighted deep (composite) polynomial approximants. The core mechanism involves a product of a composite polynomial $P(x)$ and a one-sided weight function $w(x)$.

1. Deep Polynomial Structure: The approximant $P(x)$ is defined as a composition of $L$ polynomial layers: $P(x) = p_L(p_{L-1}(\dots p_1(x)))$. To ensure a unique representation and stable optimization, inner layers are normalized to be monic with zero constant terms. This structure allows the effective (composite) degree $D = \prod d_\ell$ to grow multiplicatively with the number of layers, while the number of trainable parameters grows only additively.
2. One-Sided Weighting: The weight $w(x)$ is designed to be $1$ on the growing side (e.g., $x < 0$) and decay rapidly on the other side (e.g., $x \ge 0$). A specific example used is $w(x) = \exp(-x^2)$ for $x \ge 0$ and $1$ otherwise. This suppresses the polynomial's growth on the decaying side, effectively localizing the approximation problem.
3. Theoretical Reduction: The paper proves that the weight suppresses the error on the unbounded tail, reducing the approximation problem on a half-line to an approximation problem on a compact interval $[a, b_n]$. The length of this effective interval grows slowly with the degree (specifically $O((n \log n)^{1/\beta})$ for a field exponent $\beta$).
4. Computational Strategy:
   • End-to-End Training: Initially, the method is formulated as a trainable computational graph where all coefficients are optimized via gradient descent. However, the authors note that direct end-to-end optimization becomes ill-conditioned at high composite degrees.
   • Fine-Tuning Procedure: To address instability, a two-stage approach is introduced. The inner composition is fixed using a dictionary of pre-defined, well-conditioned monotone polynomial self-maps (e.g., smoothstep polynomials). Only the outer polynomial coefficients and the weight parameters are trained. The outer fit reduces to a convex linear program (minimax or least squares), avoiding the local minima and conditioning issues of full backpropagation.

Key Contributions

1. Quantitative Reduction: The authors establish a quantitative reduction of the half-line approximation problem to a compact interval (Proposition 6.2), demonstrating that the weight makes the problem effectively compact with a specific effective interval length dependent on the degree and field exponent.
2. Density and Existence: The paper proves the density of weighted deep polynomials in $C(K)$ and establishes the existence of best approximants in the closure of the model class. It identifies that the set of compositions with a fixed architecture is not closed, which can lead to non-attainment of the infimum within the architecture itself, though this is mitigated by the fine-tuning approach.
3. Algorithmic Framework: The authors provide a computational graph formulation for training these approximants and a specific fine-tuning algorithm (Algorithm 2) that fixes the inner layers and solves a convex optimization problem for the outer layer and weight parameters.
4. Empirical Validation: The framework is validated on Black–Scholes option-pricing functions (calls, spreads, butterflies, and option books) which exhibit the target asymmetric growth-decay structure and sharp transitions.

Results
Numerical experiments on Black–Scholes targets at matched parameter budgets (comparing trainable degrees of freedom) show:

• Superior Accuracy: The fine-tuned weighted deep polynomial achieves significantly smaller uniform ($L_\infty$) and $L_2$ errors compared to matched-budget Chebyshev polynomials, Remez minimax polynomials, and plain weighted polynomials (without deep composition).
• Tail Resolution: The weighted methods resolve the decaying tail to machine precision. In contrast, classical polynomial baselines (Chebyshev, Remez) exhibit equioscillation errors of order $10^{-3}$ across the entire domain, including the tail, because they must balance error between the growing/kinked region and the decaying region.
• Comparison with ESPRIT: In the context of the Black–Scholes butterfly spread, the method outperforms the ESPRIT algorithm (a subspace method for exponential/cosine sums). The authors attribute ESPRIT's poor performance to the non-periodic, localized nature of the target, which makes global frequency estimation via Toeplitz–Hankel pencils unstable.
• Role of Inner Composition: Experiments comparing the fine-tuned deep model against a plain weighted polynomial (same weight, no deep composition) demonstrate that the fixed inner composition is crucial for capturing sharp transitions and achieving the lowest errors, particularly on complex multi-scale targets like the option book.

Significance and Claims
The paper claims that deep weighted polynomials offer a division-free, numerically stable alternative for approximating functions with asymmetric growth and decay. By combining the expressive power of deep composition (high effective degree with few parameters) with a one-sided weight that localizes the approximation domain, the method resolves the "tail problem" inherent in unbounded approximation. The authors emphasize that their approach recovers the expressive power of deep networks while avoiding the numerical instability of end-to-end training through a convex fine-tuning procedure. The work extends classical weighted polynomial theory (Bernstein, Freud, Jackson) to the setting of composite polynomials, providing a rigorous theoretical foundation for their empirical success on financial and special function targets.