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Generalization bounds for regression and classification on adaptive covering input domains

Wen-Liang Hwang

TL;DR

The paper develops generalization bounds for regression and classification on adaptively covering input domains using a local geometry parameter $\gamma_s$. By linking $\gamma_s$ to network capacity through a refinement oracle, it shows bounds scale with $(K_f+K_{\mathcal{M}})\gamma_s$ for regression and $ (\gamma_s)^{d-1}\mathrm{vol}B_{d-1}(\mathbf{0},1)|\partial f|$ for classification, and proves that $\gamma_s$ decreases polynomially with the number of network parameters. It establishes concentration-based sample complexities for both tasks under fixed-radius deterministic and random ball coverings, yielding $m_0$ bounds and explicit dependence on dimension $d$ and boundary size. An oracle implementation via deep nets links the covering radius to parameter count and demonstrates how to halve the radius through hierarchical partitions, supporting benign overfitting in over-parameterized models. The results highlight that classification can require fewer samples than regression under these bounds and provide insights into inductive biases, network design, and potential extensions to generative modeling.

Abstract

Our main focus is on the generalization bound, which serves as an upper limit for the generalization error. Our analysis delves into regression and classification tasks separately to ensure a thorough examination. We assume the target function is real-valued and Lipschitz continuous for regression tasks. We use the 2-norm and a root-mean-square-error (RMSE) variant to measure the disparities between predictions and actual values. In the case of classification tasks, we treat the target function as a one-hot classifier, representing a piece-wise constant function, and employ 0/1 loss for error measurement. Our analysis underscores the differing sample complexity required to achieve a concentration inequality of generalization bounds, highlighting the variation in learning efficiency for regression and classification tasks. Furthermore, we demonstrate that the generalization bounds for regression and classification functions are inversely proportional to a polynomial of the number of parameters in a network, with the degree depending on the hypothesis class and the network architecture. These findings emphasize the advantages of over-parameterized networks and elucidate the conditions for benign overfitting in such systems.

Generalization bounds for regression and classification on adaptive covering input domains

TL;DR

The paper develops generalization bounds for regression and classification on adaptively covering input domains using a local geometry parameter . By linking to network capacity through a refinement oracle, it shows bounds scale with for regression and for classification, and proves that decreases polynomially with the number of network parameters. It establishes concentration-based sample complexities for both tasks under fixed-radius deterministic and random ball coverings, yielding bounds and explicit dependence on dimension and boundary size. An oracle implementation via deep nets links the covering radius to parameter count and demonstrates how to halve the radius through hierarchical partitions, supporting benign overfitting in over-parameterized models. The results highlight that classification can require fewer samples than regression under these bounds and provide insights into inductive biases, network design, and potential extensions to generative modeling.

Abstract

Our main focus is on the generalization bound, which serves as an upper limit for the generalization error. Our analysis delves into regression and classification tasks separately to ensure a thorough examination. We assume the target function is real-valued and Lipschitz continuous for regression tasks. We use the 2-norm and a root-mean-square-error (RMSE) variant to measure the disparities between predictions and actual values. In the case of classification tasks, we treat the target function as a one-hot classifier, representing a piece-wise constant function, and employ 0/1 loss for error measurement. Our analysis underscores the differing sample complexity required to achieve a concentration inequality of generalization bounds, highlighting the variation in learning efficiency for regression and classification tasks. Furthermore, we demonstrate that the generalization bounds for regression and classification functions are inversely proportional to a polynomial of the number of parameters in a network, with the degree depending on the hypothesis class and the network architecture. These findings emphasize the advantages of over-parameterized networks and elucidate the conditions for benign overfitting in such systems.
Paper Structure (23 sections, 9 theorems, 65 equations, 2 figures)

This paper contains 23 sections, 9 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related works
  4. Main results
  5. Generalization bound for covering with fixed radius deterministic balls
  6. Regression networks with Lipschitz continuity
  7. One-hot classification networks
  8. Network parameters
  9. Concentration bound and sample complexity with fixed radius random balls
  10. Covering the unit $d$-ball randomly for the first time
  11. Regression functions
  12. Classification functions
  13. The uniform probability lower bound
  14. Oracle implementation using DNNs
  15. Sample size efficiency and benign overfitting
  16. ...and 8 more sections

Key Result

Lemma 1

Suppose the target function $f: B_d(\b0, 1) \rightarrow \mathbb R^l$ is a Lipschitz function with Lipschitz constant $K_f$. The function $f$ is learned by regression network $\mathcal{M}$ using $N$ training points, $\{(\mathbf{x}_i, f(\mathbf{x}_i))\}_i$. It is assumed that the training points fulf ∎

Figures (2)

  • Figure 1: The convex polygon $p$ can be partitioned into sub-polytopes using sequences of hyperplane cuttings, leading to a binary tree structure with $p$ at the root. The root $p$ is divided into two sub-polygons by a hyperplane passing through the anchor point $\mathbf{a}_1$, the center of the maximal inscribed ball of $p$. The sub-polygon $q_1$, colored in blue, is further divided using anchor points $\mathbf{a}_2$, $\mathbf{a}_3$, and $\mathbf{a}_4$, corresponding to the centers of sub-polygons $q_2$, $q_3$, and $q_4$. It is worth noting that the volume of any sub-polygon retains a constant fraction of its parent polygon, as defined by Eq. (\ref{['fractionvol']}).
  • Figure 2: The dashed ellipsoid represents the maximum inscribed ellipsoid of the polygon. As per Eq. (\ref{['approxeff']}), the primary axis of the ellipsoid has a radius of $\eta$, while the minor axis has a radius of $\tilde{\eta}$, and their ratio is given by $\tilde{\eta}/\eta = \zeta$. The center of the ellipsoid corresponds to the highlighted anchor point of the polygon. The outer ellipsoid, shown in red and expanded by a factor of two at the boundary of the maximum inscribed ellipsoid, encloses the polygon. Enclosed within the polygon is a blue inner circle with a radius of $\tilde{\eta}$, and the polygon itself is enclosed by a blue outer circle with a radius of $2 \eta$.

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Proposition 3
  • Proposition 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Corollary 8
  • Lemma 9