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Statistically guided deep learning

Michael Kohler, Adam Krzyzak

TL;DR

This work tackles nonparametric regression with deep networks by developing a theory-guided, over-parameterized architecture that combines a parallel ensemble of depth-$L$ networks with a linear readout. It introduces data-driven initializations and a principled, adaptive scheme for selecting the learning rate and number of gradient steps, yielding provable $L_2$ convergence rates that match the minimax rate for $(p,C)$-smooth regression up to an arbitrarily small $oldsymbol psilon$. The main contributions include a general error bound decomposing into approximation and estimation terms, a practical algorithm for tuning hyperparameters, and empirical evidence showing favorable finite-sample performance on simulated univariate data, often rivaling smoothing splines. The results demonstrate that theoretical analysis can guide the design of deep-learning estimators with improved finite-sample behavior, potentially extending to higher dimensions and more complex function classes.

Abstract

We present a theoretically well-founded deep learning algorithm for nonparametric regression. It uses over-parametrized deep neural networks with logistic activation function, which are fitted to the given data via gradient descent. We propose a special topology of these networks, a special random initialization of the weights, and a data-dependent choice of the learning rate and the number of gradient descent steps. We prove a theoretical bound on the expected $L_2$ error of this estimate, and illustrate its finite sample size performance by applying it to simulated data. Our results show that a theoretical analysis of deep learning which takes into account simultaneously optimization, generalization and approximation can result in a new deep learning estimate which has an improved finite sample performance.

Statistically guided deep learning

TL;DR

This work tackles nonparametric regression with deep networks by developing a theory-guided, over-parameterized architecture that combines a parallel ensemble of depth- networks with a linear readout. It introduces data-driven initializations and a principled, adaptive scheme for selecting the learning rate and number of gradient steps, yielding provable convergence rates that match the minimax rate for -smooth regression up to an arbitrarily small . The main contributions include a general error bound decomposing into approximation and estimation terms, a practical algorithm for tuning hyperparameters, and empirical evidence showing favorable finite-sample performance on simulated univariate data, often rivaling smoothing splines. The results demonstrate that theoretical analysis can guide the design of deep-learning estimators with improved finite-sample behavior, potentially extending to higher dimensions and more complex function classes.

Abstract

We present a theoretically well-founded deep learning algorithm for nonparametric regression. It uses over-parametrized deep neural networks with logistic activation function, which are fitted to the given data via gradient descent. We propose a special topology of these networks, a special random initialization of the weights, and a data-dependent choice of the learning rate and the number of gradient descent steps. We prove a theoretical bound on the expected error of this estimate, and illustrate its finite sample size performance by applying it to simulated data. Our results show that a theoretical analysis of deep learning which takes into account simultaneously optimization, generalization and approximation can result in a new deep learning estimate which has an improved finite sample performance.

Paper Structure

This paper contains 32 sections, 4 theorems, 147 equations, 5 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Scope of this paper
  3. Nonparametric regression
  4. Least squares estimates estimates
  5. Neural networks
  6. Computation of neural network regression estimates
  7. Difficulty in the application of deep neural networks
  8. A theoretical approach to deep learning
  9. Main results
  10. Discussion of related results
  11. Notation
  12. Outline
  13. Definition of the estimate
  14. Topology of the network
  15. Initialization of the weights
  16. ...and 17 more sections

Key Result

Theorem 1

Let $n \in \mathbb{N}$, let $(X,Y)$, $(X_1,Y_1)$, …, $(X_n,Y_n)$ be independent and identically distributed $\mathbb{R}^d \times \mathbb{R}$--valued random variables such that $supp(X)$ is bounded, the regression function is bounded in absolute value, and holds. Let $K_n \in \mathbb{N}$ be such that for some $\kappa>0$, set $A=A_n$ and $B=B_n$ for some set $\beta_n= c_{12} \cdot \log n$ and def

Figures (5)

  • Figure 1: Neural network estimate with various initialization schemes, various topologies and various choices of the stepsize applied to the univariate regression problem with sample size $n=100$.
  • Figure 2: Estimate applied to a sample of size $n=100$, with parameters $K \in \{200, 400, 800, 1600\}$, $L=4$, $r=8$, $\lambda=2/K$, $t_n=K/2$, $A=1000$ and $B=20$.
  • Figure 3: Adaptive estimates applied to a sample of size $n=100$, with parameters $K \in \{200, 400, 800,1600\}$, $L=4$, $r=8$, $A=1000$ and $B=20$.
  • Figure 4: Estimate applied to a sample of size $n=100$, with parameters $K \in \{100, 200, 400, 800\}$, $L=4$, $r=8$, adaptively chosen values for $\lambda$ and $t_n$, and values of $A \in \{10, 100, 1000\}$ and $B \in \{20, 200, 2000\}$ chosen via splitting of the sample with $n_{train}=80$ and $n_{test}=20$.
  • Figure 5: Standard neural network estimates with $L=2$, $L=4$ and $L=6$ hidden layers and a smoothing spline estimate applied each time to a sample of size $n=100$.

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Lemma 2