Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A simple algorithm for output range analysis for deep neural networks

Helder Rojas, Nilton Rojas, Espinoza J. B., Luis Huamanchumo

TL;DR

The paper introduces a Simulated Annealing algorithm with reflective boundary conditions to estimate the output range of deep neural networks within a restricted domain $E$ that mirrors training data. By modeling SA as a non-homogeneous Markov process with proposals drawn from a Gaussian density and reflections at the boundary, it proves convergence to the global optimum as the temperature schedule $T_i$ decreases. The method requires no internal architectural assumptions and is demonstrated on ResNet-like networks using synthetic test functions (Ackley, Drop-Wave, and a multimodal MINIMA problem) to validate robustness to non-convex surfaces. Empirical results show efficient and stable convergence to the true minima (and maxima) within $E$, with publicly available Python implementations to ensure reproducibility and applicability to a broad class of DNN architectures.

Abstract

This paper presents a novel approach for the output range estimation problem in Deep Neural Networks (DNNs) by integrating a Simulated Annealing (SA) algorithm tailored to operate within constrained domains and ensure convergence towards global optima. The method effectively addresses the challenges posed by the lack of local geometric information and the high non-linearity inherent to DNNs, making it applicable to a wide variety of architectures, with a special focus on Residual Networks (ResNets) due to their practical importance. Unlike existing methods, our algorithm imposes minimal assumptions on the internal architecture of neural networks, thereby extending its usability to complex models. Theoretical analysis guarantees convergence, while extensive empirical evaluations-including optimization tests involving functions with multiple local minima-demonstrate the robustness of our algorithm in navigating non-convex response surfaces. The experimental results highlight the algorithm's efficiency in accurately estimating DNN output ranges, even in scenarios characterized by high non-linearity and complex constraints. For reproducibility, Python codes and datasets used in the experiments are publicly available through our GitHub repository.

A simple algorithm for output range analysis for deep neural networks

TL;DR

The paper introduces a Simulated Annealing algorithm with reflective boundary conditions to estimate the output range of deep neural networks within a restricted domain that mirrors training data. By modeling SA as a non-homogeneous Markov process with proposals drawn from a Gaussian density and reflections at the boundary, it proves convergence to the global optimum as the temperature schedule decreases. The method requires no internal architectural assumptions and is demonstrated on ResNet-like networks using synthetic test functions (Ackley, Drop-Wave, and a multimodal MINIMA problem) to validate robustness to non-convex surfaces. Empirical results show efficient and stable convergence to the true minima (and maxima) within , with publicly available Python implementations to ensure reproducibility and applicability to a broad class of DNN architectures.

Abstract

This paper presents a novel approach for the output range estimation problem in Deep Neural Networks (DNNs) by integrating a Simulated Annealing (SA) algorithm tailored to operate within constrained domains and ensure convergence towards global optima. The method effectively addresses the challenges posed by the lack of local geometric information and the high non-linearity inherent to DNNs, making it applicable to a wide variety of architectures, with a special focus on Residual Networks (ResNets) due to their practical importance. Unlike existing methods, our algorithm imposes minimal assumptions on the internal architecture of neural networks, thereby extending its usability to complex models. Theoretical analysis guarantees convergence, while extensive empirical evaluations-including optimization tests involving functions with multiple local minima-demonstrate the robustness of our algorithm in navigating non-convex response surfaces. The experimental results highlight the algorithm's efficiency in accurately estimating DNN output ranges, even in scenarios characterized by high non-linearity and complex constraints. For reproducibility, Python codes and datasets used in the experiments are publicly available through our GitHub repository.
Paper Structure (14 sections, 2 theorems, 17 equations, 9 figures, 1 algorithm)

This paper contains 14 sections, 2 theorems, 17 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Residual Neural Networks
  4. Output Range Analysis Problem
  5. Simulated Annealing with Boundary Conditions
  6. Algorithm Derivation
  7. Experimental Evaluations
  8. The Ackley Function
  9. Differences with the Classical Simulated Annealing Algorithm
  10. The Drop-Wave Function
  11. Differences with the Classical Simulated Annealing Algorithm
  12. Multiple Global Minima Function
  13. Differences with the Classical Simulated Annealing Algorithm
  14. Conclusions

Key Result

Theorem 8

Given the minimization problem $(\mathcal{F}, E)$, let $(X_i)_{i\in\mathbb{N}}$ a simulated annealing process with reflective boundary conditions on $E$. Then, for each $i\in \mathbb{N}$, the operator $P_i$ has the equilibrium distribution $\pi_i$ given by where $C_i$ is the normalization constant and $\mathcal{F}_{\min}\in M_{\mathcal{F}}$. Moreover, if we set $T_i=T_0\,\delta^i$, for some $|\de

Figures (9)

  • Figure 7.1: Graphs of the experimental evaluation using the Ackley function.
  • Figure 7.2: Red line: Sample path of Markov process $(X_i)_{i\in\mathbb{N}}$ for the Ackley function. Blue line: fitted trend line.
  • Figure 7.3: Red line: Sample path of the classical Simulated Annealing algorithm for the Ackley Function. Blue line: fitted trend line.
  • Figure 7.4: Drop-Wave function of two variables. Left: Perspective view; Right: Top View.
  • Figure 7.5: Drop-Wave surface generated by the ResNet $(E, \mathcal{F}(E))$. Left: Perspective view; Right: Top View.
  • ...and 4 more figures

Theorems & Definitions (11)

  • Definition 1: ResNet
  • Definition 2: Range Estimation Problem
  • Remark 3
  • Definition 4: Cyclic Reflection
  • Definition 5: Generating Distribution
  • Definition 6: Acceptance Probability
  • Definition 7: Simulated Annealing Process
  • Theorem 8
  • proof
  • Corollary 9
  • ...and 1 more