Random Fuzzy-Rule Foams for Explainable AI

A random foam trains several fuzzy-rule-foam function approximators and then combines them into a single rule-based approximator. The foam systems train independently on bootstrapped random samples from a trained neural classifier. The foam systems convert the neural black box into an interpretable set of rules. The fuzzy rule-based systems have an underlying probability mixture structure that gives rise to an interpretable Bayesian posterior over the rules for each input. A rule foam also measures the uncertainty in its outputs through the conditional variance of the generalized probability mixture. A random foam combines the learned additive fuzzy systems by averaging their throughputs or rule structure. A random foam is also interpretable in terms of its rules, its posterior of its rules, and its conditional variance. Thirty 1000-rule foams trained on random subsets of the MNIST digit data set. Each such foam system had about 93.5% classification accuracy. The random foam that averaged throughputs achieved \(96.80\%\) accuracy while the random foam that averaged only their outputs achieved 96.06% accuracy. The throughput-averaged random foam also slightly outperformed a standard random forest that output-averaged 30 classification trees. Thirty 1000-rule foams also trained on a deep neural classifier that had 96.26% accuracy. The random foam that averaged these foam throughputs was itself 96.14% accurate. The random foam that averaged their outputs was just 95.6% accurate. The appendix proves a Gaussian combined-foam version of the uniform approximation theorem for additive fuzzy systems.

Authors and Affiliations

1. Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, USA

   Akash Kumar Panda & Bart Kosko

Corresponding author

Correspondence to Bart Kosko .

Editors and Affiliations

1. Department of Mathematics, DigiPen Institute of Technology, Redmond, WA, USA

   Barnabás Bede

2. Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA

   Martine Ceberio

3. School of Engineering and Technology, University of Washington, Tacoma, WA, USA

   Martine De Cock

4. Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA

   Vladik Kreinovich

7 Appendix : Fuzzy Approximation Theorem for a Combination of Gaussian SAMs

Theorem 1 extends a version of the fuzzy approximation theorem [10] to the case of combined additive systems. The Gaussian structure of the if-part sets gives insight into the uniform approximation. The approximation may need ab exponentially growing total number of rules mq with respect to the number of dimensions n.

Theorem 1

A combination of q Gaussian SAMs can uniformly approximate a continuous function f over a compact space.

Proof

We combine q SAMs to approximate f(x) with F(x) by mixing their then-part sets. We want to show that the approximation error obeys \(|f(x)-F(x)| < \varepsilon \) for any \(\varepsilon >0\) using a finite number of rules. We use Gaussian if-part sets with set functions \(a_j^k(x) = \exp \{-||x-\hat{x}_j^k||^2/(d_j^k)^2\}\). We assign equal weights, equal volumes, and equal dispersions to all the rules: \(v^k = v\), \(w_j^k = w\), \(V_j^k = V\), and \(d_j^k = d\). This reduces the generalized mixing weights \(p_j(x)\) to softmax activation functions:

Multiply f(x) by \(\sum _{k=1}^q\sum _{j=1}^m p_j^k(x)=1\) and then use (7):

Rearrange the terms and use the triangle inequality:

because we center the then-part set around \(f(\hat{x}_j^k)\). So \(c_j^k = f(\hat{x}_j^k)\). The target function f is uniformly continuous because it is continuous and X is compact. So for all \(\varepsilon > 0\) there is a \(\delta > 0\) such that \(|f(x) - f(\hat{x}_j^k)| < \varepsilon /2\) if \(||x - \hat{x}_j^k|| < \delta \) . This splits the rules into the two sets \(J_1= \{ (j,k) : ||x - \hat{x}_j^k||<\delta \}\) and \(J_2= \{ (j,k) : ||x - \hat{x}_j^k||\ge \delta \}\). We assume that \(J_1 \ne \emptyset \). Then (18) becomes

Consider the first term on the right side of (19). The absolute differences obey \(\big |(f(x)-f(\hat{x}_j^k))\big | < \varepsilon /2\) because \((j,k) \in J_1\)implies that \(||x - \hat{x}_j^k||<\delta \). Add these terms over all \((j,k) \in J_1\):

because \(\sum _{(j,k)\in J_1} p_j^k(x) \le \sum _{k=1}^q\sum _{j=1}^m p_j^k(x) = 1\). Now pick a real number C large enough that \(|f(x) - f(\hat{x}_j^k)|<C\) for all j and k. Pick \((j,k) \in J_2\) and pick \((j_1, k_1) \in J_1\). Then \(||x-\hat{x}_j^k||\ge \delta \) and \(||x - \hat{x}_{j_1}^{k_1}||<\delta \). Pick a real number D small enough that \(||x - \hat{x}_j^k||^2 - ||x - \hat{x}_{j_1}^{k_1}||^2 \ge D > 0\). Choose the rule if-part dispersion d small enough that \(\exp (D/d^2)>2mqC/\varepsilon \). Then the definition of D gives

Add \(\sum \limits _{i\ne j_1}\sum \limits _{l\ne k_1}\frac{\exp {(-||x - \hat{x}_i^l||^2/d^2)}}{\exp {(-||x - \hat{x}_j^k||^2/d^2)}}\) to both sides of the above inequality:

because \(\sum \limits _{i\ne j_1}\sum \limits _{l\ne k_1}\frac{\exp {(-||x - \hat{x}_i^l||^2/d^2)}}{\exp {(-||x - \hat{x}_j^k||^2/d^2)}}>0\). Equation (16) and the definition of C give

Sum both sides over all \((j,k) \in J_2\):

because \(J_2\) has fewer than mq elements. Then (20) and (24) add to give the approximation-error bound: \(|f(x) {-} F(x)| \le \varepsilon /2 +\varepsilon /2 <\varepsilon \). So the combined SAM with mq rules uniformly approximates f. \(\square \)

The proof gives insight into a foam’s rule explosion. The proof assumes that \(J_1 \ne \emptyset \). So for all \(x \in X\) there is a \(\hat{x}_j^k \) such that \(||x - \hat{x}_j||<\delta \). Let \(N_{\delta }(\hat{x}_j^k) = \{ x : ||x - \hat{x}_j^k||<\delta \}\) denote the \(\delta \)-neighborhood around \(\hat{x}_j^k\). Then \(\forall \; x \in X :\; \exists \; \hat{x}_j^k : x \in N_{\delta }(\hat{x}_j^k)\). This gives

X is bounded because X is compact by hypothesis. Let an n-dimensional sphere of radius R contain X. The volume of this sphere is \(cR^n\) for some constant c. Each \(N_{\delta }(\hat{x}_j^k)\)’s is an n-dimensional sphere with radius \(\delta \) and volume \(c\delta ^n\). Use the \(\delta \)-neighborhoods to cover the R-sphere that contains X. The \(N_{\delta }(\hat{x}_j^k)\)’s are not disjoint. So the volume of the R-sphere is less than the volume of the mq \(N_{\delta }(\hat{x}_j^k)\)’s combined: \(cR^n \le mqc\delta ^n\). Then \( mq \ge (R/\delta )^n\). This inequality shows that the required total number of rules grows exponentially with the dimension n.

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