Glider from the game of Life, rising from the left




Topic: #simulation

Neural Networks as Function Simulators


Some of the limitations of black-box deciders become more obvious and intuitive when one recognizes the machine-learning algorithms behind them as software tools for approximating functions, using large data sets and statistical tools for calibration.

“The Delusions of Neural Networks”
Giacomo Tesio, Medium, January 18, 2018

The key points:

(A) Neural networks simulate functions and are calibrated statistically, with the assistance of large data sets comprising known argument-value pairs.

(B) Like other simulations, neural networks sometimes yield erroneous or divergent results. The functions they actually compute are usually not mathematically equal to the functions they simulate.

(C) The reason for this is that the calibration process uses only a finite number of argument-value pairs, whereas the function that the neural network is designed to simulate computes values for infinitely many arguments, or at least for many, many more arguments than are used in the calibration. (Otherwise, the simulation would be useless.) The data used in the calibration are compatible with many, many more functions than the one that the neural network is designed to simulate. The probability that calibrating the neural network results in its computing a function that is mathematically equal to the one it is designed to simulate is negligible — for practical purposes, it is zero.

(D) The problem of determining how accurately a neural network simulates the function it is designed to simulate is undecidable. There is no general algorithm to answer questions of this form. As a result, neural networks are usually validated not by proving their correctness but by empirical measurement: We apply them to arguments not used in the calibration process and compare the values they compute to the values that the functions they are designed to simulate associate with the same test arguments. When they match in a large enough percentage of cases, we pronounce the simulation a success.

(E) However, these test arguments are not generated at random, but are drawn from the same “natural” sources as the data used in the calibration of the network. The success of the simulation depends on this bias: Unless the test arguments are sufficiently similar to the data used in the calibration, the probability that the computed values will match is again negligible. This would essentially never happen if the test arguments were randomly selected.

(F) Consequently, the process of validating a neural network does not prove that it is unbiassed. On the contrary: in order to be pronounced valid, a neural network must simulate the biasses of the data set used in the calibration.

(G) In principle, it would be possible for independent judges to confirm that the data set is free from forms of bias that constitute discrimination against some protected class of persons and to provide strong empirical evidence that the function actually computed by the neural network does not actually introduce such a bias. In practice, this confirmation process would be prohibitively expensive and time-consuming.

(H) Neural networks can be used, and often are used, to simulate unknown functions. In those cases, there would be no way for a panel of independent judges even to begin the process of confirming freedom from discriminatory bias, because no one even knows whether the function that the neural network is designed to simulate exemplifies such a bias.

#black-box-deciders #neural-networks #simulation

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John David Stone (

created June 1, 2014 · last revised December 10, 2018