Identification via Functions

TL;DR

This work introduces identification via functions, framing the problem as determining whether the root of a noisy function lies within a given interval. It generalizes to a three-gap test and derives a new logarithmic lower bound on the number of observations required to reliably identify the root, via the inequality $6\lambda \ge 2\bigl(1-\frac{\varepsilon_1}{2\delta}\bigr)^n + \bigl(1-\frac{\varepsilon_2}{2\delta}\bigr)^n$. The analysis shows that, unlike Ahlswede’s message identification which achieves a double-log rate, root identification yields a slower, logarithmic growth in observation complexity, though the new bound improves prior results in several parameter regimes. The paper connects root identification to message identification, explains the limitations of coloring schemes in preserving input structure for roots, and recovers previous root-identification results as special cases while highlighting the broader implications for noisy-function identification.

Abstract

We develop a framework for studying the problem of identifying roots of a noisy function. We revisit a previous logarithmic bound on the number of observations and propose a general problem for identification of roots with three errors. As a key finding, we establish a novel logarithmic lower bound on the number of observations which outperforms the previous result across certain regimes of error and accuracy of the identification test. Furthermore, we recover the previous results for root identification as a special case and draw a connection to the message identification problem of Ahlswede.

Figures (6)

• Figure 1: Root identification setting for a stochastic given function. Given the noisy observations of an unknown function $M(x),$ a test should reliably determine whether or not a root of the function $M(x),$ belongs to the interval $[a,b].$
• Figure 2: Illustration of first and second order logarithmic lower bounds on the number of observations. The gap between the two curves increases for small values of errors.
• Figure 3: Spectrum of algorithmic bounds for message identification AD89, root identification and root computation Ahlswede87.
• Figure 4: Comparison of previous and new test for root identification across different test parameters.
• Figure 5: Illustration of identification of roots as decoding problem for the previous test.

Theorems & Definitions (11)

• Definition 1: Test with Two Errors
• Remark 1
• Definition 2: Test with Three Errors
• Definition 3: Measure
• Theorem 1: Intermediate Value Theorem for Continuous Functions; Apostol67
• Theorem 2: see KleinewaechterISIT97
• Theorem 3
• proof
• Corollary 4
• Corollary 5