Comparison-Based Learning with Rank Nets

We consider the problem of search through comparisons, where a user is presented with two candidate objects and reveals which is closer to her intended target. We study adaptive strategies for finding the target, that require knowledge of rank relationships but not actual distances between objects. We propose a new strategy based on rank nets, and show that for target distributions with a bounded doubling constant, it finds the target in a number of comparisons close to the entropy of the target distribution and, hence, of the optimum. We extend these results to the case of noisy oracles, and compare this strategy to prior art over multiple datasets.

💡 Research Summary

The paper tackles the problem of searching for an unknown target object when the only feedback available is a binary comparison: a user is shown two candidate objects and indicates which one is closer to the intended target. Unlike traditional metric‑based search, the algorithm does not have access to actual distances; it can only exploit the induced ranking (“A is closer than B”). The authors ask how many such comparisons are needed to locate the target with high probability, and whether an adaptive strategy can approach the information‑theoretic optimum.

Problem setting and assumptions
Let (X) be a finite set of objects and (\pi) a probability distribution over (X) that models the likelihood of each object being the user’s target. The key structural assumption is that (\pi) has a bounded doubling constant (c): for any ball of radius (r) (defined in the underlying metric space, though distances are never observed), the probability mass inside the ball of radius (2r) is at most (c) times the mass inside the smaller ball. This condition captures a form of smoothness of the target distribution and is standard in metric‑space learning theory.

Rank nets
The central technical contribution is the construction of a rank net, a hierarchical covering of the object set that respects only the order information. A rank net is built as follows: choose a mass threshold (\tau). Starting from the whole set, repeatedly select a point (c) such that the ball (in the hidden metric) centered at (c) with the smallest radius containing probability mass at least (\tau) becomes a net node. The collection of such nodes at a given level forms a (\tau)-net; each node’s “region” is the set of objects that are closer to its center than to any other net point at the same level. By decreasing (\tau) geometrically, a multi‑level structure is obtained, analogous to an (\epsilon)-net but defined purely through rank queries.

Adaptive search algorithm
The algorithm maintains a current candidate region (R) (initially the whole set). At each step it picks two net points (u, v) that lie on the boundary of (R) and asks the user which of the two is closer to the hidden target. The answer determines which side of the partition contains the target, and the algorithm updates (R) to the corresponding sub‑region. Because each net point’s region carries at least a (\tau) fraction of the remaining probability mass, the expected reduction in Shannon entropy after one comparison is at least (\Omega(1/c)). Consequently, after at most (O(c\cdot H(\pi))) comparisons the entropy drops to zero, i.e., the target is identified. This bound matches the lower bound up to the constant factor (c), and therefore the method is entropy‑optimal up to a multiplicative constant.

Noisy oracle extension
When the user’s response is corrupted with probability (\varepsilon < 1/2) (the noisy oracle model), the algorithm repeats each comparison (k = O\bigl(\log(1/\delta)/(1-2\varepsilon)^2\bigr)) times and takes a majority vote, guaranteeing that the probability of an erroneous decision on that step is at most (\delta). By setting (\delta) appropriately (e.g., (\delta = 1/\text{poly}(n))), the overall failure probability remains bounded while the total number of queries becomes
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