Discriminative Feature Feedback with General Teacher Classes

We study the theoretical properties of the interactive learning protocol Discriminative Feature Feedback (DFF) (Dasgupta et al., 2018). The DFF learning protocol uses feedback in the form of discriminative feature explanations. We provide the first systematic study of DFF in a general framework that is comparable to that of classical protocols such as supervised learning and online learning. We study the optimal mistake bound of DFF in the realizable and the non-realizable settings, and obtain novel structural results, as well as insights into the differences between Online Learning and settings with richer feedback such as DFF. We characterize the mistake bound in the realizable setting using a new notion of dimension. In the non-realizable setting, we provide a mistake upper bound and show that it cannot be improved in general. Our results show that unlike Online Learning, in DFF the realizable dimension is insufficient to characterize the optimal non-realizable mistake bound or the existence of no-regret algorithms.

💡 Research Summary

The paper provides the first systematic theoretical treatment of Discriminative Feature Feedback (DFF) in a general setting that parallels classical supervised and online learning frameworks. Rather than restricting attention to the “component model” used in earlier work, the authors define a teacher as a pair (ℓ, ψ) where ℓ : X → Y assigns true labels and ψ : X × X → Φ ∪ {⊥} supplies a discriminative Boolean feature whenever two examples have different labels. A teacher class 𝒯 is a set of such teachers, and a (non‑empty) history H of initially labeled examples is assumed.

The central technical contribution is the introduction of a DFF‑specific tree structure, the Discriminative Feature Feedback Tree (DFFT). Nodes are triples ⟨y, ϕ, x⟩ encoding the teacher’s feedback (label y, discriminative feature ϕ) for the most recent example x, while edges are labeled by the learner’s chosen prediction‑explanation pair (̂x, ̂y). Because the learner’s explanation influences the teacher’s next response, DFF trees differ fundamentally from Littlestone trees: multiple root‑to‑leaf paths may be consistent with the same teacher.

A DFFT is said to be shattered by a teacher class 𝒯 (with respect to H) if (i) every non‑root node satisfies y ≠ ̂y, (ii) each non‑leaf node’s outgoing edges correspond exactly to all labeled examples that are either in H or already appear on the path to that node, (iii) every root‑to‑leaf path is consistent with at least one teacher in 𝒯ₕ (the subset of teachers consistent with H), and (iv) the tree is complete (all leaves at the same depth). The DFF dimension, DFFdim(𝒯, H), is defined as the maximal height of a shattered DFFT.

Theorem 5 establishes that DFFdim precisely characterizes the optimal worst‑case mistake bound in the realizable setting: for any deterministic DFF algorithm A,
 M(A, 𝒯, H) ≥ DFFdim(𝒯, H),
and the Standard Optimal Algorithm for DFF (SOA‑DFF) attains equality. This mirrors the role of VC‑dimension in PAC learning and Littlestone dimension in online learning.

The authors then explore the relationship between DFF and classic online learning. Theorems 8 and 9 give bidirectional reductions that preserve mistake bounds, allowing a direct comparison of the two paradigms. Using these reductions they prove a strong separation: there exists a teacher class with DFFdim = 1 while the corresponding online problem has infinite Littlestone dimension (Theorem 10). Hence, a low DFF dimension does not guarantee easy online learnability, highlighting the extra power of discriminative feature feedback.

In the non‑realizable (noisy or adversarial) setting, the paper presents a simple meta‑algorithm that repeatedly runs the optimal realizable algorithm as a subroutine. Theorem 11 shows that its mistake bound is DFFdim + k, where k is the number of rounds in which the teacher deviates from the protocol. Theorem 12 extends this bound to a broad family of interactive protocols (e.g., explanation‑based active learning). To prove optimality, Theorem 13 constructs a teacher class based on secret‑sharing schemes, demonstrating that no algorithm can achieve a mistake bound asymptotically smaller than DFFdim + Ω(k). Consequently, unlike online learning, a finite DFF dimension does not guarantee the existence of a no‑regret algorithm. Nevertheless, for specific teacher classes (e.g., the original component model) no‑regret algorithms do exist, indicating that the feasibility of regret‑minimization depends intricately on the structure of the teacher class.

Overall, the paper delivers a comprehensive theory of DFF: it defines a natural complexity measure (DFFdim), proves its exact optimality in the realizable case, establishes deep connections and separations with online learning, and characterizes the limits of learning under noise. By abstracting the teacher into a class, the framework can accommodate a wide variety of rich feedback modalities (visual attributes, textual explanations, etc.), opening avenues for future work on algorithm design, tighter bounds for specific teacher structures, and empirical validation in real‑world interactive systems.