Discriminative Feature Feedback with General Teacher Classes

TL;DR

This paper develops a general theory for Discriminative Feature Feedback (DFF) with arbitrary teacher classes by introducing the Discriminative Feature Feedback Dimension, $DFFdim$, as a tree-based measure that exactly characterizes realizable mistake bounds via a Standard Optimal Algorithm (SOA-DFF). It formalizes two mappings between DFF and Online Learning, OtD and DtO, showing that $DFFdim( ext{OtD}( ext{F}))= ext{Ldim}( ext{F})$, and proves a strong separation where a problem with $DFFdim=1$ corresponds to an infinite Littlestone dimension. The non-realizable setting is analyzed; a universal upper bound of $M^L_k \\le (k+1)\\,DFFdim( ext{T},H) + k$ is established, but a matching general lower bound shows no general no-regret guarantees for all DFF problems, with tight results derived via a secret-sharing construction. The work thus reveals that realizable dimension alone does not fully predict non-realizable performance and raises open questions about factors that govern the impact of rich feedback on learning under adversarial deviations.

Abstract

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.

Figures (4)

• Figure 1: A shattered DFFT with history $H = \{(x_0, 0)\}$ and height $3$
• Figure 2: The construction of $\texttt{Tr}$ in the proof of Lemma \ref{['lem:soa-mistakes']}
• Figure 3: An example of a path prefix starting from the root in the inductive construction of the tree in the proof of Theorem \ref{['thm:AND-RM-Tdim']}, for $M=3$. The symbol '$\star$' indicates an arbitrary Boolean value.
• Figure 4: Illustrating the construction of $\texttt{Tr}'$ in the proof of Lemma \ref{['lem:ldim-to-tdim']}

Theorems & Definitions (44)

• Definition 3.1: Teacher
• Definition 4.1: DFF tree
• Definition 4.2: Shattered DFFT
• Definition 4.3: $\mathrm{DFFdim}$
• Theorem 4.4
• Lemma 4.5
• Lemma 4.6
• Lemma 4.7
• proof
• proof : Proof of Lemma \ref{['lem:soa-mistakes']}