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An action set whose linear-mean range exceeds one

Example

Consider one round with dimension $d = 1$ , regularization $λ = 1$ , feature bound $L = 2$ , parameter bound $m_{2} = 1$ , and deterministic parameter $θ^{*} = 1$ . Let the context be constant and let the two available actions have scalar features $0$ and $2$ . Use zero reward noise. Both candidate features satisfy the model's bound, but their true means are $0$ and $2$ , so the linear-mean range is $2$ .

Example references

Finite-dimensional linear contextual bandit model: all candidate features must have norm at most $L$ and the conditional noise MGF must be bounded by $e^{q^{2} / 2}$ .

LinUCB has a log-determinant high-probability pseudo-regret bound: the regret guarantee additionally assumes the true-mean range within each available action set is at most $1$ .

Tor Lattimore and Csaba Szepesvári, Bandit Algorithms, §19.3, Theorem 19.2

Abbasi-Yadkori, Pál, and Szepesvári, Theorem 13

Facts & Assumptions

Given: The one-round finite-dimensional linear contextual bandit instance in the Example.

[F1]

Finite-dimensional linear contextual bandit model: each candidate feature has Euclidean norm at most $L$ , rewards are $⟨ θ^{*}, A_{t} ⟩ + η_{t}$ , and the conditional noise MGF bound is $e^{q^{2} / 2}$ .

[F2]

LinUCB has a log-determinant high-probability pseudo-regret bound: its stated guarantee requires $\sup_{a, b \in A_{t}} ⟨ θ^{*}, a - b ⟩ \leq 1$ .

Verification

1.1givenF1algebra

The candidate features are $0$ and $2$ , so both have absolute value at most $L = 2$ . The parameter satisfies $∣ θ^{*} ∣ = 1 = m_{2}$ , and zero noise has conditional MGF $E [e^{q η_{1}} ∣ F_{1}] = 1 \leq e^{q^{2} / 2}$ for every real $q$ . Thus the instance satisfies the model hypotheses.

2.1givenF1F2step 1.1algebra∎

The true means are $θ^{*} \cdot 0 = 0$ and $θ^{*} \cdot 2 = 2$ . Consequently $\sup_{a, b \in {0, 2}} θ^{*} (a - b) = 2 > 1$ , so the additional range premise of [F2] fails. This example therefore lies outside that theorem's stated guarantee and does not refute its conclusion under its hypotheses.

Depends on

Used by

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Sources

Tor Lattimore and Csaba Szepesvári, Bandit Algorithms (standard reference, not scraped)
Yasin Abbasi-Yadkori, Dávid Pál, and Csaba Szepesvári, Online Least Squares Estimation with Self-Normalized Processes: An Application to Bandit Problems (standard reference, not scraped)