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No common confidence radius without a parameter norm bound

Example

Fix $δ = 1 / 2$ . For each $M > 0$ , take a one-round model with $d = 1$ , $λ = L = 1$ , $m_{2} = M$ , $θ^{*} = M$ , one available feature $A_{1} = 1$ , and zero noise on a one-point probability space. Consider a proposed time-1 confidence radius obtained from the stated self-normalized formula by omitting its $\sqrt{λ} m_{2}$ term. Then no single such radius covers the family as $M$ ranges over all positive reals.

Example references

Finite-dimensional linear contextual bandit model: each model in the family has a finite parameter norm bound $m_{2} = M$ .

The ridge estimate lies in a time-uniform linear confidence ellipsoid: the raw radius includes the term $\sqrt{λ} m_{2}$ .

Tor Lattimore and Csaba Szepesvári, Bandit Algorithms, §20.1, Theorem 20.5

Abbasi-Yadkori, Pál, and Szepesvári, §5

Facts & Assumptions

Given: The family of one-round models in the Example and the confidence-radius formula with its parameter-norm term omitted for comparison.

[F1]

Finite-dimensional linear contextual bandit model: $V_{0} = λ I_{d}$ , ${\hat{θ}}_{0} = 0$ , and each member of the family has $∥ θ^{*} ∥_{2} \leq m_{2}$ with $m_{2} = M$ .

[F2]

The ridge estimate lies in a time-uniform linear confidence ellipsoid: at time $1$ the confidence radius is $\sqrt{λ} m_{2} + \sqrt{2 \log (1 / δ) + \log (\det (V_{0}) / λ^{d})}$ .

Verification

1.1givenF1F2algebra

For the proposed radius with $\sqrt{λ} m_{2}$ omitted, $δ = 1 / 2$ , $λ = 1$ , $V_{0} = 1$ , and $\det (V_{0}) / λ^{d} = 1$ , the time-1 radius equals $r = \sqrt{2 \log 2}$ .

2.1givenF1step 1.1algebra∎

The time-1 estimate is ${\hat{θ}}_{0} = 0$ and its information norm is $∥ θ^{*} - {\hat{θ}}_{0} ∥_{V_{0}} = M$ . Choosing $M > r$ puts the true parameter outside the proposed interval. Since $M$ is arbitrary across the family, a radius omitting a uniform parameter-norm bound cannot cover the family.

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)