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A sequential decision model in which each round selects an action and observes its reward, balancing information gathering against immediate reward. For fixed action means $m(x)$, the pseudoregret compares selected actions with an action maximizing $m$. In models with a link function and parameter $\theta$, rewards may have means $m_{\theta }(x)=\mu (⟨x,\theta ⟩)$.

A decision principle that maximizes possible reward jointly over an action and a parameter in a confidence set. On the event that the set contains the true parameter, the reward predicted for this selected pair is at least the true optimal reward. This inequality bounds pseudoregret by the width of the confidence set in the selected action's direction. For an $L$-Lipschitz continuous reward link and a radius-$\beta$ confidence ellipsoid, Cauchy-Schwarz inequality and the triangle inequality bound that width by $2L\beta \sqrt{x^T{V}^{-1}x}$.

Use the single eventFix a realization in $\mathcal{E}$. The action space is nonempty and compact, since it is a closed subset of the finite-dimensional unit ball. The Lipschitz continuous reward function is continuous, so there is an optimal action $x_{\star }\in {\mathrm{argmax}}_{x\in \mathcal{X}}\mu (⟨x,{\theta }_{\star }⟩)$. Because the same reward function is used at every time, the pseudoregret is $R_n={\sum }_{t=1}^n{r}_t$, whereThe pair $(x_{\star },{\theta }_{\star })$ is feasible in the optimization defining $(X_t,{\theta }_t)$. This optimism in the face of uncertainty givesFor any positive-definite matrix $V$, the ordinary Cauchy-Schwarz inequality applied to $V^{-1/2}x$ and $V^{1/2}z$ provesBoth ${\theta }_t$ and ${\theta }_{\star }$ belong to the same confidence ellipsoid ${\Theta }_{t-1}$. The triangle inequality for the norm associated with this weighted Euclidean inner product therefore givesand hence $r_t\le 2L{\beta }_{t-1}(\delta )|X_t{|}_{V_{t-1}^{-1}}$. Monotonicity of ${\beta }_t(\delta )$ and another Cauchy-Schwarz inequality now yieldThis reasoning holds on $\mathcal{E}$ for every $n$ simultaneously, proving the asserted probability. In combination with part (a), it also givesOnly the supplied simultaneous confidence set coverage and the optimistic choice are needed here. No claim that the displayed linear estimator is automatically a suitable estimator for a nonlinear link is required: its coverage was an explicit hypothesis.