# A Notation

\(\defq\), equal by definition to

\(\one\{S\}\), the indicator of statement \(S\), which equals 1 if \(S\) is true and 0 otherwise.

\(O \defq (W,A,Y)\), the generic summary of how one realization of the experiments of interest unfold, our generic observation; \(W \in [0,1]\) is the context of action, \(A \in \{0,1\}\) is the action undertaken, and \(Y\in [0,1]\) is the reward of action \(A\) in context \(W\). We denote by \(\calO \defq [0,1] \times \{0,1\} \times [0,1]\) the set where a generic \(O\) takes its values.

\(P\), \(P_{0}\), \(\Pi_{0}\), \(\Pi_{h}\), \(\Pi_{0}'\), \(\Pi_{h}'\), laws (on \(\calO\)) for \(O\).

\(Pf \defq \Exp_{P} (f(O))\) for any law \(P\) for \(O\) and function \(f\) from \(\calO\) to \(\bbR^{p}\).

\(\|f\|_{P}^{2} \defq Pf^{2} = \Exp_{P} (f(O)^{2}) = \int f(o)^{2} dP(o)\), the square of the \(L^{2}(P)\)-norm of \(f\), a function from \(\calO\) to \(\bbR\).

\(\|f\|_{\infty} \defq \sup_{o \in \calO} |f(o)|\), the essential supremum of \(f\), a function from \(\calO\) to \(\bbR\).

\(P_{n}\), the empirical measure. If the observations are \(O_{1}\), , \(O_{n}\), then \(P_{n}\) is a law such that the generic random variable \(O\) drawn from \(P_{n}\) takes its values in \(\{O_{1}, \ldots, O_{n}\}\) in such a way that \(O = O_{i}\) with probability \(n^{-1}\) for each \(1 \leq i \leq n\).

\(\sqrt{n} (P_{n} - P)\), where \(P_{n}\) is the empirical measure associated to \(O_{1}, \ldots, O_{n}\) drawn independently from \(P\), the empirical process.

\(X_n = o_{P_0}(1)\) if \(X_n\), a random variable built from \(O_{1}\), , \(O_{n}\) independently drawn from \(P_0\), converges in probability to zero, that is, if \(P_0(|X_n| > t)\) converges to zero for all \(t>0\) as \(n\) goes to infinity. If \(n^{c} X_n = o_{P_0}(1)\), then one also writes \(X_n = o_{P_0}(n^{-c})\).

\(X_{n} = O_{P_0}(1)\) if \(X_n\), a random variable built from \(O_{1}\), , \(O_{n}\) independently drawn from \(P_0\), is bounded in probability, that is if, for all \(t>0\) there exists \(M >0\) such that \(\sup_{n \geq 1} P_0(|X_n| \geq M) \leq t\). If \(n^{c} X_n = O_{P_0}(1)\), then one also writes \(X_n = O_{P_0}(n^{-c})\).

\(\calM\), the model, that is, the collection of

*all*laws from which \(O\) can be drawn and that meet some constraints.\(\calM^{\text{empirical}}\), the collection of all discrete laws on \([0,1] \times \{0,1\} \times [0,1]\), of which \(P_{n}\) is a distinguished element.

\(Q_{W}\), \(Q_{0,W}\), marginal laws for \(W\) (under \(P\) and \(P_{0}\), respectively).

\(\Gbar(W) \defq \Pr_{P}(A = 1 | W)\), \(\Gbar_0(W) \defq \Pr_{P_0}(A = 1 | W)\), conditional probabilities of action \(A = 1\) given \(W\) (under \(P\) and \(P_{0}\), respectively). For each \(a \in \{0,1\}\), \(\ell\Gbar(a,W) \defq \Pr_{P}(A = a | W)\) and \(\ell\Gbar_0(a,W) \defq \Pr_{P_0}(A = a | W)\).

\(\Qbar(A,W) = \Exp_{P}(Y|A,W)\), \(\Qbar_0(A,W) = \Exp_{P_{0}}(Y|A,W)\), the conditional means of \(Y\) given \(A\) and \(W\) (under \(P\) and \(P_{0}\), respectively).

\(\calQ \defq \{\Qbar : P \in \calM\}\), the space of regression functions induced by model \(\calM\).

\(\calQ(\Qbar,\Gbar) \subset \calQ\), \(\Gbar\)-specific fluctuation model of \(\Qbar\), see (10.6).

\(q_{Y}\), \(q_{0,Y}\), conditional densities of \(Y\) given \(A\) and \(W\) (under \(P\) and \(P_{0}\), respectively).

\(\Psi : \calM \to [0,1]\), given by \(\Psi(P) \defq \int \left(\Qbar(1, w) - \Qbar(0, w)\right)dQ_{W}(w)\), the statistical mapping of interest.

\(\psi \defq \Psi(P)\), \(\psi_{0} \defq \Psi(P_{0})\).

\(\Algo\), \(\Algo_{\Gbar,1}\), \(\Algo_{\Qbar,1}\), algorithms to be trained on \(P_{n}\),

*i.e.*, mappings from \(\calM^{\text{empirical}}\) to the set where lives the feature targeted by the algorithm.\(\Algora_{\Gbar,s}\), \(\Algora_{\Qbar,s}\), \(s\)-specific oracle algorithms (\(s>0\)) that can use the true targeted features \(\Gbar_{0}\) and \(\Qbar_{0}\) to produce predictions that are almost exact, up to a \(N(0,s^{2})\) random error term.

\(L_{a}\), the contex-specific logistic (or negative binomial) loss function, given by \(-L_{a}(f)(A,W) \defq A \log f(W) + (1-A) \log (1-f(W))\) for any function \(f:[0,1]\to[0,1]\).

\(L_{y}\), the reward-specific logistic (or negative binomial) loss function, given by \(-L_{y}(f)(O) \defq Y \log f(A,W) + (1-Y) \log (1-f(A,W))\) for any function \(f:\{0,1\} \times [0,1] \to[0,1]\).