Budget-limited distribution learning in multifidelity problems

Abstract

Multifidelity methods are widely used for estimating quantities of interest (QoI) in computational science by employing numerical simulations of differing costs and accuracies. Many methods approximate numerical-valued statistics that represent only limited information, e.g., scalar statistics, about the QoI. Further quantification of uncertainty, e.g., for risk assessment, failure probabilities, or confidence intervals, requires estimation of the full distributions. In this paper, we generalize the ideas in (Xu et al. in SIAM J Sci Comput 44(1):A150–A175, 2022) to develop a multifidelity method that approximates the full distribution of scalar-valued QoI. The main advantage of our approach compared to alternative methods is that we require no particular relationships among the high and lower-fidelity models (e.g. model hierarchy), and we do not assume any knowledge of model statistics including correlations and other cross-model statistics before the procedure starts. Under suitable assumptions in the framework above, we achieve provable 1-Wasserstein metric convergence of an algorithmically constructed distributional emulator via an exploration–exploitation strategy. We also prove that crucial policy actions taken by our algorithm are budget-asymptotically optimal. Numerical experiments are provided to support our theoretical analysis.

Appendices

Appendices

Proof of Lemma 4.1

Let $|S| = s$. We first prove (28a). Recall the definition of Y and $Y'$:

$$\begin{aligned}&Y = X_S^\top \beta _S + {\varepsilon }_S&Y' = X_S^\top {\widehat{\beta }}_S + {\widehat{{\varepsilon }}}_S \end{aligned}$$

where ${\varepsilon }_S, {\widehat{{\varepsilon }}}_S$ are independent of $X_S$ by Assumption 2.2.

Define the true empirical distribution of ${\varepsilon }_S$ using exploration samples as

$$\begin{aligned} {\widetilde{{\varepsilon }}}_S\sim \frac{1}{m}\sum _{\ell \in [m]}\delta _{Y_\ell -X_{S,\ell }^\top \beta _S}. \end{aligned}$$

by the additive property of $W_1$ metric under independence [22] and the triangle inequality, we can upper bound the average $W_1$ distance between $F_Y$ and $F_{Y'}$ by conditioning on the exploration data:

$$\begin{aligned}&{{\mathbb {E}}}\left[ W_1(F_Y, F_{Y'})|Z_S, Y_{{\text {epr}}}\right] \nonumber \\&\quad \le \ {{\mathbb {E}}}[W_1(F_{X_S^\top \beta _S}, F_{X_S^\top {\widehat{\beta }}_S})|Z_S, Y_{{\text {epr}}}] +{{\mathbb {E}}}[W_1(F_{{\varepsilon }_S}, F_{{\widehat{{\varepsilon }}}_S})|Z_S, Y_{{\text {epr}}}]\nonumber \\&\quad \le \ {{\mathbb {E}}}[W_1(F_{X_S^\top \beta _S}, F_{X_S^\top {\widehat{\beta }}_S})|Z_S, Y_{{\text {epr}}}] +{{\mathbb {E}}}[W_1(F_{{\varepsilon }_S}, F_{{\widetilde{{\varepsilon }}}_S})|Z_S, Y_{{\text {epr}}}] \nonumber \\&\qquad +{{\mathbb {E}}}[W_1(F_{{\widetilde{{\varepsilon }}}_S}, F_{{\widehat{{\varepsilon }}}_S})|Z_S, Y_{{\text {epr}}}]. \end{aligned}$$

(59)

For the first term in (59), note ${{\widehat{\beta }}}_S-\beta _S$ satisfies

$$\begin{aligned} {{\widehat{\beta }}}_S-\beta _S\sim \mathcal (0, \sigma _S^2(Z^\top _SZ_S)^{-1}). \end{aligned}$$

(60)

Averaging out the randomness of exploration noise, we have that, almost surely,

$$\begin{aligned} {{\mathbb {E}}}[W_1(F_{X_S^\top \beta _S}, F_{X_S^\top {\widehat{\beta }}_S})|Z_S]&\le \left( {{\mathbb {E}}}[W^2_2(F_{X_S^\top \beta _S}, F_{X_S^\top {\widehat{\beta }}_S})|Z_S]\right) ^{1/2}\nonumber \\&\le \left( {{\mathbb {E}}}[|X_S^\top ({{\widehat{\beta }}}_S-\beta _S)|^2 |Z_S]\right) ^{1/2}\nonumber \\&=\left( \hbox {tr}({{\mathbb {E}}}[X_S X_S^\top ] {{\mathbb {E}}}[({{\widehat{\beta }}}_S-\beta _S)({{\widehat{\beta }}}_S-\beta _S)^\top | Z_S])\right) ^{1/2}\nonumber \\&= \left( \frac{\sigma _S^2}{m}\hbox {tr}(\Lambda _S(m^{-1}Z_S^\top Z_S)^{-1})\right) ^{1/2}\nonumber \\&\simeq \sqrt{\frac{s+1}{m}}\sigma _S, \end{aligned}$$

(61)

where the first step uses Jensen’s inequality, and the last step follows from the law of large numbers and Assumption 2.3.

For the second term in (59), note that ${\widetilde{{\varepsilon }}}_S$ is the empirical distribution of ${\varepsilon }_S$ based on m exploration samples, which does not depend on $Z_S$ under Assumption 2.2. Applying the nonasymptotic estimates on the convergence rate of empirical measures in Lemma 3.2 obtains

$$\begin{aligned} {{\mathbb {E}}}[W_1\left( F_{{\varepsilon }_S}, F_{{\widetilde{{\varepsilon }}}_S}\right) |Z_S] = {{\mathbb {E}}}[W_1\left( F_{{\varepsilon }_S}, F_{{\widetilde{{\varepsilon }}}_S}\right) ]\le \frac{J_1(F_{{\varepsilon }_S})}{\sqrt{m}}, \end{aligned}$$

(62)

where $J_1$ is defined in (10).

For the third term in (59), consider the natural coupling between ${\widehat{{\varepsilon }}}_S$ and ${\widetilde{{\varepsilon }}}_S$: ${\widetilde{{\varepsilon }}}^{\leftarrow }_S({\widetilde{\tau }}_\ell ) = {\widehat{{\varepsilon }}}^{\leftarrow }_S({\widehat{\tau }}_\ell )$, where $^{\leftarrow }$ denotes the preimage of a map and

$$\begin{aligned}&{\widetilde{\tau }}_\ell = Y_\ell -X_{S,\ell }^\top \beta _S&{\widehat{\tau }}_\ell = Y_\ell -X_{S,\ell }^\top {{\widehat{\beta }}}_S. \end{aligned}$$

(63)

In this case,

$$\begin{aligned}&{{\mathbb {E}}}[W_1(F_{{\widetilde{{\varepsilon }}}_S}, F_{{\widehat{{\varepsilon }}}_S})|Z_S]\le ({{\mathbb {E}}}[W^2_2(F_{{\widetilde{{\varepsilon }}}_S}, F_{{\widehat{{\varepsilon }}}_S})|Z_S])^{1/2}\le ({{\mathbb {E}}}[|{\widetilde{{\varepsilon }}}_S-{\widehat{{\varepsilon }}}_S|^2|Z_S])^{1/2}\nonumber \\&\quad =\left( \frac{1}{m}\sum _{\ell \in [m]}{{\mathbb {E}}}[(X^\top _{S,\ell }({{\widehat{\beta }}}_S-\beta _S))^2|Z_S]\right) ^{1/2}\nonumber \\&\quad = \sqrt{\frac{s+1}{m}}\sigma _S. \end{aligned}$$

(64)

Putting (61), (62), (64) together finishes the proof of (28a).

We next prove (28b). Conditioned on $Z_S$ and $Y_{\text {epr}}$, $Y'$ is a random variable with bounded r-th moments for all $r>2$. Appealing to Lemma 3.2 and averaging over the exploration noise, we have

$$\begin{aligned} {{\mathbb {E}}}\left[ W_1\left( {\widehat{F}}_{Y,S}, F_{Y'}\right) |Z_S, Y_{\text {epr}}\right]&\le \frac{{{\mathbb {E}}}[J_1(F_{Y'})|Z_S, Y_{\text {epr}}]}{\sqrt{N_S}}\Longrightarrow {{\mathbb {E}}}\left[ W_1\left( {\widehat{F}}_{Y,S}, F_{Y'}\right) |Z_S\right] \nonumber \\&\le \frac{{{\mathbb {E}}}[J_1(F_{Y'})|Z_S]}{\sqrt{N_S}}, \end{aligned}$$

(65)

where $J_1$ is defined in (10). The desired result would follow if we can show that ${{\mathbb {E}}}[J_1(F_{Y'})|Z_S]$ converges to $J_1(F_{Y})$ a.s. as $m\rightarrow \infty$. To this end, we introduce the following intermediate random variables:

$$\begin{aligned}&Y'' = X_S^\top \beta _S + {\widetilde{{\varepsilon }}}_S&Y''' = X_S^\top \beta _S + {\widehat{{\varepsilon }}}_S. \end{aligned}$$

We will prove the desired result by verifying the following convergence statements respectively:

• (a). $|{{\mathbb {E}}}[J_1(F_{Y})|Z_S]-{{\mathbb {E}}}[J_1(F_{Y''})|Z_S]|\rightarrow 0$ a.s.;

• (b). $|{{\mathbb {E}}}[J_1(F_{Y''})|Z_S]-{{\mathbb {E}}}[J_1(F_{Y'''})|Z_S]|\rightarrow 0$ a.s.;

• (c). $|{{\mathbb {E}}}[J_1(F_{Y'''})|Z_S]-{{\mathbb {E}}}[J_1(F_{Y'})|Z_S]|\rightarrow 0$ a.s.

Without loss of generality, we assume $\text {supp}({\varepsilon }_S)\subseteq [-1, 1]$ and $\Vert \beta _S\Vert _2 = 1$; the general case can be considered similarly by taking appropriate scaling involving a constant C in Assumption 2.5.

We introduce the following quantity for our analysis:

$$\begin{aligned} K_m^* = \max _{\ell \in [m]}\Vert X_{S,\ell }\Vert _2. \end{aligned}$$

(66)

It is clear that $K_m^*$ depends only on $Z_S$. Under Assumption 2.4, $X_{S,\ell }$’s are i.i.d. sub-exponential random variables with uniformly bounded sub-exponential norm. By Lemma 3.6,

$$\begin{aligned}&K_m^*\lesssim \log m&a.s., \end{aligned}$$

(67)

where the implicit constant is realization-dependent.

To prove (a), we condition on $Z_S$ and $Y_{\text {epr}}$. Using (13)–(15) in Lemma 3.5,

$$\begin{aligned}&|{{\mathbb {E}}}[J_1(F_{Y})|Z_S, Y_{\text {epr}}]-{{\mathbb {E}}}[J_1(F_{Y''})|Z_S, Y_{\text {epr}}]|\nonumber \\&\quad \le \ {{\mathbb {E}}}\left[ \int _{{\mathbb {R}}}\sqrt{| F_Y(y)-F_{Y''}(y)|} dy| Z_S, Y_{\text {epr}}\right] \nonumber \\&\quad =\ {{\mathbb {E}}}\left[ \int _{{\mathbb {R}}}\sqrt{\left| \int _{{\mathbb {R}}}F_{X_S^\top \beta _S}(y-z) dF_{{\varepsilon }_S}(z) - \int _{{\mathbb {R}}}F_{X_S^\top \beta _S}(y-z)dF_{{\widetilde{{\varepsilon }}}_S}(z)\right| } dy| Z_S, Y_{\text {epr}}\right] . \end{aligned}$$

(68)

Under Assumption 2.4, for every y, since $\Vert \beta _S\Vert _2 = 1$, $F_{X_S^\top \beta _S}(y-z)$ as a function of z, is $C_{\text {Lip}}$-Lipschitz. By the Kantorovich-Rubinstein duality (6),

$$\begin{aligned} \left| \int _{{\mathbb {R}}}F_{X_S^\top \beta _S}(y-z) dF_{{\varepsilon }_S}(z) - \int _{{\mathbb {R}}}F_{X_S^\top \beta _S}(y-z)dF_{{\widetilde{{\varepsilon }}}_S}(z)\right| \le C_{\text {Lip}}W_1({\varepsilon }_S, {\widetilde{{\varepsilon }}}_S). \end{aligned}$$

(69)

Meanwhile, note ${\text {supp}}({\varepsilon }_S) = {\text {supp}}({\widetilde{{\varepsilon }}}_S)\subseteq [-1,1]$. This combined with the fact that $X_S^\top \beta _S$ is sub-exponential implies that

$$\begin{aligned}&\left| \int _{{\mathbb {R}}}F_{X_S^\top \beta _S}(y-z) dF_{{\varepsilon }_S}(z) - \int _{{\mathbb {R}}}F_{X_S^\top \beta _S}(y-z)dF_{{\widetilde{{\varepsilon }}}_S}(z)\right| \nonumber \\&\quad =\ \left| \int _{{\mathbb {R}}}1-F_{X_S^\top \beta _S}(y-z) dF_{{\varepsilon }_S}(z) - \int _{{\mathbb {R}}}1-F_{X_S^\top \beta _S}(y-z)dF_{{\widetilde{{\varepsilon }}}_S}(z)\right| \nonumber \\&\quad \le \ \frac{1}{2}\left( M_1[F_{X_S^\top \beta _S}](y) + M_1[1-F_{X_S^\top \beta _S}](y)\right) \nonumber \\&\quad \le \ \exp \left( -\frac{\max \{|y-1|, |y+1|\}}{C}\right) \nonumber \\&\quad \le \ \exp \left( -\frac{|y|}{2C}\right)&|y|\ge 2, \end{aligned}$$

(70)

where C is an absolute constant depending only on the sub-exponential norm of $\Vert X_S\Vert _2$, and $M_1$ is the 1-local maximum operator in Definition 3.4. Substituting (69) and (70) into (68) and applying a truncated estimate,

$$\begin{aligned}&|{{\mathbb {E}}}[J_1(F_{Y})|Z_S, Y_{\text {epr}}]-{{\mathbb {E}}}[J_1(F_{Y''})|Z_S, Y_{\text {epr}}]|\nonumber \\&\le \ {{\mathbb {E}}}\left[ \int _{|y|<\max \{2, 4C\log m\}}\sqrt{W_1({\varepsilon }_S, {\widetilde{{\varepsilon }}}_S)} dy + \int _{|y|\ge \max \{2, 4C\log m\}}\exp \left( -\frac{|y|}{4C}\right) dy|Z_S, Y_{\text {epr}}\right] \nonumber \\&\le \ {{\mathbb {E}}}\left[ (4+8C\log m)\sqrt{W_1({\varepsilon }_S, {\widetilde{{\varepsilon }}}_S)} + \frac{2}{m}|Z_S, Y_{\text {epr}}\right] . \end{aligned}$$

Since $\sqrt{W_1({\varepsilon }_S, {\widetilde{{\varepsilon }}}_S)}$ is independent of $Z_S$, taking expectation over the exploration noise together with Jensen’s inequality and Lemma 3.2 yields

$$\begin{aligned}&|{{\mathbb {E}}}[J_1(F_{Y})|Z_S]-{{\mathbb {E}}}[J_1(F_{Y''})|Z_S]|\le (4+8C\log m)\nonumber \\ {}&\quad {{\mathbb {E}}}[W_1({\varepsilon }_S, {\widetilde{{\varepsilon }}}_S)]^{1/2} + \frac{2}{m}\lesssim \frac{\log m}{\sqrt{m}}\rightarrow 0. \end{aligned}$$

(71)

To prove (b), note that conditioning on $Z_S$ and $Y_{\text {epr}}$, the difference between ${\widetilde{\tau }}_\ell$ and ${\widehat{\tau }}_\ell$ is bounded as follows:

$$\begin{aligned} |{\widetilde{\tau }}_\ell -{\widehat{\tau }}_\ell |=|X_{S,\ell }^\top ({{\widehat{\beta }}}_S-\beta _S)| \le \Vert X_{S,\ell }\Vert _2\Vert {{\widehat{\beta }}}_S-\beta _S\Vert _2{\mathop {\le }\limits ^{(66)}}K_m^*\delta \delta = \Vert {{\widehat{\beta }}}_S-\beta _S\Vert _2, \end{aligned}$$

(72)

where ${\widehat{\tau }}_\ell , {\widetilde{\tau }}_\ell$ are defined in (63). Moreover, since $\text {supp}({\varepsilon }_S)\subseteq [-1, 1]$, $|{\widetilde{\tau }}_\ell |\le 1$. This combined with (72) implies

$$\begin{aligned}&{\text {supp}}({\widetilde{{\varepsilon }}}_S)\cup {\text {supp}}({\widehat{{\varepsilon }}}_S)\subseteq [-r, r]&r = 1 + K_m^*\delta . \end{aligned}$$

(73)

The rest is similar to the proof of statement (a),

$$\begin{aligned}&|{{\mathbb {E}}}[J_1(F_{Y''})|Z_S, Y_{\text {epr}}]-{{\mathbb {E}}}[J_1(F_{Y'''})|Z_S, Y_{\text {epr}}]|\nonumber \\&\quad \le \ {{\mathbb {E}}}\left[ \int _{{\mathbb {R}}}\sqrt{|F_{Y''}(y)-F_{Y'''}(y)|}dy|Z_S, Y_{\text {epr}}\right] \nonumber \\&\quad \le \ {{\mathbb {E}}}\left[ \int _{{\mathbb {R}}}\sqrt{\frac{1}{m}\sum _{\ell \in [m]}|F_{X_S^\top \beta _S}(y-{\widehat{\tau }}_\ell )-F_{X_S^\top \beta _S}(y-{\widetilde{\tau }}_\ell )|}dy|Z_S, Y_{\text {epr}}\right] . \end{aligned}$$

It is easy to verify using the Lipschitz assumption and the tail bound of $X_S^\top \beta _S$ that

$$\begin{aligned} \frac{1}{m}\sum _{\ell \in [m]}|F_{X_S^\top \beta _S}(y-{\widehat{\tau }}_\ell )-F_{X_S^\top \beta _S}(y-{\widetilde{\tau }}_\ell )|&\le C_{\text {Lip}}K_m^*\delta \\ \frac{1}{m}\sum _{\ell \in [m]}|F_{X_S^\top \beta _S}(y-{\widehat{\tau }}_\ell )-F_{X_S^\top \beta _S}(y-{\widetilde{\tau }}_\ell )|&\le \exp \left( -\frac{|y|}{2C}\right)&|y|\ge 2r. \end{aligned}$$

Thus,

$$\begin{aligned}&|{{\mathbb {E}}}[J_1(F_{Y''})|Z_S, Y_{\text {epr}}]-{{\mathbb {E}}}[J_1(F_{Y'''})|Z_S, Y_{\text {epr}}]|\\&\quad \le \ {{\mathbb {E}}}\left[ \int _{|y|<\max \{2r, 4C\log m\}}\sqrt{C_{\text {Lip}}K_m^*\delta } dy \right. \\&\left. \qquad + \int _{|y|\ge \max \{2r, 4C\log m\}}\exp \left( -\frac{|y|}{4C}\right) dy|Z_S, Y_{\text {epr}}\right] \\&\quad \le \ {{\mathbb {E}}}\left[ (4r + 8C\log m)\sqrt{C_{\text {Lip}}K_m^*\delta } +\frac{2}{m}|Z_S, Y_{\text {epr}}\right] . \end{aligned}$$

Averaging out exploration noise and applying Jensen’s inequality,

$$\begin{aligned} \begin{aligned} |{{\mathbb {E}}}[J_1(F_{Y''})|Z_S]-{{\mathbb {E}}}[J_1(F_{Y'''})|Z_S]|&\lesssim {{\mathbb {E}}}\left[ \left( K_m^*\delta \right) ^{3/2} +\log m \sqrt{K_m^*\delta } + \frac{2}{m}|Z_S\right] \\&\lesssim (K_m^*)^{3/2}{{\mathbb {E}}}[\delta ^2|Z_S]^{3/4} + \log m\sqrt{K_m^*}{{\mathbb {E}}}[\delta ^2|Z_S]^{1/4} + \frac{1}{m}\\&{\mathop {\lesssim }\limits ^{(60), (67)}}\frac{(\log m)^{3/2}}{m^{1/4}}\rightarrow 0&a.s. \end{aligned} \end{aligned}$$

To prove (c), recall from (73) that conditioning on $Z_S$ and $Y_{\text {epr}}$, ${\text {supp}}({\widehat{{\varepsilon }}}_S)\subseteq [-r ,r]$. Applying Lemma 3.5,

$$\begin{aligned}&|{{\mathbb {E}}}[J_1(F_{Y'''})|Z_S, Y_{\text {epr}}]-{{\mathbb {E}}}[J_1(F_{Y'})|Z_S, Y_{\text {epr}}]|\nonumber \\ {}&\le {{\mathbb {E}}}\left[ \left\| M_{r}[|F_{X_S^\top {{\widehat{\beta }}}_S}-F_{X_S^\top \beta _S}|]\right\| ^{1/2}_{L^{1/2}_{{\mathbb {R}}}}| Z_S, Y_{\text {epr}}\right] . \end{aligned}$$

(74)

If $\delta <1/2$, then $1/2<\Vert {{\widehat{\beta }}}_S\Vert _2< 3/2$. In this case, the $C_1, C_2, C_3$ in Lemma 3.7 are absolute constants. According to Lemma 3.7 with $p = 1/2$,

$$\begin{aligned} \left\| M_{r}[|F_{X_S^\top {{\widehat{\beta }}}_S}-F_{X_S^\top \beta _S}|]\right\| ^{1/2}_{L^{1/2}_{{\mathbb {R}}}}\lesssim (r+ 1)\delta ^{5/12}\log \left( 1/\delta \right) \le (r+1)\delta ^{1/4}, \end{aligned}$$

(75)

where we used $\log (1/\delta )<\delta ^{-1/6}$ when $\delta \le 1/2$.

If $\delta \ge 1/2$, the same result in Lemma 3.7 implies

$$\begin{aligned} \left\| M_{r}[|F_{X_S^\top {{\widehat{\beta }}}_S}-F_{X_S^\top \beta _S}|]\right\| ^{1/2}_{L^{1/2}_{{\mathbb {R}}}}\lesssim (r + 1+\delta )\delta . \end{aligned}$$

(76)

Substituting (75) and (76) into (74) yields that

$$\begin{aligned}&|{{\mathbb {E}}}[J_1(F_{Y'''})|Z_S, Y_{\text {epr}}]-{{\mathbb {E}}}[J_1(F_{Y'})|Z_S, Y_{\text {epr}}]|\\ {}&\quad \lesssim {{\mathbb {E}}}[(r+1)\delta ^{1/4} + (r + 1+\delta )\delta | Z_S, Y_{\text {epr}}]. \end{aligned}$$

Taking expectation over the exploration noise and applying Jensen’s inequality,

$$\begin{aligned}&|{{\mathbb {E}}}[J_1(F_{Y'''})|Z_S]-{{\mathbb {E}}}[J_1(F_{Y'})|Z_S]|\\&\quad \lesssim \ {{\mathbb {E}}}[\delta ^{1/4}|Z_S] + {{\mathbb {E}}}[\delta |Z_S]+K_m^*{{\mathbb {E}}}[\delta ^{5/4}|Z_S]+(K_m)^2{{\mathbb {E}}}[\delta ^2|Z_S]\\&\quad \lesssim \ {{\mathbb {E}}}[\delta ^2|Z_S]^{1/8} + {{\mathbb {E}}}[\delta ^2|Z_S]^{1/2}+K_m^*{{\mathbb {E}}}[\delta ^{2}|Z_S]^{5/8}+(K_m)^2{{\mathbb {E}}}[\delta ^2|Z_S]\\&{\mathop {\lesssim }\limits ^{60, 66}}\ \frac{1}{m^{1/8}}\rightarrow 0&a.s. \end{aligned}$$

(28b) is proved by combining statements (a), (b), (c).

A quantile regression framework

Quantile regression offers an alternative approach to simulating Y through a random coefficient interpretation [15]. For any $S\subseteq [n]$ and $\tau \in (0,1)$, we assume the conditional $\tau$-th quantile of Y on $X_S$ satisfies

$$\begin{aligned} F^{-1}_{Y|X_S}(\tau ) = X_S^\top \beta _S(\tau ), \end{aligned}$$

(77)

where $\beta _S(\tau )$ the $\tau$-th coefficient vector. (77) is a standard quantile regression formulation, and can be used to model heteroscedastic noise effects.

$$\begin{aligned}&{\widehat{\beta }}_S(\tau ) = {{\,\mathrm{arg\,min}\,}}_{\beta \in {{\mathbb {R}}}^{s+1}}\frac{1}{m}\sum _{\ell \in [m]}\rho _\tau (Y_\ell - X^\top _{{\text {epr}},\ell }\beta )&\rho _\tau (x) = x(\tau - {\varvec{1}}_{x<0}). \end{aligned}$$

Thus, (77) approximately equals

$$\begin{aligned} {\widehat{F}}^{-1}_{Y|X_S}(\tau ) = X_S^\top {\widehat{\beta }}_S(\tau ). \end{aligned}$$

(78)

As opposed to (23), (78) provides a way to simulate Y based on $X_S$ via inverse transform sampling:

(79)

In our case, $X_{{\text {epr}}, \ell }, \ell \in [m]$ are i.i.d. samples so (77) fits into a random design quantile regression framework as analyzed in [21], where the authors established a strong consistency result for ${\widehat{\beta }}_S(\tau )$ under suitable conditions. The consistency result can be further proven to hold uniformly for all $\tau \in [\delta ,1-\delta ]$ for any fixed $\delta >0$, which justifies the asymptotic behavior of the procedure in (78) as $m, N_S\rightarrow \infty$.

In the quantile regression framework, obtaining the optimal choices for m and S is much harder than in the linear regression setup. The AETC-d-q algorithm in Sect. 6 implements (79) with m set as the adaptive exploration rate given by the AETC-d, S as the corresponding model output for exploitation, and U approximated via $\frac{1}{K}\sum _{j\in [K]}\delta _{\frac{j}{K+1}}$ with $K=100$.