From e02852d86350a120ea0fe95b2dd5a9a7099db0d4 Mon Sep 17 00:00:00 2001 From: Quarto GHA Workflow Runner Date: Sun, 22 Dec 2024 02:25:02 +0000 Subject: [PATCH] Built site for gh-pages --- .nojekyll | 2 +- index.html | 34 ++++++++---------- ...hed-202407-susmann-adaptive-conformal.html | 34 ++++++++---------- ...shed-202407-susmann-adaptive-conformal.pdf | Bin 1246548 -> 1246856 bytes .../figure-pdf/fig-aci-1.pdf | Bin 133583 -> 133583 bytes .../figure-pdf/fig-agaci-1.pdf | Bin 72230 -> 72230 bytes .../figure-pdf/fig-case-study-example-1.pdf | Bin 26791 -> 26791 bytes .../figure-pdf/fig-case-study-metrics-1.pdf | Bin 11299 -> 11299 bytes .../figure-pdf/fig-dtaci-example-1.pdf | Bin 133158 -> 133158 bytes .../figure-pdf/fig-saocp-1.pdf | Bin 136042 -> 136042 bytes .../figure-pdf/fig-sf-ogd-1.pdf | Bin 135522 -> 135522 bytes .../fig-simulation-one-example-1.pdf | Bin 19292 -> 19292 bytes .../figure-pdf/fig-simulation-one-joint-1.pdf | Bin 14055 -> 14055 bytes .../fig-simulation-one-results-1.pdf | Bin 21537 -> 21537 bytes .../fig-simulation-one-widths-1.pdf | Bin 79910 -> 79910 bytes .../fig-simulation-two-example-1.pdf | Bin 131464 -> 131464 bytes .../figure-pdf/fig-simulation-two-joint-1.pdf | Bin 10938 -> 10938 bytes .../fig-simulation-two-results-1.pdf | Bin 28475 -> 28475 bytes .../simulation_one_plot_appendix-1.pdf | Bin 50054 -> 50054 bytes .../figure-pdf/unnamed-chunk-4-1.pdf | Bin 40086 -> 40086 bytes ...ting-1da17b8571ba2be2e5cc9877984c63a7.css} | 2 +- 21 files changed, 30 insertions(+), 42 deletions(-) rename published-202407-susmann-adaptive-conformal_files/libs/quarto-html/{quarto-syntax-highlighting-508bcad9dcf2c2e1342fa9dd89cfc9ba.css => quarto-syntax-highlighting-1da17b8571ba2be2e5cc9877984c63a7.css} (97%) diff --git a/.nojekyll b/.nojekyll index 8a347b0..37446b3 100644 --- a/.nojekyll +++ b/.nojekyll @@ -1 +1 @@ -234bb83b \ No newline at end of file +a4a112f1 \ No newline at end of file diff --git a/index.html b/index.html index b8f6304..5df4b09 100644 --- a/index.html +++ b/index.html @@ -2,7 +2,7 @@ - + @@ -89,7 +89,7 @@ - + @@ -283,7 +283,7 @@
Modified
-

December 15, 2024

+

December 22, 2024

@@ -567,7 +567,7 @@

3 Algorithms

For demonstration purposes we assume we have access to unbiased predictions \hat{\mu}_t = 0 for all t \in \llbracket T \rrbracket. Throughout we set the target empirical coverage to \alpha = 0.8.

3.1 Adaptive Conformal Inference (ACI)

-
+
\begin{algorithm} \caption{Adaptive Conformal Inference} \begin{algorithmic} \State \textbf{Input:} starting value $\theta_1$, learning rate $\gamma > 0$. \For{$t = 1, 2, \dots, T$} \State \textbf{Output:} prediction interval $\widehat{C}_t(\theta_t)$. \State Observe $y_t$. \State Evaluate $\mathrm{err}_t = \mathbb{I}[y_t \not\in \widehat{C}_t(\theta_t)]$. \State Update $\theta_{t+1} = \theta_t + \gamma (\mathrm{err}_t - (1 - \alpha))$. \EndFor \end{algorithmic} \end{algorithm}
@@ -619,7 +619,7 @@

3.2 Aggregated Adaptive Conformal Inference (AgACI)

-
+
\begin{algorithm} \caption{Aggregated Adaptive Conformal Inference} \begin{algorithmic} \State \textbf{Input:} candidate learning rates $(\gamma_k)_{1 \leq k \leq K }$, starting value $\theta_1$. \State Initialize lower and upper BOA algorithms $\mathcal{B}^\ell := \texttt{BOA}(\alpha \leftarrow (1 - \alpha) / 2)$ and $\mathcal{B}^u := \texttt{BOA}(\alpha \leftarrow (1 - (1 - \alpha)/2))$. \For{$k = 1, \dots, K$} \State Initialize ACI $\mathcal{A}_k = \texttt{ACI}(\alpha \leftarrow \alpha, \gamma \leftarrow \gamma_k, \theta_1 \leftarrow \theta_1)$. \EndFor \For{$t = 1, 2, \dots, T$} \For{$k = 1, \dots, K$} \State Retrieve candidate prediction interval $[\ell^k_{t}, u^k_{t}]$ from $\mathcal{A}_k$. \EndFor \State Compute aggregated lower bound $\tilde{\ell}_t := \mathcal{B}^\ell((\ell^k_t : k \in \{ 1, \dots, K \}))$. \State Compute aggregated upper bound $\tilde{u}_t := \mathcal{B}^u((u^k_t : k \in \{ 1, \dots, K \}))$. \State \textbf{Output:} prediction interval $[\tilde{\ell}_t, \tilde{u}_t]$. \State Observe $y_t$. \For{$k = 1, \dots, K$} \State Update $\mathcal{A}_k$ with observation $y_t$. \EndFor \State Update $\mathcal{B}^\ell$ with observed outcome $y_t$. \State Update $\mathcal{B}^u$ with observed outcome $y_t$. \EndFor \end{algorithmic} \end{algorithm}
@@ -650,7 +650,7 @@

3.3 Dynamically-tuned Adaptive Conformal Inference (DtACI)

-
+
\begin{algorithm} \caption{Dynamically-tuned Adaptive Conformal Inference} \begin{algorithmic} \State \textbf{Input:} starting value $\theta_1$, candidate learning rates $(\gamma_k)_{1 \leq k \leq K }$, parameters $\sigma, \eta$. \For{$k = 1, \dots, K$} \State Initialize expert $\mathcal{A}_k = \texttt{ACI}(\alpha \leftarrow \alpha, \gamma \leftarrow \gamma_k, \theta_1 \leftarrow \theta_1)$. \EndFor \For{$t = 1, 2, \dots, T$} \State Define $p_t^k := p_t^k / \sum_{i=1}^K p_t^i$, for all $1 \leq k \leq K$. \State Set $\theta_t = \sum_{k=1}^K \theta_t^k p_t^k$. \State \textbf{Output:} prediction interval $\widehat{C}_t(\theta_t)$. \State Observe $y_t$ and compute $r_t$. \State $\bar{w}_{t}^k \gets p_t^k \exp(-\eta L^\alpha(\theta_t^k, r_t))$, for all $1 \leq k \leq K$. \State $\bar{W}_t \gets \sum_{i=1}^K \bar{w}_t^i$. \State $p_{t+1}^k \gets (1 - \sigma) \bar{w}_t^k + \bar{W}_t \sigma / K$. \State Set $\mathrm{err}_t := \mathbb{I}[y_t \not\in \widehat{C}_t(\theta_t)]$. \For{$k = 1, \dots, K$} \State Update ACI $\mathcal{A}_k$ with $y_t$ and obtain $\theta_{t+1}^k$. \EndFor \EndFor \end{algorithmic} \end{algorithm}
@@ -703,7 +703,7 @@

3.4 Scale-Free Online Gradient Descent (SF-OGD)

-
+
\begin{algorithm} \caption{Scale-Free Online Gradient Descent} \begin{algorithmic} \State \textbf{Input:} starting value $\theta_1$, learning rate $\gamma > 0$. \For{$t = 1, 2, \dots, T$} \State \textbf{Output:} prediction interval $\widehat{C}_t(\theta_t)$. \State Observe $y_t$ and compute $r_t$. \State Update $\theta_{t+1} = \theta_t - \gamma \frac{\nabla L^\alpha(\theta_t, r_t)}{\sqrt{\sum_{i=1}^t} \| \nabla L^\alpha(\theta_i, r_i) \|_2^2}$. \EndFor \end{algorithmic} \end{algorithm}
@@ -742,7 +742,7 @@

3.5 Strongly Adaptive Online Conformal Prediction (SAOCP)

-
+
\begin{algorithm} \caption{Strongly Adaptive Online Conformal Prediction} \begin{algorithmic} \State \textbf{Input:} initial value $\theta_0$, learning rate $\gamma > 0$. \For{$t = 1, 2, \dots, T$} \State Initialize expert $\mathcal{A}_t = \texttt{SF-OGD}(\alpha \leftarrow \alpha, \gamma \leftarrow \gamma, \theta_1 \leftarrow \theta_{t-1})$, set weight $p_t^t = 0$. \State Compute active set $\mathrm{Active}(t) = \{ i \in \llbracket T \rrbracket : t - L(i) < i \leq t \}$ (see below for definition of $L(t)$). \State Compute prior probability $\pi_i \propto i^{-2} (1 + \lfloor \log_2 i \rfloor )^{-1} \mathbb{I}[i \in \mathrm{Active}(t)]$. \State Compute un-normalized probability $\hat{p}_i = \pi_i [p_{t,i}]_+$ for all $i \in \llbracket t \rrbracket$. \State Normalize $p = \hat{p} / \| \hat{p} \|_1 \in \Delta^t$ if $\| \hat{p} \|_1 > 0$, else $p = \pi$. \State Set $\theta_t = \sum_{i \in \mathrm{Active}(t)} p_i \theta_t^i$ (for $t \geq 2$), and $\theta_t = 0$ for $t = 1$. \State \textbf{Output:} prediction set $\widehat{C}_t(\theta_t)$. \State Observe $y_t$ and compute $r_t$. \For{$i \in \mathrm{Active}(t)$} \State Update expert $\mathcal{A}_t$ with $y_t$ and obtain $\theta_{t+1}^i$. \State Compute $g_t^i = \begin{cases} \frac{1}{D}\left(L^\alpha(\theta_t, r_t) - L^\alpha(\theta_t^i, r_t)\right) & p_t^i > 0 \\ \frac{1}{D}\left[L^\alpha(\theta_t, r_t) - L^\alpha(\theta_t^i, r_t))\right]_+ & p_t^i \leq 0 \\ \end{cases}$. \State Update expert weight $p_{t+1}^i = \frac{1}{t - i + 1}\left( \sum_{j=i}^t g_j^i \right) \left(1 + \sum_{j=i}^t p_j^i g_j^i \right)$. \EndFor \EndFor \end{algorithmic} \end{algorithm}
@@ -787,24 +787,20 @@

4 AdaptiveConformal R package

The ACI algorithms described in the previous section have been implemented in the open-source and publically available R package AdaptiveConformal, available at https://github.com/herbps10/AdaptiveConformal. CIn this section, we briefly introduce the main functionality of the package. Comprehensive documentation is, including several example vignettes, is included with the package.

-

The AdaptiveConformal package can be installed using the remotes package:

-
+

The AdaptiveConformal package can be installed using the remotes package: ::: {.cell}

Hide/Show the code 
remotes::install_github("herbps10/AdaptiveConformal")
-
-

The ACI algorithms are accessed through the aci function, which takes as input a vector of observations (y_t) and a vector or matrix of predictions (\hat{y}_t). Using the data generating process from the running example to illustrate, we can fit the original ACI algorithm with learning rate \gamma = 0.1:

-
+

:::

+

The ACI algorithms are accessed through the aci function, which takes as input a vector of observations (y_t) and a vector or matrix of predictions (\hat{y}_t). Using the data generating process from the running example to illustrate, we can fit the original ACI algorithm with learning rate \gamma = 0.1: ::: {.cell}

Hide/Show the code 
set.seed(532)
 data <- running_example_data(N = 5e2)
 fit <- aci(data$y, data$yhat, alpha = 0.8, method = "ACI", parameters = list(gamma = 0.1))
-
-

The available parameters for each method can be found in the documentation for the aci method, accessible with the command ?aci. The resulting conformal prediction intervals can then be plotted using the plot function:

-
+

::: The available parameters for each method can be found in the documentation for the aci method, accessible with the command ?aci. The resulting conformal prediction intervals can then be plotted using the plot function: ::: {.cell}

Hide/Show the code 
plot(fit)
@@ -816,9 +812,7 @@

4 AdaptiveC

-
-

The properties of the prediction intervals can also be examined using the summary function:

-
+

::: The properties of the prediction intervals can also be examined using the summary function: ::: {.cell}

Hide/Show the code 
summary(fit)
@@ -831,7 +825,7 @@

4 AdaptiveC Mean interval width: 0.354 Mean interval loss: 0.498

-

+

:::

5 Simulation Studies

diff --git a/published-202407-susmann-adaptive-conformal.html b/published-202407-susmann-adaptive-conformal.html index b8f6304..5df4b09 100644 --- a/published-202407-susmann-adaptive-conformal.html +++ b/published-202407-susmann-adaptive-conformal.html @@ -2,7 +2,7 @@ - + @@ -89,7 +89,7 @@ - + @@ -283,7 +283,7 @@
Modified
-

December 15, 2024

+

December 22, 2024

@@ -567,7 +567,7 @@

3 Algorithms

For demonstration purposes we assume we have access to unbiased predictions \hat{\mu}_t = 0 for all t \in \llbracket T \rrbracket. Throughout we set the target empirical coverage to \alpha = 0.8.

3.1 Adaptive Conformal Inference (ACI)

-
+
\begin{algorithm} \caption{Adaptive Conformal Inference} \begin{algorithmic} \State \textbf{Input:} starting value $\theta_1$, learning rate $\gamma > 0$. \For{$t = 1, 2, \dots, T$} \State \textbf{Output:} prediction interval $\widehat{C}_t(\theta_t)$. \State Observe $y_t$. \State Evaluate $\mathrm{err}_t = \mathbb{I}[y_t \not\in \widehat{C}_t(\theta_t)]$. \State Update $\theta_{t+1} = \theta_t + \gamma (\mathrm{err}_t - (1 - \alpha))$. \EndFor \end{algorithmic} \end{algorithm}
@@ -619,7 +619,7 @@

3.2 Aggregated Adaptive Conformal Inference (AgACI)

-
+
\begin{algorithm} \caption{Aggregated Adaptive Conformal Inference} \begin{algorithmic} \State \textbf{Input:} candidate learning rates $(\gamma_k)_{1 \leq k \leq K }$, starting value $\theta_1$. \State Initialize lower and upper BOA algorithms $\mathcal{B}^\ell := \texttt{BOA}(\alpha \leftarrow (1 - \alpha) / 2)$ and $\mathcal{B}^u := \texttt{BOA}(\alpha \leftarrow (1 - (1 - \alpha)/2))$. \For{$k = 1, \dots, K$} \State Initialize ACI $\mathcal{A}_k = \texttt{ACI}(\alpha \leftarrow \alpha, \gamma \leftarrow \gamma_k, \theta_1 \leftarrow \theta_1)$. \EndFor \For{$t = 1, 2, \dots, T$} \For{$k = 1, \dots, K$} \State Retrieve candidate prediction interval $[\ell^k_{t}, u^k_{t}]$ from $\mathcal{A}_k$. \EndFor \State Compute aggregated lower bound $\tilde{\ell}_t := \mathcal{B}^\ell((\ell^k_t : k \in \{ 1, \dots, K \}))$. \State Compute aggregated upper bound $\tilde{u}_t := \mathcal{B}^u((u^k_t : k \in \{ 1, \dots, K \}))$. \State \textbf{Output:} prediction interval $[\tilde{\ell}_t, \tilde{u}_t]$. \State Observe $y_t$. \For{$k = 1, \dots, K$} \State Update $\mathcal{A}_k$ with observation $y_t$. \EndFor \State Update $\mathcal{B}^\ell$ with observed outcome $y_t$. \State Update $\mathcal{B}^u$ with observed outcome $y_t$. \EndFor \end{algorithmic} \end{algorithm}
@@ -650,7 +650,7 @@

3.3 Dynamically-tuned Adaptive Conformal Inference (DtACI)

-
+
\begin{algorithm} \caption{Dynamically-tuned Adaptive Conformal Inference} \begin{algorithmic} \State \textbf{Input:} starting value $\theta_1$, candidate learning rates $(\gamma_k)_{1 \leq k \leq K }$, parameters $\sigma, \eta$. \For{$k = 1, \dots, K$} \State Initialize expert $\mathcal{A}_k = \texttt{ACI}(\alpha \leftarrow \alpha, \gamma \leftarrow \gamma_k, \theta_1 \leftarrow \theta_1)$. \EndFor \For{$t = 1, 2, \dots, T$} \State Define $p_t^k := p_t^k / \sum_{i=1}^K p_t^i$, for all $1 \leq k \leq K$. \State Set $\theta_t = \sum_{k=1}^K \theta_t^k p_t^k$. \State \textbf{Output:} prediction interval $\widehat{C}_t(\theta_t)$. \State Observe $y_t$ and compute $r_t$. \State $\bar{w}_{t}^k \gets p_t^k \exp(-\eta L^\alpha(\theta_t^k, r_t))$, for all $1 \leq k \leq K$. \State $\bar{W}_t \gets \sum_{i=1}^K \bar{w}_t^i$. \State $p_{t+1}^k \gets (1 - \sigma) \bar{w}_t^k + \bar{W}_t \sigma / K$. \State Set $\mathrm{err}_t := \mathbb{I}[y_t \not\in \widehat{C}_t(\theta_t)]$. \For{$k = 1, \dots, K$} \State Update ACI $\mathcal{A}_k$ with $y_t$ and obtain $\theta_{t+1}^k$. \EndFor \EndFor \end{algorithmic} \end{algorithm}
@@ -703,7 +703,7 @@

3.4 Scale-Free Online Gradient Descent (SF-OGD)

-
+
\begin{algorithm} \caption{Scale-Free Online Gradient Descent} \begin{algorithmic} \State \textbf{Input:} starting value $\theta_1$, learning rate $\gamma > 0$. \For{$t = 1, 2, \dots, T$} \State \textbf{Output:} prediction interval $\widehat{C}_t(\theta_t)$. \State Observe $y_t$ and compute $r_t$. \State Update $\theta_{t+1} = \theta_t - \gamma \frac{\nabla L^\alpha(\theta_t, r_t)}{\sqrt{\sum_{i=1}^t} \| \nabla L^\alpha(\theta_i, r_i) \|_2^2}$. \EndFor \end{algorithmic} \end{algorithm}
@@ -742,7 +742,7 @@

3.5 Strongly Adaptive Online Conformal Prediction (SAOCP)

-
+
\begin{algorithm} \caption{Strongly Adaptive Online Conformal Prediction} \begin{algorithmic} \State \textbf{Input:} initial value $\theta_0$, learning rate $\gamma > 0$. \For{$t = 1, 2, \dots, T$} \State Initialize expert $\mathcal{A}_t = \texttt{SF-OGD}(\alpha \leftarrow \alpha, \gamma \leftarrow \gamma, \theta_1 \leftarrow \theta_{t-1})$, set weight $p_t^t = 0$. \State Compute active set $\mathrm{Active}(t) = \{ i \in \llbracket T \rrbracket : t - L(i) < i \leq t \}$ (see below for definition of $L(t)$). \State Compute prior probability $\pi_i \propto i^{-2} (1 + \lfloor \log_2 i \rfloor )^{-1} \mathbb{I}[i \in \mathrm{Active}(t)]$. \State Compute un-normalized probability $\hat{p}_i = \pi_i [p_{t,i}]_+$ for all $i \in \llbracket t \rrbracket$. \State Normalize $p = \hat{p} / \| \hat{p} \|_1 \in \Delta^t$ if $\| \hat{p} \|_1 > 0$, else $p = \pi$. \State Set $\theta_t = \sum_{i \in \mathrm{Active}(t)} p_i \theta_t^i$ (for $t \geq 2$), and $\theta_t = 0$ for $t = 1$. \State \textbf{Output:} prediction set $\widehat{C}_t(\theta_t)$. \State Observe $y_t$ and compute $r_t$. \For{$i \in \mathrm{Active}(t)$} \State Update expert $\mathcal{A}_t$ with $y_t$ and obtain $\theta_{t+1}^i$. \State Compute $g_t^i = \begin{cases} \frac{1}{D}\left(L^\alpha(\theta_t, r_t) - L^\alpha(\theta_t^i, r_t)\right) & p_t^i > 0 \\ \frac{1}{D}\left[L^\alpha(\theta_t, r_t) - L^\alpha(\theta_t^i, r_t))\right]_+ & p_t^i \leq 0 \\ \end{cases}$. \State Update expert weight $p_{t+1}^i = \frac{1}{t - i + 1}\left( \sum_{j=i}^t g_j^i \right) \left(1 + \sum_{j=i}^t p_j^i g_j^i \right)$. \EndFor \EndFor \end{algorithmic} \end{algorithm}
@@ -787,24 +787,20 @@

4 AdaptiveConformal R package

The ACI algorithms described in the previous section have been implemented in the open-source and publically available R package AdaptiveConformal, available at https://github.com/herbps10/AdaptiveConformal. CIn this section, we briefly introduce the main functionality of the package. Comprehensive documentation is, including several example vignettes, is included with the package.

-

The AdaptiveConformal package can be installed using the remotes package:

-
+

The AdaptiveConformal package can be installed using the remotes package: ::: {.cell}

Hide/Show the code 
remotes::install_github("herbps10/AdaptiveConformal")
-
-

The ACI algorithms are accessed through the aci function, which takes as input a vector of observations (y_t) and a vector or matrix of predictions (\hat{y}_t). Using the data generating process from the running example to illustrate, we can fit the original ACI algorithm with learning rate \gamma = 0.1:

-
+

:::

+

The ACI algorithms are accessed through the aci function, which takes as input a vector of observations (y_t) and a vector or matrix of predictions (\hat{y}_t). Using the data generating process from the running example to illustrate, we can fit the original ACI algorithm with learning rate \gamma = 0.1: ::: {.cell}

Hide/Show the code 
set.seed(532)
 data <- running_example_data(N = 5e2)
 fit <- aci(data$y, data$yhat, alpha = 0.8, method = "ACI", parameters = list(gamma = 0.1))
-
-

The available parameters for each method can be found in the documentation for the aci method, accessible with the command ?aci. The resulting conformal prediction intervals can then be plotted using the plot function:

-
+

::: The available parameters for each method can be found in the documentation for the aci method, accessible with the command ?aci. The resulting conformal prediction intervals can then be plotted using the plot function: ::: {.cell}

Hide/Show the code 
plot(fit)
@@ -816,9 +812,7 @@

4 AdaptiveC

-
-

The properties of the prediction intervals can also be examined using the summary function:

-
+

::: The properties of the prediction intervals can also be examined using the summary function: ::: {.cell}

Hide/Show the code 
summary(fit)
@@ -831,7 +825,7 @@

4 AdaptiveC Mean interval width: 0.354 Mean interval loss: 0.498

-

+

:::

5 Simulation Studies

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