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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)

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\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}
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3.2 Aggregated Adaptive Conformal Inference (AgACI)

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\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}
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3.3 Dynamically-tuned Adaptive Conformal Inference (DtACI)

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

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\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}
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3.5 Strongly Adaptive Online Conformal Prediction (SAOCP)

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\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}
diff --git a/published-202407-susmann-adaptive-conformal.html b/published-202407-susmann-adaptive-conformal.html index e1ab7d2..0ca12f6 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 @@
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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)

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+
\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}
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3.2 Aggregated Adaptive Conformal Inference (AgACI)

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\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}
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3.3 Dynamically-tuned Adaptive Conformal Inference (DtACI)

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

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\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}
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3.5 Strongly Adaptive Online Conformal Prediction (SAOCP)

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\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}
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