\begin{algorithmic}[1]
\State Sort \( z \) in decreasing order \( z_{1} \geq \dots \geq z_{c} \)
\State Find \( \kappa(z) \) such that
\[
\kappa(z) = \max_{k = 1 \ldots c} \left\{ k \, \bigg| \, r + k z_{k} > \sum_{j \leq k} z_{j} \right\}
\]
\State Define 
\[
\rho(z) = \frac{\left(\sum_{j \leq \kappa(z)} z_j\right) - r}{\kappa(z)}
\]
\State \textbf{return} \( p \) such that \( p_i = \max(z_i - \rho(z), 0) \)
\end{algorithmic}

\begin{algorithmic}
[1]
\State Sort \( z \) in decreasing order \( z_{1} \geq \dots \geq z_{c} \)
\State Find \( \kappa(z) \) such that
\[
\kappa(z) = \max_{k = 1 \ldots c} \left\{ k \, \bigg| \, r + k z_{k} > \sum_{j \leq k} z_{j} \right\}
\]
\State Define
\[
\rho(z) = \frac{\left(\sum_{j \leq \kappa(z)} z_j\right) - r}{\kappa(z)}
\]
\State \textbf{return} \( p \) such that \( p_i = \max(z_i - \rho(z), 0) \)
\end{algorithmic}
{
    "title": "the lipschitz-variance-margin tradeoff for enhanced randomized smoothing",
    "id": "2309.16883",
    "abstract": "real-life applications of deep neural networks are hindered by their unsteady predictions when faced with noisy inputs and adversarial attacks. the certified radius is in this context a crucial indicator of the robustness of models. however how to design an efficient classifier with an associated certified radius? randomized smoothing provides a promising framework by relying on noise injection into the inputs to obtain a smoothed and robust classifier. in this paper, we first show that the variance introduced by the monte-carlo sampling in the randomized smoothing procedure estimate closely interacts with two other important properties of the classifier, \\textit{i.e.} its lipschitz constant and margin. more precisely, our work emphasizes the dual impact of the lipschitz constant of the base classifier, on both the smoothed classifier and the empirical variance. moreover, to increase the certified robust radius, we introduce a different way to convert logits to probability vectors for the base classifier to leverage the variance-margin trade-off. we leverage the use of bernstein's concentration inequality along with enhanced lipschitz bounds for randomized smoothing. experimental results show a significant improvement in certified accuracy compared to current state-of-the-art methods. our novel certification procedure allows us to use pre-trained models that are used with randomized smoothing, effectively improving the current certification radius in a zero-shot manner.",
    "categories": "cs.lg stat.ml",
    "doi": "",
    "created": "2023-09-28",
    "updated": "2024-02-06",
    "authors": [
        "blaise delattre",
        "alexandre araujo",
        "quentin barth\u00e9lemy",
        "alexandre allauzen"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2309.16883"
}
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