\begin{algorithmic}
\Require First stage iteration number $n,K \geq 0$, Second stage iteration number $T \geq 0$, Starting point $x_1 \in \mathcal{X}$, algorithm $\mathcal{A}$
\State Consider initial start point: $x_{1}^{0}= x_1$
\For{$ 1 \leq k \leq n$}
\State Run Algorithm $\mathcal{A}$ with $K$ iterations, obtain $(x_{K+1}^{k},y_{K+1}^{k} )$ as output. 
\State Set the new restarting point $x_{1}^{k+1} \xleftarrow{} x_{K+1}^{k}$.
\EndFor
\State Run algorithm $\mathcal{A}$ with $T$ iterations using the last  starting point $x_{1}^{n+1} = x_{K+1}^{n}$, obtain $(X_{T+1},Y_{T+1})$ and output.
\end{algorithmic}

\begin{algorithmic}
\Require First stage iteration number $n,K \geq 0$, Second stage iteration number $T \geq 0$, Starting point $x_1 \in \mathcal{X}$, algorithm $\mathcal{A}$
\State Consider initial start point: $x_{1}^{0}= x_1$
\For{$ 1 \leq k \leq n$}
\State Run Algorithm $\mathcal{A}$ with $K$ iterations, obtain $(x_{K+1}^{k},y_{K+1}^{k} )$ as output.
\State Set the new restarting point $x_{1}^{k+1} \xleftarrow{} x_{K+1}^{k}$.
\EndFor
\State Run algorithm $\mathcal{A}$ with $T$ iterations using the last starting point $x_{1}^{n+1} = x_{K+1}^{n}$, obtain $(X_{T+1},Y_{T+1})$ and output.
\end{algorithmic}
{
    "title": "optimal algorithms for stochastic complementary composite minimization",
    "id": "2211.01758",
    "abstract": "inspired by regularization techniques in statistics and machine learning, we study complementary composite minimization in the stochastic setting. this problem corresponds to the minimization of the sum of a (weakly) smooth function endowed with a stochastic first-order oracle, and a structured uniformly convex (possibly nonsmooth and non-lipschitz) regularization term. despite intensive work on closely related settings, prior to our work no complexity bounds for this problem were known. we close this gap by providing novel excess risk bounds, both in expectation and with high probability. our algorithms are nearly optimal, which we prove via novel lower complexity bounds for this class of problems. we conclude by providing numerical results comparing our methods to the state of the art.",
    "categories": "cs.lg math.oc",
    "doi": "",
    "created": "2022-11-03",
    "updated": "2024-01-23",
    "authors": [
        "alexandre d'aspremont",
        "crist\u00f3bal guzm\u00e1n",
        "cl\u00e9ment lezane"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2211.01758"
}
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