\begin{algorithm}
\caption{Non-Accelerated Composite Stochastic Mirror-Descent (NACSMD)}
\label{Algo NACSMD}
\begin{algorithmic}
\Require Number of iterations $T \geq 0$, starting point $x_1 \in \mathcal{X}$, step-sizes $(\alpha_t,\gamma_t)_t$%, proximal function $V$
\For{$ 1 \leq t \leq T$} 
\begin{equation}
 x_{t+1}  = \arg\min_{x\in {\cal X}} \{ \alpha_t [ \langle G(x_t,\xi_t) , x \rangle + H(x) ] + \gamma_t D^{H}(x,x_{t}) \}
\end{equation}
\EndFor
\State Output $x_{T+1}^{ag}:=  \frac{\sum_{t=1}^{T}  \alpha_{t} x_{t+1}}{\sum_{t=1}^{T} \alpha_{t}}$. 
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{Non-Accelerated Composite Stochastic Mirror-Descent (NACSMD)}
\begin{algorithmic}
\Require Number of iterations $T \geq 0$, starting point $x_1 \in \mathcal{X}$, step-sizes $(\alpha_t,\gamma_t)_t$%, proximal function $V$
\For{$ 1 \leq t \leq T$}
\begin{equation*}
x_{t+1} = \arg\min_{x\in {\cal X}} \{ \alpha_t [ \langle G(x_t,\xi_t) , x \rangle + H(x) ] + \gamma_t D^{H}(x,x_{t}) \}
\end{equation*}
\EndFor
\State Output $x_{T+1}^{ag}:= \frac{\sum_{t=1}^{T} \alpha_{t} x_{t+1}}{\sum_{t=1}^{T} \alpha_{t}}$.
\end{algorithmic}
\end{algorithm}
{
    "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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