Given a *strong* (not only strict) convex function $f: \mathbb{R}^n\to\mathbb{R}$. On such problems, stochastic gradient decent (SGD) has a convergence rate of $O(1/T)$, where $T$ is the number of iterations [[1]](http://jmlr.org/proceedings/papers/v28/zhang13e.pdf).
How is the convergence rate affected if a constrained is added to the problem and the projected subgradient method is used to solve it?


E.g.
\begin{align}
\text{minimize } &f(x) \text{ subject to } \\ &x\in C, 
\end{align}
where $C$ is a convex set? The projected subgradient method uses the iteration


$x^{(k+1)} = P \left(x^{(k)} - \alpha\_k g^{(k)} \right)$


where $P$ is projection on $C$ and $g^{(k)}$ is any subgradient of $f$ at $x^{(k)}$.
Is the convergence rate affected by the projection? If yes, are there better approaches to keep the convergence rate of $O(1/T)$?



Convergence rate of stochastic gradient decent with projections
See [here](http://arxiv.org/pdf/1212.2002v2.pdf) for a convergence analysis. As mentioned in the comments, the rate is still $O(\frac{1}{T})$ if the additional constraint keeps the feasible set $C$ convex and the algorithm used is projected stochastic subgradient descent with a suitable step size rule.


