====== PlayGround ======

^ Convergence rates of stochastic optimization algorithms for convex and nonconvex optimization ^^^
^ Algorithms ^ Convex Optimization ^ ^ Nonconvex Optimization ^ ^ 
|            | Constant learning rate | Diminishing learning rate | Constant learning rate | Diminishing learning rate |
| SGD \cite{sca2020}|$\displaystyle{\mathcal{O}\left( \frac{1}{T} \right) + C}$|$\displaystyle{\mathcal{O}\left( \frac{1}{\sqrt{T}} \right)}$|$\displaystyle{\mathcal{O}\left( \frac{1}{n} \right) + C}$| $\displaystyle{\mathcal{O}\left( \frac{1}{\sqrt{n}} \right)}$ |
| SGD with SPS \cite{loizou2021}|---------|$\displaystyle{\mathcal{O}\left( \frac{1}{T} \right) + C}$|---------|$\displaystyle{\mathcal{O}\left( \frac{1}{n} \right) + C}$|
| Minibatch SGD \cite{chen2020}|---------|$\displaystyle{\mathcal{O}\left( \frac{1}{T} \right) + C}$|---------|$\displaystyle{\mathcal{O}\left( \frac{1}{n} \right) + C}$| 
| Adam \cite{adam}|---------|$\displaystyle{\mathcal{O}\left( \frac{1}{\sqrt{T}} \right)}^{(*)}$|---------|---------|
| AMSGrad \cite{reddi2018}|---------|$\displaystyle{\mathcal{O}\left( \sqrt{\frac{1 + \ln T}{T}} \right)}$|---------|---------|
| GWDC \cite{liang2020}|---------|$\displaystyle{\mathcal{O}\left( \frac{1}{\sqrt{T}} \right)}$|---------|---------|
| AMSGWDC \cite{liang2020}|---------|$\displaystyle{\mathcal{O}\left( \frac{1}{\sqrt{T}} \right)}$|---------|---------|
| AMSGrad \cite{chen2019}|---------|$\displaystyle{\mathcal{O}\left( \frac{\ln T}{\sqrt{T}} \right)}$|--------|$\displaystyle{\mathcal{O}\left( \frac{\ln n}{\sqrt{n}} \right)}$|
| AdaBelief \cite{adab}|---------|$\displaystyle{\mathcal{O}\left( \frac{\ln T}{\sqrt{T}} \right)}$|---------|$\displaystyle{\mathcal{O}\left( \frac{\ln n}{\sqrt{n}} \right)}$|
| Proposed |$\displaystyle{\mathcal{O}\left( \frac{1}{T} \right)} + C_1 \alpha + C_2 \beta$|$\displaystyle{\mathcal{O}\left( \frac{1}{\sqrt{T}} \right)}$|$\displaystyle{\mathcal{O}\left( \frac{1}{n} \right)} + C_1 \alpha + C_2 \beta$|$\displaystyle{\mathcal{O}\left( \frac{1}{\sqrt{n}} \right)}$|

Note: $C$, $C_1$, and $C_2$ are constants independent of learning rates $\alpha, \beta$, number of training examples $T$, and number of iterations $n$. The convergence rate for convex optimization is measured in terms of regret as $R(T)/T$, and the convergence rate for nonconvex optimization is measured as the expectation of the squared gradient norm $\min_{k\in [n]} \mathbb{E}[\|\nabla f(x)\|^2]$. In the case of using constant learning rates, SGD \cite{sca2020} and Proposed can be applied to not only convex but also nonconvex optimization. In the case of using diminishing learning rates, SGD \cite{sca2020} and Proposed had the best convergence rates, $\mathcal{O}(1/\sqrt{n})$.  (*) Theorem 1 in \cite{reddi2018} shows that a counter-example to the \cite{adam} results exists.
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