Unbounded Iteration

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$\displaystyle \begin{aligned} \sum_{k=1}^{T-1} \frac{1}{k (k+1)}\sum_{t=T-k}^T \eta^2_t &= \alpha^2 \sum_{k=1}^{T-1} \frac{1}{k (k+1)} \sum_{t=T-k}^{T} \frac1t \\ &= \alpha^2 \sum_{k=1}^{T-1} \frac{1}{k (k+1)} \left(\frac{1}{T-k} + \sum_{t=T-k+1}^{T} \frac1t \right) \\ &\leq \frac{3}{2}\alpha^2 \sum_{k=1}^{T-1} \frac{1}{k (T-k)} \\ &= \frac{3}{2}\alpha^2 \sum_{k=1}^{T-1} \frac{1}{k T} + \frac{3}{2}\alpha^2\sum_{k=1}^{T-1} \frac{1}{T(T-k)} \\ &= \frac{3}{2}\alpha^2 \sum_{k=1}^{T-1} \frac{1}{k T} + \frac{3}{2}\alpha^2\sum_{k=1}^{T-1} \frac{1}{k T} \\ &\leq 3\alpha^2 \frac{1+\log(T-1)}{T}~. \end{aligned}$

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$\displaystyle \begin{aligned} \eta_T \left(\mathop{\mathbb E}[F({\boldsymbol x}_t)]-F({\boldsymbol x}^\star)\right) \leq \frac{1}{T} \sum_{t=1}^T \eta_t \mathop{\mathbb E}[F({\boldsymbol x}_t)-F({\boldsymbol x}^\star)] + \sum_{k=1}^{T-1} \frac{1}{k(k+1)} \sum_{t=T-k+1}^T \eta_t \mathop{\mathbb E}[F({\boldsymbol x}_t) - F({\boldsymbol x}_{T-k})]~. \end{aligned}$

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$\displaystyle k S_k = (k+1) S_{k+1} - \eta_{T-k} q_{T-k} = k S_{k+1} + S_{k+1} - \eta_{T-k} q_{T-k} \leq k S_{k+1} + \frac{1}{k+1} \sum_{t=T-k}^T \eta_t (q_t - q_{T-k}),$

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Unbounded Sequence

$\displaystyle \mathop{\mathbb E}[F(\bar{{\boldsymbol x}}_T)] \leq F({\boldsymbol x}^\star) + \frac{\|{\boldsymbol x}^\star-{\boldsymbol x}_1\|^2}{2 \alpha \sqrt{T}} + \frac{\alpha (\ln T+1)}{2\sqrt{T}}~.$

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$\displaystyle \sum_{t=T-k+1}^T \eta_t \mathop{\mathbb E}[F({\boldsymbol x}_t)-F({\boldsymbol x}_{T-k})] =\sum_{t=T-k}^T \eta_t \mathop{\mathbb E}[F({\boldsymbol x}_t)-F({\boldsymbol x}_{T-k})] \leq \frac{1}{2}\sum_{t=T-k}^T \eta^2_t \mathop{\mathbb E}[\|{\boldsymbol g}_t\|^2_2]~.$

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$\displaystyle \eta_T q_T \leq \frac{1}{T} \sum_{t=1}^T \eta_t q_t + \sum_{k=1}^{T-1} \frac{1}{k(k+1)} \sum_{t=T-k+1}^T \eta_t (q_t - q_{T-k})~.$

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Unbounded Iteration

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