AdamCB

Implements AdamCB, Adam with adaptive batch selection driven by a combinatorial bandit.

AdamCB replaces uniform mini-batch sampling with an adaptive scheme: at each step it draws \(K\) distinct samples without replacement, where the selection probabilities are maintained by a combinatorial-bandit rule that favors high-influence samples while keeping a uniform exploration floor. The resulting gradient is importance-weighted so it stays unbiased, then fed into an AMSGrad-style Adam update with a time-decaying first-moment rate \(\beta_{1,t}\) and a matching correction inside the running maximum.

\[ \begin{aligned} g_t &= \frac{1}{K}\sum_{j\in J_t}\frac{g_{j,t}}{n\,p_{j,t}} \\ m_t &= \beta_{1,t}\,m_{t-1} + (1-\beta_{1,t})\,g_t \\ v_t &= \beta_2\,v_{t-1} + (1-\beta_2)\,g_t^2 \\ \hat{v}_t &= \max\left\{\frac{(1-\beta_{1,t})^2}{(1-\beta_{1,t-1})^2}\,\hat{v}_{t-1},\; v_t\right\} \\ \theta_{t+1} &= \theta_t - \alpha_t\,\frac{m_t}{\sqrt{\hat{v}_t}+\epsilon} \end{aligned} \]

where \(J_t\) is the set of \(K\) sampled indices, \(p_{j,t}\) their selection probabilities, \(n\) the dataset size, \(g_{j,t}\) the per-sample gradient, \(\beta_{1,t}=\beta_1\lambda^{t-1}\) with \(\lambda\in(0,1)\) the decaying first-moment rate, \(\beta_2\) the second-moment rate, \(\hat{v}_t\) the AMSGrad running maximum (the \(\max\) applies for \(t\ge 2\)), \(\alpha_t=\alpha/\sqrt{t}\) the step size, and \(\epsilon\) a stability constant.

Reference: Gyu Yeol Kim, Min-hwan Oh, "Adam Optimization with Adaptive Batch Selection", arXiv 2025. https://arxiv.org/abs/2512.06795

---

Back to the Canon