Uncorrected Adam for online convex losses

Fix a finite horizon $T\ge 1$, an initial scalar $x_1\in [-1,1]$, a step parameter $\alpha >0$, moment parameters ${\beta }_1\in [0,1)$ and ${\beta }_2\in (0,1)$, a denominator parameter $ϵ\ge 0$, and a sequence of scalar gradients $g_1,\dots ,g_T$. Set $m_0=v_0=0$. At round $t$, update

$m_t={\beta }_1{m}_{t-1}+(1-{\beta }_1)g_t,v_t={\beta }_2{v}_{t-1}+(1-{\beta }_2)g_t^2\mathrm{.}$

The input is admissible only when $v_t+ϵ>0$ at every update. Set ${\alpha }_t=\alpha \mathrm{/}\sqrt{t}$, compute $y_{t+1}=x_t-{\alpha }_t{m}_t\mathrm{/}\sqrt{v_t+ϵ}$, and set $x_{t+1}=\max (-1,\min (1,y_{t+1}))$. Thus the iterate is projected onto $[-1,1]$ after the update. Return the trajectory $(x_1,\dots ,x_{T+1})$, whose final iterate is $x_{T+1}$. The denominator parameter is added inside the square root. No moment bias correction is applied.

The natural numbers $\mathbb{N}$ (von Neumann): the finite round indices are natural numbers.

Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length: $[-1,1]$ is the closed scalar interval used for initialization and clipping.

Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$: every nonnegative real radicand has a unique nonnegative square root.

Maximum and minimum of a set

Every nonempty finite set of reals has a maximum and a minimum: the displayed clipping operation is the maximum and minimum of finite real sets.

Bias-corrected Adam update: the published bias-corrected Adam procedure instead divides by $\sqrt{{\widehat{v}}_t}+ϵ$; it is a distinct update convention.

Reddi, Kale, and Kumar, Algorithm 1 and the inside-root epsilon variant

Kingma and Ba, Adam Algorithm 1

This is the uncorrected scalar online variant used for the cited regret counterexamples. Its $ϵ=0$ mode is defined only on gradient sequences with positive $v_t$ at every round; the counterexample sequences in the theorem meet that condition. The fixed-$ϵ$ mode uses the source's addition inside the square root, not the published bias-corrected denominator convention.