Hedge

Let $N,T\ge 1$ be integers, let $[N]=\{1,\dots ,N\}$ be the expert set, and let $ϵ>0$. At each round $t=1,\dots ,T$, a finite loss vector ${\mathrm{\ell }}_t=({\mathrm{\ell }}_t(1),\dots ,{\mathrm{\ell }}_t(N))\in [0,\mathrm{\infty }{)}^N$ is revealed after the learner's choice. Initialize $W_1(i)=1$ for every $i\in [N]$. At round $t$, set $x_t(i):=\frac{W_t(i)}{{\sum }_{j=1}^N{W}_t(j)},i\in [N],$ Let ${\mathcal{H}}_{t-1}$ be the pre-round history sigma-algebra generated by the previously played choices and all previously revealed loss vectors (any random loss vector is assumed measurable on the protocol's probability space). The weights and $x_t$ are ${\mathcal{H}}_{t-1}$-measurable. Draw a measurable expert choice $I_t\in [N]$ with conditional distribution $\mathbb{P}(I_t=i\mid {\mathcal{H}}_{t-1})=x_t(i)\text{almost surely},i\in [N],$ and play $I_t$. After observing the whole vector ${\mathrm{\ell }}_t$, update $W_{t+1}(i):=W_t(i)\exp (-ϵ{\mathrm{\ell }}_t(i)),i\in [N]\mathrm{.}$ The output is the distributions $x_1,\dots ,x_T$, the sampled choices $I_1,\dots ,I_T$, and the final weights. The protocol assumes a probability space carrying the stated measurable draws; no probability-space construction is claimed.

Online decision protocol: the learner chooses before current-round feedback and its randomized choices are measurable.

Adversary and feedback models: full-information feedback reveals the entire expert-loss vector after the learner's choice.

Random elements and real random variables: each sampled expert is a measurable finite-valued random element.

The real exponential function and the number $e$ by a power series: the update uses the real exponential function.

Hedge's second-order and static-regret bounds ↗: proof step 2.1 establishes positive finite weights and normalizers by induction for every finite nonnegative loss sequence, independently of the fixed-loss assumption used later for expected regret.