SlowGuess/NeurIPS_2023_OCR huggingface dataset:
- OCO with long-term constraints [16, 11], where  $m$  convex constraints of the form  $f_{i}(\cdot) \leq 0$  for  $i \in [m]$  should be satisfied in the long run, that is, the goal is to minimize the cumulative loss while keeping low constraint violation  $\sum_{t=1}^{T} f_{i}(y_{t})$  for each  $i \in [m]$ .
- OCO with long-term and time-varying constraints [20, 14] which compared to the previous extension, considers time-varying convex constraints of the form  $f_{t,i}(\cdot) \leq 0$  where  $f_{t,i}$  is revealed at the end of time period  $t$ . Even as long-term constraints, this learning task is known to be unsolvable in general (see e.g. [28, Proposition 2.1] or [18, Proposition 4] for more precise statement), thus, restricted notions of regret have been considered in this context.
- OCO with ramp constraints [2], where at each time period  $t \in \{2,3,\ldots\}$  the decision-maker should choose  $y_{t} \in \mathcal{V}$  such that  $|y_{t,i} - y_{t-1,i}| \leq \kappa_{i}$  for all  $i \in [n]$ .
We argue that OIO problems are different from these extensions. Indeed, our feasibility constraints (1) are neither long-term constraint since we do not allow for violations, nor ramp constraints since the bounds are time-varying. Also, our task is further challenging since we aim at bounding the regret (3) based on a competitor  $y \in \mathcal{V}$  which does not suffer from the feasibility constraint (1).
D.2 On the notion of pseudo-regret
There exists an alternative notion of regret we call the pseudo-regret  $\bar{R}_T$  which is in fact more common in the literature of online inventory problems [9, 3, 25, 32]. It is defined as the difference between the expected cumulative loss of the algorithm and that of the best fixed constant strategy, that is, formally
$$
\bar {R} _ {T} = \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \ell_ {t} (y _ {t}) \right] - \inf  _ {y \in \mathcal {Y}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \ell_ {t} (y) \right].
$$
The difference between the expected regret  $\mathbb{E}[R_T]$  and the pseudo-regret  $\bar{R}_T$  is in the competitor  $y$ . In the former, the competitor is random and depends on the realization of the losses, whereas, in the latter the competitor is fixed and depends only on the distribution of the losses. Notice that we always have  $\bar{R}_T \leq \mathbb{E}[R_T]$ , thus, an upper bound obtained on the expected regret applies directly to the pseudo-regret.