SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In the second phase (Lines 6-9), TPE starts leveraging the predictions. More precisely, TPE needs to set the speeds for three different types of jobs: (1) the remaining jobs that were correctly predicted (i.e.,  $\mathcal{J}_{\geq t_{\lambda}} \cap \hat{\mathcal{J}}_{\geq t_{\lambda}}$ ) (2) the remaining jobs that were not predicted (i.e.,  $\mathcal{J}_{\geq t_{\lambda}} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}}$ ) (3) the jobs that were not correctly scheduled in the first phase and still have work remaining at the switch point  $t_{\lambda}$  (which are a subset of  $\mathcal{J}_{< t_{\lambda}}$ ). To schedule these jobs, TPE combines two different schedules. The first one is the offline schedule  $\hat{S} \coloneqq \mathrm{OFFLINEALG}(\hat{\mathcal{J}}_{\geq t_{\lambda}})$  for the jobs  $\hat{\mathcal{J}}_{\geq t_{\lambda}}$  that are predicted to arrive in the second phase. Each future job in the true set that was correctly predicted (i.e.,  $j \in \mathcal{J}_{[t_{\lambda}, t]} \cap \hat{\mathcal{J}}_{\geq t_{\lambda}}$  on Line 9) will then be scheduled by following  $\hat{S}$ . The second schedule is an online schedule for the set of jobs  $\mathcal{J} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}} = \mathcal{J}_{< t_{\lambda}} \cup \mathcal{J}_{\geq t_{\lambda}} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}}$ , which includes all jobs that have not been completed during the first phase ( $\subseteq \mathcal{J}_{< t_{\lambda}}$ ) and the incorrectly predicted jobs that are released during the second phase ( $\mathcal{J}_{[t_{\lambda}, t]} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}}$ ). This online schedule is constructed by running ONLINEALG on the set  $\mathcal{J} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}}$  (Line 8). Note that the total speed of the machine at each time step is the sum of the speeds of these two online and offline schedules.
3.2 Analysis of the algorithm
We analyze the competitive ratio of TPE as a function of the prediction error  $\eta$ , from which the consistency and robustness bounds follow. Missing proofs are provided in Appendix B. We separately bound the cost of the algorithm due to jobs in  $\mathcal{I}_{<t_{\lambda}}$ ,  $\mathcal{J}_{\geq t_{\lambda}} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}}$  and  $\mathcal{J}_{\geq t_{\lambda}} \cap \hat{\mathcal{J}}_{\geq t_{\lambda}}$ . We do this by analyzing the costs of schedules  $S^{on} := \text{ONLINEALG}(\mathcal{J} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}})$  and  $\hat{S} := \text{OFFLINEALG}(\hat{\mathcal{J}}_{\geq t_{\lambda}})$ . In the next lemma, we first analyze the cost of combining, i.e., summing, two arbitrary schedules.
Lemma 3.1. Let  $\mathcal{J}_1$  be a set of jobs and  $S_{1}$  be a feasible schedule for  $\mathcal{I}_1$ , let  $\mathcal{J}_2$  be a set of jobs and  $S_{2}$  be a feasible schedule for  $\mathcal{J}_2$ . Consider the schedule  $S := S_1 + S_2$  for  $\mathcal{J}_1 \cup \mathcal{J}_2$  which, at each time  $t$ , runs the machine at total speed  $s(t) = s_1(t) + s_2(t)$  and processes each job  $j \in \mathcal{J}_1$  at speed  $s_{1,j}(t)$  and each job  $j \in \mathcal{J}_2$  at speed  $s_{2,j}(t)$ . Then,  $\mathrm{cost}(S,\mathcal{J}_1 \cup \mathcal{J}_2) \leq \left(\mathrm{cost}(S_1,\mathcal{J}_1)^{\frac{1}{\alpha}} + \mathrm{cost}(S_2,\mathcal{J}_2)^{\frac{1}{\alpha}}\right)^{\alpha}$ .
We next upper bound the cost of the schedule output by TPE as a function of the prediction error  $\eta$ , which we decompose into  $\eta_{1} = \frac{\mathrm{OPT}(\mathcal{J}\backslash\hat{\mathcal{J}})}{\mathrm{OPT}(\hat{\mathcal{J}})}$  and  $\eta_{2} = \frac{\mathrm{OPT}(\hat{\mathcal{J}}\backslash\mathcal{J})}{\mathrm{OPT}(\hat{\mathcal{J}})}$ . The proof uses the previous lemma repeatedly, first to analyze the cost of the schedule  $S^{on} := \mathrm{ONLINEALG}(\mathcal{J}\backslash \hat{\mathcal{J}}_{\geq t_{\lambda}})$  for the set of jobs  $\mathcal{J}\setminus \hat{\mathcal{J}}_{\geq t_{\lambda}} = (\mathcal{J}_{< t_{\lambda}})\cup (\mathcal{J}_{\geq t_{\lambda}}\setminus \hat{\mathcal{J}}_{\geq t_{\lambda}})$ , then to analyze the cost of the final schedule, which combines  $S^{on}$  and  $\hat{S} := \mathrm{OFFLINEALG}(\hat{\mathcal{J}}_{\geq t_{\lambda}})$ .
Lemma 3.2. Assume that OFFLINEAlg is  $\gamma_{off}$ -competitive and that ONLINEAlg is  $\gamma_{on}$ -competitive. Then, for all  $\lambda \in (0,1]$ , the schedule  $S$  output by TPE run with confidence parameter  $\lambda$  satisfies cost(S, J)  $\leq OPT(\hat{\mathcal{J}})\left(\gamma_{off}^{\frac{1}{\alpha}} + \gamma_{on}^{\frac{1}{\alpha}}((\lambda \gamma_{off})^{\frac{1}{\alpha}} + \eta_1^{\frac{1}{\alpha}})\right)^{\alpha}$ .
Proof. We start by upper bounding  $\cos(S^{on}, \mathcal{I} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}})$ . First, by the algorithm, we have that  $\mathrm{OFF}(\mathcal{J}_{<t_{\lambda}}) \leq \lambda \cdot \mathrm{OFF}(\hat{\mathcal{J}})$ . Since OFFLINEALG is  $\gamma_{\mathrm{off}}$ -competitive, we get
$$
\mathrm {O P T} \left(\mathcal {J} _ {<   t _ {\lambda}}\right) \leq \mathrm {O F F} \left(\mathcal {J} _ {<   t _ {\lambda}}\right) \leq \lambda \cdot \mathrm {O F F} (\hat {\mathcal {J}}) \leq \lambda \gamma_ {\text {o f f}} \cdot \mathrm {O P T} (\hat {\mathcal {J}}).
$$
We also have that  $\mathsf{OPT}(\mathcal{I}_{\geq t_{\lambda}}\setminus \hat{\mathcal{I}}_{\geq t_{\lambda}})\leq \mathsf{OPT}(\mathcal{I}\setminus \hat{\mathcal{I}})\leq \eta_1\mathsf{OPT}(\hat{\mathcal{I}})$  where the first inequality is since  $\mathcal{I}_{\geq t_{\lambda}}\setminus \hat{\mathcal{I}}_{\geq t_{\lambda}}\subseteq \mathcal{I}\setminus \hat{\mathcal{I}}$  and the second is by definition of  $\eta_{1}$ . Recall that  $S^{*}(.)$  denotes the optimal
offline schedule for the problem and consider the schedule  $S' = S^{\star}(\mathcal{J}_{<t_{\lambda}}) + S^{\star}(\mathcal{J}_{\geq t_{\lambda}} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}})$  for  $\mathcal{J} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}} = \mathcal{J}_{<t_{\lambda}} \cup (\mathcal{J}_{\geq t_{\lambda}} \setminus \hat{\mathcal{J}}_{\geq t_{\lambda}})$ . We obtain that
$$
\begin{array}{l} \mathsf {O P T} (\mathcal {J} \setminus \hat {\mathcal {J}} _ {\geq t _ {\lambda}}) \leq \operatorname {c o s t} (S ^ {\prime}, \mathcal {J} \setminus \hat {\mathcal {J}} _ {\geq t _ {\lambda}}) \leq \left(\mathsf {O P T} (\mathcal {J} _ {<   t _ {\lambda}}) ^ {\frac {1}{\alpha}} + \mathsf {O P T} (\mathcal {J} _ {\geq t _ {\lambda}} \setminus \hat {\mathcal {J}} _ {\geq t _ {\lambda}}) ^ {\frac {1}{\alpha}}\right) ^ {\alpha} \\ \leq \left(\left(\lambda \gamma_ {\text {o f f}} \mathsf {O P T} (\hat {\mathcal {J}})\right) ^ {\frac {1}{\alpha}} + \left(\eta_ {1} \mathsf {O P T} (\hat {\mathcal {J}})\right) ^ {\frac {1}{\alpha}}\right) ^ {\alpha} \\ = \mathsf {O P T} (\hat {\mathcal {J}}) \left(\left(\lambda \gamma_ {\text {o f f}}\right) ^ {\frac {1}{\alpha}} + \eta_ {1} ^ {\frac {1}{\alpha}}\right) ^ {\alpha}, \\ \end{array}
$$
where the second inequality is by Lemma 3.1. Since we assumed that ONLINEAlg is  $\gamma_{\mathrm{on}}$ -competitive,
$$
\operatorname {c o s t} \left(S ^ {\text {o n}}, \mathcal {J} \setminus \hat {\mathcal {J}} _ {\geq t _ {\lambda}}\right) \leq \gamma_ {\text {o n}} \cdot \operatorname {O P T} \left(\mathcal {J} \setminus \hat {\mathcal {J}} _ {\geq t _ {\lambda}}\right) \leq \gamma_ {\text {o n}} \cdot \operatorname {O P T} (\hat {\mathcal {J}}) \left(\left(\lambda \gamma_ {\text {o f f}}\right) ^ {\frac {1}{\alpha}} + \eta_ {1} ^ {\frac {1}{\alpha}}\right) ^ {\alpha}.
$$
We now bound the cost of schedule  $S$ . First, note that  $\mathrm{cost}(\hat{S},\hat{\mathcal{J}}_{\geq t_\lambda}) = \mathrm{OFFLINEALG}(\hat{\mathcal{J}}_{\geq t_\lambda})\leq \gamma_{\mathrm{off}}\cdot \mathrm{OPT}(\hat{\mathcal{J}}_{\geq t_\lambda})\leq \gamma_{\mathrm{off}}\cdot \mathrm{OPT}(\hat{\mathcal{J}})$ , where the first inequality is since OFFLINEALG is  $\gamma_{\mathrm{off}}$ -competitive and the last one since  $\hat{\mathcal{J}}_{\geq t_{\lambda}}\subseteq \hat{\mathcal{I}}$ . Therefore, by applying again Lemma 3.1, we get:
$$
\begin{array}{l} \operatorname {c o s t} (S, \mathcal {J}) \leq \left(\operatorname {c o s t} (\hat {S}, \hat {\mathcal {J}} _ {\geq t _ {\lambda}}) ^ {\frac {1}{\alpha}} + \operatorname {c o s t} (S ^ {o n}, \mathcal {J} \setminus \hat {\mathcal {J}} _ {\geq t _ {\lambda}}) ^ {\frac {1}{\alpha}}\right) ^ {\alpha} \\ \leq \left(\left(\gamma_ {\text {o f f}} \cdot \mathsf {O P T} (\hat {\mathcal {J}})\right) ^ {\frac {1}{\alpha}} + \left(\gamma_ {\text {o n}} \cdot \mathsf {O P T} (\hat {\mathcal {J}})\right)\left(\left(\lambda \gamma_ {\text {o f f}}\right) ^ {\frac {1}{\alpha}} + \eta_ {1} ^ {\frac {1}{\alpha}}\right) ^ {\alpha}\right) ^ {\frac {1}{\alpha}}\left. \right) ^ {\alpha} \\ = \mathsf {O P T} (\hat {\mathcal {J}}) \left(\gamma_ {\mathrm {o f f}} ^ {\frac {1}{\alpha}} + \gamma_ {\mathrm {o n}} ^ {\frac {1}{\alpha}} ((\lambda \gamma_ {\mathrm {o f f}}) ^ {\frac {1}{\alpha}} + \eta_ {1} ^ {\frac {1}{\alpha}})\right) ^ {\alpha}. \\ \end{array}
$$
We next state a simple corollary of Lemma 3.1.
Corollary 3.3.  $OPT(\mathcal{J} \cap \hat{\mathcal{J}}) \geq \left(1 - \eta_2^{\frac{1}{\alpha}}\right)^{\alpha} OPT(\hat{\mathcal{J}})$ , and, assuming that OFFLINEAlg is  $\gamma_{off}$ -competitive, we have: if  $OFF(\mathcal{J}) \leq \lambda OFF(\hat{\mathcal{J}})$ , then  $\eta_2 \geq \left(1 - (\lambda \gamma_{off})^{\frac{1}{\alpha}}\right)^{\alpha}$ .
We are ready to state the main result of this section, which is our upper bound on the competitive ratio of TPE.
Theorem 3.4. For any  $\lambda \in (0,1]$ , TPE with a  $\gamma_{on}$ -competitive algorithm ONLINEAlg and a  $\gamma_{off}$ -competitive offline algorithm OFFLINEAlg achieves a competitive ratio of
$$
\left\{ \begin{array}{l l} \gamma_ {o n} & i f O F F (\mathcal {J}) \leq \lambda O F F (\hat {\mathcal {J}}) \\ \left(\gamma_ {o f f} ^ {\frac {1}{\alpha}} + \gamma_ {o n} ^ {\frac {1}{\alpha}} \left((\lambda \gamma_ {o f f}) ^ {\frac {1}{\alpha}} + \eta_ {1} ^ {\frac {1}{\alpha}}\right)\right) ^ {\alpha} \\ \hline \max  \left\{\frac {\lambda}{\gamma_ {o f f}}, \eta_ {1} + \left(1 - \eta_ {2} ^ {\frac {1}{\alpha}}\right) ^ {\alpha} \right\} & o t h e r w i s e. \end{array} \right.
$$
The consistency and robustness immediately follow (for simplicity, we present the results in the case where OFFLINEAlg is optimal). Additional discussion on this competitive ratio is provided in Appendix 3.4.
Corollary 3.5. For any  $\lambda \in (0,1)$ , TPE with a  $\gamma_{on}$ -competitive algorithm ONLINEAlg and an optimal offline algorithm OFFLINEAlg is  $1 + \gamma_{on}2^{\alpha}\lambda^{\frac{1}{\alpha}}$  competitive if  $\eta_1 = \eta_2 = 0$  (consistency) and  $\max \{\gamma_{on},\frac{1 + \gamma_{on}2^{2\alpha}\lambda^{\frac{1}{\alpha}}}{\lambda}\}$ -competitive for all  $\eta_1,\eta_2$  (robustness). In particular, for any constant  $\epsilon >0$ , with  $\lambda = (\frac{\epsilon}{\gamma_{on}2^{\alpha}})^{\alpha}$ , TPE is  $1 + \epsilon$ -consistent and  $O(1)$ -robust.
3.3 Results for well-studied GES problems
We apply the general framework detailed in Section 3 to derive smooth, consistent and robust algorithms for a few classically studied objective functions.
Energy plus flow time minimization. Recall that  $c_{S}^{j}$  denote the completion time of job  $j$ . The quality cost function is defined as:  $F(S, \mathcal{J}) = \sum_{j \in \mathcal{J}} (c_{S}^{j} - r_{j})$ , with total objective cost  $(S, \mathcal{J}) = F(S, \mathcal{J}) + E(S)$ . By a direct application of Corollary 3.5, we get that for all  $\lambda \in (0,1]$ , Algorithm 1
run with the 2-competitive online algorithm from [3] and confidence parameter  $\lambda$  is  $(1 + 2^{\frac{1}{\alpha}}\lambda^{\frac{1}{\alpha}})^{\alpha}$  consistent and  $\frac{1 + 2\cdot 2^{2\alpha}\lambda^{\frac{1}{\alpha}}}{\lambda}$ -robust.
Energy plus fractional weighted flow time minimization. In this setting, each job has a weight  $v_{j}$ . The quality cost is  $F(S,\mathcal{J}) = \sum_{j\in \mathcal{J}}v_{j}\int_{t\geq r_{j}}w_{j}^{S}(t)\mathrm{d}t$ . We can use as ONLINEALG the 2-competitive algorithm from [14].
Energy plus integral weighted flow time minimization. In this setting, each job has a weight  $v_{j}$ . The quality cost function is defined as:  $F(S,\mathcal{J}) = \sum_{j\in \mathcal{J}}v_{j}(c_{S}^{j} - r_{j})$ . We can use as ONLINEALG the  $O((\alpha /\log \alpha)^2)$ -competitive algorithm from [12].
Energy minimization with deadlines. In this setting, there is also a deadline  $d_{j}$  for the completion of each job. By writing the quality cost as  $F(S,\mathcal{J}) = \sum_{j\in \mathcal{J}}\delta_{c_S^j >d_j}$ , where  $\delta_{c_S^j >d_j} = +\infty$  if  $c_S^j >d_j$  and 0 otherwise, the total objective can be written as  $\mathrm{cost}(S,\mathcal{J}) = E(S) + F(S,\mathcal{J})$ . We can use as ONLINE ALG the AVERAGE RATE heuristic [28] (which is  $2^{\alpha}$ -competitive for uniform deadlines [7]). In particular, for uniform deadlines, and for all  $\epsilon \in (0,1]$ , by setting  $\lambda = (\frac{\epsilon}{C^{2\alpha}})^{\alpha}$ , we obtain a consistency of  $(1 + \epsilon)$  for a robustness factor of  $O(4^{\alpha^2} / \epsilon^{\alpha -1})$ .
3.4 Discussion on the competitive ratio
We assume in this section that OFFLINEAlg is optimal. Note that for small  $\eta_{1}$  and  $\eta_{2}$ , the competitive ratio is upper bounded as  $\frac{\left(1 + \gamma_{\mathrm{on}}^{\frac{1}{\alpha}}\left(\lambda^{\frac{1}{\alpha}} + \eta_{1}^{\frac{1}{\alpha}}\right)\right)^{\alpha}}{\eta_{1} + \left(1 - \eta_{2}^{\frac{1}{\alpha}}\right)^{\alpha}}$ , which smoothly goes to  $(1 + \gamma_{\mathrm{on}}^{\frac{1}{\alpha}}\lambda^{\frac{1}{\alpha}})^{\alpha}$  (consistency case) when  $\eta_{1}, \eta_{2}$  go to 0. Moreover, our upper bound distinguishes the effect of two possible sources of errors on the algorithm: (1) when removing jobs from the prediction  $(\eta_{1} = 0$  and  $\eta_{2}$  goes to 1), the upper bound degrades monotonically to  $O\left(\frac{1}{\lambda}\right)$ . (2) when adding jobs to the prediction  $(\eta_{2} = 0$  and  $\eta_{1}$  goes to  $+\infty$ ), the upper bound first degrades, then improves again, with an optimal asymptotic rate of  $\gamma_{\mathrm{on}}$ . This is since our algorithm mostly follows the online algorithm when the cost of the additional jobs dominates.
4 The Extension to Small Deviations
Note that in the definition of the prediction error  $\eta$ , a job  $j$  is considered to be correctly predicted only if  $r_j = \hat{r}_j$  and  $p_j = \hat{p}_j$ . In this extension, we consider that a job is correctly predicted even if its release time and processing time are shifted by a small amount. We also allow each job to have some weight  $v_j > 0$ , that can be shifted as well. Assuming an additional smoothness condition on the quality cost function  $F(:,.)$ , which is satisfied for the energy plus flow time minimization problem and its variants, we propose and analyze an algorithm that generalizes the algorithm from the previous section.
The algorithm, called TPE-S and formally described in Appendix C, takes the same input parameters as Algorithm TPE, with some additional shift tolerance parameter  $\eta^{\mathrm{shift}}\in [0,1)$  that is chosen by the algorithm designer. Two main ideas are to artificially increase the predicted processing time  $\hat{p}_j$  of each job  $j$  (because the true processing time  $p_j$  of job  $j$  could be shifted and be slightly larger than  $\hat{p}_j$ ) and to introduce small delays for the job speeds (because the true release time  $r_j$  of some jobs  $j$  could be shifted and be slightly later than  $\hat{r}_j$ ). Details can be found in Appendix C.
We empirically evaluate the performance of Algorithm TPE-S on both synthetic and real datasets. Specifically, we consider the energy plus flow time minimization problem where  $F(S, \mathcal{J}) = \sum_{j \in \mathcal{J}} c_j - r_j$  and consider unit-work jobs (i.e.,  $p_j = 1$  for all  $j$ ) and fix  $\alpha = 3$ .
5.1 Experiment settings
Benchmarks. TPE-S is Algorithm 2 with the default setting  $\lambda = 0.02$ ,  $\eta^{\mathrm{shift}} = 1$  and  $\sigma = 0.4$ , where  $\sigma$  is a parameter that controls the level of prediction error, that we call the error parameter.
Figure 1: The competitive ratio achieved by our algorithm, TPE-S, and the benchmark algorithm, as a function of the error parameter  $\sigma$  (from left-most to the second from the right), and the competitive ratio of TPE-S for a larger range of  $\sigma$ , as a function of  $\sigma$  (right-most).
Figure 2: The competitive ratio achieved by our algorithm, TPE-S, as a function of the shift tolerance parameter  $\eta^{\mathrm{shift}}$  (left) and as a function of the confidence parameter  $\lambda$  (right).
2-COMPETITIVE is the 2-competitive online algorithm from [3] that sets the speed at each time  $t$  to  $n(t)^{\frac{1}{\alpha}}$ , where  $n(t)$  is the number of jobs with  $r_j \leq t$  unfinished at time  $t$ , and uses the Shortest Remaining Processing Time rule.
Data sets. We consider two synthetic datasets and a real dataset. For the synthetic data, we first generate a predicted set of jobs  $\hat{\mathcal{I}}$  and we fix the value of the error parameter  $\sigma > 0$ . To create the true set of jobs  $\mathcal{J}$ , we generate, for each job  $j \in \hat{\mathcal{I}}$ , some error  $err(j)$  sampled i.i.d. from  $\mathcal{N}(0, \sigma)$ . The true set of jobs is then defined as  $\mathcal{J} = \{(j, \hat{r}_j + err(j)): j \in \hat{\mathcal{I}}\}$ , which is the set of all predicted jobs, shifted according to  $\{err(j)\}$ . Note that for all  $j \in \mathcal{J}$ ,  $j \in \mathcal{J}^{\mathrm{shift}}$  only if  $|\hat{r}_j - r_j| = |err(j)| < \frac{\eta^{\mathrm{shift}}}{\beta(\hat{\mathcal{I}})} \cdot \frac{\mathrm{OPT}(\hat{\mathcal{I}})}{|\hat{\mathcal{I}}|}$ . Hence, a larger  $\sigma > 0$  corresponds to a larger prediction error  $\eta^g$ . For the first synthetic dataset, called the periodic dataset, the prediction is a set of  $n = 300$  jobs, with  $i^{th}$  job's arrival  $r_i = i / \alpha$ . For the second synthetic dataset, we generate the prediction by using a power-law distribution. More precisely, for each time  $t \in \{1, \dots, T\}$ , where we fix  $T = 75$ , the number of jobs' arrivals at time  $t$  is set to  $M(1 - p(a))$ , where  $p(a)$  is sampled from a power law distribution of parameter  $a$ , and  $M$  is some scaling parameter. In all experiments, we use the values  $a = 100$ ,  $M = 500$ .
We also evaluate the two algorithms on the College Message dataset from the SNAP database [26], where the scheduler must process messages that arrive over 9 days, each with between 300 and 500 messages. We first fix the error parameter  $\sigma > 0$ , then, for each day, we define the true set  $\mathcal{J}$  as the arrivals for this day, and we create the predictions  $\hat{\mathcal{I}}$  by adding some error  $err(i)$  to the release time of each job  $i$ , where  $err(i)$  is sampled i.i.d. from  $\mathcal{N}(0,\sigma)$ .
5.2 Experiment results
For each of the synthetic datasets, the competitive ratio achieved by the different algorithms is averaged over 10 instances generated i.i.d., and for the real dataset, it is averaged over the arrivals for each of the 9 days.
Experiment set 1. We first evaluate the performance of the algorithms as a function of the error parameter  $\sigma$ . In Figure 1, we observe that TPE-S outperforms 2-COMPETITIVE when the error parameter is small. In the right-most figure of Figure 1, the competitive ratio of TPE-S plateaus when the value of  $\sigma$  increases, which is consistent with our bounded robustness guarantee.
Experiment set 2. In the second set of experiments, we study the impact of the parameters  $\eta^{\mathrm{shift}}$  and  $\lambda$  of the algorithm for the periodic dataset (results for the other datasets can be found in Appendix E) and fix  $\sigma = 0.4$ . In the left plot of Figure 2, we observe the importance of allowing some shift in the predictions as the performance of our algorithms first rapidly improves as a function of  $\eta^{\mathrm{shift}}$
and then slowly deteriorates. The rapid improvement is because an increasing number of jobs are treated by the algorithm as being correctly predicted when  $\eta^{\mathrm{shift}}$  increases. Next, in the right plot, we observe that the competitive ratio deteriorates as a function of  $\lambda$ , which implies that the algorithm can completely skip the first phase that ignores the predictions and run the second phase that combines the offline and online schedules when the prediction error is not too large. Note, however, that a larger value of  $\lambda$  leads to a better competitive ratio when the predictions are incorrect. Hence, there is a general trade-off here.
The results in Section 3 and Section 4 require the quality cost function  $F$  to be monotone subadditive, which holds for the flow time problem and the problem with deadlines but might not hold for some other energy-efficient scheduling problems. The results in Section 4 require an additional smoothness assumption on  $F$ , which holds for the flow time problem but not for the problem with deadlines. Finally, we have only tested our algorithm on the three datasets described in Section 5.
Eric Balkanski was supported by NSF grants CCF-2210502 and IIS-2147361. Clifford Stein was supported in part by NSF grant CCF-2218677 and ONR grant ONR-13533312, and by the Wai T. Chang Chair in Industrial Engineering and Operations Research.
[18] Tak-Wah Lam, Lap-Kei Lee, Isaac KK To, and Prudence WH Wong. Speed scaling functions for flow time scheduling based on active job count. In European Symposium on Algorithms, pages 647–659. Springer, 2008.
[19] Silvio Lattanzi, Thomas Lavastida, Benjamin Moseley, and Sergei Vassilvitskii. Online scheduling via learned weights. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1859-1877, 2020.
[20] Alexander Lindermayr and Nicole Megow. Permutation predictions for non-clairvoyant scheduling. In Proceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures, pages 357-368, 2022.
[21] Alexander Lindermayr, Nicole Megow, and Martin Rapp. Speed-oblivious online scheduling: Knowing (precise) speeds is not necessary. arXiv preprint arXiv:2302.00985, 2023.
[22] Thodoris Lykouris and Sergei Vassilvitskii. Competitive caching with machine learned advice. J. ACM, 68(4):24:1-24:25, 2021. doi: 10.1145/3447579. URL https://doi.org/10.1145/3447579.
[23] Mohammad Mahdian, Hamid Nazerzadeh, and Amin Saberi. Allocating online advertisement space with unreliable estimates. In Proceedings of the 8th ACM conference on Electronic commerce, pages 288-294, 2007.
[24] Michael Mitzenmacher. Scheduling with Predictions and the Price of Misprediction. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), volume 151 of Leibniz International Proceedings in Informatics (LIPics), pages 14:1-14:18, 2020. ISBN 978-3-95977-134-4.
[25] Trevor Mudge. Power: A first-class architectural design constraint. Computer, 34(4):52-58, 2001.
[26] Pietro Panzarasa, Tore Opsahl, and Kathleen M Carley. Patterns and dynamics of users' behavior and interaction: Network analysis of an online community. Journal of the American Society for Information Science and Technology, 60(5):911-932, 2009.
[27] Manish Purohit, Zoya Svitkina, and Ravi Kumar. Improving online algorithms via ml predictions. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems. Curran Associates, Inc., 2018.
[28] Frances Yao, Alan Demers, and Scott Shenker. A scheduling model for reduced cpu energy. In Proceedings of IEEE 36th annual foundations of computer science, pages 374-382. IEEE, 1995.