Scheduling-Aware Loss (SAL)

• Scheduling-Aware Loss (SAL) is a family of loss functions that integrate scheduling information into training objectives to reflect asymmetric operational or statistical costs.
• SAL adapts its formulation across contexts, from profit-asymmetric forecasting in cloud-edge platforms to dynamic weighting in multi-objective and multimodal frameworks.
• Implementations of SAL yield practical benefits such as reduced SLA violations, more stable multi-objective optimization, and enhanced performance in federated and contrastive learning.

Scheduling-Aware Loss (SAL) is a label used in several recent arXiv works for objective functions that embed scheduling structure into learning. In one line of work, SAL is an asymmetric profit-impact loss for time-series forecasting in Crowdsourced Cloud-Edge Platforms (CCPs); in another, it denotes online scheduling of multipliers in a surrogate multi-objective penalty; in a third, it is the expected excess loss used to optimize age-aware differentially-private federated learning (DP-FL); and in a fourth, it is a variance-driven weighting rule for bidirectional contrastive loss in multimodal alignment. This suggests that SAL is not a single canonical loss, but a family of scheduling-coupled objective constructions whose common purpose is to align optimization with operational or statistical asymmetries (Zhang et al., 18 Aug 2025, Sun et al., 2024, Lin et al., 2024, Pillai, 5 Mar 2025).

1. Terminological scope and domain-specific meanings

The term SAL has been introduced in distinct problem settings, with materially different mathematical definitions and optimization roles. The shared element is not a uniform formula, but the use of scheduling information, scheduling variables, or scheduled loss weights as part of the learning objective.

Setting	SAL formulation	Scheduling element
CCP load forecasting	Piecewise asymmetric loss in terms of revenue $R$, penalty $P$, and cost $C$	Supports Predict-then-Schedule decisions (Zhang et al., 18 Aug 2025)
Multi-objective optimization	Time-varying scalarized loss $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$	Online loss-landscape scheduling via multipliers and setpoints (Sun et al., 2024)
DP federated learning	Expected excess loss $$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$	Schedules client data-collection ages and privacy parameters (Lin et al., 2024)
Multimodal alignment	Weighted bidirectional contrastive loss $L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$	Schedules loss weights from similarity variance (Pillai, 5 Mar 2025)

A plausible implication is that SAL functions as a problem-class descriptor rather than a standardized loss family in the narrow sense used for, for example, cross-entropy or InfoNCE. Consequently, interpretation requires attention to the surrounding application, the control variables being scheduled, and the downstream objective being protected.

2. Profit-asymmetric SAL in Crowdsourced Cloud-Edge Platforms

In "HRS: Hybrid Representation Framework with Scheduling Awareness for Time Series Forecasting in Crowdsourced Cloud-Edge Platforms" (Zhang et al., 18 Aug 2025), SAL formalizes the asymmetric economic effect of forecasting error in CCPs. Let $Y\in\mathbb{R}^T$ be the ground-truth demand over a forecast horizon of length $T$, and let $\hat{Y}\in\mathbb{R}^T$ be the prediction. SAL is defined sample-wise as

$$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$

Here $P$0 is the unit revenue coefficient, $P$1 is the fixed penalty for an SLA violation caused by under-provisioning, and $P$2 is the per-unit resource cost of over-provisioning. The paper states that in most CCP deployments one finds $P$3, reflecting that SLA fines dominate over pure resource expenses.

The derivation is explicitly operational. Classical losses such as $P$4 treat overshoot and undershoot symmetrically, whereas CCP forecasting does not. Under-forecasting by $P$5 forces routers or servers into overload, drops packets or rejects sessions, forfeits revenue $P$6, and incurs the SLA penalty $P$7. Over-forecasting by $P$8 ties up spare CPU, memory, or bandwidth and costs $P$9. The resulting objective measures loss directly in unified profit terms.

The training objective averages this quantity over the dataset:

$C$0

The paper adds standard regularization such as weight decay, but states that no auxiliary forecasting loss is needed; SAL alone guides the model to err on the side of safe over-provisioning.

A central tuning quantity is the Under-vs-Over loss ratio,

$C$1

A large ratio heavily penalizes under-forecasting and drives safe over-provisioning when perfect accuracy is unattainable. For streaming traffic forecasting on ECT1/ECT2, the reported setting is $C$2, $C$3, and $C$4, yielding $C$5. The stated guideline is to estimate average revenue per unit demand, average penalty, and average idle-capacity cost from historical logs; if persistent under-provisioning is observed, increase $C$6 or $C$7, and if over-provisioning becomes too expensive, increase $C$8. Ratios between $C$9 and $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$0 are reported as effective.

The computation is deliberately simple. For a batch of size $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$1 and horizon $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$2, one iterates over sample $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$3 and time step $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$4, computes $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$5, adds $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$6 when $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$7, adds $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$8 when $$\mathcal{L}(\theta;\lambda(t))=\sum_{i=1}^m \lambda_i(t)\ell_i(\theta)$$9, and averages to obtain $$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$0 before back-propagation. The paper notes that this branch-wise computation can be vectorized efficiently in any deep-learning library.

Empirically, the paper evaluates models under Average Profit Loss (APL), defined as $$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$1 on held-out data. Training with SAL reduces APL by $$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$2 on average across four datasets. Figure 1 shows that SAL-trained models make far fewer under-forecasts; across all horizons, HRS + SAL cuts SLA violations by $$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$3–$$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$4 compared to baseline models trained with MSE. In ablation, training HRS with MSE and then manually adding an offset $$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$5 to all predictions does not match the APL of directly trained SAL, which the authors interpret as evidence that SAL forces the network to internalize the asymmetric cost trade-off. As the ratio $$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$6 grows from $$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$7 to $$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$8, the proportion of under-predictions falls smoothly and stabilizes once the ratio exceeds approximately $$E[L(W,D(t_{\mathrm{agg}}))-L(w^*,D(t_{\mathrm{agg}}))]$$9. In a one-month simulation on $L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$0 servers, HRS + SAL reduces total profit loss by $L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$1 compared to MSE-trained baselines, and the paper’s headline experimental summary reports reduction of SLA violation rates by $L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$2 and total profit loss by $L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$3.

3. SAL as online loss-landscape scheduling in multi-objective optimization

In "M-HOF-Opt: Multi-Objective Hierarchical Output Feedback Optimization via Multiplier Induced Loss Landscape Scheduling" (Sun et al., 2024), SAL is described as a framework also referred to as "M-HOF-Opt" in the paper. Here SAL is not a piecewise penalty on prediction error. Instead, it is a controlled mechanism for scheduling the weights of multiple loss terms during optimization.

The formulation begins with a probabilistic graphical model over model parameters $L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$4 and multipliers $L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$5. At iteration $L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$6, the current multipliers configure the low-level optimizer through

$L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$7

while the observed loss vector is

$L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$8

The next multiplier vector is generated by a controller

$L_{\mathrm{total}(t)}=w_I(t)L_{I2T}+w_T(t)L_{T2I}$9

To promote joint descent of all $Y\in\mathbb{R}^T$0 losses, the paper defines a hypervolume-based likelihood

$Y\in\mathbb{R}^T$1

where $Y\in\mathbb{R}^T$2 is the hypervolume of the region dominated by $Y\in\mathbb{R}^T$3 with respect to a reference point $Y\in\mathbb{R}^T$4.

Operationally, the multi-objective problem

$Y\in\mathbb{R}^T$5

is replaced at each epoch by the scalarized surrogate

$Y\in\mathbb{R}^T$6

The defining feature is that $Y\in\mathbb{R}^T$7 is scheduled online rather than fixed in advance. The high-level controller sets a bound vector $Y\in\mathbb{R}^T$8, and the optimization is interpreted as approximately solving

$Y\in\mathbb{R}^T$9

Whenever $T$0 component-wise, the bound is shrunk according to

$T$1

for each satisfied constraint, so the setpoint moves inward according to Pareto dominance.

The low-level controller is PI-like. With error signal

$T$2

the multiplier update is

$T$3

The exponential preserves positivity, and clipping maintains a safe numerical range. Algorithm 1 interleaves evaluation of $T$4, computation of $T$5, multiplier updates, one epoch of low-level training on the weighted loss, and conditional setpoint shrinkage.

The empirical case study uses the Domain-Invariant Variational Autoencoder on the PACS domain generalization benchmark, leave-one-out on the sketch domain. Figure 2 shows that each regularizer is driven toward its setpoint and that the corresponding multiplier increases whenever the regularizer exceeds the setpoint and decays once it falls below. Figure 3 reports more consistent sketch-domain accuracy, approximately $T$6–$T$7, across a wide range of controller gains and initial $T$8; warm-up and fixed-multiplier baselines can drop to approximately $T$9 when the multiplier choice is suboptimal. The paper’s interpretation is that SAL automatically corrects the weighting on each loss term without the excessive memory requirements and extra computational burden of existing multi-objective deep learning methods.

4. Age-dependent SAL in differently-private federated learning

In "Age Aware Scheduling for Differentially-Private Federated Learning" (Lin et al., 2024), SAL is defined as an expected excess loss for a federated system with time-varying databases, age-dependent privacy, and scheduled local data collection times. Let $\hat{Y}\in\mathbb{R}^T$0 be the number of clients, $\hat{Y}\in\mathbb{R}^T$1 the global aggregation time, and $\hat{Y}\in\mathbb{R}^T$2 the scheduled local-data collection time for client $\hat{Y}\in\mathbb{R}^T$3. The aggregated model is

$\hat{Y}\in\mathbb{R}^T$4

where

$\hat{Y}\in\mathbb{R}^T$5

Under the Laplace mechanism, the server observes

$\hat{Y}\in\mathbb{R}^T$6

with $\hat{Y}\in\mathbb{R}^T$7. The SAL is then

$\hat{Y}\in\mathbb{R}^T$8

where $\hat{Y}\in\mathbb{R}^T$9 is the ERM optimum trained on fresh data without DP constraints.

Theorem 1 upper-bounds this quantity by a sum involving clientwise mutual information terms, a term $$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$0, and the expected excess-risk term $$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$1, with $$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$2. The bound depends on age through $$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$3, on privacy through the noise parameters $$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$4, and on the learned model through aggregation and excess risk. The paper’s derivation uses a Donsker-Varadhan variational representation of KL-divergence, cumulant-generating-function bounds, and data-processing inequalities.

The optimization problem chooses the collection times and client privacy parameters to minimize SAL subject to an age-dependent DP constraint:

$$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$5

subject to

$$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$6

Here $$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$7 is the total-variation mixing distance for client $$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$8 over age $$SAL(Y,\hat{Y}) = \begin{cases} R\cdot(Y-\hat{Y}) + P & \text{if } \hat{Y}<Y,\ 0 & \text{if } \hat{Y}=Y,\ C\cdot(\hat{Y}-Y) & \text{if } \hat{Y}>Y. \end{cases}$$9. The rationale is explicit: by delaying collection and increasing age, $P$00 decreases, allowing smaller $P$01 and hence smaller noise for fixed global $P$02; however, greater age also degrades freshness and can increase the utility loss captured by SAL.

The scheduler evaluates candidate age vectors $P$03, computes $P$04, sets

$P$05

computes the Laplace-noise parameter $P$06, and ranks schedules using the SAL upper bound. In each round, client $P$07 collects data at $P$08, trains a local model $P$09, adds Laplace noise, and uploads $P$10 to the server.

The simulation uses three clients, each with a 4-state birth-death Markov model for household electricity consumption, local sample size $P$11, ERM with MSE loss, quantized consumption to 4 bins, and transition probabilities $P$12. Six schemes are compared: random scheduling with constant or adaptive noise, proposed scheduling with constant or adaptive noise, and optimal scheduling via brute-force simulation with constant or adaptive noise. The main reported result is that proposed scheduling plus adaptive noise achieves the lowest loss, and that the theoretical bound accurately predicts the performance gap and guides scheduling.

5. Variance-aware SAL for multimodal alignment

In "Variance-Aware Loss Scheduling for Multimodal Alignment in Low-Data Settings" (Pillai, 5 Mar 2025), SAL denotes a dynamic reweighting scheme for the two directional terms of the symmetric contrastive objective. On a batch of $P$13 image-text pairs $P$14, the paper defines

$P$15

and

$P$16

where $P$17 is cosine similarity and the reported temperature is $P$18. The total loss at epoch $P$19 is

$P$20

The scheduling signal is derived from the variance of positive cosine similarities. For the image-to-text side,

$P$21

and

$P$22

An analogous definition is used for the text-to-image side. The paper interprets low variance as similarities being bunched together, hence more uncertainty, and high variance as greater separation between true and false matches.

Raw weights are computed as

$P$23

so that the direction with lower variance receives larger weight. To reduce jitter, the variances are smoothed with an EMA,

$P$24

with example smoothing coefficient $P$25, and the per-epoch weight change is clipped to within $P$26 of the previous weights, with $P$27 in the reported experiments.

The training loop initializes $P$28, computes embeddings and cosine similarities for each minibatch, backpropagates the weighted contrastive loss, then updates positive-pair variances and the scheduled weights at the end of each epoch. Reported hyperparameters are batch size $P$29, learning rate $P$30 with Adam, embedding dimension $P$31, and number of epochs $P$32. Image features are from a frozen ResNet-50 penultimate layer, the text encoder is a GRU with 256-D output and fine-tuned GloVe word embeddings, and Flickr8k standard splits of $P$33k/$P$34k/$P$35k are used.

On Flickr8k, averaged over two runs, the fixed-weight baseline obtains $P$36 and $P$37 for image-to-text R@1 and R@5 and $P$38 and $P$39 for text-to-image R@1 and R@5, whereas variance-aware SAL reports $P$40, $P$41, $P$42, and $P$43, respectively. The paper states that SAL yields an absolute gain of approximately $P$44 points in image-to-text R@1, approximately $P$45 relative. It also reports that entropy-based adaptive weighting and cosine-spread adaptive weighting are worse than the variance-aware schedule on the reported metrics, that t-SNE visualizations show tighter image-caption clusters and clearer boundaries, and that under noisy-data stress tests with $P$46 corrupted captions or images, SAL’s R@5 degrades by approximately $P$47 relative versus approximately $P$48 for the fixed baseline.

6. Cross-cutting interpretation and points of distinction

Across these four uses, SAL always changes the training criterion so that optimization reflects an asymmetry not captured by a static generic loss. In CCP forecasting, the asymmetry is economic and is encoded directly through $P$49, $P$50, and $P$51. In M-HOF-Opt, the asymmetry is multi-objective and is handled by output-feedback scheduling of multipliers and setpoints. In age-aware DP-FL, the asymmetry is a three-way interaction among age, accuracy, and privacy, expressed through expected excess loss and an age-dependent privacy constraint. In multimodal alignment, the asymmetry is statistical and directional, with the more uncertain contrastive direction receiving more weight (Zhang et al., 18 Aug 2025, Sun et al., 2024, Lin et al., 2024, Pillai, 5 Mar 2025).

Several distinctions follow directly. First, “scheduling” does not always mean resource scheduling in the systems sense. It can also mean online scheduling of loss multipliers, client data-collection ages, or direction-specific contrastive weights. Second, SAL is not always a scalar penalty on raw prediction error: it may be an expected excess-loss functional, a surrogate upper-bound objective, or a dynamically weighted aggregate of pre-existing losses. Third, post-hoc correction is not equivalent to direct SAL optimization in at least one setting: in the CCP study, manually adding a bias $P$52 to MSE-trained predictions does not match the APL of direct SAL training.

This suggests a broad conceptual taxonomy. One branch of SAL methods embeds downstream operational cost directly into the loss, as in forecasting for CCP scheduling. A second branch uses feedback or statistics to schedule how existing loss terms are weighted during training, as in multi-objective optimization and multimodal contrastive learning. A third branch uses SAL as an analysis and optimization target over scheduling decisions external to gradient descent, as in age-aware DP-FL. The term therefore denotes a design philosophy—align the objective with the consequences of when, how, or in what proportion decisions are scheduled—rather than a single transferable equation.