Training Loss for Cost Predictors

• Training Loss for Cost Predictors is a framework that designs loss functions to align predicted costs with actual operational or decision-making outcomes in diverse applications.
• The approach uses techniques like regret-reweighted MSE and bilevel optimization to directly minimize decision regret and integrate cost-sensitive metrics.
• Empirical findings indicate that tailored, task-aware losses can reduce operational costs and improve calibration in cost prediction models.

A training loss for cost predictors is a formulation used to fit models that predict future or realized costs, typically with the goal of optimizing downstream decisions or directly predicting incurred losses under task- or resource-sensitive conditions. This family of losses spans a broad set of applications in contextual optimization, reinforcement learning (RL), neural architecture search (NAS), autoregressive generation, calibration auditing, and value-based forecasting, with key technical differences depending on the problem’s structure and the eventual cost metric of interest.

1. Conceptual Foundations and Formulation

Cost predictors are regression or scoring models trained to forecast not merely ground-truth labels but the actual or anticipated cost that a prediction will induce in a real-world pipeline. The design of the training loss depends fundamentally on:

• The structure of the downstream cost function (linear, piecewise-linear, combinatorial, stochastic, etc.).
• The level of alignment required between the predictive loss and the cost incurred by automated decisions, policies, or resource allocations.
• Theoretical desiderata such as multicalibration or small-cost sample complexity.

The canonical form, exemplified in neural architecture search and direct loss prediction, is the mean-squared error (MSE) loss between predicted and true costs:

$$L(\theta) = \frac{1}{|D|} \sum_{(x_i, y_i) \in D} (f(x_i;\theta) - y_i)^2$$

where $y_i$ represents the measured cost variable, possibly abstract (e.g., inference latency, realized regret, resource allocation loss) (Akhauri et al., 2023, Gollakota et al., 27 Feb 2025).

However, in contexts where costs are intrinsically policy-embedded (e.g., contextual LPs, batch RL, or operational forecasting), the training loss often becomes a surrogate objective directly designed for decision-aware risk or task-aligned loss (Lawless et al., 2022, Zhang et al., 2023, Ayoub et al., 2024).

2. Task-Aware and Decision-Regret Losses

In stochastic linear optimization with contextual costs, training losses reweight prediction error by the empirical decision-regret associated with each example (Lawless et al., 2022). Given a cost vector predictor $\hat{c}(x)$ and the regret

$$\mathrm{Regret}(\hat{c}, c) := c^T[z^*(\hat{c}) - z^*(c)]$$

where $z^*(\cdot)$ is the optimal decision under the given cost vector, the reweighted MSE objective is

$$L(f) = \mathbb{E}_{x,c}\left[w(x,c)\Vert f(x)-c\Vert^2\right]$$

$$w(x, c) := c^T[z^*(\hat{c}_{\mathrm{pilot}}(x)) - z^*(c)]$$

To mitigate zero-inflated weights, a mixture with unweighted MSE is employed:

$$L_\nu(f) = \mathbb{E}\left[(1-\nu)\Vert f(x)-c\Vert^2 + \nu w(x,c) \Vert f(x)-c\Vert^2\right]$$

with $\nu \in [0,1]$ (Lawless et al., 2022).

This surrogate loss preserves convexity and tractability, and empirical results demonstrate substantial regret reduction over standard "predict-then-optimize" MSE predictors when the model is misspecified.

3. Surrogate and Bilevel Losses for Operational Costs

When cost is determined through a subsequent operational decision (e.g., in power dispatch), bilevel programs naturally lead to a training loss that minimizes the downstream operational cost, not prediction error:

$$\min_{\theta} \frac{1}{M}\sum_{m=1}^M \left\{ c_D^T x_{m,D}^*(\hat{y}_m, l_m) + c_R^T z_{m,R}^*(\hat{y}_m, y_m) \right\}$$

where $\hat{y}_m = g(s_m; \theta)$ and the lower-level decision variables $x^*_{m,D}, z^*_{m,R}$ are solutions to LPs parameterized by the forecast. By applying multiparametric programming, the training problem is collapsed to empirical minimization of a piecewise-linear function of the forecast output, allowing efficient training of differentiable predictors with direct operational alignment (Zhang et al., 2023).

This value-oriented loss produces models with lower actual operational costs compared to MSE-trained models, confirming the benefit of direct cost alignment in practical deployments.

4. Losses for Cost Prediction in Sequence and Structured Prediction

In learning-to-search (L2S) for autoregressive structured tasks, the cost predictor models the expected downstream metric (e.g., BLEU, Kendall-τ distance) conditional on each next-token choice. Losses for cost predictors in this setting include:

• KL divergence to cost-induced target distribution:

$$L_{KL}^{(t)} = -\sum_{i=1}^k p_t^{cost}(i)\log p_t^{model}(i)$$

where $p_t^{cost}(i)$ is the softmax over negated true costs at a fixed temperature.

• Ordering-based KL and ListMLE losses: focus learning on cost-rank, using geometric decay or explicit listwise ranking of candidates.
• Pairwise hinge losses: enforce that lower-cost tokens receive higher model scores by a margin (Saparina et al., 2019).

These losses explicitly encode the structure of the test metric and provide empirical gains, particularly when combined with alignment-based reference rollouts to define cost targets.

5. Calibration, Loss Prediction, and Surrogate Training Losses

A distinct but increasingly relevant use of cost predictors is the task of "loss prediction": training a secondary model to forecast the actual loss a predictor will incur on an instance. The central result shows that the MSE objective for the loss predictor directly measures failures in multicalibration of the base model:

$$\mathcal{R}(g) = \mathbb{E}_{(x, y)}\left[ (g(\phi(p, x)) - \ell(y, p(x)))^2 \right]$$

and that any reduction of this risk below the baseline "self-entropy predictor" constitutes a certificate for calibration or fairness failure. Empirical results establish a strong correspondence between loss prediction advantage and multicalibration error measured via group-wise smoothed expected calibration error (Gollakota et al., 27 Feb 2025).

6. Reinforcement Learning: Loss Selection Induces Cost-Dependent Sample Complexity

In batch RL, the choice of training loss for Q-function predictors dictates the scaling of sample complexity bounds with the optimal achievable cost. Specifically:

• FQI with log-loss ($\ell_{\log}$) achieves an $O(1/n)$ rate in the small-cost regime due to variance reduction at the optimal policy, with sample complexity scaling as $\sim \bar v^*/(\epsilon^2 (1-\gamma)^4)$ (Ayoub et al., 2024).
• FQI with squared loss lacks this instance-dependent benefit, as its worst-case variance remains constant regardless of the optimal cost.

This illustrates the critical role played by the choice of the training loss in controlling the sample efficiency of cost-predicting RL algorithms, especially when optimal policies render the cost vanishingly small.

7. Practical Methodologies and Empirical Considerations

The following summarizes practical recommendations and empirical behaviors across contexts:

• In NAS and standard regression settings, MSE remains the default for training both cost and latency predictors. Transfer and few-shot adaptation gains arise through better input encoding, not loss modification (Akhauri et al., 2023).
• For task-aware or downstream-cost loss, one should prefer surrogate objectives directly aligned to ultimate decision regret or operational cost, using regret-reweighted MSE, bilevel-optimized piecewise-linear losses, or end-to-end differentiable proxies as technically appropriate (Lawless et al., 2022, Zhang et al., 2023).
• In structured sequence prediction, the alignment between cost predictor training loss and evaluation metric is pivotal. Ranking-based and order-focused losses saturate or exceed performance of strict KL or MLE, and are robust to cost scale and candidate set size (Saparina et al., 2019).
• Empirical gains from loss predictors are conditional on capacity, input feature design, and alignment of the loss with the real or proxy cost; their success in auditing or improving predictors' calibration hinges on the multicalibration error of the base predictor (Gollakota et al., 27 Feb 2025).

Summary Table: Training Losses for Cost Predictors by Context

Application	Loss Formulation	Empirical/Practical Note
Contextual Linear Opt.	Regret-reweighted MSE	Improves regret under misspecification (Lawless et al., 2022)
RL (FQI)	Log-loss ($\ell_{\log}$), MSE	Log-loss enables small-cost bounds (Ayoub et al., 2024)
Value-oriented Forecasting	Piecewise-linear loss (bilevel)	Directly minimizes operational cost (Zhang et al., 2023)
NAS Cost/Latency	MSE (with input engineering)	Few-shot/sample-efficient prediction with no loss modification (Akhauri et al., 2023)
Autoregressive/Seq. Prediction	KL, Order-KL, ListMLE, hinge loss	Metric-aligned training for downstream cost; robust via ranking losses (Saparina et al., 2019)
Calibration Auditing	MSE (loss prediction)	Empirically tracks multicalibration error (Gollakota et al., 27 Feb 2025)

• "A Note on Task-Aware Loss via Reweighing Prediction Loss by Decision-Regret" (Lawless et al., 2022)
• "Multi-Predict: Few Shot Predictors For Efficient Neural Architecture Search" (Akhauri et al., 2023)
• "Switching the Loss Reduces the Cost in Batch (Offline) Reinforcement Learning" (Ayoub et al., 2024)
• "Deriving Loss Function for Value-oriented Renewable Energy Forecasting" (Zhang et al., 2023)
• "Cost-Sensitive Training for Autoregressive Models" (Saparina et al., 2019)
• "When does a predictor know its own loss?" (Gollakota et al., 27 Feb 2025)