# Large-Deviation Rare-Event Augmentation

- ID: 824
- Canonical URL: https://synthcore.org/idea/824/large-deviation-rare-event-augmentation
- API JSON: https://synthcore.org/api/idea/824.json
- API Markdown: https://synthcore.org/api/idea/824.md
- Verification status: unverified
- Source: [arXiv:2607.12666](https://arxiv.org/abs/2607.12666)
- Category: sampling
- Solves: sample-efficiency, accuracy, scalability
- ML areas: data-augmentation, world-model, training, rl
- Math tags: probability, stochastic-processes, optimization, measure-theory
- Ratings: usefulness 5/10; difficulty 5/10; novelty 7/10

## Idea description

Train a neural queue or event-sequence predictor using trajectories generated under an exponentially tilted scheduled-arrival law that makes rare overloads common. Reweight each tilted trajectory by its likelihood ratio, while optionally oversampling the rare-event subset to improve prediction of tail behavior.

## Mathematical statement

The paper considers scheduled-arrival paths A(t), where scheduled times are perturbed by independent unpunctuality variables, and the empirical mean of n independent copies, \(\bar A^{(n)}(t)=n^{-1}\sum_{i=1}^n A_i(t)\). The path-level LDP has speed n and good rate function \(I_T(\phi)=\sup_{m\ge1}\sup_{0<t_1<\cdots<t_m\le T} I_{t_1,\ldots,t_m}(\phi(t_1),\ldots,\phi(t_m))\), so a trajectory with probability roughly \(\exp[-nI_T(\phi)]\) can be sought by minimizing its rate over an overload set. For continuous \(\theta:[0,T]\to\mathbb R\) and \(\gamma\in\mathbb R\), the paper defines \(F_{\theta,\gamma}(\phi)=\int_0^T\theta(t)\phi(t)\,dt+\gamma\phi(T)\) and \(\Lambda_T(\theta,\gamma)=\log\mathbb E\exp(F_{\theta,\gamma}(A))\). Use the exponential tilt \(dP_{\theta,\gamma}/dP=\exp(F_{\theta,\gamma}(A)-\Lambda_T(\theta,\gamma))\). For a discretized path with bins k=1,...,K, let \(\theta_k\) be piecewise-constant values and \(\gamma\) a terminal coefficient; then the sampling weight for a path a is \(w(a)=\exp[-\sum_k\theta_k a_k\Delta t-\gamma a_K+\Lambda_T(\theta,\gamma)]\). The tilt parameters are optimized empirically so that the overload functional G(a), such as maximum workload or terminal queue length, exceeds a target threshold with moderate probability.

## Key formulas

- $$I_T(\phi)=\sup_{m\geq 1}\sup_{0<t_1<\cdots<t_m\leq T}I_{t_1,\dots,t_m}\bigl(\phi(t_1),\dots,\phi(t_m)\bigr).$$
- $$F_{\theta,\gamma}(\phi):=\int_{0}^{T}\theta(t)\phi(t)\,dt+\gamma\phi(T),\qquad \Lambda_T(\theta,\gamma):=\log\mathbb{E}\exp\left(\int_{0}^{T}\theta(t)A(t)\,dt+\gamma A(T)\right).$$
- $$\frac{dP_{\theta,\gamma}}{dP}(A)=\exp\left(F_{\theta,\gamma}(A)-\Lambda_T(\theta,\gamma)\right),\qquad w(A)=\exp\left(-F_{\theta,\gamma}(A)+\Lambda_T(\theta,\gamma)\right).$$
- $$\mathcal L(\psi)=\frac{1}{M}\sum_{j=1}^{M}w(a^{(j)})\,\ell\bigl(f_\psi(a^{(j)}),y^{(j)}\bigr),\qquad a^{(j)}\sim P_{\theta,\gamma}.$$

## Implementation notes

(1) Integrate at the data-generation layer for a neural model that predicts workload, queue length, overflow probability, or future event counts from scheduled-arrival histories. Represent each trajectory on a grid of K time bins. Generate a nominal trajectory by sampling scheduled times s_r and independent unpunctualities U_r, then setting A_k=\sum_r 1\{s_r+U_r\le t_k\}. Compute workload recursively with the chosen service model. (2) Add a cross-entropy tilting loop. Initialize piecewise-constant tilt parameters theta[1:K] and terminal gamma to zero. Repeatedly sample B trajectories under the current tilted unpunctuality law, compute overload scores G(A), retain the top rho fraction, and update theta and gamma by gradient ascent on the elite mean of F_{theta,gamma}(A)-Lambda_T(theta,gamma). Estimate Lambda_T by log-mean-exp of F over nominal samples, or by the analytic moment-generating function of the unpunctuality distribution when available. Use theta <- theta + eta*(mean_elite[A_k]-mean_nominal[A_k]) and gamma <- gamma + eta*(mean_elite[A_K]-mean_nominal[A_K]); clip parameters to avoid degenerate sampling. (3) The paper supplies the path functional, log-Laplace transform, exponential tilt, and rate interpretation. Estimate finite-dimensional moments and overload-conditioned targets empirically. (4) For training, mix nominal and tilted trajectories, retaining importance weights w=exp(-F+Lambda) in the loss; alternatively reserve a fixed fraction of unweighted rare-event examples for an auxiliary tail loss. First test on a 10-minute synthetic appointment queue with 50-200 scheduled arrivals, an MLP or GRU baseline, and a 1% overload event. Compare nominal-only training, naive oversampling, and LDP-guided tilting at equal simulator calls. Success is lower Brier score and recall on held-out 0.1%-tail overloads, or at least 2x lower tail prediction error at equal data-generation cost, without degrading nominal calibration.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.