This project develops a formal account of long-horizon deployment instability for frozen predictors under dynamic covariate drift. The central object is the Jacobian-velocity interaction \(J_f(X_t)\dot X_t\): environmental motion only becomes dangerous when it passes through directions where the model is locally steep.
The practical implication is to concentrate robustness where deployment drift is expected to travel. When that estimated subspace may be incomplete, anisotropic DTR adds a weaker orthogonal guardrail instead of smoothing every direction equally. The same geometry also yields a matched monitoring signal.
The empirical arc first verifies the time-domain bound, directional advantage, and conditional-risk remainder in controlled settings. The central field study then tests a strict predeployment drift basis on a 128-dimensional gas-sensor array, with Air Quality and Tetouan City power consumption providing complementary regression evidence.
In the rank-1 setting, a short bookkeeping proposition makes the monitoring proxy gap explicit: block averaging, residual drift, and angular misalignment determine how the hazard proxy departs from the leading low-rank term.
The repository also includes a dedicated proof-verification suite: symbolic identities, numerical stress tests, and cached-summary checks are collected into an HTML report that follows the same visual language as this site.
The paper studies frozen-model deployment under continuous covariate drift: the predictor is fixed, the environment evolves, and the object of interest is a deployment-risk trajectory rather than a single train-test shift gap.
The model \(f_\theta\) stays fixed after training while covariates evolve as \(X_t\). Instability is therefore temporal and geometric.
The same drift path can be benign for one model and harmful for another. What matters is whether motion aligns with locally steep predictor directions.
Once the drift geometry is estimated, robustness should be enforced in those directions instead of spread uniformly across the ambient space.
The formal story controls temporal volatility by derivative energy, bounds the model-mediated component by Jacobian-velocity energy, retains conditional-risk variation as an explicit remainder, and then specializes the controllable term to low-rank drift.
Large volatility requires large derivative energy somewhere along deployment time.
Deployment instability is driven by accumulated tangent amplification of realized motion.
In low-rank regimes, the leading term is directional Jacobian energy in the drift subspace.
Conditional-risk variation outside the score geometry remains explicit rather than being attributed to the predictor Jacobian.
In the rank-1 monitoring setting, block averaging contributes residual drift energy, while angular misalignment mixes aligned, orthogonal, and overlap energies. This bookkeeping result explains how the monitoring proxy departs from the leading low-rank term.
The mathematical claims are paired with an explicit verification layer: exact symbolic checks for the theorem chain, conditional-risk remainder, and hazard-score bookkeeping; numerical stress tests for the low-rank and anisotropic inequalities; and consistency checks against the cached experiment summaries.
SymPy checks the core algebra exactly: the Poincaré sharpness example, a deterministic equality case for the Jacobian-velocity theorem, the composition and conditional-risk remainder identities, the rank-1 hazard-score bookkeeping identity, and the Bernoulli cross-entropy derivative bound.
Some parts of the argument are more naturally checked by dense quadrature or randomized linear-algebra trials. The suite therefore stress-tests both the low-rank and anisotropic corollaries and a smooth expectation-based example of the full theorem chain.
Running the verifier writes a machine-readable JSON summary and an HTML report that reuses this site's fonts, palette, card layout, and MathJax setup.
proof_verification/ keeps the checks separate from the training scriptsThe experiments isolate the theorem, alignment, and remainder mechanisms before testing anisotropic tangent control on a strict predeployment gas-sensor benchmark. Air Quality and Tetouan then test transfer to regression.
The deployment path is identical in both panels. Only tangent geometry changes. Drift is dangerous only when it aligns with steep local directions of the predictor.
Each point is one trained model evaluated over the drifting synthetic path. DTR moves the cloud toward lower derivative energy and lower volatility.
At matched \(\lambda=0.03\), DTR improves on both the standard model and isotropic Jacobian regularization across derivative energy, volatility, directional energy, and terminal risk.
When the fixed label conditional varies along the covariate path, a small directional Jacobian need not imply stable risk. The explicit remainder restores the bound in every tested nonconstant-conditional run.
A five-gas classifier is trained on the first two chronological batches, tuned on batches 3–5, and frozen for deployment on batches 6–10. The rank-4 basis is estimated entirely before deployment from concentration-adjusted calibration motion in 128 sensor dimensions.
The CE-rule hybrid improves deployment CE, volatility, and terminal CE over pure DTR in every matched seed. The stability-rule hybrid gives the best mean CE and terminal CE while retaining substantially more accuracy than isotropic smoothing.
This site-only companion plot visualizes the Air Quality deployment and monitoring trajectories that support the real-data table. The real-data study freezes the regressor, estimates a target-orthogonal sensor-drift subspace from unlabeled deployment covariates, and evaluates blockwise MSE over 20 biweekly deployment blocks. With that more focused nuisance-drift subspace, validation-selected DTR improves deployment MSE and volatility against standard training.
This frozen-deployment study predicts Zone 1 load from weather and diffuse-flow covariates across six monthly deployment blocks in late 2017. Validation-selected DTR improves both deployment MSE and volatility relative to both baselines.
The repo is organized to be readable both as a paper companion and as a reproducible verification package: manuscript, cached summaries, figures, proof checks, and training scripts are all included.
scripts/Training, evaluation, reporting, and plotting code for the synthetic theorem benchmark, remainder stress test, directional ablation, regression studies, prospective subspaces, and monitoring ablations.benchmark_package/Gas-sensor and Tetouan benchmark assets, including the gas rank sweep, random-subspace controls, anisotropic hybrid sweep, matched-seed outputs, and dedicated runners.proof_verification/SymPy-based checks, numerical stress tests, and the generated HTML/JSON verification reports.figures/Rendered manuscript figures and cached CSV/JSON summaries consumed by the plotting scripts.data/Cached Air Quality data; the gas-sensor and Tetouan data are contained in their benchmark-package directories.Install the Python dependencies, regenerate the manuscript figures and reports, run the verification suite, and use the benchmark-package runners for the gas-sensor and Tetouan studies.