# Directional Conformal Residual Sets for Neural Dynamics - ID: 1593 - Canonical URL: https://synthcore.org/idea/1593/directional-conformal-residual-sets-for-neural-dynamics - API JSON: https://synthcore.org/api/idea/1593.json - API Markdown: https://synthcore.org/api/idea/1593.md - Verification status: beats_baseline - Source: [arXiv:2607.29344](https://arxiv.org/abs/2607.29344) - Category: regularization - Solves: stability, accuracy, generalization - ML areas: world-model, rl, regularization, training - Math tags: probability, statistics, control-theory, dynamical-systems - Ratings: usefulness 8/10; difficulty 4/10; novelty 6/10 ## Idea description Augment a neural dynamics model with a separately trained discrepancy predictor and calibrate an asymmetric conformal residual score. Use the resulting state- and input-dependent uncertainty set to reject, damp, or regularize neural rollouts when they leave a calibrated region, rather than treating all residual directions as equally uncertain. ## Mathematical statement Let the observed transition be $y_i=x_{i+1}$, the nominal neural predictor be $f_\theta(z_i)$ with $z_i=(x_i,u_i)$, and the learned signed discrepancy be $d_\phi(z_i)$ in the scalar-output case. Define the residual $e_i=y_i-f_\theta(z_i)$ and directional score $S_i=\max\{0,|e_i|-\operatorname{sign}(e_i)d_\phi(z_i)\}$. A positive discrepancy reduces the score for positive residuals and increases it for negative residuals; the opposite occurs for negative discrepancy. Given $N$ exchangeable calibration examples and miscoverage level $\alpha$, let $q$ be the $\lceil(N+1)(1-\alpha)\rceil$-th split-conformal order statistic of the scores. Define $\tilde q(z)=\max\{q,|d_\phi(z)|\}$ to guarantee that the interval contains zero, and return $C(z)=\{y:S(y-f_\theta(z),z)\leq\tilde q(z)\}$. Under exchangeability, the marginal coverage satisfies $\Pr[y\in C(z)]\geq1-\alpha$; the useful regime is where the directional score quantile is smaller than the symmetric residual quantile. ## Key formulas - $$e_i=y_i-f_\theta(z_i),\qquad S_i=\max\left\{0,\left|e_i\right|-\operatorname{sign}(e_i)d_\phi(z_i)\right\}.$$ - $$q=\operatorname{Quantile}_{\lceil(N+1)(1-\alpha)\rceil/N}\left(S_1,\ldots,S_N\right),\qquad \tilde q(z)=\max\{q,|d_\phi(z)|\}.$$ - $$C(z)=\left\{y:\max\left(0,|y-f_\theta(z)|-\operatorname{sign}(y-f_\theta(z))d_\phi(z)\right)\leq\tilde q(z)\right\},\qquad \Pr(y\in C(z))\geq1-\alpha.$$ - $$C(z)=[d_\phi(z)-\tilde q(z),\ d_\phi(z)+\tilde q(z)]\quad\text{when }\tilde q(z)\geq|d_\phi(z)|;\qquad |C(z)|=2\tilde q(z).$$ ## Implementation notes 1. Integration point: use this as an uncertainty head around a neural one-step dynamics model in a world model, model-based RL agent, or neural MPC system. Split transition data into three disjoint sets: fit $f_\theta$ on the first set, fit the discrepancy network $d_\phi(z)$ to predict the signed residual on the second set, and perform conformal calibration only on the third set. Freeze both networks and the calibration quantile during evaluation. For vector states, apply the scalar construction componentwise initially; a later implementation can replace absolute value by a Minkowski gauge for joint sets. 2. Pseudocode: compute $e_i=y_i-f_\theta(z_i)$ on calibration data; compute $S_i=\max(0,|e_i|-\operatorname{sign}(e_i)d_\phi(z_i))$; sort scores; select index $k=\lceil(N+1)(1-\alpha)\rceil$; at inference compute $d=d_\phi(z)$ and $q_z=\max(q,|d|)$; return the interval $[d-q_z,d+q_z]$. During rollout, calculate normalized violation $v_t=\max(0,|\hat x_{t+1}-f_\theta(z_t)|-q_{z_t})$ and either terminate the rollout, increase MPC constraint tightening, or multiply the training loss by $1+\lambda v_t$. 3. Computed from the paper: the directional score, order-statistic calibration, exchangeability-based coverage guarantee, and origin-containing interval construction. Estimated empirically: the discrepancy network, local coverage, violation frequency, and whether the calibration distribution remains exchangeable under policy shift. 4. First cheap experiment: train a one-step MLP dynamics model on CartPole, Pendulum, or a low-dimensional MuJoCo transition dataset. Compare symmetric split-conformal residual intervals with directional intervals at $\alpha=0.1$ over 10-step open-loop rollouts. Measure marginal coverage, conditional coverage in bins of $z$, average interval width, and rollout divergence. The quantitative prediction is coverage at least approximately $90\%$ on IID test transitions for both methods, while directional width should shrink in regions where the discrepancy predictor captures a persistent signed bias; specifically, if the symmetric residual 90th percentile is $q_0$ and the directional score percentile is $q_d$, the mean width should approach $2q_d$ and satisfy $q_d