# Midpoint Consistent-Gain Cancellation

- ID: 2251
- Canonical URL: https://synthcore.org/idea/2251/midpoint-consistent-gain-cancellation
- API JSON: https://synthcore.org/api/idea/2251.json
- API Markdown: https://synthcore.org/api/idea/2251.md
- Verification status: unverified
- Source: [arXiv:2608.13651](https://arxiv.org/abs/2608.13651)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: rnn, ssm, world-model
- Math tags: control-theory, dynamical-systems, optimization, linear-algebra
- Ratings: usefulness 6/10; difficulty 6/10; novelty 8/10

## Idea description

Augment a recurrent or diagonal state-space neural block with online interval estimates for persistent transition gains. At every step, intersect the current parameter interval with the set compatible with the latest transition and bounded residual, then use its midpoint for certainty-equivalent cancellation. The method learns passively and avoids the transient spikes caused by exploratory probing or endpoint selection.

## Mathematical statement

The source system is x at time t plus 1 equals a times x at time t plus u at time t plus w at time t, where the unknown constant gain a belongs to the interval from minus Delta to Delta and the disturbance satisfies absolute value of w at time t at most 1. For nonzero x at time t, the observation implies that a belongs to the interval with endpoints (x at time t plus 1 minus u at time t minus 1) divided by x at time t and (x at time t plus 1 minus u at time t plus 1) divided by x at time t; the endpoints must be sorted if x is negative. The consistent set is the intersection of this interval with the previous set. If the current set is A at time t equals [l at time t, r at time t], its midpoint is mu at time t equals (l at time t plus r at time t) divided by 2. The controller uses u at time t equals minus mu at time t times x at time t. The paper proves that the exact worst-case peak is 1 plus Delta. For a neural state coordinate z, use z at time t plus 1 equals a times z at time t plus q at time t plus e at time t, where q is the learned input contribution and the residual e is bounded by epsilon. Estimate a with the same interval update and cancel the estimated recurrent drift.

## Key formulas

- $x_{t+1}=a x_t+u_t+w_t,\quad a\in[-\Delta,\Delta],\quad |w_t|\leq 1$
- $A_{t+1}=A_t\cap\left[\frac{x_{t+1}-u_t-1}{x_t},\frac{x_{t+1}-u_t+1}{x_t}\right],\quad x_t\neq 0$
- $A_t=[\ell_t,r_t],\qquad \mu_t=\frac{\ell_t+r_t}{2},\qquad u_t=-\mu_t x_t$
- $\gamma^\star(\Delta)=\inf_K\sup_{a,w}\|x\|_\infty=1+\Delta$

## Implementation notes

Use a small one-dimensional RNN or diagonal state-space model as the first integration point. Decompose each state transition as z[t+1,i] = a[i] z[t,i] + q[t,i] + e[t,i], where z is the hidden state, q is the learned input and nonlinear contribution, a[i] is an unknown persistent gain, and the residual e has a known or conservatively estimated bound epsilon[i]. Initialize an interval A[i] = [-Delta[i], Delta[i]]. At each step compute mu[i] = (left[i] + right[i]) / 2 and use the transition q[t,i] = learned_input[t,i] - mu[i] z[t,i], thereby canceling the estimated recurrent drift. After z[t+1] is available, and whenever absolute z[t,i] exceeds a threshold tau, construct the candidate interval with endpoints (z[t+1,i] - learned_input[t,i] - epsilon[i]) / z[t,i] and (z[t+1,i] - learned_input[t,i] + epsilon[i]) / z[t,i]. Sort the endpoints, intersect with A[i], and skip the update if the intersection is empty. The paper supplies the interval logic, midpoint policy, and exact peak law; the implementation must estimate Delta and epsilon from calibration data. First test a scalar recurrence with fixed random gain a, unit-bounded adversarial noise, and Delta swept over 0.1, 0.5, 1, 2, and 5. Compare midpoint selection with zero-gain cancellation, random exploration, and endpoint selection. Then test a tanh RNN on noisy sine-wave prediction. The predicted signature is a worst hidden-state peak approximately equal to 1 plus Delta in the isolated scalar experiment, with slope one versus Delta and a midpoint peak lower than endpoint selection. Agreement within 20 percent is the initial acceptance criterion.

## Disclaimer

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