# Reciprocal Feasibility-Preserving Optimizer

- ID: 2840
- Canonical URL: https://synthcore.org/idea/2840/reciprocal-feasibility-preserving-optimizer
- API JSON: https://synthcore.org/api/idea/2840.json
- API Markdown: https://synthcore.org/api/idea/2840.md
- Verification status: unverified
- Source: [arXiv:2608.29628](https://arxiv.org/abs/2608.29628)
- Category: optimization
- Solves: stability, accuracy, generalization
- ML areas: optimizer, training, rl, regularization
- Math tags: optimization, dynamical-systems, control-theory, convex-analysis
- Ratings: usefulness 6/10; difficulty 5/10; novelty 5/10

## Idea description

Replace ordinary parameter updates for a constrained neural network with an annealed reciprocal-manifold flow. Each differentiable inequality constraint remains strictly satisfied during the optimization trajectory, avoiding projection or a per-step quadratic program. This is most useful for safety-critical policy learning, bounded network outputs, parameter-budget constraints, or training with explicit robustness inequalities.

## Mathematical statement

The paper defines the reciprocal multiplier manifold for an inequality constraint \(g(x)<0\) as \(\lambda(x)=-k/g(x)\), where \(x\) is the optimization state, \(g:\mathbb{R}^d\to\mathbb{R}\) is a differentiable constraint function, \(k>0\) is the annealed barrier parameter, and \(\lambda>0\) is its associated multiplier. The equivalent logarithmic-barrier objective is \(F_k(x)=f(x)-k\log(-g(x))\), where \(f\) is the neural-network loss; its gradient flow is \(\dot{x}=-\nabla f(x)-\lambda\nabla g(x)\). For multiple constraints \(g_i(x)<0\), use \(\lambda_i=-k/g_i(x)\) and \(\dot{x}=-\nabla f(x)-\sum_i\lambda_i\nabla g_i(x)\). The exact tangent multiplier dynamics induced by the manifold are \(\dot{\lambda}_i=-\dot{k}/g_i+(k/g_i^2)\nabla g_i(x)^\top\dot{x}\). The reciprocal relation supplies the repulsive force that diverges near a boundary; annealing \(k\to0\) reduces barrier bias and targets the original constrained optimum. In code, the algebraic relation \(\lambda_i=-k/g_i\) is safer than independently integrating \(\lambda_i\), while the differentiated update is useful for testing the invariant-manifold dynamics.

## Key formulas

- $$\lambda(x)=-\frac{k}{g(x)}$$
- $$F_k(x)=f(x)-k\log\bigl(-g(x)\bigr),\qquad g(x)<0$$
- $$\dot{x}=-\nabla f(x)-\sum_{i=1}^{m}\lambda_i\nabla g_i(x),\qquad \lambda_i=-\frac{k}{g_i(x)}$$
- $$\dot{\lambda}_i=-\frac{\dot{k}}{g_i(x)}+\frac{k}{g_i(x)^2}\nabla g_i(x)^\top\dot{x}$$

## Implementation notes

1. Integration point: modify the optimizer update applied to neural-network parameters \(\theta\), or to policy parameters in constrained RL. Define a batch of differentiable inequalities \(g_i(\theta)<0\); examples include \(\|\theta\|_2^2-B<0\), an estimated worst-case loss minus a safety threshold, action-limit violations, or a differentiable FLOP/memory budget. Start from a strictly feasible initialization. 2. Pseudocode: set \(k=k_0\), compute the task loss \(L(\theta)\), constraints \(g_i\), and gradients; set \(\lambda_i=-k/(g_i+\epsilon)\); form \(v=-\nabla_\theta L-\sum_i\lambda_i\nabla_\theta g_i\); choose an adaptive step \(h\), update \(\theta\leftarrow\theta+h v\), then recompute every constraint. Use \(k\leftarrow\max(k_{\min},k\exp(-h/\tau))\). Before accepting a step, cap \(h\) using \(h\le \eta\min_i[-g_i/(|\nabla g_i^\top v|+\epsilon)]\), with \(0<\eta<1\); reject and halve \(h\) if any constraint is nonnegative. 3. The reciprocal multiplier, barrier force, and annealing schedule come from the paper; the step cap and finite-step rejection are engineering safeguards because the guarantee concerns continuous time. Estimate constraint gradients with automatic differentiation and optionally clip \(\lambda_i\) to prevent floating-point overflow. 4. First experiment: train a small PPO policy on a constrained continuous-control task such as Safety Gymnasium, comparing Adam, projected-gradient updates, and this optimizer at equal environment interactions. Log task return, cumulative constraint violation, minimum constraint margin, rejected steps, and wall-clock cost. Success means zero or substantially lower intermediate violations with comparable final return, and fewer expensive safety QP calls than the projected baseline. A second cheap test can impose a differentiable parameter-norm budget during CIFAR-10 MLP training and measure accuracy at fixed budget and the fraction of steps crossing the boundary.

## Disclaimer

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