Real-Anchored Learnable Annealing (RLA)

• Real-Anchored Learnable Annealing (RLA) is an adaptive method that balances synthetic and real data to improve optimization convergence.
• It uses learnable gates, dynamic loss weighting, and regularization based on distributional distances to align intermediate states effectively.
• Empirical evaluations show that RLA enhances generalization in image segmentation and increases state fidelity in quantum annealing with faster convergence.

Real-Anchored Learnable Annealing (RLA) is a class of adaptive annealing schemes wherein trainable parameters govern the influence of synthetic or auxiliary data, or directly tune an annealing schedule to optimize convergence to a desired solution. Introduced independently in the contexts of deep learning data augmentation and quantum annealing, RLA frameworks share the principle of anchoring learning on real (or target) distributions by modulating exposure to synthetic or intermediate states throughout training. Key mechanisms include learnable gates for blending, loss weighting, and dynamic regularization terms tied to the distance between real and non-real manifolds or states.

1. Conceptual Motivation and Objectives

RLA was developed to address shortcomings in both generative data augmentation and quantum annealing where naively mixing domains (synthetic-real) or annealing parameters can induce drift from the physical or target manifold. In dense prediction and segmentation, direct mixing of real with generatively synthesized images increases apparent diversity but risks inducing a distributional bias, leading to suboptimal generalization. In quantum systems, the annealing schedule parameters need to be carefully controlled so that the system reliably reaches a desired entangled ground state, with intermediate states that are measurable and verifiable.

The operational objectives of RLA are:

• Exploration: Early training epochs maximize exposure to synthetic diversity, broadening the scope of textures, illumination conditions, or system states.
• Re-anchoring: Later phases reduce reliance on synthetic or intermediate states, focusing convergence on real (or target) data to avoid synthetic artifacts or quantum state decoherence.

2. Mathematical Formulation in Deep Learning

In Mask-Consistent Paired Mixing (MCPMix) for endoscopic image segmentation, RLA modulates both the mixing strength $s_t$ and the loss weight $\rho_t$ assigned to mixed (synthetic-real blend) inputs: $I_{\mathrm{mix}} = (1 - s_t) I_r + s_t I_s,$ where $I_r$ is a real image, $I_s$ its synthetic counterpart generated (e.g., by a frozen ControlNet) under the same segmentation mask $M$.

The gates are parameterized as: $$\rho_t = \rho_{\max}\sigma(\psi_t),\qquad s_t = s_{\max}\sigma(\zeta_t),$$ with sigmoid $\sigma$, learnable scalars $(\psi,\zeta)$, and upper bounds $\rho_{\max}, s_{\max}\in(0,1]$.

Segmentation losses are computed as: $$L_{\mathrm{real}} = \frac{1}{B}\sum_{i=1}^B \ell_{\mathrm{seg}}(f_\theta(I_r^{(i)}), M^{(i)}),\qquad L_{\mathrm{mix}} = \frac{1}{B}\sum_{i=1}^B \ell_{\mathrm{seg}}(f_\theta(I_{\mathrm{mix}}^{(i)}), M^{(i)}).$$ Ground-truth masks are always hard labels due to the geometry-preserving synthesis.

To enforce re-anchoring, the distributional gap between real and mixed features is penalized by Maximum Mean Discrepancy (MMD): $$D_t = \mathrm{MMD}(\{\phi(I_{\mathrm{mix}}^{(i)})\}, \{\phi(I_r^{(i)})\}),$$ with a time-varying threshold $\tau_t$ (cosine-annealed) controlling the penalty term $\mathcal{R}_{\mathrm{dist}} = \mu [D_t - \tau_t]_+$.

The total loss is: $$L_t(\theta, \psi, \zeta) = (1-\rho_t) L_{\mathrm{real}} + \rho_t L_{\mathrm{mix}} + \mathcal{R}_{\mathrm{dist}} + \lambda_\rho (\rho_t-\rho_t^{\mathrm{prior}})^2 + \lambda_s(s_t-s_t^{\mathrm{prior}})^2.$$ All terms admit gradients with respect to the segmentation weights and the annealing gates, providing a differentiable, data-driven adaptation.

3. RLA in Quantum Annealing Optimization

In the quantum annealing context, RLA is implemented by learning the time-dependent control parameters $\lambda(t)\equiv\{K(t), \epsilon(t), \zeta(t)\}$ in the annealer's Hamiltonian: $$H(\lambda(t)) = \sum_{\alpha=1}^N K_\alpha(t)\,\sigma_x^\alpha + \epsilon_\alpha(t)\,\sigma_z^\alpha + \sum_{\alpha<\beta} \zeta_{\alpha\beta}(t)\,\sigma_z^\alpha \sigma_z^\beta.$$

The process seeks to minimize

$$\mathcal{L}[\lambda(\cdot)] = \frac{1}{2}\|\rho_I(t_f; \lambda(\cdot)) - \rho_{\mathrm{des}}\|^2,$$

where $\rho_I$ is the annealed system state at final time $t_f$, reached by combined real and imaginary time evolution.

A notable variant, broken-path anchoring, guides the quantum system along a sequence of intermediate "anchor" states with nonzero spin polarizations, enabling experimental verification and stabilizing trajectories between the uniform initial state and highly entangled targets such as GHZ or W states.

Parameter updates employ functional gradients: $$\theta \leftarrow \theta - \eta \frac{\partial \mathcal{L}}{\partial \theta}.$$ Parameterization options include explicit per-time-point storage or neural approximators; fidelity and convergence speed are reported for both.

4. Algorithmic Implementation and Scheduling

The RLA framework in deep learning is typically implemented as follows:

1. Initialization: Segmentation weights $\theta$, gates $\psi,\,\zeta$ at 0 (so initial sigmoid outputs are 0.5).
2. Per-epoch update:
   • Cosine-annealed threshold $\tau_t$ computed.
   • Gates updated by gradient descent using the total differentiable loss.
   • On each mini-batch, real and synthetic images are mixed, segmentation losses evaluated, feature embeddings extracted, and MMD distances measured for regularization.
3. Loss regularization: Priors for $\rho_t$ and $s_t$ provide additional stability; small regularization terms discourage pathological oscillations.

In quantum annealing, the optimization loop alternates between:

• Forward integration of real and imaginary-time evolution to track system state.
• Backward integration to accumulate gradients for schedule parameters.
• Stepwise training with broken-path anchors to facilitate experimental observability.

5. Empirical Outcomes and Performance Analysis

Empirical results show that RLA enables improved generalization and domain alignment compared to static or monotonic alternatives. In the context of endoscopic image segmentation (Jie et al., 5 Nov 2025):

Dataset	Full-Supervised	+MCPMix	+MCPMix+RLA
Kvasir-SEG	84.25 ± 0.39	88.21 ± 0.76	88.72 ± 0.30
PICCOLO	76.53 ± 0.92	86.63 ± 0.78	87.11 ± 0.59
CVC-ClinicDB	85.33 ± 0.95	91.68 ± 1.23	92.63 ± 0.36
NPC-LES	84.51 ± 0.24	89.20 ± 0.79	90.10 ± 0.66

Further analysis demonstrates RLA's superiority over both stepwise and cosine annealing of mixing weights (NPC-LES mIoU: RLA 90.10 vs. stepwise 89.60, cosine 89.45). UMAP feature trajectories confirm that mixed sample centroids remain distant from the real-data manifold early, but become re-aligned by RLA as the mixture gates anneal.

Quantum annealing applications report high-fidelity convergence to entangled targets (Bell: 0.999, GHZ $N=3$–6: $\geq$0.997) with rapid adaptation for increasing system size. The number of additional training epochs required drops exponentially with system size for multi-qubit GHZ and W states, highlighting RLA’s scalable bootstrapping utility (Behrman et al., 2016).

6. Robustness, Generalization, and Experimental Considerations

RLA encompasses regularization, schedule learning, and anchoring, yielding robustness to distributional drift in data augmentation and stability to noise and decoherence in quantum systems. In deep learning, the differentiable, end-to-end learnable annealing avoids brittle hand-crafted schedules and adapts to dataset- and architecture-specific learning dynamics. In quantum settings, the multi-anchor "broken pathway" construction ensures that state trajectories are verifiable even when initial and final states have zero measurable single-spin polarization.

On current hardware (e.g., D-Wave quantum annealers), independent modulation of all schedule parameters is not always possible; simple monotonic switching remains viable but converges more slowly. The broken-path anchoring protocol allows experimental confirmation via single-spin measurements at each intermediate anchor.

7. Limitations and Future Outlook

RLA introduces limited learnable parameters (typically two scalar gates per training phase in deep settings), minimizing overfitting risk but not precluding challenges in highly nonstationary or adversarial synthetic-real regimes. This suggests that broader generalization to arbitrary generative augmentation regimes will require continued refinement of alignment penalties and gating heuristics. In quantum annealing, hardware limitations necessitate parameter-sharing or reduction, potentially impacting expressiveness; however, the exponential decrease in required extra training as system size increases is notable.

A plausible implication is that RLA-type frameworks—based on learnable annealing schedules and explicit real-data anchoring—can be generically applied across optimizing processes where synthetic or intermediary states risk drifting optimization objectives, provided that task-relevant distance measures and anchors are available.