silva_delta_equilibrium Reproduction Dossier

delta-cached equilibrium inference. This dossier connects the source mechanism to its SILVA implementation, compact evidence, replaceable components, and source-scale route. Existing tests and notebooks remain the executable authority.

Evidence boundary

The mechanism is compact-verified in the package suite. The final source-scale stage remains planned until the cited data, complete optimization budget, checkpoints, and evaluation protocol have actually run.

Identity and Sources

Field	Value
Domain	optimization and certified equilibria
Task contract	B,D or B,C,H,W input -> equilibrium with per-iteration activity diagnostics
Source relation	paper-adaptation
References	[80]
Repositories	https://github.com/ZuowenWang0000/Delta-Deep-Equilibrium-Models
Editable scale plan	experiments/reproduction/configs/silva_delta_equilibrium.json

Governing Equation

The domain-level state contract is

\[ 0\in\mathcal A_\theta(z^\star;x)+\mathcal B(z^\star),\qquad y=Q_\psi(z^\star). \]

The implementation registry specializes it operationally as

c_k=c_(k-1)+W mask(|z_k-z_(k-1)|>tau)(z_k-z_(k-1))

Define the root residual

\[ R_\theta(z;x)=z-T_\theta(z;x). \]

At a regular equilibrium, differentiating \(R_\theta(z^\star;x)=0\) gives

\[ \frac{\partial z^\star}{\partial x} = \left(I-\frac{\partial T_\theta}{\partial z}\right)^{-1} \frac{\partial T_\theta}{\partial x}. \]

This identity explains why the forward residual, the conditioning derivative, and the adjoint linear solve must be diagnosed separately from the task metric.

What Is Preserved

• cached linear or convolutional recurrent output updated from thresholded state deltas
• zero-threshold algebraic equivalence to full recurrent evaluation
• full-map training with independently selectable delta-cached inference

What Can Be Replaced

Each item below is an explicit control rather than an undocumented modification:

• wrap linear or convolutional recurrent operators in the delta cache
• train with the full transition and enable thresholded cached evaluation independently
• state size
• delta threshold
• operator sparsity
• solver budget

Constructor and Shape Contract

silva_delta_equilibrium(in_dim: 'int', state_dim: 'int', out_dim: 'int', *, recurrent: 'nn.Module | None' = None, source: 'nn.Module | None' = None, activation: 'Callable[[Tensor], Tensor]' = <built-in method tanh of type object>, readout: 'nn.Module | None' = None, delta_threshold: 'float' = 0.0, config: 'SolverConfig | None' = None) -> 'None'

The transition must preserve the declared equilibrium-state shape even when the encoder, branch operators, constraints, solver, and readout are replaced. Test the transition by itself before testing the complete root solve.

Progressive Experiment Ladder

1. Equation and tensor contract

Objective: Make the state, conditioning variables, operator, and readout explicit.

Procedure:

• Write and evaluate the family equation: c_k=c_(k-1)+W mask(|z_k-z_(k-1)|>tau)(z_k-z_(k-1))
• Declare every tensor axis, boundary, mask, graph, or physical unit.
• Check the transition output has exactly the same shape as the equilibrium state.

Acceptance checks:

• finite transition values
• shape-preserving state update
• all conditioning variables affect the intended branch

Evidence target: contract-verified.

2. Primitive mechanism reconstruction

Objective: Build the retained source mechanism from replaceable modules.

Procedure:

• cached linear or convolutional recurrent output updated from thresholded state deltas
• zero-threshold algebraic equivalence to full recurrent evaluation
• full-map training with independently selectable delta-cached inference

Acceptance checks:

• primitive modules expose trainable parameters and gradients
• mechanism-specific invariance or constraint check passes
• direct transition evaluation is deterministic under a fixed seed

Evidence target: compact-verified.

3. Public abstraction equivalence

Objective: Verify that the assembled family evaluates the same transition as its primitives.

Procedure:

• Copy the primitive module parameters into the public family constructor.
• Evaluate one transition and one complete equilibrium with identical inputs.
• Compare outputs, residuals, and parameter gradients with declared tolerances.

Acceptance checks:

• transition outputs agree
• equilibrium residual is finite and decreases
• primitive and assembled gradients agree on the compact case

Evidence target: compact-verified.

4. Compact real or analytic task

Objective: Exercise training, evaluation, diagnostics, and serialization end to end.

Procedure:

• make_delta_heterogeneous_dataset gives an exact affine equilibrium with coordinates converging at different rates.

Acceptance checks:

• record task metric
• record active fraction
• record exact full-map residual
• record latency and memory traffic
• checkpoint reload reproduces the recorded prediction
• result record contains data and configuration fingerprints

Evidence target: compact-verified.

5. Official-data subset

Objective: Validate the complete source data path before spending the full budget.

Procedure:

• Load a source-compatible checkpoint and reproduce the task data preprocessing and ordinary full-map evaluation first.
• Freeze preprocessing, split logic, metric code, and checkpoint format.
• Run a deterministic subset large enough to expose batching and memory failures.

Acceptance checks:

• dataset receipt and checksum are stored
• resume and evaluation paths reproduce the same subset metric
• memory and runtime are measured rather than estimated

Evidence target: subset-verified.

6. Source-scale reproduction or declared extension

Objective: Run the cited protocol, or change it explicitly as a SILVA extension.

Procedure:

• Load a source-compatible checkpoint and reproduce the task data preprocessing and ordinary full-map evaluation first.
• Wrap supported recurrent linear or convolutional operators, begin at zero threshold, and verify prediction/state equivalence and exact residual.
• Sweep thresholds and report task degradation, active fraction, wall time, memory traffic, solver evaluations, and hardware details.
• source model checkpoint, recurrent operators, data preprocessing, and evaluation sequence
• threshold policy, warm starts, solver tolerances, hardware, precision, and seeds
• task metric, active fraction, exact residual, latency, memory traffic, and source baseline

Acceptance checks:

• all required artifacts are archived
• reported metrics use the cited evaluation protocol
• every architectural or training deviation is listed
• claims match the achieved evidence status

Evidence target: planned.

Data, Access, and Storage

Candidate datasets:

• FlyingChairs
• Sintel
• KITTI
• compact heterogeneous-rate equilibria

Authoritative routes:

• https://papers.nips.cc/paper_files/paper/2024/hash/69f5b860d6dc469ac6e52f03866b73c4-Abstract-Conference.html
• https://github.com/ZuowenWang0000/Delta-Deep-Equilibrium-Models

Access obligations:

• FlyingChairs, Sintel, and KITTI retain their own download and evaluation terms.
• Store the base checkpoint separately from delta thresholds and report whether the evaluation route uses warm starts or cached states.

Storage planning:

• The cache stores one previous state and one recurrent output per wrapped operator in addition to the ordinary solver state.
• For image or flow evaluation, log activity summaries rather than full boolean masks unless a detailed profiling shard is required.

Preprocessing record:

• record dataset version, split, normalization, shape convention, and seed
• preserve masks, graph indices, boundaries, or physical units required by the domain

Metrics and Current Evidence

Required metrics:

• task metric
• active fraction
• exact full-map residual
• latency and memory traffic

The vector compact suite ran this family on the same task and data as the other compatible families in that suite.

Measure	Recorded value
Initial loss	0.163513
Final loss	0.00607382
Fractional loss reduction	0.963
Residual or final increment norm	9.17086e-07
Iterations or tied increments	19
Parameter count	73
Final gradient norm	0.0662025

These values are compact-verified evidence. They establish finite optimization, gradient flow, and diagnostic reporting; they are not a publication ranking.

Executed notebook paths:

• notebooks/package_api/41_silva_delta_equilibrium.ipynb

Mechanism tests:

• tests/test_structured_equilibria.py

Compact Defaults

Option	Value
tier	'smoke'
config	SolverConfig(solver='anderson', max_iter=12, tol=1e-05, alpha=1.0, history=3, ridge=0.0001, beta=1.0, stop_mode='relative', relative_eps=1e-08, anderson_batch_dims=1, track_residuals=True, reengage=True, backward_mode='implicit', backward_solver='gmres', backward_max_iter=20, backward_tol=1e-05, backward_stop_mode='relative', backward_relative_eps=1e-08, phantom_steps=1, phantom_tau=1.0, neumann_terms=5, shine_refine_steps=0, indexing=(), return_best=True)

Full Defaults

Option	Value
tier	'full'
config	SolverConfig(solver='anderson', max_iter=60, tol=1e-05, alpha=1.0, history=6, ridge=0.0001, beta=1.0, stop_mode='relative', relative_eps=1e-08, anderson_batch_dims=1, track_residuals=True, reengage=True, backward_mode='implicit', backward_solver='gmres', backward_max_iter=80, backward_tol=1e-05, backward_stop_mode='relative', backward_relative_eps=1e-08, phantom_steps=1, phantom_tau=1.0, neumann_terms=5, shine_refine_steps=0, indexing=(), return_best=True)

Defaults establish a starting budget; the cited source protocol takes precedence whenever reproduction is the claim.

Source-Scale Checklist

• Load a source-compatible checkpoint and reproduce the task data preprocessing and ordinary full-map evaluation first.
• Wrap supported recurrent linear or convolutional operators, begin at zero threshold, and verify prediction/state equivalence and exact residual.
• Sweep thresholds and report task degradation, active fraction, wall time, memory traffic, solver evaluations, and hardware details.

Benchmark-specific requirements:

• source model checkpoint, recurrent operators, data preprocessing, and evaluation sequence
• threshold policy, warm starts, solver tolerances, hardware, precision, and seeds
• task metric, active fraction, exact residual, latency, memory traffic, and source baseline

Required archived artifacts:

• machine-readable model and solver configuration
• dataset receipt with source revision, split, license, and checksum
• preprocessing and normalization record
• seeded training and evaluation log
• checkpoint and optimizer-resume state for trained experiments
• task metrics and equilibrium diagnostics in a machine-readable result
• runtime, peak-memory, device, precision, and dependency record
• declared deviations from the cited protocol

Reporting Rule

Report the achieved evidence status, not the intended one. A compact or subset run may validate the implementation and data path, but only a completed cited protocol supports a source-scale reproduction statement. Modified operators are valuable SILVA extensions when every deviation is named and measured.

Where to Go Next

Question	Page
Where are all family dossiers?	Family Dossier Index
How is a custom family assembled?	Advanced Extension Handbook
How are experiment stages represented in the API?	Research-Depth API
Which lab inspects every dossier?	Family Dossier Lab