silva_hyper_deq Reproduction Dossier

learned initializer and learned Anderson equilibrium solver. 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	core equilibrium
Task contract	condition plus replaceable transition -> accelerated equilibrium and task readout
Source relation	paper-adaptation
References	[87]
Repositories	https://github.com/locuslab/deq
Editable scale plan	experiments/reproduction/configs/silva_hyper_deq.json

Governing Equation

The domain-level state contract is

\[ z^\star=T_\theta(z^\star;x),\qquad y=Q_\psi(z^\star). \]

The implementation registry specializes it operationally as

z_0=h_phi(x); alpha_k,beta_k=H_phi(r_(k-m+1:k),x); z_(k+1)=beta_k sum_i alpha_(k,i) f(z_i,x)+(1-beta_k) sum_i alpha_(k,i) z_i

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

• learned condition-to-state initialization and learned Anderson coefficients/mixing
• high-precision teacher equilibrium and weighted trajectory supervision

What Can Be Replaced

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

• replace the transition, initializer, residual compressor, controller, or readout
• train the solver around a frozen task model or jointly with the SILVA transition
• teacher trajectory cache
• learned solver steps
• residual history
• compressor and controller width

Constructor and Shape Contract

silva_hyper_deq(state_shape: 'int | Sequence[int]', condition_dim: 'int', *, transition: 'nn.Module | None' = None, initializer: 'nn.Module | None' = None, controller: 'SILVAHyperAndersonController | None' = None, readout: 'nn.Module | None' = None, learned_steps: 'int' = 5, history: 'int' = 5, teacher_config: 'SolverConfig | None' = None, state_scale: 'float' = 0.2) -> '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: z_0=h_phi(x); alpha_k,beta_k=H_phi(r_(k-m+1:k),x); z_(k+1)=beta_k sum_i alpha_(k,i) f(z_i,x)+(1-beta_k) sum_i alpha_(k,i) z_i
• 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:

• learned condition-to-state initialization and learned Anderson coefficients/mixing
• high-precision teacher equilibrium and weighted trajectory supervision

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:

• The learned-solver lab generates seeded affine-tanh conditions, high-precision teacher roots, and complete learned Anderson trajectories without an external download.

Acceptance checks:

• record task metric
• record teacher-state error
• record fixed-point residual by learned step
• record operator evaluations
• 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:

• Choose one source task, reproduce its ordinary equilibrium transition and checkpoint, and verify the unaccelerated task metric 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:

• Choose one source task, reproduce its ordinary equilibrium transition and checkpoint, and verify the unaccelerated task metric first.
• Generate high-precision roots and solver trajectories with fixed tolerances, then train the initializer and learned Anderson controller against that immutable teacher cache.
• Compare equal-budget classical and learned solvers on residual, task metric, operator evaluations, latency, memory, and failure rate before testing transfer to new inputs or transitions.
• source task model/checkpoint, teacher solver budget, training split, latency protocol, and task metric

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:

• WikiText-103
• ImageNet
• Cityscapes
• compact contractive teacher trajectories

Authoritative routes:

• https://openreview.net/forum?id=B0oHOwT5ENL
• https://github.com/locuslab/deq
• https://www.salesforce.com/blog/the-wikitext-long-term-dependency-language-modeling-dataset/
• https://www.image-net.org/
• https://www.cityscapes-dataset.com/

Access obligations:

• WikiText-103 is publicly distributed under its stated terms; ImageNet and Cityscapes require their respective registrations and licenses.
• Record the task checkpoint, teacher-solver revision, cached trajectory indices, preprocessing, split, and learned-controller checkpoint independently.

Storage planning:

• Teacher-cache bytes = samples * retained states * state elements * bytes per element; projected residuals and task labels add separate arrays.
• For 1,000,000 samples, eight retained 512-float32 states require about 15.3 GiB before conditions, labels, optimizer state, and checkpoints.
• Shard teacher trajectories by task split and checkpoint hash so controller training can stream states without loading the full cache.

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
• teacher-state error
• fixed-point residual by learned step
• operator evaluations
• latency, memory, and failure rate

This family is verified through its listed mechanism tests and executed notebook. It is not included in a same-task comparison when another family does not share its input, state, output, and loss contract. The absence of a comparison row is therefore a scope decision, not missing implementation evidence.

Executed notebook paths:

• notebooks/package_api/48_silva_learned_solvers.ipynb
• notebooks/package_api/51_equilibrium_expansion_atlas.ipynb

Mechanism tests:

• tests/test_solver_learning.py

Compact Defaults

Option	Value
tier	'smoke'
teacher_config	SolverConfig(solver='broyden', 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=0, 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'
teacher_config	SolverConfig(solver='broyden', 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=0, 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

• Choose one source task, reproduce its ordinary equilibrium transition and checkpoint, and verify the unaccelerated task metric first.
• Generate high-precision roots and solver trajectories with fixed tolerances, then train the initializer and learned Anderson controller against that immutable teacher cache.
• Compare equal-budget classical and learned solvers on residual, task metric, operator evaluations, latency, memory, and failure rate before testing transfer to new inputs or transitions.

Benchmark-specific requirements:

• source task model/checkpoint, teacher solver budget, training split, latency protocol, and task metric

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