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silva_monotone_operator_equilibrium Reproduction Dossier

monotone-operator equilibrium with certified splitting. 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 input -> B,H equilibrium and task readout
Source relation paper-adaptation
References [75]
Repositories https://github.com/locuslab/monotone_op_net
Editable scale plan experiments/reproduction/configs/silva_monotone_operator_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

0 in (I-W)z_star-Ux-b+partial f(z_star); W=(1-m)I-A^T A+B-B^T

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

  • strongly monotone parameterization W=(1-m)I-A^T A+B-B^T
  • forward-backward and Peaceman-Rachford operator splittings
  • proximal nonlinearities and implicit differentiation at the solved equilibrium

What Can Be Replaced

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

  • replace the monotone operator, source, proximal map, splitting, or readout
  • compose structured convolutions while retaining the monotonicity certificate
  • state width
  • operator factor rank
  • splitting iterations
  • implicit backward budget

Constructor and Shape Contract

silva_monotone_operator_equilibrium(in_dim: 'int', state_dim: 'int', out_dim: 'int', *, operator: 'nn.Module | None' = None, source: 'nn.Module | None' = None, prox: 'Callable[[Tensor], Tensor]' = <function relu>, readout: 'nn.Module | None' = None, splitting: 'MonotoneSplitting' = 'forward_backward', step_size: 'float' = 1.0, margin: 'float' = 1.0, 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: 0 in (I-W)z_star-Ux-b+partial f(z_star); W=(1-m)I-A^T A+B-B^T
  • 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:

  • strongly monotone parameterization W=(1-m)I-A^T A+B-B^T
  • forward-backward and Peaceman-Rachford operator splittings
  • proximal nonlinearities and implicit differentiation at the solved equilibrium

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_monotone_operator_dataset gives a seeded affine source and known monotone-ReLU equilibrium.

Acceptance checks:

  • record task accuracy
  • record monotonicity certificate
  • record fixed-point residual
  • 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:

  • Acquire one source benchmark and reproduce its split, normalization, augmentation, and architecture dimensions.
  • 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:

  • Acquire one source benchmark and reproduce its split, normalization, augmentation, and architecture dimensions.
  • Choose the forward-backward or Peaceman-Rachford route and match the monotone factorization, proximal map, step, and solver tolerances.
  • Validate the compact known-solution case, then report task accuracy, certificate, residual, evaluations, runtime, and memory at source scale.
  • source architecture width/depth, convolutional parameterization, data split, and augmentation
  • splitting step size, forward/backward tolerances, optimizer, regularization, and seeds
  • task accuracy, residual, evaluation count, memory, and source baselines

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:

  • MNIST
  • CIFAR-10
  • SVHN
  • compact known-solution monotone inclusions

Authoritative routes:

  • https://arxiv.org/abs/2006.08591
  • https://github.com/locuslab/monotone_op_net

Access obligations:

  • MNIST, CIFAR-10, and SVHN have established public acquisition routes under their stated terms.
  • Record the source repository revision, data split, augmentation, and any pretrained checkpoint checksum.

Storage planning:

  • Dense operator storage scales quadratically with state width; structured convolutions replace that term with kernel parameters and feature maps.
  • Budget activations, solver history, checkpoints, and optimizer state separately even when implicit differentiation avoids storing every iteration.

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 accuracy
  • monotonicity certificate
  • fixed-point residual
  • operator evaluations
  • runtime and memory

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 0.161426
Final loss 0.00628446
Fractional loss reduction 0.961
Residual or final increment norm 0.0362642
Iterations or tied increments 20
Parameter count 103
Final gradient norm 0.176981

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/36_silva_monotone_operator_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

  • Acquire one source benchmark and reproduce its split, normalization, augmentation, and architecture dimensions.
  • Choose the forward-backward or Peaceman-Rachford route and match the monotone factorization, proximal map, step, and solver tolerances.
  • Validate the compact known-solution case, then report task accuracy, certificate, residual, evaluations, runtime, and memory at source scale.

Benchmark-specific requirements:

  • source architecture width/depth, convolutional parameterization, data split, and augmentation
  • splitting step size, forward/backward tolerances, optimizer, regularization, and seeds
  • task accuracy, residual, evaluation count, memory, and source baselines

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