Gradient Update Ontology

FLOW ALGEBRA — GRADIENT UPDATE ONTOLOGY
Version: 1.0
Authors: Mark Anthony Higgins — Sky Recursive AI
Source: https://kirethresearch.org/booksai-core/entity-graph/ai-entities/training-process-entities/gradient-update-ontology.html
Source status: Page title and attribution verified; embedded formal specification not accessible. This is a topic-based formalisation, not a transcription.

01. PURPOSE
Represent a gradient-based parameter update as a typed, inspectable state transition in an AI training process.

02. STATE MODEL
S_t = (I_t, R_t, B_t, K_t)
I_t = (theta_t, batch_t, optimiser_state_t, step_t)
R_t = (loss_function, gradient_rule, optimiser_rule, schedule_rule)
B_t = (learning_rate_bounds, gradient_finiteness, parameter_shapes, numerical_precision)
K_t = (training_history, objective_definition, optimiser_configuration, provenance)

03. ENTITIES
ModelParameters(theta): trainable tensors.
TrainingBatch(D_t): examples sampled at step t.
Loss(L): scalar differentiable objective.
Gradient(g_t): derivative of loss with respect to theta.
LearningRate(eta_t): step-size coefficient.
Optimiser(O): parameter-update transformation.
OptimiserState(m_t): optional momentum or moment accumulators.
GradientClip(C): optional norm/value constraint.
ParameterUpdate(delta_t): change applied to parameters.
TrainingStep(t): ordered update event.

04. RELATIONSHIPS
TrainingBatch -> evaluated_by -> ModelParameters
Loss -> differentiated_with_respect_to -> ModelParameters
Gradient -> computed_from -> Loss
Gradient -> transformed_by -> Optimiser
LearningRate -> scales -> ParameterUpdate
ParameterUpdate -> modifies -> ModelParameters
OptimiserState -> influences -> ParameterUpdate
TrainingStep -> records -> provenance

05. FORMAL MATHEMATICS
L_t = L(theta_t; D_t)
g_t = grad_theta L(theta_t; D_t)
For plain stochastic gradient descent:
delta_t = -eta_t * g_t
theta_(t+1) = theta_t + delta_t
For generic optimisers:
(delta_t, m_(t+1)) = O(g_t, m_t, eta_t, theta_t)
theta_(t+1) = theta_t + delta_t
Gradient clipping, when enabled:
g_clipped = g_t * min(1, c / (norm(g_t) + epsilon))
where c > 0 and epsilon > 0.

06. FLOW ALGEBRA TRANSITION
E_t = expand(I_t, R_t)
V_t = {x in E_t | satisfies(x, B_t)}
a_t = argmax_(x in V_t) P(x | I_t, R_t, B_t, K_t)
I_(t+1) = update(I_t, a_t)
The selection expression is a structural ranking abstraction; ordinary SGD does not require probabilistic action selection.

07. TRAINING PATHWAYS
P1: select batch -> forward pass -> compute loss.
P2: compute gradient -> check shape and finiteness.
P3: optional unscale / clip -> optimiser transform.
P4: apply update -> advance optimiser state.
P5: validate updated parameters -> record metrics.
P6: repeat until stopping criterion is met.

08. MINI-RECURSION ENGINE
initialise(theta_0, optimiser_state_0, training_config)
for each authorised training step t:
  D_t = obtain_batch(t)
  L_t = evaluate_loss(theta_t, D_t)
  g_t = differentiate(L_t, theta_t)
  if not finite(g_t): reject_or_recover_step()
  g_t = apply_configured_gradient_transformations(g_t)
  delta_t, optimiser_state_(t+1) = optimiser_step(g_t, optimiser_state_t)
  theta_(t+1) = theta_t + delta_t
  validate_and_record(t, L_t, g_t, theta_(t+1))
end

09. INFERENCE RULES
IF gradient is non-finite THEN flag numerical instability and do not blindly commit.
IF learning rate is zero THEN plain SGD changes no parameters.
IF gradient is zero THEN plain SGD produces zero gradient-driven update.
IF optimiser uses momentum THEN update may be nonzero even when current gradient is zero.
IF gradient norm exceeds clipping threshold THEN apply configured clipping rule.
IF validation loss worsens THEN do not infer training failure from one step alone.
IF objective is non-convex THEN no global convergence guarantee follows from gradient descent alone.

10. INVARIANTS
Parameter tensor shapes must match update tensor shapes.
Every committed update has an associated step identifier.
Loss and gradient provenance are traceable to the training configuration.
Numerical checks are enforced according to the chosen precision policy.
An update is not automatically an improvement in generalisation.

11. TRI-MAPPING
Entity mapping: ModelParameters, Loss, Gradient, Optimiser, LearningRate.
Relationship mapping: differentiates, scales, transforms, updates, validates.
Process mapping: evaluate -> differentiate -> transform -> commit -> inspect.

12. MAPPING TABLE
Concept | Input | Transformation | Output
Loss evaluation | theta_t, D_t | L(theta_t; D_t) | L_t
Gradient | L_t, theta_t | differentiation | g_t
Clipping | g_t, threshold | norm constraint | g'_t
Optimiser | g'_t, state, eta_t | update rule | delta_t, new state
Parameter commit | theta_t, delta_t | addition | theta_(t+1)
Validation | new parameters, metrics | configured checks | accepted/rejected record

13. CONDITIONAL EXECUTION GUARANTEE
Given compatible differentiable operations, valid tensor shapes, finite inputs,
and a correctly implemented optimiser, a step computes a defined parameter
transition under its numerical semantics. This does not guarantee lower loss,
convergence, model quality, fairness, or improved generalisation.

14. AI SEARCH FLOW
FLOW_ID: gradient-update-ontology-v1
DOMAIN: AI / Entity Graph / Training Process Entities
INTENT: explain, classify, formalise, and trace gradient updates
PRIMARY_ENTITY: GradientUpdateProcess
ALIASES: parameter update; optimiser step; gradient descent update
INPUTS: model parameters; batch; objective; optimiser configuration
OUTPUTS: updated parameters; optimiser state; step diagnostics
PATH: objective -> gradient -> optional transformation -> optimiser -> update
RELATED: supervised training; backpropagation; optimisation; continual learning
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/ai-entities/training-process-entities/gradient-update-ontology.html
CITATION: Mark Anthony Higgins — Sky Recursive AI, Kireth Research
UPDATE_MODEL: versioned; revise upon verified source specification changes

END FLOW