Risk Optimization

Risk Optimization

Trust as Objective Function

Frame trust minimization as optimization:

minimize: Σᵢ DR(component_i)
subject to:
    Capability(system) ≥ required_capability
    Σᵢ Cost(component_i) ≤ budget
    Architecture_constraints

Decision variables:

• Component selection (which implementations)
• Architecture (how components connect)
• Trust bounds (what each component can access)
• Verification investment (how much to verify each component)

Trust Gradient Computation

For differentiable trust functions, compute gradients:

∂DR_system / ∂capability_i = marginal trust cost of capability i

Tip

High gradient capabilities are expensive in trust terms—candidates for removal or externalization. Focus optimization effort on the capabilities that dominate your delegation risk budget.

Example: System with components A, B, C

Delegation Risk = DR_A(cap_A) + DR_B(cap_B) + DR_C(cap_C) + interaction_terms

∂DR/∂cap_A = ∂DR_A/∂cap_A + Σⱼ ∂interaction_Aj/∂cap_A

If ∂DR/∂cap_A >> ∂DR/∂cap_B, reducing A’s capability gives more trust reduction per unit capability lost.

Trust-Capability Pareto Frontier

Plot achievable (capability, trust) pairs:

quadrantChart
    title Trust-Capability Tradeoff
    x-axis Low Capability --> High Capability
    y-axis Low Trust --> High Trust
    quadrant-1 High Cap, High Trust
    quadrant-2 Low Cap, High Trust
    quadrant-3 Low Cap, Low Trust
    quadrant-4 Infeasible Zone
    Pareto Optimal: [0.3, 0.3]
    Current Design: [0.5, 0.6]
    Target: [0.7, 0.4]

Pareto efficient designs: Can’t reduce trust without reducing capability.

Dominated designs: Another design has both lower trust and higher capability.

Goal: Find designs on or near Pareto frontier.

Trust Optimization Algorithms

Algorithm 1: Greedy Trust Reduction

while trust_budget_exceeded:
    compute ∂DR/∂capability for all capabilities
    find capability c with highest gradient
    if removing c maintains minimum_capability:
        remove c
    else:
        find alternative implementation of c with lower trust

Algorithm 2: Trust-Aware Architecture Search

for architecture in architecture_space:
    compute DR(architecture)
    compute Capability(architecture)
    if dominates_current_best:
        update_best(architecture)
return pareto_frontier

Algorithm 3: Simulated Annealing for Trust

current = initial_architecture
temperature = high
while not converged:
    neighbor = random_modification(current)
    Δ_trust = DR(neighbor) - DR(current)
    Δ_capability = Capability(neighbor) - Capability(current)

    if Δ_trust < 0 and Δ_capability >= 0:
        current = neighbor  # Strictly better
    elif random() < exp(-Δ_trust / temperature):
        current = neighbor  # Accept worse with probability

    temperature *= cooling_rate

Convexity and Trust Optimization

Question: Is trust minimization convex?

If Delegation Risk is convex in capabilities, optimization is tractable (gradient descent finds global optimum).

Arguments for convexity:

• Individual component Delegation Risk often convex (more capability → more risk, increasing rate)
• Sum of convex functions is convex

Arguments against:

• Interaction terms might be non-convex
• Discrete choices (which implementation) make problem combinatorial
• Verification effectiveness might have non-convex relationship with investment

Practical Approach

Assume local convexity, use multiple random starts, verify solutions. Most real architectures have enough structure that local search finds good solutions.

Trust Sensitivity Analysis

How robust is optimal trust allocation to parameter changes?

Parameters to vary:

• Damage estimates (what if 2× higher?)
• Probability estimates (what if 2× more likely?)
• Verification effectiveness (what if 50% less effective?)
• Correlation structure (what if agents more correlated?)

Sensitivity: ∂(optimal_DR) / ∂(parameter)

High sensitivity parameters need better estimation or robust optimization.

Robust trust optimization:

minimize: max_{parameters ∈ uncertainty_set} DR(architecture, parameters)

Find architecture that minimizes worst-case trust over parameter uncertainty.