Prover Optimization Problem

The goal is to find a set of design parameters that optimize a weighted combination of the following objectives while respecting the constraints. Multi-objective optimization algorithms can be applied to solve this complex problem and find a set of design choices that best meet the desired objectives.

Decentralization

The mechanism should avoid unintentionally leading to centralization or monopolization. Some selection mechanisms might inadvertently favor well-funded or resource-rich participants, leading to centralization. The mechanism should aim to reduce such biases.

1. Maximize Geographic Diversity

Geographic distribution shows the spread of prover network across different physical locations. The Geographic Diversity Index (GDI) is a measure that can quantify the degree of decentralization or geographic diversity in a network.

Geographic Diversity Index (GDI) = Number of distinct geographic regions or countries where provers are located / Total number of provers in the network

Maximize:
Objective = GDI

A GDI of 1 represents maximum geographic diversity, indicating that each prover is located in a different distinct geographic region or country.

Constraints:

• Specify a minimum number of distinct geographic regions or countries where provers must be located.
  • Number of distinct regions or countries ≥ Minimum required diversity threshold
• Specify a minimum number of provers available in each geographic region or country.
• Specify a minimum distance or separation between provers in the same geographic region to ensure that they are not concentrated in a small area within the region.

2. Minimize the Gini Coefficient of Resource Distribution

The Gini coefficient measures the inequality in the distribution of resource concentration among provers in the network. It considers two primary factors:

• Computing Power Concentration (CPC) as the Gini coefficient for computing power distribution.
• Stake Concentration (SC) as the Gini coefficient for stake distribution.

Minimize:
Objective = w1 * CPC + w2 * SC

w1 and w2 are weights assigned to each Gini coefficient based on their relative importance.

Constraints:

• Set limits on the maximum computing power that each prover can contribute to the network.
• Implement caps or limits on the maximum stake that any individual or entity can hold in the network.
• Specify hardware compatibility requirements to ensure that participants use widely available and affordable hardware. This constraint can encourage accessibility and reduce entry barriers.

Liveness

The mechanism should ensure that there’s always a prover ready to generate proofs, ensuring the continual operation of the rollup without delays and no downtime if some provers are temporarily unavailable.

1. Maximize Prover Availability

Ensure that there is always a prover ready to generate proofs.

Maximize:
Objective = Total number of provers

Constraint:

• Consider a constraint that accounts for the maximum number of provers the system can support. This can help in planning for system scalability.

2. Maximize Prover Reputation Score

Prover Reputation Score (PRS) ensure that there is always a prover ready to generate proofs based on factors such as uptime and reliability.

• Prover Uptime (PU) represents the amount of time a prover is available and actively participating in the network. It can be measured in units of time or epochs?.
• Prover Reliability (PR) factor accounts for the reliability of each prover in terms of their ability to generate proofs accurately and without errors. PR = 1 represents a reliable prover.
  • PR = Total amount of accurate proofs generated by a prover/ Total amount of proof generated by a prover

Maximize:
Objective = PRS = w1 * PU * w2 * PR

Constraint:

• Require that provers maintain a certain level of reputation score (uptime and reliability) to ensure network security.

3. Minimize Downtime and Delays

The primary goal is to minimize any downtime and delays in the network. This can be quantified as the sum of downtime and delay costs over time.

Cost of Downtime and Delays represents the cost associated with any periods during which provers are unavailable or fail to generate proofs. The cost of downtime can be quantified in terms of lost revenue or potential user frustration.

Minimize:
Objective = Lost Revenue Due to Cost of Downtime and Delays

Constraints:

• Ensure a minimum number of provers are available to handle temporary unavailability. Include rewards or penalties for their availability based on the network activity. This can encourage provers to stay online and prioritize proof generation.
• Implement Service Level Agreements as constraints to specify the maximum allowable downtime and delay in generating proofs.
• Integrate real-time monitoring and alerting mechanisms to detect and respond to prover unavailability or excessive delays promptly.
• Implement a priority queue for transactions to ensure that critical transactions are processed with higher priority. This can help in reducing delays for important transactions.
• Consider including a mechanism for backup provers or redundancy to handle situations where primary provers may become temporarily unavailable.
• Consider resource constraints when selecting provers such as the computational power and storage capacity of each prover. These constraints can ensure that the system doesn't overwhelm provers, which could lead to downtime or delays.

Censorship Resistance

The mechanism should encourage more provers to participate, which should increase censorship resistance. A small set of provers may refuse to prove certain types of transactions.

Competition

The mechanism should encourage competition between large sets of provers, which could strengthen market pressures to generate faster and cheaper proofs.

1. Maximize Competition

Competition is essential to enhance market pressures and incentivize the generation of faster and more cost-effective proofs among a large set of provers participating in the network.

• Proof Cost (PC) represents the cost associated with generating a proof. It can include computation costs, storage costs and gas fees.
• Proof Speed/ Latency (PL) measures the time taken by a prover to generate a proof.
• Prover Reliability (PR) = refers to the accuracy of the generated proof.
• Number of Provers (N) represents the count of active provers in the system.

Maximize:
Objective = w1 * (1 / PC) + w2 * (1 / PL) + w3 * PR + w4 * N

w1, w2, w3, w4 are weights assigned to each coefficient based on their relative importance.

• Maximizing (1 / PC) encourages provers to reduce proof generation costs.
• Maximizing (1 / PL) encourages provers to minimize proof generation time (latency).
• Maximizing PR encourages provers to generate accurate proofs.
• Maximizing N encourages more provers to participate, fostering competition.

Constraints:

• Specify a minimum level of participation from each prover to prevent free-riding and ensure an active and competitive environment.
• Set a maximum allowed time for generating a proof to ensure that proofs are produced quickly.
• Ensure a minimum level of prover reliability.
• Ensure that provers have access to the necessary computational resources, such as CPU, memory, and storage to generate proofs efficiently.
• Place lower barriers to enter the network to promote higher competition.
• Implement penalties for provers engaged in anti-competitive behavior, such as collusion, fraud, or malicious activities.

Permissionless

The mechanism should allow any eligible participant to join without requiring special permissions or approvals, ensuring open access and inclusivity.

Transparency and Fairness

The mechanism should maintain transparency in its operation and ensure that all participants are treated fairly, without any discriminatory practices or bias in participant selection and reward distribution.

The incentive mechanisms in place for zkRollup provers can affect decentralization. Systems that provide fair and attractive rewards for participation are more likely to attract a diverse set of participants.

Incentives

The mechanism should provide appropriate incentives to encourage participation and honest behavior among provers, aligning their interests with the overall success of the system.

1. Maximize Honest Provers

Maximize the overall success of the system while ensuring that provers are appropriately incentivized to participate honestly. The objective function should strike a balance between rewarding provers for their contributions and deterring malicious actions, ultimately aligning their interests/ utility with the long-term success and security of the zk network.

Reward = (V * Base Reward) + (C * Complexity Bonus)
Penalty = I * Base Penalty * Penalty Severity

Where:

• V is the number of valid proofs submitted by the prover.
• Base Reward is the fixed reward assigned to each valid proof.
• C is the complexity of the computations involved.
• Complexity Bonus is a coefficient that rewards more complex computations.
• I represents the number of incorrect or invalid proofs.
• Base Penalty is the fixed penalty assigned to each invalid proof.
• Penalty Severity is a coefficient that determines the severity of the penalty based on the degree of misbehavior (e.g., 0 for no misbehavior, 1 for minor errors, 2 for major errors, 3 for deliberate fraud).

Maximize for each prover:
Objective = Reward - Penalty

The system should be designed to maximize R−P for each prover. This means that honest provers who submit valid proofs and handle complex computations will be rewarded, while dishonest provers who submit incorrect or invalid proofs will face penalties proportionate to the severity of their misbehavior.

Constraint:

• Define limits on the maximum rewards that can be earned by honest provers. This prevents overly generous rewards that might strain the system.

Security

The mechanism should prioritize security measures to protect against potential attacks, vulnerabilities, malicious behavior, or errors, possibly using staking and slashing as mechanisms to deter and penalize wrongful actions, ensuring the safety of the network.

Cost

The mechanism should aim to minimize operational costs, making it economically feasible for provers to participate while keeping transaction costs low for users.

1. Minimize Operational Cost

Minimize the total operational cost while maintaining low transaction costs for users. The operational cost can be represented as a weighted sum of various components, including:

• Prover's Computational Cost (PCC): The cost associated with the computation performed by provers.
• Prover's Storage Cost (PSC): The cost of storing data required for proof generation and verification.
• Prover's Bandwidth Cost (PBC): The cost of transmitting proofs to the blockchain network.
• Calldata Cost (CC): The cost of submitting transactions to Layer 1.

The objective function can be represented as:

Minimize:

Objective = w1 * PCC + w2 * PSC + w3 * PBC + w4 * CC

w1, w2, w3 and w4 are weighting factors that reflect the relative importance of each cost component. These weights can be determined based on the specific requirements and priorities.

Constraints:

• Ensure that the operational costs do not exceed the revenue generated by the system, making it economically viable.
  • Revenue ≥ Total Operational Cost
• The transaction costs for users should be kept low. This can be defined as a constraint on the total transaction cost for users, which is a combination of the fees charged by the network operator and any fees paid to the provers.
  • Total Transaction Cost for Users ≤ User's Maximum Acceptable Cost
• The operational costs should be kept within a range that ensures economic feasibility for the provers. This means that the prover should be able to cover their costs while potentially making a profit.
  • Prover i Operational Costs ≤ Prover i Earnings
  • Limit the available computational resources, storage capacity, and bandwidth.
    • Computational Resources ≤ Max Computational Resources
    • Storage Capacity ≤ Max Storage Capacity
    • Bandwidth ≤ Max Bandwidth
  • Encourage provers to optimize their proof generation process to minimize computational and storage costs.
  • Ensure the use of future contracts for calldata price volatility risk?

Sybil Attack Resistance

The mechanism should be resistant to Sybil attacks, where an attacker creates many pseudonymous identities to increase their chances of being selected.

Scalability

The mechanism should accommodate growth in transaction volume and participant numbers, ensuring that the increase in transaction volume doesn’t result in bottlenecks.

Efficiency

The mechanism should ensure timely proof generation without excessive redundancy.

Simplicity

The mechanisms should not be overly complex, minimizing potential vulnerabilities or inefficiencies.