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Yuma Consensus

Introduction

Yuma Consensus (YC) is a critical algorithmic process within Bittensor, which runs on-chain within Subtensor. Its responsibility is to compute validator and miner emissions from validators' rankings of miners' performances.

Each of a subnet's validators periodically submits a vector of weights ranking the value of the work of each miner they've evaluated. The YC algorithm resolves this matrix of rankings into two emissions vectors—one each for miners and validators. These emissions vectors allocate emissions over participants based on their performances.

The algorithm is designed to more heavily weight the inputs of more trusted validators, in order to ignore the portion of the validation signal that is less reliable. By disregarding unreliable weight-settings, YC incentivizes validators to be worthy of trust by working hard to give consistent, fast, honest evaluations of miners that predict the eventual convergence of other validators' evaluations. YC incentivizes miners to work hard for the highest combined evaluation by the community of validators.

See:

• How Yuma Consensus 3 Makes Bittensor More Fair
• Yuma Consensus 3 Migration Guide
• Emissions
• Subtensor Docs: Yuma Consensus

Clipping

Clipping is designed to punish innacurate miner evaluation, especially in patterns that could constitute collusion to manipulate the accuracy of consensus to favor certain miners.

To achieve this, the judgment of the most trusted validators (as measured by stake) serves as a benchmark. Weights that exceed this benchmark are clipped, meaning neither the miner nor the validator receives emissions for them.

This clipping protects against erroneous or collusive over-evaluation of miners by validators.

See how it's calculated

To compute the benchmark $\overline{W_j}$ for miner $j$ and set $\mathbb{V}$ of the subnet's validators, we gather all validator weights $W_{ij}$ for any validator $i \in \mathbb{V}$, sort them by validator’s stake $S_i$, and then find the maximum weight level $w$ supported by at least a fraction $\kappa$ of total stake.

$$\overline{W_j} = \arg \max_{w} \Bigl(\, \sum_{i \in \mathbb{V}} S_i \,\cdot\, \bigl\{\,W_{ij}\,\ge w \bigr\} \ge \kappa \Bigr).$$

Any validator’s original weight $W_{ij}$ above $\overline{W_j}$ is clipped to:

$$\overline{W_{ij}} = \min( W_{ij}, \overline{W_j} )$$

note

Kappa is a configurable hyperparameter with default: $\kappa = 0.5$.

This means that if least generous half (0.5) of the validators (with each validator measured by stake, not 1 per validator) set weights for a given miner to no more than x, then the weights of the other, more generous, half of the validators for that miner are going to be clipped down to x. 'Generous' here refers to giving a high weight to the miner in question.

Miner emissions

Miner emissions are based on an aggregate ranking which is the summed rankings of validators, weighted by validators' stake, where $\overline{W_{ij}}$ is the post-clip weight.

See how it's calculated

$$R_j = \sum_{i \in \mathbb{V}} S_i \,\cdot\, \overline{W_{ij}}$$

Each miner $j$’s share $M_j$ of the subnet's miner-emissions (41% of each subnet's overall emissions) is equal to their proportion of the total aggregate miner ranking. Where $\mathbb{M}$ is the subnet's miners:

$$M_j = \frac{\,R_j\,}{\sum_{k \in \mathbb{M}} R_k}$$

Penalizing out-of-consensus bonds

If a validator's evaluation of a miner is too high, it is penalized. If a submitted weight $W_{ij}$ by validator $i$ for miner $j$ exceeds the $j$'s consensus evaluation, $\overline{W_j}$, its bond value is penalized by factor $\beta$.

See how it's calculated

Bond-weight $\widetilde{W_{ij}}$ is:

$$\widetilde{W_{ij}} = (1-\beta)\,W_{ij} +\beta\,\overline{W_{ij}}$$

note

Penalty factor $\beta$ is a configurable hyperparameter.

Bonding mechanics

The instant bond $\Delta B_{ij}$ of validator $i$ to miner $j$ is equal to $i$’s stake, $\,S_i$ times $i$'s bond-weight for $j$ normalized by $j$'s total bond-weight for all $\mathbb{V}$ of the subnet's validators:

See how it's calculated

$$\Delta B_{ij} = \frac{\,S_i \,\cdot\, \widetilde{W_{ij}}\,}{ \sum_{k \in \mathbb{V}} S_k \,\cdot\, \widetilde{W_{kj}}}$$

This then updates an exponential moving average (EMA) bond:

$$B_{ij}^{(t)} = \alpha \,\Delta B_{ij} + (1-\alpha)\,B_{ij}^{(t-1)}$$

note

The $\alpha$ variable here is unrelated to the concept of subnet specific currencies, referred to as alpha $\alpha$ tokens. Here $\alpha$ refers to a factor used in this EMA smoothing function—see consensus-based weights, a.k.a. liquid alpha.

Validators who stay near consensus build stronger EMA bonds and thus extract more emissions, while any attempt to overstate a particular miner’s performance is penalized. The EMA smooths out abrupt swings in validator behavior and incentivizes consistent alignment with the consensus.

Validator emissions

Each validator $i$’s share $V_i$ of validator emissions (41% of each subnet's total emissions) is the sum of all of its bonds to miners, weighted by the miner's total emissions:

See how it's calculated

$$V_i = \sum_{j \in \mathbb{M}} \Bigl(\,B_{ij} \,\times\, M_j\Bigr)$$