SUBMIT A PRSUBMIT AN ISSUElast edit: Jun 24, 2026

Exponential Moving Averages (EMAs) in Bittensor

The exponential moving average (EMA) is a mathematical technique for tracking a dynamic quantity, such as a token price, over time. Specifically, EMA is a weighted moving average that exponentially decreases the weight of older data point. This extracts a signal reflecting where the value has spent most of its time most recently, stabilizing or 'smoothing' the constant noise of rapid, largely random fluctuations.

Bittensor uses EMAs to smooth two critical dynamical values during the emission process:

• TAO emissions to each subnet are determined by a three-factor weighted share of the fixed block emission: each subnet's EMA price (SubnetMovingPrice), scaled by its root proportion and reduced by any miner-burn penalty.
• Alpha emissions to participants of each subnet are determined by EMAs of instantaneous validator-miner bond-strengths. This plays an important role in ensuring that validators and miners are fairly rewarded for innovation, as measured by eventual consensus (rather than immediate consensus) about miner weights.

Mathematical definition

The EMA of a changing value at a given time is determined by weighted average of the current value and the EMA at the last time step. The parameter factor, or 'smoothing factor' is called $\alpha$.

$$\mathrm{EMA}^{(t)} = \alpha \times \mathrm{current} + (1 - \alpha) \times \mathrm{EMA}^{(t-1)}$$

The alpha parameter controls how quickly the EMA responds to changes:

• Small $\alpha$ (e.g., 0.01): Very slow response, high stability, takes many periods for significant changes
• Large $\alpha$ (e.g., 0.5): Fast response, lower stability, quickly incorporates new information
• $\alpha$ = 1: No smoothing (immediate response to current value)

tip

Note that this alpha parameter is distinct from and unrelated to the usage of 'alpha' to refer to subnet-specific currencies.

Subnet Price EMA Smoothing

EMA smoothing protects the network's economic model from manipulation by making TAO emission shares extremely slow to respond to short-term price fluctuations.

How It Works: The price-based model uses an EMA of each subnet's token price (SubnetMovingPrice) to determine emission shares, rather than the live spot price. This means a sudden price spike or crash has minimal immediate effect on a subnet's emission share:

$$\mathrm{EMA}^{(t)} = \alpha \times \mathrm{current\_price} + (1 - \alpha) \times \mathrm{EMA}^{(t-1)}$$

Unlike the previous flow-based model which used a fixed smoothing factor, the price EMA uses a dynamic alpha that varies by subnet age. New subnets start with a near-zero alpha (extremely slow adaptation) and gradually approach base_alpha as they mature:

$$\alpha = \frac{ \mathrm{base\_alpha} \times \mathrm{blocks\_since\_start}}{\mathrm{blocks\_since\_start} + \mathrm{halving\_blocks}}$$

This dynamic alpha prevents:

• Short-term gaming through temporary price pumps, especially on newly registered subnets
• Coordinated buy pressure to capture emission shares
• Flash attacks on emission shares

info

Unlike the previous flow-based model, there is no zero-emission floor tied to net flows. A subnet's emission share tracks its relative EMA price — a subnet with a very low or declining EMA price receives a correspondingly small but non-zero share of block emissions (subject to the miner-burn penalty and root proportion weighting).

See:

• Subnet emissions source code
• Default alpha value for subnet price smoothing

Validator-Miner Bond Smoothing

This smoothing function ensures that relationships between validators and miners evolve gradually, preventing sudden manipulation while rewarding validators who discover promising miners early.

Basic Bond EMA (Liquid Alpha Disabled)

Default Mode: Single $\alpha$ for all validator-miner pairs

• Default $\alpha$: ~0.1 (10%)
• Response Time: 7-22 blocks for significant changes (~1-4 minutes)
• Formula The EMA of the bond (BondEMA)of a validator i for a miner j, at time t, is the $\alpha$-weighted average of the instantaneous bond and the previous timestep's BondEMA: $$BondEMA_{ij}^{(t)} = \alpha \times \, InstantBond_{ij} + (1-\alpha)\,BondEMA_{ij}^{(t-1)}$$

Advanced Bond EMA (Liquid Alpha Enabled)

Consensus-Based Mode: Dynamic $\alpha$ per validator-miner pair based on consensus alignment

• $\alpha$ Range: Dynamic between $\alpha$_low and $\alpha$_high (default: 0.7 to 0.9)
• Sigmoid Steepness: Controls transition rate between $\alpha$_low and $\alpha$_high (default: 1000)
• Individual Alpha: Each validator-miner pair gets its own $\alpha$ value
• Response Time: 1-13 blocks depending on consensus alignment (~12 seconds to 2.6 minutes)

See Liquid Alpha/Consensus-Based Weights