ml-stable-diffusion/swift/StableDiffusion/pipeline/Scheduler.swift

// For licensing see accompanying LICENSE.md file.
// Copyright (C) 2022 Apple Inc. All Rights Reserved.

import CoreML

@available(iOS 16.2, macOS 13.1, *)
public protocol Scheduler {
    /// Number of diffusion steps performed during training
    var trainStepCount: Int { get }

    /// Number of inference steps to be performed
    var inferenceStepCount: Int { get }

    /// Training diffusion time steps index by inference time step
    var timeSteps: [Int] { get }

    /// Schedule of betas which controls the amount of noise added at each timestep
    var betas: [Float] { get }

    /// 1 - betas
    var alphas: [Float] { get }

    /// Cached cumulative product of alphas
    var alphasCumProd: [Float] { get }

    /// Standard deviation of the initial noise distribution
    var initNoiseSigma: Float { get }

    /// Compute a de-noised image sample and step scheduler state
    ///
    /// - Parameters:
    ///   - output: The predicted residual noise output of learned diffusion model
    ///   - timeStep: The current time step in the diffusion chain
    ///   - sample: The current input sample to the diffusion model
    /// - Returns: Predicted de-noised sample at the previous time step
    /// - Postcondition: The scheduler state is updated.
    ///   The state holds the current sample and history of model output noise residuals
    func step(
        output: MLShapedArray<Float32>,
        timeStep t: Int,
        sample s: MLShapedArray<Float32>
    ) -> MLShapedArray<Float32>
}

@available(iOS 16.2, macOS 13.1, *)
public extension Scheduler {
    var initNoiseSigma: Float { 1 }
}

@available(iOS 16.2, macOS 13.1, *)
public extension Scheduler {
    /// Compute weighted sum of shaped arrays of equal shapes
    ///
    /// - Parameters:
    ///   - weights: The weights each array is multiplied by
    ///   - values: The arrays to be weighted and summed
    /// - Returns: sum_i weights[i]*values[i]
    func weightedSum(_ weights: [Double], _ values: [MLShapedArray<Float32>]) -> MLShapedArray<Float32> {
        assert(weights.count > 1 && values.count == weights.count)
        assert(values.allSatisfy({ $0.scalarCount == values.first!.scalarCount }))
        var w = Float(weights.first!)
        var scalars = values.first!.scalars.map({ $0 * w })
        for next in 1 ..< values.count {
            w = Float(weights[next])
            let nextScalars = values[next].scalars
            for i in 0 ..< scalars.count {
                scalars[i] += w * nextScalars[i]
            }
        }
        return MLShapedArray(scalars: scalars, shape: values.first!.shape)
    }
}

/// How to map a beta range to a sequence of betas to step over
@available(iOS 16.2, macOS 13.1, *)
public enum BetaSchedule {
    /// Linear stepping between start and end
    case linear
    /// Steps using linspace(sqrt(start),sqrt(end))^2
    case scaledLinear
}


/// A scheduler used to compute a de-noised image
///
///  This implementation matches:
///  [Hugging Face Diffusers PNDMScheduler](https://github.com/huggingface/diffusers/blob/main/src/diffusers/schedulers/scheduling_pndm.py)
///
/// This scheduler uses the pseudo linear multi-step (PLMS) method only, skipping pseudo Runge-Kutta (PRK) steps
@available(iOS 16.2, macOS 13.1, *)
public final class PNDMScheduler: Scheduler {
    public let trainStepCount: Int
    public let inferenceStepCount: Int
    public let betas: [Float]
    public let alphas: [Float]
    public let alphasCumProd: [Float]
    public let timeSteps: [Int]

    // Internal state
    var counter: Int
    var ets: [MLShapedArray<Float32>]
    var currentSample: MLShapedArray<Float32>?

    /// Create a scheduler that uses a pseudo linear multi-step (PLMS)  method
    ///
    /// - Parameters:
    ///   - stepCount: Number of inference steps to schedule
    ///   - trainStepCount: Number of training diffusion steps
    ///   - betaSchedule: Method to schedule betas from betaStart to betaEnd
    ///   - betaStart: The starting value of beta for inference
    ///   - betaEnd: The end value for beta for inference
    /// - Returns: A scheduler ready for its first step
    public init(
        stepCount: Int = 50,
        trainStepCount: Int = 1000,
        betaSchedule: BetaSchedule = .scaledLinear,
        betaStart: Float = 0.00085,
        betaEnd: Float = 0.012
    ) {
        self.trainStepCount = trainStepCount
        self.inferenceStepCount = stepCount

        switch betaSchedule {
        case .linear:
            self.betas = linspace(betaStart, betaEnd, trainStepCount)
        case .scaledLinear:
            self.betas = linspace(pow(betaStart, 0.5), pow(betaEnd, 0.5), trainStepCount).map({ $0 * $0 })
        }
        self.alphas = betas.map({ 1.0 - $0 })
        var alphasCumProd = self.alphas
        for i in 1..<alphasCumProd.count {
            alphasCumProd[i] *= alphasCumProd[i -  1]
        }
        self.alphasCumProd = alphasCumProd
        let stepsOffset = 1 // For stable diffusion
        let stepRatio = Float(trainStepCount / stepCount )
        let forwardSteps = (0..<stepCount).map {
            Int((Float($0) * stepRatio).rounded()) + stepsOffset
        }

        var timeSteps: [Int] = []
        timeSteps.append(contentsOf: forwardSteps.dropLast(1))
        timeSteps.append(timeSteps.last!)
        timeSteps.append(forwardSteps.last!)
        timeSteps.reverse()

        self.timeSteps = timeSteps
        self.counter = 0
        self.ets = []
        self.currentSample = nil
    }

    /// Compute a de-noised image sample and step scheduler state
    ///
    /// - Parameters:
    ///   - output: The predicted residual noise output of learned diffusion model
    ///   - timeStep: The current time step in the diffusion chain
    ///   - sample: The current input sample to the diffusion model
    /// - Returns: Predicted de-noised sample at the previous time step
    /// - Postcondition: The scheduler state is updated.
    ///   The state holds the current sample and history of model output noise residuals
    public func step(
        output: MLShapedArray<Float32>,
        timeStep t: Int,
        sample s: MLShapedArray<Float32>
    ) -> MLShapedArray<Float32> {
        
        var timeStep = t
        let stepInc = (trainStepCount / inferenceStepCount)
        var prevStep = timeStep - stepInc
        var modelOutput = output
        var sample = s

        if counter != 1 {
            if ets.count > 3 {
                ets = Array(ets[(ets.count - 3)..<ets.count])
            }
            ets.append(output)
        } else {
            prevStep = timeStep
            timeStep = timeStep + stepInc
        }

        if ets.count == 1 && counter == 0 {
            modelOutput = output
            currentSample = sample
        } else if ets.count == 1 && counter == 1 {
            modelOutput = weightedSum(
                [1.0/2.0, 1.0/2.0],
                [output,  ets[back: 1]]
            )
            sample = currentSample!
            currentSample = nil
        } else if ets.count == 2 {
            modelOutput = weightedSum(
                [3.0/2.0,      -1.0/2.0],
                [ets[back: 1], ets[back: 2]]
            )
        } else if ets.count == 3 {
            modelOutput = weightedSum(
                [23.0/12.0,    -16.0/12.0,   5.0/12.0],
                [ets[back: 1], ets[back: 2], ets[back: 3]]
            )
        } else {
            modelOutput = weightedSum(
                [55.0/24.0,    -59.0/24.0,   37/24.0,      -9/24.0],
                [ets[back: 1], ets[back: 2], ets[back: 3], ets[back: 4]]
            )
        }

        let prevSample = previousSample(sample, timeStep, prevStep, modelOutput)
        counter += 1
        return prevSample
    }

    /// Compute  sample (denoised image) at previous step given a current time step
    ///
    /// - Parameters:
    ///   - sample: The current input to the model x_t
    ///   - timeStep: The current time step t
    ///   - prevStep: The previous time step t−δ
    ///   - modelOutput: Predicted noise residual the current time step e_θ(x_t, t)
    /// - Returns: Computes previous sample x_(t−δ)
    func previousSample(
        _ sample: MLShapedArray<Float32>,
        _ timeStep: Int,
        _ prevStep: Int,
        _ modelOutput: MLShapedArray<Float32>
    ) ->  MLShapedArray<Float32> {

        // Compute x_(t−δ) using formula (9) from
        // "Pseudo Numerical Methods for Diffusion Models on Manifolds",
        // Luping Liu, Yi Ren, Zhijie Lin & Zhou Zhao.
        // ICLR 2022
        //
        // Notation:
        //
        // alphaProdt       α_t
        // alphaProdtPrev   α_(t−δ)
        // betaProdt        (1 - α_t)
        // betaProdtPrev    (1 - α_(t−δ))
        let alphaProdt = alphasCumProd[timeStep]
        let alphaProdtPrev = alphasCumProd[max(0,prevStep)]
        let betaProdt = 1 - alphaProdt
        let betaProdtPrev = 1 - alphaProdtPrev

        // sampleCoeff = (α_(t−δ) - α_t) divided by
        // denominator of x_t in formula (9) and plus 1
        // Note: (α_(t−δ) - α_t) / (sqrt(α_t) * (sqrt(α_(t−δ)) + sqr(α_t))) =
        // sqrt(α_(t−δ)) / sqrt(α_t))
        let sampleCoeff = sqrt(alphaProdtPrev / alphaProdt)

        // Denominator of e_θ(x_t, t) in formula (9)
        let modelOutputDenomCoeff = alphaProdt * sqrt(betaProdtPrev)
        + sqrt(alphaProdt * betaProdt * alphaProdtPrev)

        // full formula (9)
        let modelCoeff = -(alphaProdtPrev - alphaProdt)/modelOutputDenomCoeff
        let prevSample = weightedSum(
            [Double(sampleCoeff), Double(modelCoeff)],
            [sample, modelOutput]
        )

        return prevSample
    }
}

/// Evenly spaced floats between specified interval
///
/// - Parameters:
///   - start: Start of the interval
///   - end: End of the interval
///   - count: The number of floats to return between [*start*, *end*]
/// - Returns: Float array with *count* elements evenly spaced between at *start* and *end*
func linspace(_ start: Float, _ end: Float, _ count: Int) -> [Float] {
    let scale = (end - start) / Float(count - 1)
    return (0..<count).map { Float($0)*scale + start }
}

extension Collection {
    /// Collection element index from the back. *self[back: 1]* yields the last element
    public subscript(back i: Int) -> Element {
        return self[index(endIndex, offsetBy: -i)]
    }
}