DPMSolverMultistepScheduler.swift

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

import Accelerate
import CoreML

/// A scheduler used to compute a de-noised image
///
///  This implementation matches:
///  [Hugging Face Diffusers DPMSolverMultistepScheduler](https://github.com/huggingface/diffusers/blob/main/src/diffusers/schedulers/scheduling_dpmsolver_multistep.py)
///
/// It uses the DPM-Solver++ algorithm: [code](https://github.com/LuChengTHU/dpm-solver) [paper](https://arxiv.org/abs/2211.01095).
/// Limitations:
///  - Only implemented for DPM-Solver++ algorithm (not DPM-Solver).
///  - Second order only.
///  - Assumes the model predicts epsilon.
///  - No dynamic thresholding.
///  - `midpoint` solver algorithm.
@available(iOS 16.2, macOS 13.1, *)
public final class DPMSolverMultistepScheduler: 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]

    public let alpha_t: [Float]
    public let sigma_t: [Float]
    public let lambda_t: [Float]
    
    public let solverOrder = 2
    private(set) var lowerOrderStepped = 0
    
    /// Whether to use lower-order solvers in the final steps. Only valid for less than 15 inference steps.
    /// We empirically find this trick can stabilize the sampling of DPM-Solver, especially with 10 or fewer steps.
    public let useLowerOrderFinal = true
    
    // Stores solverOrder (2) items
    private(set) var modelOutputs: [MLShapedArray<Float32>] = []

    /// Create a scheduler that uses a second order DPM-Solver++ algorithm.
    ///
    /// - 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

        // Currently we only support VP-type noise shedule
        self.alpha_t = vForce.sqrt(self.alphasCumProd)
        self.sigma_t = vForce.sqrt(vDSP.subtract([Float](repeating: 1, count: self.alphasCumProd.count), self.alphasCumProd))
        self.lambda_t = zip(self.alpha_t, self.sigma_t).map { α, σ in log(α) - log(σ) }

        self.timeSteps = linspace(0, Float(self.trainStepCount-1), stepCount+1).dropFirst().reversed().map { Int(round($0)) }
    }
    
    /// Convert the model output to the corresponding type the algorithm needs.
    /// This implementation is for second-order DPM-Solver++ assuming epsilon prediction.
    func convertModelOutput(modelOutput: MLShapedArray<Float32>, timestep: Int, sample: MLShapedArray<Float32>) -> MLShapedArray<Float32> {
        assert(modelOutput.scalars.count == sample.scalars.count)
        let (alpha_t, sigma_t) = (self.alpha_t[timestep], self.sigma_t[timestep])
        
        // This could be optimized with a Metal kernel if we find we need to
        let x0_scalars = zip(modelOutput.scalars, sample.scalars).map { m, s in
            (s - m * sigma_t) / alpha_t
        }
        return MLShapedArray(scalars: x0_scalars, shape: modelOutput.shape)
    }

    /// One step for the first-order DPM-Solver (equivalent to DDIM).
    /// See https://arxiv.org/abs/2206.00927 for the detailed derivation.
    /// var names and code structure mostly follow https://github.com/huggingface/diffusers/blob/main/src/diffusers/schedulers/scheduling_dpmsolver_multistep.py
    func firstOrderUpdate(
        modelOutput: MLShapedArray<Float32>,
        timestep: Int,
        prevTimestep: Int,
        sample: MLShapedArray<Float32>
    ) -> MLShapedArray<Float32> {
        let (p_lambda_t, lambda_s) = (Double(lambda_t[prevTimestep]), Double(lambda_t[timestep]))
        let p_alpha_t = Double(alpha_t[prevTimestep])
        let (p_sigma_t, sigma_s) = (Double(sigma_t[prevTimestep]), Double(sigma_t[timestep]))
        let h = p_lambda_t - lambda_s
        // x_t = (sigma_t / sigma_s) * sample - (alpha_t * (torch.exp(-h) - 1.0)) * model_output
        let x_t = weightedSum(
            [p_sigma_t / sigma_s, -p_alpha_t * (exp(-h) - 1)],
            [sample, modelOutput]
        )
        return x_t
    }

    /// One step for the second-order multistep DPM-Solver++ algorithm, using the midpoint method.
    /// var names and code structure mostly follow https://github.com/huggingface/diffusers/blob/main/src/diffusers/schedulers/scheduling_dpmsolver_multistep.py
    func secondOrderUpdate(
        modelOutputs: [MLShapedArray<Float32>],
        timesteps: [Int],
        prevTimestep t: Int,
        sample: MLShapedArray<Float32>
    ) -> MLShapedArray<Float32> {
        let (s0, s1) = (timesteps[back: 1], timesteps[back: 2])
        let (m0, m1) = (modelOutputs[back: 1], modelOutputs[back: 2])
        let (p_lambda_t, lambda_s0, lambda_s1) = (Double(lambda_t[t]), Double(lambda_t[s0]), Double(lambda_t[s1]))
        let p_alpha_t = Double(alpha_t[t])
        let (p_sigma_t, sigma_s0) = (Double(sigma_t[t]), Double(sigma_t[s0]))
        let (h, h_0) = (p_lambda_t - lambda_s0, lambda_s0 - lambda_s1)
        let r0 = h_0 / h
        let D0 = m0
        
        // D1 = (1.0 / r0) * (m0 - m1)
        let D1 = weightedSum(
            [1/r0, -1/r0],
            [m0, m1]
        )
        
        // See https://arxiv.org/abs/2211.01095 for detailed derivations
        // x_t = (
        //     (sigma_t / sigma_s0) * sample
        //     - (alpha_t * (torch.exp(-h) - 1.0)) * D0
        //     - 0.5 * (alpha_t * (torch.exp(-h) - 1.0)) * D1
        // )
        let x_t = weightedSum(
            [p_sigma_t/sigma_s0, -p_alpha_t * (exp(-h) - 1), -0.5 * p_alpha_t * (exp(-h) - 1)],
            [sample, D0, D1]
        )
        return x_t
    }

    public func step(output: MLShapedArray<Float32>, timeStep t: Int, sample: MLShapedArray<Float32>) -> MLShapedArray<Float32> {
        let stepIndex = timeSteps.firstIndex(of: t) ?? timeSteps.count - 1
        let prevTimestep = stepIndex == timeSteps.count - 1 ? 0 : timeSteps[stepIndex + 1]

        let lowerOrderFinal = useLowerOrderFinal && stepIndex == timeSteps.count - 1 && timeSteps.count < 15
        let lowerOrderSecond = useLowerOrderFinal && stepIndex == timeSteps.count - 2 && timeSteps.count < 15
        let lowerOrder = lowerOrderStepped < 1 || lowerOrderFinal || lowerOrderSecond
        
        let modelOutput = convertModelOutput(modelOutput: output, timestep: t, sample: sample)
        if modelOutputs.count == solverOrder { modelOutputs.removeFirst() }
        modelOutputs.append(modelOutput)
        
        let prevSample: MLShapedArray<Float32>
        if lowerOrder {
            prevSample = firstOrderUpdate(modelOutput: modelOutput, timestep: t, prevTimestep: prevTimestep, sample: sample)
        } else {
            prevSample = secondOrderUpdate(
                modelOutputs: modelOutputs,
                timesteps: [timeSteps[stepIndex - 1], t],
                prevTimestep: prevTimestep,
                sample: sample
            )
        }
        if lowerOrderStepped < solverOrder {
            lowerOrderStepped += 1
        }
        
        return prevSample
    }
}