# adam(learningRate:beta1:beta2:epsilon:beta1Power:beta2Power:values:momentum:velocity:maximumVelocity:gradient:name:) Creates operations to apply Adam optimization. ``` func adam(learningRate learningRateTensor: MPSGraphTensor, beta1 beta1Tensor: MPSGraphTensor, beta2 beta2Tensor: MPSGraphTensor, epsilon epsilonTensor: MPSGraphTensor, beta1Power beta1PowerTensor: MPSGraphTensor, beta2Power beta2PowerTensor: MPSGraphTensor, values valuesTensor: MPSGraphTensor, momentum momentumTensor: MPSGraphTensor, velocity velocityTensor: MPSGraphTensor, maximumVelocity maximumVelocityTensor: MPSGraphTensor?, gradient gradientTensor: MPSGraphTensor, name: String?) -> [MPSGraphTensor] ``` ## Parameters `learningRateTensor` scalar tensor which indicates the learning rate to use with the optimizer `beta1Tensor` beta1Tensor `beta2Tensor` beta2Tensor `beta1PowerTensor` `beta1^t` beta1 power tensor `beta2PowerTensor` `beta2^t` beta2 power tensor `valuesTensor` values to update with optimization `momentumTensor` momentum tensor `velocityTensor` velocity tensor `maximumVelocityTensor` optional maximum velocity tensor `gradientTensor` partial gradient of the trainable parameters with respect to loss `name` name for the operation ## Return Value if maximumVelocity is nil array of 3 tensors (update, newMomentum, newVelocity) else array of 4 tensors (update, newMomentum, newVelocity, newMaximumVelocity) ## Discussion The adam update ops are added current learning rate: ```md lr[t] = learningRate * sqrt(1 - beta2^t) / (1 - beta1^t) m[t] = beta1 * m[t-1] + (1 - beta1) * g v[t] = beta2 * v[t-1] + (1 - beta2) * (g ^ 2) maxVel[t] = max(maxVel[t-1], v[t]) variable = variable - lr[t] * m[t] / (sqrt(maxVel) + epsilon) ``` --- Copyright © 2026 Apple Inc. All rights reserved. | [Terms of Use](https://www.apple.com/legal/internet-services/terms/site.html) | [Privacy Policy](https://www.apple.com/privacy/privacy-policy)