Moreau Envelope and Huber Duality

The Moreau envelope provides a way to smooth non-differentiable convex functions while preserving their minimum. By adding a quadratic penalty scaled by a parameter λ, the proximal minimization operation yields a continuously differentiable envelope whose derivative represents a bounded slope, directly linking to the Huber loss.

Experiment

To visualize the smoothing effect and the duality gap in the interactive Moreau Envelope and Huber Duality module, follow these steps to adjust the parameters and observe the effect on the Moreau envelope.

1. Select Input x from the dropdown to set the evaluation point of the envelope.
2. Select Smoothing lambda to control the width of the quadratic bowl (Huber loss transition region). Observe how a larger λ creates a wider smoothed tip.
3. Select Trial z to evaluate the trial objective g(z) independently of the optimal proximal point.