Implemented Delayed Preconditioners for noise reduction (#88) Split PR 4

[debanganghosh08]

This PR completes the implementation suite for Issue #88 by adding an example of Delayed Preconditioners (DP²), based Research Paper, "Differentially Private Adaptive Optimization with Delayed Preconditioners".

Technical Logic:

1. Standard private adaptive optimizers suffer from rapid noise accumulation in the second-moment estimate. This implementation introduces the alternating-phase protocol, keeping the preconditioner stale for a window of 's' steps while accumulating noised gradients.
2. The preconditioner is then updated using the average of this window, significantly increasing the signal-to-noise ratio of the adaptive update.

✅ Correctness Verification: I have implemented an automated state-snapshot check within the script:
Phase 1 (Staleness): The script saves a snapshot at Step 4 and asserts that the preconditioner remains identical to the initial state.
Phase 2 (Update): It saves a snapshot at Step 5 and verifies that a successful update occurred following the accumulation window.
These assertions pass consistently, proving the "Snap" update logic is functioning correctly.

This is the last PR for the Split section for Issue #88 as you asked.
Kindly review and give a feedback! Thank you.