feat: Implement Adaptive Sampling loop (Algorithm 2) for Online Mode

[Mox301]

Summary

This PR implements the Adaptive Sampling logic described in Algorithm 2 and Appendix B.1 of the paper.

Previously, the _deepthink_online method generated the entire trace budget in a single batch after warmup. This implementation updates the logic to generate traces iteratively, checking for consensus ($\beta \ge \tau$) before proceeding, ensuring the model stops generation early when a confident consensus is reached.

Changes

• Modified deepconf/wrapper.py:
  • Updated _deepthink_online to replace bulk generation with a while loop.
  • Implemented the consensus check $\frac{V(\hat{a})}{\sum V(a)} \ge \tau$.
  • Added consensus_threshold (default 0.95 ) and adaptive_step_size (default 1 to strictly follow Appendix B.1 logic) to the deepthink interface.

Motivation

The paper describes "Online Thinking" as a dynamic process that halts trace generation once consensus is reached to save tokens. The previous implementation lacked this iterative check, generating the full total_budget regardless of consensus. This change aligns the codebase with the paper's efficiency claims (e.g., Figure 6).

Test Plan

Verified using the existing examples/example_online.py.

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