An Empirical Audit of k-NAF Budget Accounting for Anchored Decoding

While reinforcement learning has enabled LLM-based search agents to invoke external tools, existing methods train under fixed budgets and cannot adapt when constraints vary at deployment. We propose AnySearch, a framework that enables a single policy to perform budget-aware search under any budget constraint through a training scaffold and curriculum reinforcement learning. In the first phase, we train the agent with explicit budget state injection and structured reasoning prompts that guide efficient allocation under linearly decaying budgets. In the second phase, the scaffold is removed and the agent learns to operate autonomously under adaptively sampled budget constraints, matching inference conditions. Both phases are optimized with a composite reward that couples answer accuracy with budget efficiency through absolute and relative signals, where an adaptive weight amplifies the efficiency signal for high-accuracy queries and attenuates it for low-accuracy ones. Extensive experiments on seven general and multi-hop QA benchmarks show that our method outperforms baselines across all budget scales, generalizes to unseen constraints beyond the training range, and achieves superior tool productivity without excessive token overhead. Our code is available at https://github.com/xwsun01/AnySearch.

Speculative decoding speeds up autoregressive decoding by using a drafter to propose multiple tokens that a verifier validates in parallel. In resource-constrained deployments, the drafter uses a sparse KV cache to limit peak GPU memory and end-to-end latency under a fixed KV budget, while the verifier keeps a full KV cache. Mid-to-long context inference (4K--16K context length) is common in real applications. However, naive sparse/full speculative decoding suffers from the sparse/full mismatch as context length grows, causing the acceptance rate to drop quickly. We propose BudgetDraft, a multi-view sparse training method for sparse drafting in mid-to-long inference. The drafter is exposed to multiple sampled KV budgets during training and learns to align each sparse view with one shared full-cache teacher target. BudgetDraft combines an acceptance-aware loss on a full-cache branch with a multi-view loss on a sparse-cache branch, producing a single budget-robust drafter that recovers acceptance across sparsity levels without extra inference-time components. Experimental results on PG-19, LongBench, and LWM show that BudgetDraft achieves up to 6.55x, 4.46x, and 2.10x end-to-end speedup vs AR at 4K, 8K, and 16K context lengths, while keeping the inference pipeline memory-friendly.

Large language models increasingly support scientific and algorithmic discovery through inference-time search over evaluated candidates. Existing adaptive discovery controllers assign credit based only on score progress, even though prompt length, retries, and guidance calls cause search actions to incur different token costs. We prove that cost-blind credit can forfeit all but a vanishing fraction of attainable quality as frontiers multiply and costs diverge. Under a fixed search-side token budget, the controller must decide which frontier is improving and whether its gain justifies the realized cost before the budget is exhausted. We introduce \textbf{CostAda}, a cost-calibrated adaptive controller built around \emph{cost-calibrated frontier utility}. The utility values frontier progress relative to realized action cost and conditions that credit on the remaining budget. CostAda uses this signal to control local exploration intensity, frontier allocation, and budgeted tactic intervention. Cost and remaining budget therefore shape the search rather than serving only as accounting variables or a stopping rule. CostAda reaches the strongest baseline's full-budget quality with at most half the budget on twelve of sixteen benchmark--backbone pairs while achieving the strongest mean final quality on all eight benchmarks under GLM-5 and GPT-5.4.