Certified Sparse Attention via Perturbed Retrieval

Large Language Model inference over long contexts is increasingly bottlenecked by the cost of attending to a growing Key-Value (KV) cache. Sparse attention reduces this cost by accessing only a fraction of the cache, but existing methods face a trade-off between deterministic retrieval, which captures high-scoring tokens but discards residual attention mass, and sampling, which improves coverage at the cost of higher variance. Hybrid methods combine top-$k$ retrieval with uniform sampling, yet still exhibit a substantial gap from dense attention at aggressive sparsity. We introduce Stratified Sparse Attention (SSA), a unified framework combining deterministic retrieval with score-guided sampling. SSA deterministically attends to high-scoring tokens while sampling the residual using proxy attention scores and correcting for their inclusion probabilities, thereby recovering attention mass beyond the retrieval set. SSA encompasses top-$k$, top-$p$, and hybrid top-$k$+sampling as special cases. Across RULER, LongBench, LOFT, and AIME, and five models ranging from 3B to 27B parameters, SSA consistently improves over the baselines at matched sparsity. On RULER-HARD with Ministral-3-8B, SSA improves accuracy over the strongest practical baseline from 81.8% to 84.2% at 32K and from 63.8% to 69.3% at 128K under $50\times$ attention sparsity, recovering 94% and 85% of dense-attention performance, respectively. With optimized CUDA kernels, SSA achieves up to $2.96\times$ decode speedup over FlashInfer at 1M context, demonstrating that its accuracy gains translate into practical long-context inference acceleration. We open-source our code at https://github.com/skylight-org/SSA-Paper-Code.

Massive KV caches can cause severe memory-bandwidth bottlenecks during long-context decoding. Sparse attention methods mitigate this problem, but often drop necessary context, leading to quality degradation. We introduce Elastic Threshold Attention (ETA), an end-to-end trainable architecture that rivals dense model quality under hardware-aligned block-sparse decoding. ETA predicts dynamic, contextual thresholds directly from query representations, adjusting context retention depending on the task at hand. To learn this policy from scratch while enabling near lossless KV cache pruning at inference time, we filter attention logits through a shifted SiLU gate during training. We show theoretically and empirically that this creates a smooth, near-uniform attention floor that neutralizes sub-threshold value contributions while simultaneously causing localized attention sinks on initial tokens to disappear. To materialize these advantages, we implement a fused inference-time kernel in Triton that screens KV blocks in $O(1)$ time using cached geometric-probabilistic bounds. Across language modeling, reasoning, and RULER benchmarks, our 1.45B ETA model matches or exceeds dense quality, outperforming alternative fast decoding methods (Quest, H$_2$O, NSA) while achieving higher sparsity levels. Our kernel also achieves up to $2.15\times$ end-to-end speedups over FlashAttention-2 at context lengths of up to $512$K tokens. Finally, we introduce an offline calibration algorithm for domain-specific deployments that freezes per-head constant thresholds, cutting attention compute by an additional 27% at no quality cost.

Sparse attention improves LLM inference efficiency by selecting a subset of key-value entries, but at the cost of potential accuracy degradation. In particular, omitting critical KV entries can induce substantial errors in model outputs. Existing methods typically operate under fixed or adaptive token budgets and provide empirical robustness or partial theoretical guarantees, yet they do not ensure zero false negatives in decoding steps, particularly since the set of relevant tokens is both query- and step-dependent. Our empirical observations confirm that missing even one critical key can lead to sharp error spikes, especially in long reasoning tasks where the set of important tokens varies throughout decoding. This observation motivates the need for indexing methods that dynamically adapt to these variations across decoding steps while guaranteeing a full recall of the relevant keys above a certain threshold. We address this challenge by reformulating sparse attention as the halfspace range searching problem. However, existing range searching indices are not suitable for modern LLM inference due to their computational and implementation overheads. To overcome this, we introduce Louver, a novel index structure tailored for efficient KV cache retrieval. Louver (i) guarantees zero false negatives with respect to a specified threshold in both theory and practice, (ii) is lightweight to integrate into existing LLM pipelines, and (iii) incorporates hardware-aware optimizations for both CPU and GPU executions. Our experiments demonstrate that Louver outperforms prior sparse attention methods in both accuracy and runtime, and is faster than highly optimized dense attentions such as FlashAttention. These results highlight that recall guarantees are a critical and overlooked dimension of sparse attention, and open a new direction for building theoretically grounded, efficient KV cache indices.