Ablation, Statistical Inference, and Validation for KV-Cache Compression

We analyse three KV cache quantization schemes under a fair bit budget: \textbf{KV} (scalar MSE baseline), \textbf{KQV} (WHT + MSE on $K$; WHT + MSE + QJL on $V$), and \textbf{QKQV} (WHT + MSE + QJL on both). Starting from the Beta distribution on the hypersphere, we trace how QJL on $K$ inflates inner product variance by $π/2$, which softmax amplifies nonlinearly via Jensen's inequality, and we present statistical inference and information metrics to highlight practical differences. Three empirical findings emerge. (1)~At $n=4$ (the practically dominant budget), KQV wins on every measure -- KL divergence, geometric $K$ error, and 6D distance -- across all distributions and ranks tested. (2)~The K--V asymmetry is unconditional: QKQV is consistently worse than KQV in KL divergence at every budget and distribution. (3)~A budget-dependent crossover exists: QKQV achieves better geometric $K$ reconstruction at $n \in \{2,3,5\}$, KQV at $n \in \{4,6\}$, invariant to rank and tail weight -- an open rate-distortion problem. $\mathrm{KL}(p_{\mathrm{ref}} \| p_{\mathrm{quant}})$, K-only by construction, bridges K direction error to routing corruption and output collapse. We present a sufficient condition when the Jensen mechanism amplifies superlinearly through the softmax. At $n \in \{2,3,5\}$, QKQV wins geometrically because this assumption does not bind. At $n=4$, elevated K error and KL divergence for QKQV strongly suggest the Jensen mechanism is the operative cause of the crossover, providing a new perspective and explanation.

Transformer inference on long sequences is expensive because softmax attention repeatedly reads from a large KV cache. The prevalent approach to this bottleneck is KV cache compression, which replaces the full cache with a compact summary. Despite its practical importance, the design of such summaries is largely driven by empirical experimentation. On the theoretical side, existing results show that KV cache compression can be impossible in the worst case, but offer little systematic guidance for designing algorithms in regimes where accurate compression is possible. We bridge this gap by characterizing the minimax risk of KV cache compression in terms of the intrinsic compressibility of a cache, revealing when and how accurate compression is possible. These results yield novel design principles for KV cache compression under causal masking that map efficiently to prefill and autoregressive decoding while achieving minimax-optimal risk. We instantiate these principles in a practical algorithm and report promising performance on LongBench in targeted experiments. Overall, our results provide a principled avenue for practical KV cache compression with theoretical guarantees.

What limits KV-cache compression at extreme bit-rates? We argue that it is not the choice of compression scheme, but how its budget is allocated across attention heads. Existing methods apply rank and bit-width uniformly, ignoring that each head has a different optimal mix of rank truncation and quantization. We show that co-optimizing rank and bit-width per head, using only standard low-rank projection and scalar quantization, dominates uniform allocation, with the largest gains at low bit-rates. Our method, KV-COBRA (Co-Optimized Bit-Rank Allocation), formalizes this as a resource-allocation problem: it balances rank-truncation loss against quantization loss within each head, then redistributes budget across heads to minimize total distortion. A fused Hadamard rotation equalizes per-channel variance, and reordering the SVD basis by attention-KL importance makes the solver query-aware. The same allocator extends to joint $K{+}V$ compression. On perplexity, zero-shot, and long-context benchmarks from $0.5$ to $4$ bits per dimension (bpd), KV-COBRA shows the smallest accuracy degradation among evaluated methods at low bpd, with no per-token overhead.