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.

The key-value (KV) cache dominates memory bandwidth and footprint in long-context autoregressive inference. Recent rotation-preconditioned codecs (TurboQuant, PolarQuant) show that a structured random rotation followed by a per-coordinate scalar quantizer matched to an analytically tractable marginal is a near-optimal recipe for KV compression. OCTOPUS advances this paradigm through joint quantization of rotated coordinate triplets. Each triplet's direction is mapped to a square via an octahedral parameterization, and the two resulting coordinates and the triplet norm are Lloyd-Max quantized against implementation-matched marginals. Optimizing the per-triplet squared error gives a strictly non-uniform bit allocation depending only on the total dimensionality of the keys. We find the finite-dimensional quality optimum with sweeps to be constant on every real decoder we test. The codec is data-oblivious, online, and deterministic given a seed. Across text, video, and audio, OCTOPUS matches or beats every prior rotation codec at every reported bit width and metric, with a lead that grows as bits drop for extreme compression. Furthermore, a fused Triton implementation reconstructs keys on the fly without materializing the uncompressed key, so the codec adds no decode-time bandwidth or latency over the existing dequantization. Project Page: https://octopus-quant.github.io/