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Abstract: Full-parameter fine-tuning of large language models (LLMs) incurs substantial optimizer state memory overhead, limiting the model sizes that fit on modern GPUs. Existing approaches either compress optimizer state, abandon first-order gradients, or change the update geometry while retaining dense state. The recently introduced Muon optimizer reduces optimizer memory through matrix-valued updates. Still, its geometry differs from AdamW and can lead to performance degradation when fine-tuning AdamW-pretrained models. To reduce optimizer memory without sacrificing accuracy or computational efficiency in LLM fine-tuning, we propose Ternary Absolute-max Column-wise One-sparse optimizer, or TACO, which follows Muon's operator-norm steepest-descent view but takes the geometric route further. TACO computes the exact steepest-descent direction under a dimension-normalized $1\to1$ operator norm by selecting the sign of the largest magnitude entry in each column of two-dimensional weight matrices. This retains first-order gradients while making optimizer state memory nearly negligible. Our practical TACO optimizer maintains only a small set of low precision gradient components per column, reducing persistent optimizer state by $174\times$ relative to AdamW8bit (from 27.7 GB to 0.16 GB) and peak training memory by $2.9\times$ (from 80.6 GB to 27.5 GB) on OPT-13B, while achieving comparable accuracy and runtime. TACO further enables full-parameter fine-tuning of 30-32B-parameter models on a single 80 GB H100 GPU across multiple model families and tasks.

Abstract: When inference demand exceeds available compute capacity, model providers must decide which requests should be served first. Users have different tolerances for delay from an LLM API, but current priority pricing schemes compress these differences into coarse fixed-price service tiers. We design an inference auction that allows users to bid for faster service. Our auction allocates priority in an economically efficient way without sacrificing latency, and we develop fast algorithms for implementing prices that incentivize truthful bidding. We also design an autobidding agent for our inference auction, where users specify an inference budget and the autobidder dynamically adjusts its bids over time to maximize user utility subject to the budget constraint. Experiments validate the practicality of our auction: it increases system welfare while maintaining the cache utilization and latency advantages of SGLang, a state-of-the-art inference serving framework.

Abstract: Neural scaling, in which loss falls as a power law with training, is central to large language models, and one recent proposal is that a $1/3$ exponent emerges from learning peaked distributions. That account describes SGD, but models in practice are trained with adaptive optimizers. Here we separate two exponents the $1/3$ account does not distinguish: how fast the loss falls with training steps along a single run, and how fast the optimally tuned loss falls with dataset size $D$. We show that the first, a dynamic exponent, is optimizer-specific while the second, an optimal data exponent, converges to $1/3$ across optimizers. In an online teacher-student model we decompose the loss into norm growth (radial) and alignment toward the teacher direction (tangential), each decaying as a power law with dynamic exponents $α_{r}$ and $α_{t}$. Under SGD, both are close to $1/3$, so the data exponent is also $1/3$ across different learning rates. Under Adam the two separate: $α_{r} \simeq 0.48$ but $α_{t} \simeq 0.08$. Since the total loss is minimized when these two parts are balanced, the optimal learning rate is optimizer-dependent: $D$-independent for SGD but falls with $D$ for Adam. Yet tuned to that optimum, the loss returns to $D^{-1/3}$ for both. A stochastic-dynamics analysis explains why: the optimizers can trade decay speed between the two channels, but they all fall on a single dynamic exponent relation, $2α_{r}+ α_{t} = 1$, which fixes the optimal data exponent at $1/3$. Across seven optimizers, including Muon, the measured exponents are consistent with this relation, and the optimal-loss envelopes agree with $D^{-1/3}$ across them. The optimizer sets how fast a model learns per step; tuned optimally, it changes the prefactor but not the rate at which loss falls per sample.

Abstract: Modern LLMs with QK-normalization, gated attention, learned attention sinks, or logit softcapping exhibit weaker persistent attention sinks, on which existing KV cache eviction methods primarily rely. We observe that across these models, weaker sinks co-occur with greater value-vector dispersion relative to key-vector dispersion. Motivated by this value-side dispersion, we present ValueDiff, a value-geometric eviction that ranks tokens by the L2 deviation of their value vectors from the cache mean. The same score arises as the minimal-disturbance eviction under a max-entropy assumption about future attention. We evaluate under fixed cache budgets, with eviction at every block boundary during prefill and at every decoding step during generation. On RULER at a tight 2k token budget, ValueDiff retains 88--99\% of dense across seven sink-suppressed models (best on 6 out of 7). On LongBench at the 4k budget, ValueDiff averages 92\% retention across sink-suppressed models versus 83\% for the strongest prior baseline. On MATH-500, ValueDiff is the strongest non-dense method on every sink-suppressed model tested at the 25\% cache budget, outperforming prior methods by up to $\sim$20 points on gated-attention models. Across all three benchmarks, value geometry emerges as the more reliable query-invariant eviction signal for sink-suppressed models.