Underscoring the Problem: Why Softpick Fails at Initialization

Low-precision Transformer systems increasingly quantize attention matrix multiplications, while softmax often remains at higher precision. During pretraining, an approximate softmax changes the gradients that train the model as well as its forward computation. We study this interaction with K-interval attention, which approximates the exponential using K+1 grid values. We vary per-row grid calibration, interpolation versus hard rounding, and the placement of a straight-through surrogate relative to normalization. We derive the corresponding backward rules, including calibration derivatives, and compare these choices in pretraining experiments matched on model, data, and optimizer. Detaching the row extrema leaves the forward computation unchanged but produces a delayed increase in validation loss. With hard rounding at K=4, min-max calibration and a pre-normalization surrogate incur a large loss gap; changing either choice substantially reduces it. At 124M parameters and 2.5B training tokens, fixed-window calibration with a post-normalization surrogate yields a validation loss gap of +0.019 nats relative to softmax at K=4, and with a pre-normalization surrogate yields +0.004 nats at K=16.

Low-bit attention accelerates Transformer inference by moving the $QK^\top$ and $PV$ matrix multiplications to FP8 or FP4 matrix engines. However, the softmax path often evaluates shifted-score exponentials in higher precision, forms a temporary probability block, and quantizes it before low-bit $PV$ multiplication. This exp-then-quantize path creates a mismatch between a high-precision probability producer and a low-bit matrix consumer. We propose EFQ-Softmax (Exp-Free Quantization for Softmax), a low-bit probability-generation method that directly maps shifted attention scores to block-scaled E2M1 operands. For each microscaling block, EFQ-Softmax selects an exponent-only scale from the local maximum, maps the shifted scores to a normalized residual domain, and generates nonnegative E2M1 probability codes using a single affine rule. The resulting operand is used consistently in both the $\widetilde{P}V$ numerator update and the $\widetilde{P}\mathbf{1}$ denominator update. The FlashAttention-style row-maximum update, historical rescaling, high-precision accumulation, and final normalization remain unchanged. We evaluate end-to-end quality on Qwen3-8B, Qwen3-VL-8B-Instruct, and WAN2.2-TI2V-5B, and separately measure kernel-level performance on the A5 vector unit. EFQ-Softmax improves the Qwen3-8B seven-task mean from 0.6749 with MXFP4 to 0.6773 and the Qwen3-VL nine-task mean from 0.7826 to 0.8000. On WAN2.2, it maintains temporal consistency and visual quality comparable to the FP16 and MXFP4 baselines under VBench. On the A5 vector unit, EFQ-Softmax reduces the vector-stage latency of the fused probability-generation kernel by 40.33% on average across sequence lengths from 16K to 128K. These results show that direct low-bit probability generation can replace the conventional exp-then-quantize path while preserving end-to-end model quality.

Most post-training quantization pipelines fit each weight matrix to its pretrained counterpart, one matrix at a time. Whether that proxy tracks what an attention block actually computes, or how errors in the Q, K and V projections compound inside the softmax, is rarely checked. We write the objective on the attention output instead, over all three projections at once, and reuse it throughout the pipeline. JAB defines one scalar loss over the joint Q, K, V weights of a block, evaluated against the block's real causally-masked attention output, and uses it twice: to fit the quantized weights (GPTQ warm start, then STE with learnable scales), and to score the block for a multiple-choice knapsack allocation. On attention-only quantization of Mistral-7B this works. At 3 bits JAB recovers 77-90% of the gap between uniform GPTQ and full precision, and its sensitivity estimate tracks an oracle costing 73 forward passes to within a fraction of a point. It stops working once MLP layers enter the allocation. A role-aware offset rule needing no sensitivity estimate at all beats JAB on GPT-2's MLP and on the full Mistral-7B model: with a 3-bit floor it quantizes 96.4% of the weights to 4.5 bits per parameter at 6.933 perplexity, within 4.4% of full precision (6.643) at 3.56x compression, against 7.158 for JAB at the same budget. Which matrix a weight sits in matters more than any sensitivity estimate we computed. Two things came out sideways. Block-local reconstruction is an unreliable proxy for end-to-end perplexity: one run improved a block's own objective 4.6x while perplexity rose 32x, which is why every allocation here is validated end-to-end. And on attention-only quantization, fine-tuning moved weights farther from their pretrained values while pulling attention outputs closer, with net gains. Post-training seems to recover attention behavior, not weights.