Batch-Invariant Kernels for LLM Reproducibility

• Batch-Invariant Kernels (BIO) are computational primitives that enforce a fixed reduction order over feature dimensions per sample, ensuring consistent outputs regardless of batch size.
• They mitigate nondeterminism by avoiding non-associative IEEE 754 floating-point addition issues in matrix multiplications and normalization operations in LLMs.
• Empirical findings show BIO reduces output variability and probability divergence, though it incurs moderate latency and requires pairing with TBIK for full determinism in tensor parallelism.

Batch-Invariant Kernels (BIO) are computational primitives designed to eliminate nondeterminism that arises from varying batch sizes in LLM inference and training workloads. The core objective of BIO is to guarantee that all reductions over feature dimensions for each sample in a batch are performed in a fixed order, independent of the batch’s cardinality. This approach targets the non-associativity of IEEE 754 floating‐point addition, which is a primary source of divergent numerical results across different computational settings. BIO achieves bit-wise reproducibility of outputs when batch size changes but does not address nondeterminism originating from other system aspects such as tensor parallel (TP) configuration (Zhang et al., 21 Nov 2025).

1. Nondeterminism in Floating-Point Computation and Motivation for BIO

Floating-point arithmetic, specifically addition as per the IEEE 754 standard, is non-associative: $(a + b) + c \neq a + (b + c)$ due to cumulative rounding errors. In the context of deep learning, batched inference typically utilizes matrix-multiply (MatMul) kernels that split the reduction (K-) dimension into tiles. These tiles are accumulated in orders that vary with both batch size and system configuration. Altering the order of floating-point summations—even with greedy decoding and fixed seeds—leads to discrepancies in hidden states, logits, and model outputs, causing nondeterministic behavior when batch or TP size changes (Zhang et al., 21 Nov 2025).

2. Fundamental Principles and Operation of Batch-Invariant Kernels

BIO kernels explicitly structure intra-sample reductions to ensure invariance to batch size. During MatMul, conventional implementations compute $C = A \cdot B$ by dividing the K-dimension into $T$ tiles and summing partial results in an arbitrary parenthesization, introducing order-dependent rounding. BIO replaces “split-K” partial reductions with per-tile full reductions (“non-split-K”), enforcing a fixed reduction topology per sample that remains unaltered across different batch sizes.

In normalization operations such as RMSNorm, BIO assigns each sample’s feature-dimension reduction (for example, $\sum_{i=1}^D x_i^2$) to a single thread/core. The transformation

$$y_j = \frac{x_j}{\sqrt{\epsilon + (1/D) \sum_{i=1}^D x_i^2}}$$

is computed with a consistent summation tree over $i = 1 \ldots D$, ensuring result invariance regardless of batch cardinality (Zhang et al., 21 Nov 2025).

3. Application and Integration Scope

BIO kernels are deployed in all operations where intra-sample reduction order can otherwise depend on batch partitioning. Typical LLM architectures require BIO for:

• Column-parallel layers: Query, Key, Value (QKV) projections, gate projections, up projections, and language modeling (lm_head) projections.
• Normalization and rotary positional encodings: RMSNorm and RoPE both employ BIO methods to ensure reproducibility of feature-dimension reductions.

BIO kernels are integrated into distributed inference/training frameworks such as vLLM and FSDP, where maintaining batch-size-invariant reduction order is necessary for reproducibility as batch scheduling and cardinality fluctuate across hardware configurations (Zhang et al., 21 Nov 2025).

4. Empirical Analysis and Observed Determinism with BIO

Experimental evaluation reveals that when only BIO is applied:

• The number of unique output sequences ($U$) for K=12 configuration combinations (TP=1/2/4/8, Batch Size = 8/16/32) reduces to approximately 7–8, compared to 12 for baseline BF16 MatMul kernels.
• Maximum probability divergence ($D$), defined as the average positional gap in top-5 token probabilities across configurations, is reduced (BIO $D\approx0.006$–0.02 vs. baseline $D\approx0.006$–0.03).
• BIO ensures batch-size invariance only when TP is fixed at 1 or 2; it does not guarantee bit-wise consistency against changes in TP size, indicating its scope is limited to batch-induced nondeterminism (Zhang et al., 21 Nov 2025).

Inference Configuration	Unique Outputs U	Max Probability Divergence D
Vanilla BF16	~12	0.006 – 0.03
BIO only	~7 – 8	0.006 – 0.02
BIO + TBIK	1	0

The table summarizes reproducibility metrics for various kernel strategies on multi-TP/multi-batch LLM inference (Zhang et al., 21 Nov 2025).

5. System Impact and Limitations

BIO is effective against batch-size–related nondeterminism but insufficient for eliminating TP-size–induced artifacts. In distributed training and serving environments with varying TP, output discrepancies persist under BIO alone; full precision alignment across parallel configurations requires additional order-invariant reduction schemes such as Tree-Based Invariant Kernels (TBIK). BIO also introduces moderate computational overhead due to serialization and thread assignment for per-sample reductions:

• BIO alone adds 18–99% end-to-end inference latency overhead over baseline MatMul workflows.
• A plausible implication is that scaling bio-infused kernels for ultra-large batches may necessitate further architectural or scheduling optimizations to control performance regressions (Zhang et al., 21 Nov 2025).

6. Relationship to Tree-Based Invariant Kernels and Future Extensions

BIO addresses batch-size parallelism invariance but cannot address nondenominism induced by changing TP size. For bit-wise identical outputs in multi-GPU or multi-node inference and training, BIO is paired with TBIK, which implements hierarchical binary-tree reductions for MatMul and all-reduce. This extended strategy aligns intra- and inter-GPU summation trees, providing reproducibility across both batch and TP variations (Zhang et al., 21 Nov 2025).

Future extensions pertain to integrating batch- and TP-invariant reductions for quantized data types (FP8, INT4), generalizing tree topologies to non-power-of-two TP, and optimizing BIO’s underlying reduction strategies to mitigate the observed inference latency overhead in high-throughput distributed environments.

7. Significance and Ongoing Challenges

BIO kernels represent a robust solution to a key reproducibility challenge in the deployment of LLM systems, supporting traceable, batch-size–invariant inference and training. However, guaranteeing determinism in all parallel strategies, especially as underlying hardware and software frameworks evolve, remains an open engineering challenge. In heterogeneous clusters, extending BIO to operate seamlessly alongside advanced networking topologies (like NVLink and hierarchical PCIe fabrics) is identified as critical for multi-node reproducible AI workflows (Zhang et al., 21 Nov 2025).