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Image / Video Generation 训练优化

Solvability of Approximate Agreement on Graphs and Simplicial Complexes

arXiv 2026-06-23

Approximate agreement tasks on graphs are discrete relaxations of consensus, where each process in a distributed system is given as input a vertex on a graph \(G\), and processes have to output vertices that lie on a clique of \(G\) contained in the convex hull of the input vertices. Although such tasks have been widely studied in a variety of models, graph classes and notions of convexity, it remains largely open for which classes of graphs these problems are solvable in asynchronous systems. In this work, we give a complete topological characterisation of the \(t\)-resilient solvability of approximate agreement on graphs and simplicial complexes in asynchronous shared-memory systems with read-write registers. As a result, we answer several open problems related to different variants of approximate agreement on graphs. For example, we give the first proof of Ledent's conjecture [PODC 2021] about the wait-free solvability of clique agreement. In fact, we show a more general result: clique agreement is \(t\)-resilient solvable on a graph \(G\) if and only if its clique complex is \((t-1)\)-connected in the homotopical sense. We also show that clique and monophonic agreement are solvable on the same class of graphs, but there exists a separation between monophonic and geodesic agreement, answering a question by Alistarh et al. [TCS 2023]. In the message-passing setting, our results imply new resilience bounds for asynchronous approximate agreement and round lower bounds for synchronous approximate agreement on graphs.

Less is More: Quality-Aware Training Data Selection for Scientific Summarization

arXiv 2026-06-23

Scientific long-document summarization datasets commonly treat author-written abstracts as gold reference summaries, although their quality and alignment with the source article vary. At the same time, publicly available scientific summarization datasets remain limited in scale and structure for modern long-context models. In this work, we address both challenges by a) constructing and releasing one of the largest biomedical and life science datasets for long-document summarization, containing 1.88 million PMC articles, and b) analyzing the reference quality of author-written abstracts with source-grounded and model-based metrics. We show that author-written abstracts vary in their alignment with the full article and that these quality signals can guide training-data selection. Training on selected high-quality subsets outperforms random sampling at matched training sizes and can match or exceed larger random subsets on factuality-oriented metrics. Our findings suggest that reference quality is an important factor in scientific summarization and that quality-aware data selection can improve training efficiency.

Holistic Data Scheduler for LLM Pre-training via Multi-Objective Reinforcement Learning

arXiv 2026-06-23

The composition of training data, governed by the diversity of sources and their mixing strategy, is a cornerstone of Large Language Model (LLM) pre-training. Online Data Mixing (ODM), the technique of adaptively adjusting data mixtures during training, has emerged as a promising direction to improve efficiency. However, existing methods are constrained by their reliance on a singular optimization perspective, which fundamentally overlooks the need for complex LLM pre-training to consider the dynamic data composition from multiple dimensions. To overcome this limitation, we introduce the Holistic Data Scheduler (HDS), a novel online data mixing framework. HDS formulates the data scheduling challenge as a reinforcement learning problem in a continuous control space and leverages the Soft Actor-Critic (SAC) algorithm for its stability and sample efficiency in exploring the high-dimensional policy space. At the core of HDS lies a novel multi-objective, holistic reward function that integrates three critical perspectives: a data-driven reward for quality, a loss-driven reward capturing inter-domain influence, and a model-driven reward based on weight norms. To validate our design and determine its optimal configuration, we conducted systematic experiments on LLMs of various sizes. On The Pile benchmark, HDS reaches the final validation perplexity of the next best method with 44% fewer training iterations. Furthermore, it achieves a 7.2% improvement on the MMLU 0-shot task along with consistent gains on other benchmarks, showcasing its ability to enhance both training efficiency and final model capability.

Semi-asynchronous Federated Learning in Flower: Framework Extension and Performance Assessment

arXiv 2026-06-23

This paper presents an extension of the Flower federated learning framework to support Semi-Asynchronous Federated Learning. The proposed approach adapts the traditional synchronous paradigm to better handle client heterogeneity and straggler effects. By introducing a semi-asynchronous training strategy, the system allows partial synchronization among clients while maintaining training efficiency and scalability. We implement and evaluate the proposed modification within Flower, instantiated as the FedSaSync strategy, demonstrating improved robustness and reduced idle time compared to fully synchronous baselines in heterogeneous environments. The results show that SAFL can balance convergence stability and system efficiency in heterogeneous environments typical of edge and distributed learning scenarios.

cuSBF: A Minimizer-Aware Bloom Filter for Genomic Sequence Data on Modern GPUs

arXiv 2026-06-23

Efficient genomic k-mer indexing depends on approximate membership query (AMQ) structures that must deliver high throughput, low false-positive rates (FPR), and modest memory footprints. The Super Bloom filter (SBF) is attractive for this scenario because minimizer-guided sharding and the Findere scheme exploit the redundancy of overlapping k-mers. However, those same features cause high per-k-mer compute cost, severe register pressure, and irregular memory accesses, which hinder an effective GPU implementation. We present cuSBF, an open-source, header-only CUDA library that implements SBF for sequence-native workloads. cuSBF's design merges sectorized shards, cooperative shared-memory tiling, warp-level shard sharing, and segmented warp reductions, turning super-k-mer locality into scalable GPU parallelism. Across real genomic workloads on RTX PRO 6000 Blackwell and GH200 systems, cuSBF achieves the highest throughput among all evaluated sequence-capable baselines. On the RTX PRO 6000, it surpasses the cuCollections blocked Bloom filter baseline by up to 9.1x for insertion and 7.7x for query, while reaching up to 92x and 234x speedups over the multi-threaded CPU Super Bloom reference implementation. It also outperforms GPU-based dynamic AMQs (Cuckoo, Two-Choice, Quotient filters) by 1.5-3400x depending on workload characteristics. A parameter sweep identifies (s = 28, m = 16, H = 4) as Pareto-optimal for k = 31, yielding significantly lower FPR than cuCollections at matched memory budgets. Crucially, cuSBF's architecture-aware design sustains 85% streaming multiprocessor utilization even for out-of-cache filters - proving that sequence locality, not raw bandwidth, is the key to GPU-accelerated genomic indexing.

When Top-1 Fails: Calibrating LoRA Monitors for Masked Diffusion LMs

arXiv 2026-06-23

Discrete diffusion language model (DLM) fine-tuning inherits inexpensive diagnostics from denoising-time confidence monitors, but their PEFT-training meaning is untested. We test top-1 argmax concentration as a collapse warning. Across 816 LoRA/PEFT configurations from three DLM families, the warning fires for every configuration while logs record 0/816 actual collapses at the 200 step horizon, giving zero precision. The cause is pre-equilibrium saturation: top-1 concentration is already high before optimization and quickly becomes insensitive to final training stability. We then evaluate max LoRA gradient norm, a parameter-side signal that samples gradient routing rather than token concentration. On a pooled held-out LLaDA-family split, a train-optimized threshold identifies top-decile final-loss configurations with precision 0.68 and F1=0.79, above the all-positive top-1 baseline even at the lower split-bootstrap confidence bound. Autoregressive controls and cross-family threshold failures bound the result to short-horizon DLM-LoRA inspection rather than a universal collapse detector. Workflow: drop top-1 as a PEFT alarm, log max-gradient early in training, and calibrate thresholds per DLM family before routing runs for inspection.

BluTrain: A C++/CUDA Framework for AI Systems

arXiv 2026-06-23

Progress in deep learning is, at scale, more a matter of systems engineering than of modelling: the behaviour of a model in training (its throughput, its memory footprint, and the numerical fidelity of the result) is determined less by the architecture itself than by how that architecture is expressed on the hardware. To achieve absolute control over this hardware expression while abstracting away systems complexity to make modelling seamless and eliminating the need for repetitive orchestration logic, BluTrain was architected from first principles as a robust, lightweight, and architecture-general training framework in standard C++ and the core CUDA programming model. Every layer is implemented natively: a typed tensor module with reverse-mode autograd, a linear-algebra library, a caching allocator, a multi-mode distributed-execution module, and an MLIR-based deep-learning compiler. In formal evaluations training a 124M-parameter GPT-2 baseline in FP32 on an 8-GPU 6000 Ada system, BluTrain outperforms industry-standard baselines in both throughput (sustaining an average of 407K tokens/s versus PyTorch's 395K tokens/s) and memory efficiency (achieving up to a 22% footprint reduction), while strictly preserving numerical fidelity and converging to a marginally lower final validation loss. With every layer explicitly open to native tuning, the performance ceiling is the framework's own to raise.

PixJail: Self-Evolving Paper-to-Pipeline Reproduction for Text-to-Image Jailbreak Evaluation

arXiv 2026-06-23

As Text-to-Image (T2I) jailbreak techniques evolve rapidly, existing benchmarks and reproduction workflows often struggle to keep pace. More importantly, T2I jailbreak evaluation is not a single prompt-level test, but a pipeline-level problem shaped by multiple stages, including prompt transformation, image generation, safety filtering, and multimodal judging. This makes results across papers difficult to reliably reproduce and fairly compare. To bridge this gap, we propose PixJail, a self-evolving paper-to-pipeline agent framework for reproducible T2I jailbreak evaluation. Given a T2I jailbreak paper and optional reference code, PixJail rapidly constructs a paper-specific attack module and a runnable evaluation pipeline under a unified contract, while faithfully reproducing the original experimental results. PixJail further maintains a memory bank that stores paper digests, attack evolution patterns, reusable templates, failure cases, and versioned artifacts, enabling future reproduction efforts to reuse prior experience. We reproduce eleven representative T2I jailbreak methods, including both code-available and code-unavailable papers. Under their original settings, our framework accurately recovers prior results with minimal error (2.1\% average, 0\% median). We hope that PixJail can serve as a unified foundation for future T2I jailbreak reproduction and evaluation, significantly reducing manual effort.

Accelerating Disaggregated RL for Visual Generative LLMs with Diffusion-Based Parallelism and Trainer-Assisted Generation

arXiv 2026-06-23

Reinforcement learning (RL) has become a dominant post-training paradigm, driving the emergence of high-performance RL systems such as veRL for autoregressive large language models (LLMs). In parallel, diffusion-oriented RL algorithms, e.g., DanceGRPO and FlowGRPO, have rapidly expanded the scope of RL from language reasoning to diffusion-based visual and flow-based generation. However, efficient RL systems for diffusion generative LLMs remain underexplored. Existing implementations, e.g., veRL-Omni, still rely on colocated execution, which simplifies synchronization but couples rollout and training resources, limits heterogeneous deployment, and constrains independent scaling. To this end, we introduce DigenRL, a disaggregated RL framework for diffusion-based generative LLMs that supports flexible resource allocation, accommodates heterogeneous GPUs, and facilitates efficient task scheduling. To maximally reduce the execution bubbles in the disaggregated architecture, we propose: 1) a generation-axis pipeline (GAP) and time-step parallelism (TSP) in the diffusion architecture to enable finer-grained pipelining between rollout and training; 2) an elastic trainer-assisted generation (TAG) approach to enable the trainer GPU resources to dynamically assist in executing rollout generations; and 3) a tightly one-step constrained asynchronous strategy to further utilize the tail bubble in the pipeline. Extensive experiments are conducted on three hardware testbeds with 16-32 GPUs using HunyuanVideo-13B, Wan2.1-14B, FLUX.1-12B, and QwenImage-20B generative models. Experimental results show that DigenRL achieves 1.56-2.10x throughput improvements over state-of-the-art diffusion RL systems, veRL-Omni and GenRL.

DBT-Bleed: Dual-Branch Temporal Modeling with Key-Frame Selection for Surgical Bleeding Detection

arXiv 2026-06-22

Intraoperative Adverse Events (IAEs) detection is critical for improving surgical safety, with bleeding being among the most frequent events across many surgery types. Existing methods struggle to distinguish bleeding IAE from visually similar residual blood due to limited temporal reasoning. Moreover, modeling long surgical videos while preserving fine-grained temporal dynamics remains computationally challenging. We propose DBT-Bleed, a dual-branch multi-scale temporal modeling framework disentangling bleeding and normal representations using layer-wise temporal adapters for short- and long-term bleeding progression. To efficiently process long surgical videos without sacrificing fine-grained temporal information, we introduce HiRED, a Hierarchical Entropy-Driven frame selection strategy that retains temporally informative segments while removing redundancy. Experiments on the MultiBypass dataset demonstrate gains of 6.53% in F1, 5.62% in Recall and 9% in MCC values for bleeding IAE detection, consistently outperforming video-level baselines. Additionally, we evaluate cross-procedure generalization on a newly curated dataset from a different surgical procedure type, where DBT-Bleed demonstrates robust transferability by achieving gain of 6% in F1 and 8% in MCC under zero-shot setting. To support this evaluation, we introduce EndoPit-IAE, an Endonasal Pituitary Surgery dataset annotated for IAEs, representing the first IAE-annotated dataset in neurosurgery. Code will be made publicly available upon acceptance.

HiL-ResRL: A Model-Agnostic Finetuning Adapter via Human-in-the-loop Residual Reinforcement Learning

arXiv 2026-06-22

Recent advancements in generative imitation learning have significantly propelled the field of robotic manipulation. However, the majority of existing models rely heavily on Behavior Cloning (BC), a paradigm that suffers from compounding errors and distributional shift. Consequently, the efficacy of these models in practical industrial deployments remains limited. To address these challenges, we introduce a novel, plug-and-play fine-tuning pipeline designed to facilitate the robust deployment of Vision-Language-Action (VLA) models in real-world environments. In contrast to contemporary reinforcement learning (RL) fine-tuning strategies, which are often constrained by specific model architectures, our proposed framework is model-agnostic and adaptable to a diverse range of VLA models. We conceptualize VLA-generated actions as a unified interface, upon which we train a residual policy. This policy is designed to rectify suboptimal actions and address the distributional shift inherent in imitation learning. Additionally, we incorporate human-in-the-loop guidance to ensure safe exploration and maximize training efficiency. We conduct experiments directly in real-world robotic settings. The results demonstrate that within only 1.5 hour of real-world online RL training, the average success rate exceeds 95% on real robots. Our work presents a practical solution for deploying behavior cloning models in industrial scenarios.

Litmus: Zero-Label, Code-Driven Metric Specification for Evaluating AI Systems

arXiv 2026-06-22

As agentic LLM systems move from prototypes to deployment across increasingly diverse domains, evaluating them has become both more important and more difficult. The challenge is not only that individual metrics may be unreliable, but that evaluation goals are often left implicit. Without a clear account of what a system is expected to do, how it can fail, and which failures matter, metric choices become difficult to justify, interpret, or validate. We present Litmus, a zero-label system that designs evaluation and monitoring metrics for AI pipelines by eliciting evaluation intent from source code and targeted interrogation. Instead of assuming that the evaluation target is already known, Litmus first identifies what must be measured and why, then converts those answers into constraints for constructing a justified, per-stage metric portfolio. We evaluate Litmus on three real, code-defined AI pipelines - financial account grouping, scientific QA, and inherent risk assessment - against AutoMetrics and three DynamicRubric baselines. Litmus achieves the broadest or tied-broadest concern coverage, spans more pipeline stages, produces a near-zero-redundancy portfolio, and ranks first in validity against per-row quality labels on all three pipelines - decisively on scientific QA (Spearman \(ρ=0.72\) vs. less than \(0.47\) for every baseline), and within overlapping confidence intervals in relation to two components of the audit framework despite using no labels during metric design. Our results support a shift from automatic metric implementation to automatic metric specification: before asking which metric to compute, evaluation systems should ask what must be measured and why.

Factored Gossip DiLoCo: Reducing Blocking Communication in DiLoCo

arXiv 2026-06-22

To make large-scale distributed training practical outside high-bandwidth datacenters, we must reduce blocking, high-volume synchronization. While DiLoCo communicates infrequently, its outer synchronization remains bandwidth-heavy and brittle to stragglers and transient failures. We relax exact synchronization to approximate synchronization via mixing/gossip, which degrades gracefully under delays and communication failures. This allows us to factorize DiLoCo synchronization into a non-blocking mixing step that overlaps computation with no staleness, and a blocking mixing step that tightens worker agreement, yielding a tunable trade-off between compute utilization and optimization stability. On up to billion-parameter language models in low-bandwidth settings, our framework substantially improves compute utilization compared to DiLoCo, with training progress ranging from comparable to closely matching it, and is more robust to failures.

LMS-AR: LMS Prediction-based Adaptive Regulator for Memory Bandwidth in Multicore Systems

arXiv 2026-06-22

Memory bandwidth contention in multi-core systems severely impacts application performance and quality-of-service (QoS) guarantees. Regulating the shared memory bandwidth mitigates the memory performance uncertainty thereby making it a manageable resource and improving trustworthiness of multi-core systems. In this work we propose a memory bandwidth regulation mechanism LMS-AR, i.e., LMS Prediction-based Adaptive Regulator within a Linux kernel module to distribute the memory bandwidth as a resource among the CPU cores. We describe a design in which both monitoring and regulation is enforced from outside by a master core - which is not a dedicated controller for regulation. This allows for plugging in computationally heavy prediction and regulation algorithms without interfering with the regulated core. An adaptive filtering technique was employed for prediction of per-core bandwidth requirement. We conducted several experiments with SPEC CPU 2017 benchmarks distributed across multiple cores. Our proposed approach demonstrated significant improvement over Memguard with respect to slowdown ratios caused due to memory contention. Our solution is hosted publicly at \(\href{https://github.com/ss22ongithub/LMSAdaptiveRegulator}{https://github.com/ss22ongithub/LMSAdaptiveRegulator}\).

Node-Level Performance and Energy Characterization of Flagship Science Applications on SuperMUC-NG Phase 2

arXiv 2026-06-22

We present a systematic performance and energy-efficiency characterization of five flagship scientific workloads on SuperMUC-NG phase 2, the 28 PetaFLOPs system at the Leibniz Supercomputing Center (LRZ) equipped with Intel Xeon Platinum 8480+ and Intel Data Center GPU Max 1550 (Ponte Vecchio, PVC) accelerators. The selected codes span molecular dynamics (gromacs, lammps), astrophysics and cosmology (OpenGadget3, AthenaK), and finite-element PDE solvers from the dealii-X Center of Excellence. For each code we measure throughput and energy efficiency expressed as compute-elements per wall-clock second (or per Joule of consumed energy) on a single compute node, comparing CPU-only (SPR) against combined CPU+GPU (SPR+PVC) configurations where available. Energy measurements rely on lightweight code instrumentation with p3em, or the Energy Aware Runtime (EAR) present on the system. Our results show that GPU offload yields \(4-12\times\) higher throughput and up to \(15\times\) better energy efficiency compared to CPU-only execution, with lammps and AthenaK benefiting most. However, both throughput and energy gains are sensitive to problem granularity: insufficient work per GPU tile erodes the accelerator advantage, as clearly observed in AthenaK at small mesh-block sizes. The power-budget utilization is systematically lower for CPUs than it is for GPUs, indicating that even at peak useful-work rate, most applications running on CPUs leave a significant fraction of the node's thermal envelope unused.

The Energy Consumption of Transformer Fine-Tuning: A Roofline-Inspired Scaling Model

arXiv 2026-06-22

Transformer-based models underpin modern natural language processing but incur rapidly growing computational and energy costs. As training scales in both model size and parallelism, accurately predicting energy consumption has become critical for sustainable and cost-aware system design. We present a framework for modeling the energy consumption of Transformer training on multiple GPUs. Using controlled architectural sweeps of BERT models, we relate measured energy to lightweight proxies for compute, memory traffic, and hardware efficiency. Inspired by roofline models, our approach incorporates a speedup-based hardware-efficiency factor that captures the effects of tensor parallelism and fully sharded data parallelism. We derive a scaling law model that accurately predicts training energy across heterogeneous configurations.

Flowing With Purpose: Latent Action Guided Flow Matching Policies For Robotic Manipulation

arXiv 2026-06-22

Flow matching has recently become a new standard for behavior cloning in robotic manipulation. However, state-of-the-art flow matching policies suffer from a systematic structural mismatch: they rely on a globally fixed isotropic source distribution despite the strongly fragmented and heteroscedastic structure of robotic action spaces. This agnostic initialization forces the model to learn highly entangled vector fields, bottlenecking training efficiency and limiting overall policy performance. To address this limitation, we introduce Latent Action Guided Flow Matching (LAFM), a novel framework that replaces the monolithic Gaussian with an adaptive library of learned prior distributions. By grounding these distributions using a latent action model, LAFM maps current observations to discrete motion primitives, selecting a specialized base distribution that provides an informed, structurally aligned initialization for the denoising process. This dynamic adaptivity naturally accommodates heteroscedasticity in human demonstrations and makes transport trajectories shorter and less entangled. Empirically, LAFM substantially outperforms standard flow matching formulations, increasing task success rates by 23.4% in real-world robotic deployments and by 10.4% on the LIBERO-90 benchmark. Furthermore, we demonstrate that LAFM achieves state-of-the-art results, surpassing massively pre-trained vision-language-action models while utilizing significantly smaller architectures.

Learning to See While Learning to Act: Diffusion Models for Active Perception in Robot Imitation

arXiv 2026-06-22

Most imitation learning methods assume full observability in table-top settings. In practice, objects are often occluded, requiring robots to both search and act, and learning this coupled behavior from limited demonstrations remains challenging. We propose See2Act, an imitation learning approach that conditions action prediction on a sequence of actively-inferred viewpoints at test time, by coupling action denoising with viewpoint refinement. The policy is trained using camera poses anchored to keyframe actions from offline demonstrations, enabling implicit learning of where to see, while learning how to act. We empirically demonstrate that in Ravens the policy recovers informative viewpoints under severe occlusions, and on RLBench tasks it improves performance by up to 34% over prior methods. In the real world, we collect 50 demonstrations in a digital twin and achieve zero-shot sim-to-real transfer on pick-and-place tasks using depth observations. The policy handles significant occlusions, showing that learned viewpoint reasoning enables robust manipulation under partial observability.

GeoFidelity-Bench: Evaluating Segment-Level Geographic Fidelity in Text-to-Image Street-View Generation

arXiv 2026-06-22

Text-to-image models can generate visually plausible city streets, but whether their outputs correspond to a requested road segment rather than a generic city prior remains unclear. We introduce GeoFidelity-Bench, a reference-panel benchmark for segment-conditioned geographic fidelity in street-view generation. It contains 7,117 curated Mapillary images covering 109 named OpenStreetMap road segments in 25 cities across six continents. For each generated panel, the benchmark ranks the target reference panel against panels from the nearest segment in the same city, other segments in the same city, and segments from other cities, making local discrimination rather than absolute target similarity the primary test. We evaluate six open-weight text-to-image generators under city-only, street-and-neighborhood, and GPS-augmented prompts. Adding street and neighborhood names is associated with an increase of 5.5 percentage points in top-1 retrieval accuracy over city-only prompts, with a 95% confidence interval from 3.4 to 7.7 percentage points. However, the similarity margin between the target and the nearest segment in the same city remains near zero, indicating that local names improve broad local plausibility more than exact segment identity. Prompts that keep the city fixed but use incorrect street or neighborhood names further show that only part of the gain depends on the correct local names, while appending raw GPS coordinates as ordinary text yields no statistically clear additional benefit. Held-out real-image queries successfully recover segment identity, showing that the curated references contain usable segment-level signal. GeoFidelity-Bench thus reveals a persistent gap between city- or neighborhood-plausible street-view generation and faithful generation for a specific road segment.

FORGE: Fused On-Register Gradient Elimination for Memory-Efficient LLM Training

arXiv 2026-06-22

Reverse-mode differentiation computes every weight gradient, writes it to memory, and only then lets the optimizer read it back. This two-phase schedule sets the memory ceiling of modern training: at the seam between the phases, every layer's gradient is live at once. We argue that this materialized gradient is an artifact of how differentiation is staged, not a quantity that learning requires -- and we eliminate it. FORGE folds the optimizer step into the backward pass and applies it one tile at a time, entirely in registers, so each gradient tile is consumed the instant it is produced and never becomes a tensor. The fusion changes only when the update happens, not what it computes: in full precision the fused step is provably exact -- the identical optimizer update, for every element-wise rule -- and that exactness survives tensor- and sequence-parallel sharding; in the bf16 and 8-bit regimes used in practice it is faithful rather than bit-identical, its deviation bounded and, for the weight store, rendered unbiased by stochastic rounding. Because each gradient tile is born and consumed in the same registers, it is never converted down to bf16 to be stored and read back; FORGE thus preserves the full-precision fidelity that both bf16 and 8-bit optimizers lose to that conversion. Nor is the method tied to one architecture or one optimizer: linear layers are ubiquitous, and FORGE reclaims the gradient memory of any of them under any element-wise rule. Empirically FORGE more than halves the memory of an optimizer step and, at the small batch sizes typical of fine-tuning and continued pretraining, runs about 1.5x faster; integrated into tensor-parallel Megatron-LM it fits 8B training at four times the micro-batch a standard optimizer allows on the same GPUs.