Image / Video Generation 训练优化
FlowTrain: Flow-Based Decoupled Training for Industrial-Grade Vision-Language Models
Industrial-grade distributed training of vision-language models (VLMs) remains far less efficient than that of unimodal LLMs. Existing solutions either follow a monolithic design that assigns uniform parallelism to heterogeneous modules or adopt a disaggregated deployment that separates modules while executing them as a batch-synchronized pipeline. In this paper, we highlight that the above solutions are still not sufficient, and VLM training can be further decoupled. To this end, we present FlowTrain, a flow-based decoupled training framework that reformulates VLM training as a producer-consumer dataflow coordinated through a unified memory pool. The encoder and backbone can progress independently over a global virtual address space. Since this execution decoupling fundamentally changes the optimization objective of allocation and scheduling, FlowTrain further introduces a heterogeneous parallel allocator that assigns module-specific parallelism strategies by solving a throughput matching problem. The dynamic packing scheduler is used to construct balanced microbatches at runtime according to the actual LLM-side computation cost. Extensive experiments on real-world workloads show that FlowTrain achieves over 50% MFU and up to 1.7x throughput improvement, narrowing the efficiency gap to LLM-only training.
Same question, different history: language, national identity, and credit in large language models
Who invented the radio, Russia's Alexander Popov or Italy's Guglielmo Marconi? Was the telephone the achievement of Bell in the United States or Meucci in Italy? Does printing belong to China's Bi Sheng or Germany's Gutenberg? The answer depends not only on historical record but also on language and perspective. We analyse eleven widely used large language models across 21 disputed inventions and discoveries, evaluated in twelve languages and 75,896 responses. While models generally acknowledge that credit is contested, query language systematically affects which claimant is surfaced. Lower-status claimants are more likely to appear when questions are asked in their associated language, whereas dominant Anglophone figures remain stable across languages. These patterns persist after controlling for response length, model differences, historical prominence, and levels of national commemoration. Language thus acts as a switch that activates different national versions of the same history, producing systematically different national memories from the same question. We interpret this as evidence that large language models function as distributed systems of cultural memory, where language conditions which histories become visible, contributing to a computational form of banal nationalism.
PeLAP-A: Adaptive Latent Pruning for Lightweight Latent Diffusion Models
Latent diffusion models achieve strong generative performance by operating in a compressed latent space produced by a variational autoencoder (VAE). However, it remains unclear whether all latent channels contribute equally to the diffusion process, or whether significant redundancy exists. We introduce PeLAP-A (Adaptive Latent Pruning for Diffusion), a lightweight framework that augments a standard latent diffusion pipeline with a learnable channel-wise importance predictor. A two-layer MLP operating on globally pooled latent features produces a soft mask that suppresses unimportant latent channels before they enter the denoising UNet. The entire system is trained jointly on CIFAR-10 under a combined diffusion, reconstruction, and sparsity loss. Experiments reveal a striking result: under aggressive sparsity regularization (lambda = 0.01), the importance predictor drives all latent channels to near-zero yet the denoising UNet achieves lower diffusion loss (0.0236 vs. 0.0240) and lower VAE reconstruction MSE (22.59 vs. 24.67) compared to the unpruned baseline. We term this the sparsity collapse phenomenon and provide an analysis of why it occurs and what it reveals about the information requirements of latent diffusion models. These findings constitute an exploratory study of sparsity dynamics in latent diffusion training, and demonstrate that denoising UNets can remain remarkably robust to latent channel suppression even under aggressive regularization. Code is available at: https://github.com/kissasium/PeLAP-A.git.
InteractiveAvatar: Real-Time Streaming Video Generation for Consistent and Intent-Aware Avatars
Recent diffusion-based models have enabled realistic audio-driven avatar generation in real-time streaming. However, existing approaches struggle to maintain visual temporal consistency and fail to explicitly perceive user intent in complex interactive streaming scenarios. To address these challenges, we propose InteractiveAvatar, a real-time infinite-streaming video generation framework that supports visually consistent avatar video generation and intent-aware interactions. With autoregressive distillation, InteractiveAvatar achieves real-time str-eaming generation of human avatars over arbitrarily long durations. For visual consistency, we introduce a Long-Short Visual Memory (LSVM) mechanism that flexibly compresses historical visual information into compact tokens, preserving both short-range coherence and long-term consistency. To generate avatars with speeches and actions aligned with user intent, we propose a Reasoning-Reaction Module (RRM), which incorporates a State-Cycling strategy and a Cache-Switching mechanism. Extensive experimental results over diverse scenarios demonstrate that our method achieves state-of-the-art visual consistency in long-duration generation, while enabling complex user-avatar interaction in real time.
Distribution-Aware Diffusion-LLM for Robust Ultra-Long-Term Time Series Forecasting
Time series forecasting is a fundamental machine learning task. Recent work has explored Large Language Models (LLMs) for this purpose due to their strong generalization, pattern recognition, and zero-shot or few-shot capabilities. Despite their suitability for long-context learning, LLMs face challenges in multimodal settings: they lack calibrated probabilistic modeling for non-text data and struggle to align heterogeneous representations. To address these issues, we propose a new framework Diffusion-LLM that integrates a conditional diffusion model into an LLM-based forecasting pipeline. This joint design enables learning the conditional distribution of future data while improving semantic alignment in a shared latent space. We evaluate Diffusion-LLM on six long-term forecasting benchmarks, including ETT, Weather, and ECL. Our method consistently outperforms existing LLM-based baseline, achieving notable gains in ultra-long-term and few-shot forecasting and demonstrating the value of distribution-aware regularization for enhancing robustness and generalization in time series LLMs.
When Is a Columnar Scan Bandwidth-Bound? A Decode-Throughput Law and Its Cross-Hardware Validation
A columnar scan that decompresses, filters, and aggregates should be limited only by memory bandwidth (the roofline floor T >= BytesRead/beta), yet real kernels are often compute-bound and leave bandwidth idle. We give a predictive answer to when a scan is bandwidth-bound. Across encodings, predicate selectivities, and two very different machines, a decoder's value throughput T_dec (values decoded per second) is essentially independent of bit-width b: it is set by the decode layout/strategy, not by how many bits each value occupies. Hence the achieved bandwidth fraction obeys a one-parameter law, f = min(1, T_dec * b / (8beta)), with the compute-to-bandwidth ridge at b = 8beta/T_dec. Fitting one T_dec per strategy reproduces measured bandwidth fractions with median error 0.027 on x86/AVX2 and 0.003 on a held-out Apple M4/NEON machine, and the ridge b shifts correctly with each machine's bandwidth. Inserting FastLanes' reported decode throughput into the law reproduces its "decode is free at three bits" headline as the large-T_dec limit, unifying our portable decoder and hand-tuned state of the art in one curve. We add two crossovers, validated on both machines: branch-free predicate evaluation beats branchy in a mid-selectivity band (the sigma(1-sigma) misprediction parabola), and zone-map skipping is clustering-gated rather than selectivity-gated. We release the micro-benchmark, the correctness oracle, and a one-command reproduction. This is a baseline and a model, not a faster kernel: our portable C decoders reach ~2 values/cycle, far below hand-tuned SOTA, and the law holds precisely because it is parameterized by the measured T_dec.
Architecture for Health Initiative (Arch4Health): Computational Challenges in Health-Related Applications and the Role of Computer Architecture in Addressing Them
Recent biotechnological advances enable high-throughput, low-cost, and accurate biological data generation. This wealth of data enables unique opportunities for advancing healthcare. Despite these opportunities, efficiently analyzing large-scale biological data poses significant challenges for conventional computing systems. These systems often cannot keep up with the high-throughput rate at which data is generated, and they face additional constraints related to energy efficiency, scalability, privacy, and security. Therefore, to facilitate the wide adoption of recent advances in healthcare, there is a need to optimize the computing systems to enable high-performance, energy-efficient, low-cost, private, and secure analysis of biological data. We introduce the Architecture for Health (Arch4Health) initiative, which aims to (i) identify and analyze key computational challenges in current and future health- and life science-related applications and (ii) explore how computer architects and computing system designers can advance healthcare by addressing these challenges. In this short paper, we first present the motivations behind the Arch4Health initiative and, second, elaborate on its vision and goals, related topics, Arch4Health workshops, and future outlooks.
Efficient Document Tampering Localization with Multi-Level Discrepancy Features and Unified DCT-Quantization Embedding
Localizing document tampering is extremely challenging, as manipulations are crafted to appear visually consistent and often leave only subtle traces that are nearly invisible to the human eye. In prior work, evaluation has been largely dominated by synthetic benchmarks that closely match the training distribution, and methods have shown steady progress under this setting. However, these gains often translate poorly to human-made forgeries and to cross-domain evaluation, where both the source documents and the tampering pipeline can change, leading to a distribution shift. In addition, since the introduction of the Frequency Perception Head for the discrete cosine transform (DCT) modality, it has become a standard choice, and subsequent work has largely focused on downstream modules and fusion strategies rather than revisiting the backbone itself. To help close this gap in cross-domain performance and improve the DCT backbone design, we propose \textbf{DiffNet}, a relatively simple yet effective RGB--DCT early-fusion architecture driven by two key design choices. First, to ensure that the decoder aggregates multi-scale inconsistency evidence rather than operating on raw, content-heavy activations, we apply a lightweight multi-level discrepancy transformation at the output of each backbone stage, replacing features with magnitude-only responses to learned zero-sum filters. Second, we design an efficient DCT-domain backbone that relies on a lightweight frequency-index-aware DCT--quantization joint embedding. Our approach achieves state-of-the-art performance on cross-domain and human-made document tampering localization, outperforming prior methods by around 30\%, with up to \(7\times\) higher throughput than the previous best model.
ZeroGVC: Zero-Shot Generative Video Compression with Autoregressive Diffusion Priors
Recent generative video compression methods leverage powerful generative priors to achieve perceptually pleasing reconstructions. However, most existing approaches require additional training to adapt generative models to produce realistic reconstructions from compact representations. In this paper, we propose ZeroGVC, a zero-shot generative video compression framework that leverages pretrained autoregressive diffusion priors for low-delay video reconstruction. ZeroGVC encodes the first frame of each group of pictures (GOP) with an image codec and represents subsequent P-frames through Codebook-Guided Autoregressive Latent Compression. This design is motivated by our observation that the compression scheme of denoising diffusion codebook models is effective in few-step consistency sampling. By selecting compact combinations of reproducible codebook noise vectors, ZeroGVC steers the latent denoising trajectory toward the target P-frame while allowing the decoder to reproduce the same trajectory in only a few denoising steps. In addition, we design an optional bidirectional reference mode that mitigates error propagation by leveraging the next I-frame context without introducing any additional bitrate overhead. Extensive experiments on standard video compression benchmarks demonstrate that ZeroGVC achieves superior perceptual reconstruction quality at ultra-low bitrates without any additional training.
Keyless Attention: Value-Space Routing and Value-Only Caching for Efficient Transformers
We propose Keyless Attention, an attention mechanism that eliminates the key projection entirely, operating over queries and values only. This yields a Value-Only Cache that reduces KV cache memory and access overhead by exactly 50% over standard attention, while matching or exceeding standard attention's decode throughput. Beyond efficiency, we introduce Depth-\(m\) Attention Factorization: standard attention computes a depth-2 factorization of the attention bilinear form, while Keyless Attention realizes a depth-\(m\) instance of this family. At m=3, Keyless Attention matches the projection matrix count of standard attention via a value-space routing matrix that replaces the key projection and introduces a coupling between routing and retrieval. Experiments across five models and four architectures (GPT-2 280M, GPT-2 557M, Pythia 410M, Qwen2 1.5B, and Llama 3.2 1B) show that Keyless Attention matches or outperforms standard QKV attention on perplexity in 4 out of 5 models. On downstream zero-shot evaluation (GPT-2 557M), Keyless Attention outperforms on 4 out of 5 commonsense reasoning benchmarks, while achieving 50% KV cache reduction throughout.
L20-Edu-135M: An Auditable Single-GPU Study of Data-Efficient Small Language Modeling
Small language models are cheap to serve and feasible on local hardware, but strong public 135M-class systems are commonly trained with hundreds of billions to trillions of tokens on large clusters. We study a sharply resource-constrained regime: a complete 134.5M-parameter language-model pipeline executed on one NVIDIA L20 GPU. The released checkpoint, L20-Edu-135M, receives approximately 13B pretraining tokens: 10B FineWeb-Edu tokens followed by a 3B-token educational, mathematics, code, and reasoning mixture. We document the architecture, data gates, cross-source MinHash/LSH near-deduplication, segment deduplication, benchmark-overlap removal, throughput optimization, supervised fine-tuning (SFT) with weight interpolation, and reinforcement learning from verifiable rewards (RLVR) on GSM8K. In a self-run zero-shot six-task harness, L20-Edu-135M obtains a mean score of 0.4150. It trails SmolLM-135M (0.4767) and SmolLM2-135M (0.4917), but its mean is 87.1% of SmolLM-135M's while its nominal token count is 2.17% as large. This ratio is descriptive, not evidence of statistical equivalence or a controlled scaling law. The model exceeds several older 100M-160M public baselines under the same harness. Direct GRPO-style RLVR decreases GSM8K exact-match accuracy from 1.82% to 1.59% (192-token completions) and 1.21% (320-token completions). These single-run results identify a concrete failure mode rather than establishing a general lower bound on RLVR. The contribution is an auditable resource-constrained case study, not a state-of-the-art claim.
TALAS: Teacher-Anchored Layer Alignment with Adaptive Sharpness-Aware Minimization for Embedding Distillation
Knowledge Distillation (KD) has established itself as a pivotal technique for compressing large pre-trained language models. However, existing methods that force a student to strictly mimic the teacher's sentence embeddings or internal features often incur prohibitive computational costs and yield suboptimal performance due to the inherent capacity gap. To address these challenges, we propose TALAS (Teacher-Anchored Layer Alignment with Sharpness-aware minimization), a unified framework that synergizes hierarchical (multi-layer) alignment with robust optimization. First, we introduce a Teacher-Anchored mechanism that selectively distills final sentence embeddings only into the student's upper layers, thereby reducing overhead while respecting capacity constraints. Second, we bridge the semantic gap in lower layers via Layer-Aligned Self-Distillation, which propagates knowledge top-down using internal geometric relational constraints in the embedding space. Finally, to prevent the student from memorizing point-wise teacher noise, we integrate Adaptive Sharpness-Aware Minimization (ASAM) into the training objective, guiding the model towards flat minima for enhanced generalization. Empirical results on standard sentence embedding benchmarks demonstrate that TALAS consistently outperforms strong distillation baselines while achieving superior training efficiency in terms of computational cost and memory footprint.
Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers
Solving high-dimensional and high-order PDEs is challenged by the coupled growth of spatial dimensionality and derivative order. Recent stochastic derivative estimators reduce this cost by replacing full derivative tensors with randomized dimension or Taylor estimators, but they are mostly designed for fixed physical parameters and require retraining for each new parameter. We show that direct conditional parameterization of such solvers entangles physical parameters with the high-order automatic differentiation graph, causing extra memory growth and parameter-induced variance amplification. We propose Parameterized Representations via Implicit Stochastic Modulation (PRISM), a plug-and-play framework for parameterized high-dimensional and high-order stochastic neural PDE solvers. PRISM uses a hyper-generator to map physical parameters to affine modulators that scale and shift a purely spatial latent manifold, while keeping parameter branches value-connected but spatial-tangent-disconnected. This design preserves unbiased stochastic dimension and Taylor estimators, removes the parameter encoder from high-order spatial AD, and provides a variance-aware Lipschitz envelope over the parameter space. We prove parameterized unbiasedness, estimation-error bounds, and convergence under bounded stochastic variance. Experiments with PRISM-STDE and PRISM-SDGD on nonlinear parameterized PDEs show stable zero-shot generalization, reduced memory usage, and scalability up to 100,000 dimensions on a single GPU, with efficient low-rank SVD adaptation for unseen parameters.
Constant-Time Certificate Selection for Local Broadcast Repair in Dense Gaussian and Eisenstein--Jacobi Networks
Dense Gaussian and Eisenstein--Jacobi (EJ) networks are algebraic interconnection networks with compact coordinate balls, fixed degree, and simple modular addressing. A source-centered coordinate-reduction tree gives a non-redundant one-to-all broadcast in the fault-free network, but processor faults can split the tree into multiple healthy components. Unlike search-based repair methods that require a linear scan of the network to select the repair plan, the certificate selectors introduced here operate in \(O(1)\) time and \(O(1)\) memory, consulting only the fault coordinates. This paper develops this stronger formulation for the one- and two-fault regime: a constant-time certificate selector. Given only the faulty coordinates, the selector classifies the relative fault geometry, chooses a coordinate-reduction orientation, and returns a bounded ordered set of component-crossing repair edges. For dense Gaussian networks \(G_k\), every source-free fault set with \(|F|\le2\) is repaired with depth at most \(k+2\) and with exactly \(c-1\) external component-crossing edges for the selected fault-pruned orientation. For dense EJ networks \(H_t\), every one-fault placement is repaired within depth \(t+1\), and every two-fault placement is repaired within depth \(t+2\), again with exactly \(c-1\) external repair edges. Exhaustive strict validation confirms the Gaussian selector over \(146{,}156\) one- and two-fault cases for \(k=5,\ldots,12\) and the EJ selector over \(52{,}395\) cases for \(t=2,\ldots,8\), with zero failures in connectivity, acyclicity, exact repair count, or depth bound.
Fast-TurboQuant: A Multiplier-Free Online Vector Quantization Approach
As large language models scale, memory bandwidth for key-value caches and retrieval-augmented generation systems becomes a critical bottleneck. While 1-bit quantization addresses this constraint, recent TurboQuant relies on dense random rotation matrices to condition the vector distribution before quantization. This projection demands millions of floating-point multiplications per embedding, making it difficult to deploy on constrained edge silicon. We introduce Fast-TurboQuant, a multiplier-free projection architecture that replaces the dense matrix with a structured fast Johnson-Lindenstrauss transform. By applying a Rademacher phase inversion followed by a fast Walsh-Hadamard transform (FWHT), the method leverages sub-Gaussian concentration to satisfy the prerequisites of scalar Lloyd-Max quantization without Gaussian projections. This substitution reduces the arithmetic complexity to only additions, eliminating hardware multipliers. Evaluation on DBpedia OpenAI-3 Large embeddings demonstrates a 19.7 times algorithmic speedup under sequential execution. Furthermore, the dimension expansion due to the FWHT zero-padding reduces the mean squared error and improves Recall@10.
Single-Event Upsets in 3D Gaussian Splatting Rendering: Bit-Level Criticality, Spatial Extent, and a Parallel Support Guard
Three-dimensional Gaussian splatting is a standard real-time scene representation increasingly deployed on hardware exposed to transient faults, such as spaceborne processors and robotic edge devices where silent data corruption occurs. A trained model is a large array of floating-point parameters in GPU memory, where a single-event upset corresponds to a single flipped bit. This paper measures these effects and constructs a defense. A GPU-resident parallel fault-injection engine applies over 3.8 million controlled single-bit upsets across four scenes, six fields, all bit positions, and three numeric formats (fp32, fp16, bf16), using 5.3 GPU-hours. The effect is highly concentrated: most upsets leave the image perceptually unchanged due to high redundancy, but a small set of high-order bits principally the logarithmic scale's sign bit enlarge a single primitive to cover up to 75.7% of the frame. A closed-form perturbation bound derived from the IEEE-754 layout and pipeline activations predicts this per-bit ordering. This concentration motivates a support guard: a per-primitive clamp of each parameter to the coordinate box observed during training, costing 76 us per frame. Over 768,000 guarded upsets, the worst corruption footprint is restricted to 11.68% of the frame. We prove the guard leaves clean models unchanged and prevents frame-covering corruption. Under an accumulated dose of 20,000 simultaneous upsets, the unguarded renderer degrades to 10.6 dB, whereas the guarded renderer remains at 21.8 dB. The corruption footprint also dictates the number of tile/compositing nodes contaminated in distributed renderers, where the per-node guard contains it.
Bridging Design and Execution: A Visual Graph Editor for Edge and Cloud Workflows
Designing modular applications for edge and cloud computing environments involves coordinating multiple computational kernels, shared data, and event-driven execution. This paper presents a domain-specific visual graph editor that enables users to model such applications using a unified interface. The editor supports three first-class abstractions: kernels, representing computational units; shared memory nodes, modeling distributed data; and event triggers, capturing execution dependencies. Users can construct graphs visually, configure node properties, and connect elements to define data and control flow. The resulting graphs are automatically serialized into machine-readable representations (JSON/XML) and can be passed to an execution API, bridging the gap between design-time modeling and runtime deployment. The editor's graph-based model improves reasoning about data sharing, execution order, and dependencies, particularly in distributed edge and cloud scenarios. To demonstrate its applicability, we discuss a federated learning workflow, where local training kernels interact with a shared global model through event-driven coordination. Compared to traditional workflow editors and general-purpose diagramming tools, the proposed system provides explicit execution semantics, modularity, and direct deployability. This work lays the foundation for visual orchestration of modular computation in distributed environments and offers extensibility for user-defined kernels, event types, and alternative execution backends, enabling future exploration of complex distributed applications.
KineticSim: A Lightweight, High-Performance Execution Engine for Real-Time Market Simulators
Simulating financial markets at scale with multi-agent (Agent-Based) models is critical for market design, regulatory stress-testing, and reinforcement learning, but traditional CPU simulators are bottlenecked by sequential processing while vectorized GPU frameworks suffer from kernel-launch overhead and redundant global-memory round-trips. We formalize, analyze, and evaluate a reusable parallel design pattern: persistent, state-carrying clearing for iterative multi-agent reductions. By caching mutable simulation state in thread-block shared memory across step boundaries, aggregating agent actions via shared-memory atomics, and resolving the clearing function cooperatively, the pattern reduces the per-step critical-path depth from Theta(L+A) for sequential clearing (L price-grid ticks, A agents) to Theta(log L + ceil(A/L)) and makes global-memory traffic independent of the step count. We implement this in KineticSim, a lightweight GPU execution engine that simulates massive ensembles of limit-order books in parallel, reaching a peak throughput of over 54.7 billion agent-events per second. On a fixed workload it delivers speedups of 3406x over CPU (NumPy), 27.8x over PyTorch GPU, 42.8x over JAX GPU, and 8.4x over a naive custom CUDA baseline, while using roughly an order of magnitude less GPU memory than PyTorch. Across 53 configurations the two custom CUDA engines produce bitwise-identical order books, and aggregate statistics match the CPU reference to within 0.1%. The pattern generalizes to other iterative multi-agent workloads requiring state-persistent, block-localized reductions.
Quantum ring all-reduce: communication and privacy advantages for distributed learning
Machine learning models have scaled to unprecedented sizes, making training across distributed devices the de facto standard in the field. In this work, we explore how quantum communications can make distributed training both more communication-efficient and information-theoretically private, for both classical and quantum learning models. Ring all-reduce is the foundational communication primitive for large-scale distributed training. We present a quantum version that reduces per-link online communication by a provably optimal factor of two using pre-shared entanglement and superdense coding, without requiring the learning model or gradient computation to change. Beyond bandwidth, the primitive enables privacy guarantees that are information-theoretically impossible for any classical protocol, achieving composable ε-secure aggregation, via verified entanglement, at a 2x overhead in GHZ copies. Our hybrid quantum-classical communication architecture yields simultaneous communication and security advantages for large scale distributed training, regardless of whether the learning itself is quantum or classical. Finally, we characterise quantum advantages in gradient conflict detection for server-to-client communication under bandwidth constraints, a setting that arises after ring all-reduce is completed, when full gradient broadcast to external clients is infeasible. Two variants of the problem admit different separations. For margin-based alignment testing (\textsc{GapIP}_τ), the quantum advantage is quadratic in the margin parameter: \widetilde{O}(τ^{-1}\log P) qubits versus \widetilde{O}(\min(\τ^{-2},P)) bits. For sign-consistency auditing against a private parameter matching (\textsc{TieAudit}_ε), the advantage represents an exponential separation in communication complexity: Ω(\sqrt{P}) bits whereas O(ε^{-2}\log P) qubits suffice.
Data Bias Mitigation under Coverage Constraints & The Price of Fairness
Machine learning models have been shown to exhibit discriminatory outcomes or degraded performance for individuals at the intersection of multiple sensitive attributes, such as race and gender. This stems in part from two interrelated challenges: the lack of principled measures for quantifying bias (potentially intersectional), and insufficient representation of intersectional subgroups in training data. We extend a recent bias mitigation framework to incorporate coverage constraints that enforce sufficient representation across groups, including intersectional subgroups. Since achieving exactly zero bias for all groups may not be data efficient (meaning it may require large amounts of data), our solution trades small approximation errors in bias for greater data efficiency while satisfying coverage constraints. We also formulate bias mitigation as an integer linear program that optimizes over all mitigation strategies, and characterize the price of fairness, the minimum data modification cost, as a function of fairness tolerance. This is essential both for legal compliance, where regulations may mandate specific fairness thresholds, and for data governance, enabling practitioners to make informed trade-offs between bias reduction and data modification (particularly, data purchasing) costs. We evaluate our techniques on publicly available datasets, demonstrating that bias mitigation via our framework preserves predictive accuracy across multiple classifiers, and that coverage constraints, while motivated by statistical considerations, are essential for preserving downstream ML performance.