Loss Function

Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both function values and derivatives and are the natural metrics for PDE well-posedness, stability, and generalization. We develop a functional-analytic framework for operator learning in Sobolev spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical PDE. First, for a continuous nonlinear operator $\mathcal{G}: H^{s}(D)\to H^{t}(D')$ with $s > d/2$ and inputs restricted to a compact subset of $H^{s}(D)$, we prove that $\mathcal{G}$ can be uniformly approximated in $H^{t}$-norm by a neural operator with $\mathcal{O}(\varepsilon^{-d/s})$ trainable parameters. This yields an explicit complexity--error relation of the form $\|\mathcal{G}-\mathcal{G}_θ\|_{H^{t}} \lesssim C N^{-s/d}$. We then study the one-dimensional viscous Burgers solution operator $\mathcal{G}: u_{0}\mapsto u(\cdot,1)$ on a bounded $H^{1}$-ball and train FNOs with an $H^{1}$-loss. Across a sweep of model sizes, we obtain test $H^{1}$-errors down to $\mathcal{O}(10^{-7})$ and relative errors of order $10^{-3}$, with predictions accurately matching both solutions and spatial derivatives on held-out data. A log-log plot of Sobolev error versus parameter count exhibits an approximate power law $\|\mathcal{G}-\mathcal{G}_θ\|_{H^{1}} \approx C N^{-α}$ with empirical exponent $α\approx 1.4$, and long-horizon training reveals optimization instabilities in large FNOs, providing quantitative evidence that Sobolev-space approximation theory meaningfully predicts neural-operator scaling behavior.

Medical experts often manually segment images to obtain diagnostic statistics and discard the resulting annotations. We aim to train segmentation models to alleviate this burden, but constrained to the retained summary statistics (e.g., the area of the annotated region). Empirical results suggest that statistics alone are insufficient for this task, but adding weak information in the form of a few pixels within the area of interest significantly improves performance. We use a novel loss function that combines terms for image reconstruction quality, matching to summary statistics, and overlap between the predicted foreground and the weak supervisory signal. Experiments on standard image, ultrasound (breast cancer), and Computed Tomography (CT) scan (kidney tumors) data demonstrate the utility and potential of the approach.

Barrier certificates are scalar functions over the state space of dynamical systems that separate all unsafe states from all reachable states. The existence of a barrier certificate formally verifies the safety of the dynamical system. Recent approaches synthesize barrier certificates by iteratively training a neural network. In each iteration, the candidate is formally verified - if successful, the barrier certificate is found. Instead, we propose a set-based training approach that tightly integrates verification into training via a set-based loss function that soundly encodes all barrier certificate properties. A loss of zero formally proves the validity of the barrier certificate, collapsing the iterative training and verification into a single training procedure. Our experiments demonstrate that our set-based training approach scales well with the system dimension and naturally handles complex nonlinear dynamics.

In sequential search, alternatives are tested until the true class is found. Standard proper scoring rules like log loss are local, ignoring the ranking of competitors and misaligning model evaluation with search utility. We show that sequential search induces a pairwise structure that overcomes this. By analyzing the expected cost of optimal search under varying testing costs, we derive Pandora's Regret: a closed-form, pairwise-additive, and strictly proper scoring rule. Pandora's Regret both elicits true probabilities and penalizes rank-reversing miscalibrations where distractors outrank the true class. Our construction yields a one-parameter Beta family that balances penalties for rank-swapping versus probability magnitude, while retaining a grounded interpretation as expected search cost. We prove that log loss, accuracy, and macro-F1 rely on implicit decision models misaligned with sequential search. Across 597 MedMNIST models, Pandora-based metrics better predict clinical diagnostic costs than standard alternatives, extending decision-theoretic scoring rule construction to the multiclass setting.

Theoretical studies show that for any differentiable function on a compact domain, there exists a neural network that approximates both the function values and gradients. However, such a result cannot be used in practice since it assumes real parameters and exact internal operations. In contrast, real implementations only use a finite subset of reals and machine operations with round-off errors. In this work, we investigate whether a similar result holds for neural networks under floating-point arithmetic, when the gradient with respect to the input is computed by the automatic differentiation algorithm $D^\mathtt{AD}$. We first show that given a floating-point function $φ$ (e.g., a loss function), arbitrary function values and gradients can be represented by a floating-point network $f$ and $D^\mathtt{AD}(φ\circ f)$, respectively. We further extend this result: given $φ_1,\dots,φ_n$, $D^\mathtt{AD}(φ_i\circ f)$ can simultaneously represent arbitrary gradients while $f$ represents the target values, under mild conditions. Our results hold for practical activation functions, e.g., $\mathrm{ReLU}$, $\mathrm{ELU}$, $\mathrm{GeLU}$, $\mathrm{Swish}$, $\mathrm{Sigmoid}$, and $\mathrm{tanh}$.

Unsupervised deraining has attracted attention for its ability to learn the real-world distribution of rain without paired supervision. However, the lack of strong constraints makes it difficult for the network to converge, especially with the complex diversity of rain degradation. A key motivation is that high-quality deraining results occasionally emerge during training, which can be leveraged to guide the optimization process. To overcome these challenges, we introduce RGSUD (Reward-Guided Self-Reinforcement Unsupervised Image Deraining), comprising two key stages: reward recycling and self-reinforcement (SR) training. For the former stage, we propose an Image Quality Assessment (IQA)-based dynamic reward recycling mechanism that selects optimal derained outputs during training and continuously collects high-quality deraining images. In latter stage, we incorporate these rewards into the model's optimization process, constraining the optimization space and improving alignment between derained outputs and clean images. By leveraging IQA-based self-reinforced loss and dynamically updated rewards, we enhance the quality of synthesized pseudo-paired data and stabilize the optimization. Extensive experiments demonstrate that our method achieves SOTA performance across multiple datasets, including paired synthetic, paired real, and unpaired real images, outperforming existing unsupervised deraining approaches in both subjective and objective IQA metrics. Additionally, we show that the self-reinforcement strategy is adaptable to other unsupervised deraining methods and our deraining framework demonstrates strong generalization across existing supervised deraining networks.

Assigning one of K options to each of N groups under a total cost budget is a recurring problem in efficient AI, including mixed-precision quantization, non-uniform pruning, and expert selection. The objective, typically model loss, depends jointly on all assignments and does not decompose across groups, preventing combinatorial solvers from directly optimizing the true objective and forcing reliance on proxy formulations. Methods such as evolutionary search evaluate the actual loss but lack gradient information, while penalty-based approaches enforce the budget only approximately and often require extensive hyperparameter tuning. We present a new approach by showing that, under softmax relaxation, the budget constraint defines a smooth Riemannian manifold in logit space with unusually simple geometry. The normal vector admits a closed-form expression, shifting logits along the cost vector changes expected cost monotonically, and vector transport reduces to a single inner product. Building on these properties, we propose Riemannian Constrained Optimization (RCO), which augments a standard Adam step with tangent projection, binary-search retraction, and momentum transport. Combined with Gumbel straight-through estimation and budget-constrained dynamic programming for discrete feasibility, RCO enables first-order optimization of the actual loss under exact budget enforcement without introducing constraint-specific hyperparameters. Across both synthetic benchmarks and realistic LLM compression settings, RCO matches or exceeds state-of-the-art methods while often requiring substantially less wall-clock time. Source code is available at https://github.com/IST-DASLab/RCO.

The choice of loss function in classification involves a fundamental trade-off: smooth losses (like Cross-Entropy) enable fast optimization rates but yield slow square-root consistency bounds, while piecewise-linear losses (like Hinge) offer fast linear consistency rates but suffer from non-differentiability. We propose Linear-Core (LC) Surrogates, a new family of convex loss functions that resolve this tension by stitching a linear core to a smooth tail. We prove that these surrogates are differentiable everywhere while retaining strict linear $H$-consistency bounds, effectively combining the optimization benefits of smoothness with the statistical efficiency of margin-based losses. In the structured prediction setting, we show that this smoothness unlocks a massive computational and energy advantage: it allows for an unbiased stochastic gradient estimator that bypasses the quadratic complexity $O(|\mathscr{Y}|^2)$ of exact inference (e.g., Viterbi). Empirically, our method achieves a 23$\times$ speedup over Structured SVMs on large-vocabulary sequence tagging tasks and demonstrates superior robustness to instance-dependent label noise, outperforming Cross-Entropy by 2.6% on corrupted CIFAR-10.

Federated learning is a machine learning paradigm in which multiple devices collaboratively train a model under the supervision of a central server while ensuring data privacy. However, its performance is often hindered by redundant, malicious, or abnormal samples, leading to model degradation and inefficiency. To overcome these issues, we propose novel sample selection methods for image classification, employing a multitask autoencoder to estimate sample contributions through loss and feature analysis. Our approach incorporates unsupervised outlier detection, using one-class support vector machine (OCSVM), isolation forest (IF), and adaptive loss threshold (AT) methods managed by a central server to filter noisy samples on clients. We also propose a multi-class deep support vector data description (SVDD) loss controlled by a central server to enhance feature-based sample selection. We validate our methods on CIFAR10 and MNIST datasets across varying numbers of clients, non-IID distributions, and noise levels up to 40%. The results show significant accuracy improvements with loss-based sample selection, achieving gains of up to 7.02% on CIFAR10 with OCSVM and 1.83% on MNIST with AT. Additionally, our federated SVDD loss further improves feature-based sample selection, yielding accuracy gains of up to 0.99% on CIFAR10 with OCSVM. These results show the effectiveness of our methods in improving model accuracy across various client counts and noise conditions.

Neural architectures trained with losses inspired by differential conditions are the basis for PINN models. Since many constructions in differential geometry may be framed as minimisation of a differential functional, these functionals can be coded as loss functions to align the AI loss-minimisation goal with that of solving the geometric problem. This contribution to the Recent Progress in Computational String Geometry workshop proceedings introduces the PINN architecture defining principles, motivates how they are well suited for problems in differential geometry, and demonstrates their use via summaries of three works at this intersection.

Training large neural networks with data-parallel stochastic gradient descent allocates N GPU replicas to compute effectively identical updates -- a practice that leaves the rich space of learning rate configurations entirely unexplored during training. We propose Hyperparameter-Divergent Ensemble Training (HDET), a method that repurposes these replicas for simultaneous learning rate exploration at negligible communication overhead. HDET operates in alternating phases: a fan-out stage in which replicas train independently under a structured, symmetric spread of learning rates, and a converge stage in which parameters are averaged across all replicas via AllReduce every T steps. Building on this ensemble substrate, we further propose an automatic learning rate (auto-LR) controller that treats the relative training loss across replicas as a performance signal, updating the shared base schedule toward higher-performing configurations via a momentum-based gradient-free meta-update. The combined method produces a self-adapting learning rate schedule that improves both optimization quality and generalization without additional hyperparameter sweeps or training budget. Crucially, the framework generalizes beyond learning rate: any scalar hyperparameter that does not alter model architecture -- such as dropout rate, attention scale temperature, or weight-decay coefficient -- can be explored across replicas using the same fan-out/converge protocol, with inter-replica loss differences serving as zero-order hypergradients that guide the search direction. HDET is implemented as a drop-in replacement for PyTorch's OneCycleLR scheduler, requiring no changes to model architecture, optimizer, or data pipeline.

Lattice Hamiltonian systems underpin models across condensed matter, nonlinear optics, and biophysics, yet learning their dynamics from data is obstructed by two unknowns: the interaction topology and whether node dynamics are homogeneous. Existing graph-based approaches either assume the graph is given or, as in $α$-separable graph Hamiltonian network, infer it only for separable Hamiltonians with homogeneous node dynamics. We introduce the Hamiltonian Graph Inference Network (HGIN), which jointly recovers the interaction graph and predicts long-time trajectories from state data alone, for both separable and non-separable Hamiltonians and under heterogeneous node dynamics. HGIN couples a structure-learning module -- a learnable weighted adjacency matrix trained under a Hamilton's-equations loss -- with a trajectory-prediction module that partitions edges into physically distinct subgraphs via $k$-means clustering, assigning each subgraph its own encoder and thereby breaking the parameter-sharing bottleneck of conventional GNNs. On three benchmarks -- a Klein--Gordon lattice with long-range interactions and two discrete nonlinear Schrödinger lattices (homogeneous and heterogeneous) -- HGIN reduces long-time energy prediction error and trajectory prediction error by six to thirteen orders of magnitude relative to baselines. A symmetry argument on the Hamiltonian loss further shows that the learned weights encode the parity of the underlying pair potential, yielding an interpretable readout of the system's interaction structure.

Dynamic Tanh (DyT) removes LayerNorm by bounding activations with a learned tanh(alpha x). We show that this bounding is a regime-dependent implicit regularizer, not a uniformly beneficial replacement. Across GPT-2-family models spanning 64M to 3.78B parameters and 1M to 118M tokens, with Llama and ViT cross-checks, DyT improves validation loss by 27.3% at 64M/1M but worsens it by 18.8% at 64M/118M; the 1M benefit vanishes with capacity (+1.7% at 3.78B), while the 118M penalty reaches +27.9%. The mechanism is measurable: 49% of DyT activations saturate at 1M versus 23% at 118M, and a 500-step saturation heuristic classifies DyT's sign with 75% raw in-sample accuracy on the 12-cell GPT-2 calibration set (AUC 0.75; 64% when adding Scale 5 stress cells), correctly labels 3/3 Llama checks, but only reaches 50% raw leave-one-scale-out accuracy. Three interventions support the bounding explanation: HardTanh reproduces the regime pattern, increasing alpha at 118M monotonically reduces DyT's penalty, and vanilla+dropout(p=0.5) matches DyT's data-rich loss. We also localize Llama-DyT collapse to SwiGLU gating, where saturation separates collapse from convergence in a 3-seed component ablation (r=0.94). Scope: all experiments are compute-limited (T/P < 1.84), below Chinchilla-optimal training.

Reconstructing high-fidelity fluid dynamics from sparse temporal observations is quite challenging, mainly due to the chaotic and non-linear nature of fluid transport. Standard deep learning-based interpolation methods often tend to regress to the mean, which results in spatial blurring and temporal strobing, especially noticeable around the observed anchor frames where transitions become discontinuous. In this work, we propose a novel Temporal U-Net architecture that integrates a VGG-based perceptual loss along with a Physics-Informed Bridge to overcome these issues. By introducing time-weighted feature blending and enforcing a parabolic boundary condition defined by t(1 - t), the model ensures smooth transitions while also maintaining perfect consistency at the endpoints. Experimental results on multi-channel RGB fluid data show that our method clearly outperforms standard models, both in terms of structural fidelity and texture preservation. In particular, the model achieves a Mean Absolute Error of 0.015, compared to 0.085 for a standard L1 baseline. Further Spatial Power Spectral Density (PSD) analysis reveals that the model is able to retain high-frequency turbulent details that are usually lost in deterministic reconstructions.

Audio-text retrieval enables semantic alignment between audio content and natural language queries, supporting applications in multimedia search, accessibility, and surveillance. However, current state-of-the-art approaches struggle with long, noisy, and weakly labeled audio due to their reliance on contrastive learning and large-batch training. We propose a novel multimodal retrieval framework that refines audio and text embeddings using a cross-modal embedding refinement module combining transformer-based projection, linear mapping, and bidirectional attention. To further improve robustness, we introduce a hybrid loss function blending cosine similarity, $\mathcal{L}_{1}$, and contrastive objectives, enabling stable training even under small-batch constraints. Our approach efficiently handles long-form and noisy audio (SNR 5 to 15) via silence-aware chunking and attention-based pooling. Experiments on benchmark datasets demonstrate improvements over prior methods.

Physical adversarial patch attacks critically threaten pedestrian detection, causing surveillance and autonomous driving systems to miss pedestrians and creating severe safety risks. Despite their effectiveness in controlled settings, existing physical attacks face two major limitations in practice: they lack systematic disruption of the multi-stage decision pipeline, enabling residual modules to offset perturbations, and they fail to model complex physical variations, leading to poor robustness. To overcome these limitations, we propose a novel pedestrian adversarial patch generation method that combines multi-stage collaborative attacks with robustness enhancement under physical diversity, called TriPatch. Specifically, we design a triplet loss consisting of detection confidence suppression, bounding-box offset amplification, and non-maximum suppression (NMS) disruption, which jointly act across different stages of the detection pipeline. In addition, we introduce an appearance consistency loss to constrain the color distribution of the patch, thereby improving its adaptability under diverse imaging conditions, and incorporate data augmentation to further enhance robustness against complex physical perturbations. Extensive experiments demonstrate that TriPatch achieves a higher attack success rate across multiple detector models compared to existing approaches.

K-plane clustering (KPC), hyperplane clustering, and mixture regression all essentially fall within the same class of problems. This problem can be conceptualized as clustering in relatively high-dimensional K subspaces or K linear manifolds. Traditional KPC or fuzzy KPC models demonstrate a pronounced susceptibility to outliers, as they presuppose that the projection distance between data points and the plane normal vector adheres to the L2 distance. Meanwhile, the assumption of infinitely extending clusters adversely affects clustering performance. To solve these problems, this paper proposed a new robust fuzzy local k-plane clustering (RFLkPC) method that combines the mixture distance of hinge loss and L1 norm. The RFLkPC model assumes that each plane cluster is bounded to a finite area, which can flexibly and robustly handle plane clustering tasks with outliers or not. The corresponding model and optimization algorithms of RFLkPC were provided. Compared to other related models on this topic, a large number of experiments verify the efficiency of RFLkPC on simulated data and real data. The source code for the proposed RFLkPC method is publicly available at https://github.com/xuelin-xie/RFLkPC.

Recent advances in distilling expensive diffusion models into efficient few-step generators show significant promise. However, these methods typically demand substantial computational resources and extended training periods, limiting accessibility for resource-constrained researchers, and existing supplementary loss functions have notable limitations. Regression loss requires pre-generating large datasets before training and limits the student model to the teacher's performance, while GAN-based losses suffer from training instability and require careful tuning. In this paper, we propose Embedding Loss (EL), a novel supplementary loss function that complements existing diffusion distillation methods to enhance generation quality and accelerate training with smaller batch sizes. Leveraging feature embeddings from a diverse set of randomly initialized networks, EL effectively aligns the feature distributions between the distilled few-step generator and the original data. By computing Maximum Mean Discrepancy (MMD) in the embedded feature space, EL ensures robust distribution matching, thereby preserving sample fidelity and diversity during distillation. Within distribution matching distillation frameworks, EL demonstrates strong empirical performance for one-step generators. On the CIFAR-10 dataset, our approach achieves state-of-the-art FID values of 1.475 for unconditional generation and 1.380 for conditional generation. Beyond CIFAR-10, we further validate EL across multiple benchmarks and distillation methods, including ImageNet, AFHQ-v2, and FFHQ datasets, using DMD, DI, and CM distillation frameworks, demonstrating consistent improvements over existing one-step distillation methods. Our method also reduces training iterations by up to 80%, offering a more practical and scalable solution for deploying diffusion-based generative models in resource-constrained environments.

Standard physics-informed neural networks (PINNs) struggle to simulate highly oscillatory Helmholtz solutions in heterogeneous media because pointwise minimization of second-order PDE residuals is computationally expensive, biased toward smooth solutions, and requires artificial absorbing boundary layers to restrict the solution. To overcome these challenges, we introduce a Green-Integral (GI) neural solver for the acoustic Helmholtz equation. It departs from the PDE-residual-based formulation by enforcing wave physics through an integral representation that imposes a nonlocal constraint. Oscillatory behavior and outgoing radiation are encoded directly through the integral kernel, eliminating second-order spatial derivatives and enforcing physical solutions without additional boundary layers. Theoretically, optimizing this GI loss via a neural network acts as a spectrally tuned preconditioned iteration, enabling convergence in heterogeneous media where the classical Born series diverges. By exploiting FFT-based convolution to accelerate the GI loss evaluation, our approach substantially reduces GPU memory usage and training time. However, this efficiency relies on a fixed regular grid, which can limit local resolution. To improve local accuracy in strong scattering regions, we also propose a hybrid GI+PDE loss, enforcing a lightweight Helmholtz residual at a small number of nonuniformly sampled collocation points. We evaluate our method on seismic benchmark models characterized by structural contrasts and subwavelength heterogeneity at frequencies up to 20Hz. GI-based training consistently outperforms PDE-based PINNs, reducing computational cost by over a factor of ten. In models with localized scattering, the hybrid loss yields the most accurate reconstructions, providing a stable, efficient, and physically grounded alternative.

We measure how much one recurrence is worth to a looped (depth-recurrent) transformer, in equivalent unique parameters. From an iso-depth pretraining sweep across recurrence counts $r \in \{1, 2, 4, 8\}$ spanning ${\sim}50\times$ in training compute, we fit a joint scaling law $L = E + A\,(N_\text{once} + r^{\varphi} N_\text{rec})^{-α} + B\,D^{-β}$ and measure a recurrence-equivalence exponent $\varphi = 0.46$. Intuitively, $\varphi$ tells us whether looping a block $r$ times is equivalent in validation loss to $r$ unique blocks of a non-looped model (full equivalence, $\varphi{=}1$) or to a single block run repeatedly with no capacity gain ($\varphi{=}0$). Our $\varphi = 0.46$ sits in between, so replacing unique blocks with shared recurrences increases validation loss at matched training compute. For example, at $r{=}4$ a 410M looped model performs on par with a 580M non-looped model, but incurs the training cost of a 1B non-looped one. We demonstrate the utility of $\varphi$ as a diagnostic tool on two case studies: commonly used truncated backpropagation lowers $\varphi$ to $0.38$, indicating that the loop mechanism is poorly trained under truncation, even though validation loss decreases. Conversely, hyperconnections raise $\varphi$ to $0.65$, a genuine capacity gain. Our method separates true loop improvements from training-side gains, a distinction raw validation loss cannot make.

Training vision-language models (VLMs) for medical report generation is often hindered by the scarcity of high-quality annotated data. This work evaluates the use of a weighted loss function to improve data efficiency. Compared to standard cross-entropy loss, which treats all token prediction errors equally, the reweighted loss shifts the focus to semantically salient tokens with outsized clinical importance. In experiments on ophthalmological report generation, we show that this simple method improves efficiency across multiple data scales, achieving similar report quality with up to ten times less training data.

Transformer-based OCR models have shown strong performance on Latin and CJK scripts, but their application to African syllabic writing systems remains limited. We present the first adaptation of TrOCR for printed Tigrinya using the Ge'ez script. Starting from a pre-trained model, we extend the byte-level BPE tokenizer to cover 230 Ge'ez characters and introduce Word-Aware Loss Weighting to resolve systematic word-boundary failures that arise when applying Latin-centric BPE conventions to a new script. The unmodified model produces no usable output on Ge'ez text. After adaptation, the TrOCR-Printed variant achieves 0.22% Character Error Rate and 97.20% exact match accuracy on a held-out test set of 5,000 synthetic images from the GLOCR dataset. An ablation study confirms that Word-Aware Loss Weighting is the critical component, reducing CER by two orders of magnitude compared to vocabulary extension alone. The full pipeline trains in under three hours on a single 8 GB consumer GPU. All code, model weights, and evaluation scripts are publicly released.

This study aims to develop and improve machine learning-based post-processing models for precipitation, temperature, and wind speed predictions using the Mesoscale Model (MSM) dataset provided by the Japan Meteorological Agency (JMA) for 18 locations across Japan, including plains, mountainous regions, and islands. By incorporating meteorological variables from grid points surrounding the target locations as input features and applying feature selection based on correlation analysis, we found that, in our experimental setting, the LightGBM-based models achieved lower RMSE than the specific neural-network baselines tested in this study, including a reproduced CNN baseline, and also generally achieved lower RMSE than both the raw MSM forecasts and the JMA post-processing product, MSM Guidance (MSMG), across many locations and forecast lead times. Because precipitation has a highly skewed distribution with many zero cases, we additionally examined Tweedie-based loss functions and event-weighted training strategies for precipitation forecasting. These improved event-oriented performance relative to the original LightGBM model, especially at higher rainfall thresholds, although the gains were site dependent and overall performance remained slightly below MSMG.

Semantic Tube Prediction (STP) leverages representation geometric to regularize LLM hidden-state trajectories toward locally linear geodesics during fine-tuning, thereby greatly improving data efficiency. The original STP recipe samples random token sub-spans, which is compatible with the base large language model (LLM) training architecture. Inspired by STP, we are interested to investigate whether the sampling position can further enhance the semantic structure of multi-step reasoning, and hence affect its geometric impact. We applied STP at consecutive semantic reasoning step boundaries and achieved 168x more accurate multi-step latent prediction than frozen baselines on ProcessBench (3,400 samples), compared to only 4x for the random-token STP. Probing the latent manifold with a learned non-linear predictor reveals that STP-shaped trajectories are smooth curves, not straight lines: a 3-layer MLP reduces prediction error by a further 3-12x over linear extrapolation on step-boundary models. Removing the language modeling loss yields trajectories that are 2x more MLP-predictable than the combined loss, revealing a tradeoff between generation quality and geometric purity. Our results identify sampling position as the critical variable in geometric regularization and establish multi-step latent prediction MSE as a new evaluation metric for this class of methods.

Recent advances in summarization research focus on improving summary quality across multiple criteria, such as completeness, conciseness, and faithfulness, by jointly optimizing these dimensions. However, these efforts largely overlook the challenge of controlling summary generation with respect to individual criteria, especially in the presence of their inherent trade-offs. For example, enhancing conciseness can compromise completeness, and vice versa. In this work, we address this gap by proposing a loss function that aligns model outputs with fine-grained, model-based evaluation scores (e.g., from FineSurE), enabling both improvement in summary quality and dimension-specific control. Our approach improves the overall quality of summaries while maintaining the ability to selectively prioritize one criterion over others. Experiments on three pretrained models (LLaMA, Qwen, and Mistral) demonstrate that our method achieves performance comparable to state-of-the-art summarizers, while uniquely offering strong controllability over individual quality dimensions.

Diffusion policies are becoming mainstream in robotic manipulation but suffer from hard negative class imbalance due to uniform sampling and lack of sample difficulty awareness, leading to slow training convergence and frequent inference timeout failures. We propose VADF (Vision-Adaptive Diffusion Policy Framework), a vision-driven dual-adaptive framework that significantly reduces convergence steps and achieves early success in inference, with model-agnostic design enabling seamless integration into any diffusion policy architecture. During training, we introduce Adaptive Loss Network (ALN), a lightweight MLP-based loss predictor that quantifies per-step sample difficulty in real time. Guided by hard negative mining, it performs weighted sampling to prioritize high-loss regions, enabling adaptive weight updates and faster convergence. In inference, we design the Hierarchical Vision Task Segmenter (HVTS), which decomposes high-level task instructions into multi-stage low-level sub-instructions based on visual input. It adaptively segments action sequences into simple and complex subtasks by assigning shorter noise schedules with longer direct execution sequences to simple actions, and longer noise steps with shorter execution sequences to complex ones, thereby dramatically reducing computational overhead and significantly improving the early success rate.

While large language models (LLMs) show strong performance in natural language understanding and generation, their evaluation and adaptation to domain-specific constraints in telecommunications customer support remain limited. In addition, data sovereignty, regulatory constraints, and the handling of sensitive customer and network information complicate the use of externally hosted foundation models in this domain. We present a systematic study of parameter-efficient fine-tuning (PEFT) using Low-Rank Adaptation (LoRA) applied to Qwen2.5-3B to build a domain-specific conversational assistant. We introduce a combinatorial synthetic data generation approach based on a glossary of 52 industry-specific terms, producing approximately 30,000 training examples across 1,560 distinct problem scenarios via a generative pipeline powered by Gemini 2.0 Flash. We evaluate 16 LoRA configurations by varying hyperparameters and target modules. Our evaluation extends beyond standard metrics by incorporating energy consumption analysis and qualitative assessment using an LLM-as-a-judge framework with GPT-5.2 and Claude 4.5 Sonnet. Results show a clear divergence between quantitative and qualitative performance: models achieving the lowest validation loss do not necessarily obtain the best human-aligned rankings. The best validation loss (0.5024) ranks only 6th-7th in qualitative evaluation, while the worst loss (0.6807) ranks first according to both judges. This work contributes (1) a combinatorial method for synthetic dataset construction, (2) insights into the impact of target module selection for LoRA injection, (3) evidence that validation loss alone is insufficient for selecting fine-tuning configurations in conversational AI, and (4) an energy-performance trade-off analysis for sustainable LLM deployment.

With the rise of pre-trained models in the 3D point cloud domain for a wide range of real-world applications, adapting them to downstream tasks has become increasingly important. However, conventional full fine-tuning methods are computationally expensive and storage-intensive. Although prompt tuning has emerged as an efficient alternative, it often suffers from overfitting, thereby compromising generalization capability. To address this issue, we propose Prototypical Point-level Prompt Tuning (P$^3$T), a parameter-efficient prompt tuning method designed for pre-trained 3D vision-language models (VLMs). P$^3$T consists of two components: 1) \textit{Point Prompter}, which generates instance-aware point-level prompts for the input point cloud, and 2) \textit{Text Prompter}, which employs learnable prompts into the input text instead of hand-crafted ones. Since both prompters operate directly on input data, P$^3$T enables task-specific adaptation of 3D VLMs without sacrificing generalizability. Furthermore, to enhance embedding space alignment, which is key to fine-tuning 3D VLMs, we introduce a prototypical loss that reduces intra-category variance. Extensive experiments demonstrate that our method matches or outperforms full fine-tuning in classification and few-shot learning, and further exhibits robust generalization under data shift in the cross-dataset setting. The code is available at \textcolor{violet}{https://github.com/gyjung975/P3T}.

We consider the problem of approximating a function by an element of a nonlinear manifold which admits a differentiable parametrization, typical examples being neural networks with differentiable activation functions or tensor networks. Natural gradient descent (NGD) for the optimization of a loss function can be seen as a preconditioned gradient descent where updates in the parameter space are driven by a functional perspective. In a spirit similar to Newton's method, a NGD step uses, instead of the Hessian, the Gram matrix of the generating system of the tangent space to the approximation manifold at the current iterate, with respect to a suitable metric. This corresponds to a locally optimal update in function space, following a projected gradient onto the tangent space to the manifold. Still, both gradient and natural gradient descent methods get stuck in local minima. Furthermore, when the model class is a nonlinear manifold or the loss function is not ideally conditioned (e.g., the KL-divergence for density estimation, or a norm of the residual of a partial differential equation in physics informed learning), even the natural gradient might yield non-optimal directions at each step. This work introduces a natural version of classical inertial dynamic methods like Heavy-Ball or Nesterov and show how it can improve the learning process when working with nonlinear model classes.

This paper is dedicated to lossless data compression with probability estimation using neural networks. First, we propose a probability estimation architecture based on a chain of neural predictors, so that each unit of the chain is defined as a neural network with the minimum possible number of weights, which is sufficient for efficient compression of data generated by Markov sources of a given order. We show that this architecture allows us to minimize the overall number of weights participating in the probability estimation process depending on the statistical properties of the input data. Second, in order to improve compression efficiency, we introduce an information inheritance mechanism, where the probability estimate obtained by a low-order unit is used at the next higher-order unit. Experimental results show that the proposed lossless data compressor equipped with the chained probability estimation architecture provides compression ratios close to the state-of-the-art PAC compressor. At the same time, it outperforms PAC by a factor of 1.2 to 6.3 in encoding throughput and by a factor of 2.8 to 12.3 in decoding throughput on a consumer GPU.

The use of synthetic data in machine learning applications and research offers many benefits, including performance improvements through data augmentation and privacy preservation of original samples. This work proposes an efficient synthetic data generation method based on a fully connected neural network that transforms a high-dimensional random Gaussian distribution to approximate a target real-world dataset. The proposed solution combines data preprocessing designed for tabular data with distribution modeling and PCA dimensionality reduction to further enhance data privacy. The work also defines two dedicated randomized loss functions based on Wasserstein distance combined with feature Covariance and a randomized pairwise error reduction loss function. The experiments conducted on 25 diverse tabular real-world datasets confirm that the proposed solution obtains similarity and privacy scores relative to the state-of-the-art generative methods and achieves reference MMD scores orders of magnitude faster than modern deep learning solutions. The experiments involved analyzing distributional similarity, privacy protection, and the utility of synthetic data in classification tasks.

Pixel-space diffusion has recently re-emerged as a strong alternative to latent diffusion, enabling high-quality generation without pretrained autoencoders. However, standard pixel-space diffusion models receive relatively weak semantic supervision and are not explicitly designed to capture high-level visual structure. Recent representation-alignment methods (e.g., REPA) suggest that pretrained visual features can substantially improve diffusion training, and visual co-denoising has emerged as a promising direction for incorporating such features into the generative process. However, existing co-denoising approaches often entangle multiple design choices, making it unclear which are truly essential. We therefore present V-Co, a systematic study of visual co-denoising in a unified JiT-based framework. This controlled setting allows us to isolate the ingredients that make visual co-denoising effective. Our study reveals two main ingredients. First, co-denoising benefits from preserving feature-specific computation while enabling flexible cross-stream interaction, which leads to a fully dual-stream architecture together with a structurally defined unconditional prediction for classifier-free guidance. Second, it requires both stronger semantic supervision and proper cross-stream calibration, which we realize through a perceptual-drifting hybrid loss and RMS-based feature rescaling. Together, these findings yield a simple recipe for visual co-denoising. Experiments on ImageNet-256 show that, at comparable model sizes, V-Co outperforms the underlying pixel-space diffusion baseline and strong prior pixel-diffusion methods while using fewer training epochs, offering practical guidance for future representation-aligned generative models.

Medical image analysis depends on accurate segmentation and controllable synthesis, but both tasks face severe spatial imbalance: lesions occupy small regions against large backgrounds. We study adaptive spatial weighting as a task-level design principle and instantiate it in two adapters. LAW learns per-pixel loss weights for mask-conditioned diffusion by modulating a ratio prior with a feature-dependent delta map, with normalization, clamping, and Dice regularization for stability. ORDER improves lightweight segmentation by adding selective bidirectional skip attention with stage-wise confidence gating. On held-out diffusion test sets, LAW lowers FID from 158.13$\pm$0.15 to 108.43$\pm$0.71 on Polyps, from 144.13$\pm$0.31 to 89.51$\pm$0.96 on KiTS19, and from 139.22$\pm$0.38 to 112.58$\pm$0.68 on BRISC, while improving held-out mask-recovery Dice from 0.681$\pm$0.013 to 0.825$\pm$0.003 on Polyps. When the resulting images are added to nnUNet training, downstream Polyps mDice rises from 71.7$\pm$0.4 to 74.1$\pm$0.8. On the cleaned Polyps segmentation protocol, the reported ORDER configuration reaches 76.3$\pm$1.9 mDice and 67.2$\pm$2.0 mIoU at 42K parameters and 0.11 GFLOPs, versus 70.3$\pm$1.5 mDice and 59.9$\pm$1.7 mIoU for matched MK-UNet. On BRISC under the same training recipe, ORDER reaches 77.4$\pm$0.8 mDice and 68.1$\pm$0.7 mIoU. These results position adaptive spatial weighting as a practical design idea for both medical diffusion and efficient segmentation.

Until recently, numerical weather prediction (NWP) models have stood rivalless in operational forecasting despite a few limitations. Namely, physically-based models are computationally demanding and struggle at short lead times, reducing their utility for nowcasting. Motivated by these shortcomings, recent work proposes AI-weather prediction (AI-WP) alternatives that emulate analysis data with neural networks. While these data-driven approaches have achieved high skill for medium-range forecasting-applications of AI-WP to precipitation and to nowcasting are less explored. To these ends, this paper discusses \textit{SaTformer}: a video transformer adapted for precipitation nowcasting. To ameliorate some problems related to what is essentially a fat-tailed regression task, we find it prudent to formulate nowcasting as a classification problem and employ a frequency-weighted loss. This straightforward approach scored first on the NeurIPS Weather4Cast 2025 ``Cumulative Rainfall'' challenge. Code and model weights are available: \texttt{\href{github.com/leharris3/w4c-25}{github.com/leharris3/satformer}}.

The accurate forecasting of complex, high-dimensional dynamical systems from observational data is a fundamental task across numerous scientific and engineering disciplines. A significant challenge arises from noise-corrupted measurements, which severely degrade the performance of data-driven models. In chaotic dynamical systems, where small initial errors amplify exponentially, it is particularly difficult to develop a model from noisy data that achieves short-term accuracy while preserving long-term invariant properties. To overcome this, we consider the weak formulation as a complementary approach to the classical $L2$-loss function for training models of dynamical systems. We empirically verify that the weak formulation, with a proper choice of test function and integration domain, effectively filters noisy data. This insight explains why a weak form loss function is analogous to fitting a model to filtered data and provides a practical way to parameterize the weak form. Subsequently, we demonstrate how this approach overcomes the instability and inaccuracy of standard Neural ODE (NODE) in modeling chaotic systems. Through numerical examples, we show that our proposed training strategy, the Weak Penalty NODE, is computationally efficient, solver-agnostic, and yields accurate and robust forecasts across benchmark chaotic systems and a real-world climate dataset.

We investigate robust federated learning, where a group of workers collaboratively train a shared model under the orchestration of a central server in the presence of Byzantine adversaries capable of arbitrary and potentially malicious behaviors. To simultaneously enhance communication efficiency and resilience against such adversaries, we propose a Byzantine-resilient Nesterov-accelerated federated learning (Byrd-NAFL) algorithm. Byrd-NAFL seamlessly integrates Nesterov's momentum into the federated learning process alongside Byzantine-resilient aggregation rules to achieve fast and safe convergence against gradient corruption. We establish a finite-time convergence guarantee for Byrd-NAFL under non-convex and smooth loss functions with relaxed assumptions on the aggregated gradients. Extensive numerical experiments validate the effectiveness of Byrd-NAFL and demonstrate the superiority over existing benchmarks in terms of convergence speed, accuracy, and resilience to diverse malicious attacks.

We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle. Such problems arise in learning costs, energies, and dynamics from distributional data, but the associated forward solution map is typically nonlinear and implicit. We show that its optimality gap nevertheless yields convex empirical objectives for finite-dimensional potential classes, and we introduce sharpened Fenchel--Young losses that add a data-dependent discrepancy inside the forward problem. This keeps the estimator calibrated while improving the local geometry of the loss. Our main stability theorem separates the inverse error analysis into measurement error, forward perturbation, and empirical curvature. We instantiate this principle for inverse entropic unbalanced optimal transport and for inverse Jordan--Kinderlehrer--Otto (JKO) learning from independent snapshot samples, obtaining high-probability parameter recovery bounds. JKO schemes discretize Wasserstein gradient flows through a sequence of variational problems over measures, making them a natural language for population dynamics observed through snapshots. In this JKO case, the sharpened objective reduces to an unbalanced transport problem, which also clarifies the connection between variational gap losses and quadratic iJKO$^\star$ surrogates. Numerical experiments illustrate the conditioning effect of sharpening and its benefits for sparse inverse-gradient-flow recovery.

We study a general class of optimization problems with decision variable $\boldsymbolΘ \in \mathbb{R}^{p \times k}$ and cost function which is the sum of $n$ terms, each dependent on $\boldsymbolΘ$ through the $k$-dimensional projection $\boldsymbolΘ^\top \boldsymbol{x}_i$, where $\boldsymbol{x}_i$, $i \leq n$ are i.i.d. random vectors. This setting is general enough to include examples of current interest in statistical physics, high-dimensional statistics, and statistical learning theory. We consider the proportional asymptotics $n, p \to \infty$, with $n/p = Θ(1)$, and prove that, whenever there exists a minimizer satisfying a suitable generalization of a "delocalization" condition, the minimum value is universal. Namely, (for subgaussian $\boldsymbol{x}_i$) it depends on the distribution of $\boldsymbol{x}_i$ only through its asymptotic mean and covariance. This delocalization condition is essentially necessary. Earlier universality results for such problems were limited to strongly convex loss functions. We derive applications of our theory to statistical learning and prove general universality results both for train and (under additional conditions) test error. In particular, we establish universality for vectors $\boldsymbol{x}_i$ generated by random 1-layer neural networks (random features models) and first-order Taylor approximations of 2-layer networks (neural tangent models). Finally, we establish that the delocalization property holds for a class of statistical learning problems under a condition that is easy to verify.

The problem of linear predictions has been extensively studied for the past century under pretty generalized frameworks. Recent advances in the robust statistics literature allow us to analyze robust versions of classical linear models through the prism of Median of Means (MoM). Combining these approaches in a piecemeal way might lead to ad-hoc procedures, and the restricted theoretical conclusions that underpin each individual contribution may no longer be valid. To meet these challenges coherently, in this study, we offer a unified robust framework that includes a broad variety of linear prediction problems on a Hilbert space, coupled with a generic class of loss functions. Notably, we do not require any assumptions on the distribution of the outlying data points ($\mathcal{O}$) nor the compactness of the support of the inlying ones ($\mathcal{I}$). Under mild conditions on the dual norm, we show that for misspecification level $ε$, these estimators achieve an error rate of $O(\max\left\{|\mathcal{O}|^{1/2}n^{-1/2}, |\mathcal{I}|^{1/2}n^{-1} \right\}+ε)$, matching the best-known rates in literature. This rate is slightly slower than the classical rates of $O(n^{-1/2})$, indicating that we need to pay a price in terms of error rates to obtain robust estimates. Additionally, we show that this rate can be improved to achieve so-called "fast rates" under additional assumptions.

Classification outperforms regression across matched machine learning models in portfolio construction. A stacking ensemble of gradient boosted tree, random forest, and neural network yields a value-weighted annualized Sharpe ratio of 1.83 for classification and 1.11 for regression. This outperformance persists in multiclass settings, across subsamples, and after transaction costs. Spanning tests show that classification retains economically large alphas after we control for regression, whereas regression alphas shrink substantially once we control for classification. These results indicate that classification extracts more return information than matched regression. Our diagnostics trace classification's advantage to sharper and more precise separation of return deciles.