Comparing two probability distributions is a basic building block of statistics and machine learning, and the right family is well understood: the Rényi divergences of order $α\in[0,\infty]$ are the unique family monotone under data processing and additive on independent products. Many problems instead compare more than two distributions at once -- multi-population fairness, multi-prior PAC-Bayes bounds, multi-hypothesis testing -- and the right multi-distribution generalization of the Rényi family has been an open question. We characterize it. Every functional of $W$-tuples of distributions that is monotone under data processing and additive on independent products is a positive integral of multi-way coincidence divergences $C_α(π_1,\dots,π_W) := -\log\int π_1^{α_1}\cdotsπ_W^{α_W}$ (with $\sum_k α_k = 1$) over a parameter space with four strata: the simplex interior; mixed-sign exponent cones (the analogue of Rényi orders $>1$); a tropical boundary at infinity carrying max-divergences; and pairwise Kullback-Leibler edges at the simplex vertices. Each stratum is necessary -- the destination of an explicit data-processing-monotone, product-additive divergence the others cannot reproduce -- and each is a clean limit of simplex-interior atoms. The same family arises from several independent routes -- the structural axioms, Kolmogorov-Nagumo means with Rényi's entropy axiomatics, classical entropy characterizations, multi-hypothesis testing error exponents, and a multi-lottery betting interpretation -- structural evidence that this is the canonical multi-distribution Rényi calculus rather than an artefact of any one axiomatic input. The two-prior case recovers the standard Rényi result; a worked $W=3$ instance, numerical verification, and a conditional extension round out the treatment.

When two or more parameters or labels produce similar data, they are degenerate, or hard to distinguish. Degeneracies render both label prediction and inverse problems difficult, since both machine learning algorithms and probabilistic samplers rely on the distinguishability of data and its gradients with respect to parameters. However, identifying degeneracies in physical models or real-world datasets can be elucidating about the choice of model or the underlying process that produces the data. We present the degeneracy distillery, a method that (1) detects and (2) resolves degenerate parameter combinations (a) automatically and (b) symbolically, from parameter-data (or parameter-simulation) pairs alone, through estimation and flattening of the Fisher information matrix. By exploring the information geometry of the likelihood, we characterize degeneracies as an intrinsic property of the physical model, requiring no realised data observation. We demonstrate our approach on a range of synthetic and real-world problems, discovering symbolic coordinate transformations that identify the combinations of parameters of a model which yield independent effects on the data. The resulting coordinates flatten the Fisher information in expectation globally, in contrast to posterior-based methods that flatten only at a single point, and substantially reduce the simulation budget required for downstream neural posterior estimation. In test cases we require up to $10\times$ fewer simulations for posterior estimation at matched validation calibration whilst simultaneously gaining physical insight on the system.

Mean Field Variational Inference (MFVI) is widely understood to underestimate posterior variance. By analysing conjugate Bayesian Linear Regression (BLR), we show that this characterization is incomplete: while MFVI underestimates the variance in parameter space, it can overestimate the predictive variance compared to the exact posterior. We show that if the MFVI posterior underestimates predictive variances in some directions, it necessarily overestimates them in others. Crucially, this overestimation occurs in directions where the training data concentrates. This leads to the surprising result that, for a test point drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior. We demonstrate a pathological case of this effect, where the MFVI posterior fails to reduce predictive variance compared to the prior on in distribution data. We connect these results to the Cold Posterior Effect, arguing that varying the temperature can correct this overestimation, yielding predictions closer to those of the exact posterior. We validate our theory on synthetic and real-world regression tasks.

To improve detection robustness in adverse conditions (e.g., haze and low light), image restoration is commonly applied as a pre-processing step to enhance image quality for the detector. However, the functional mismatch between restoration and detection networks can introduce instability and hinder effective integration--an issue that remains underexplored. We revisit this limitation through the lens of Lipschitz continuity, analyzing the functional differences between restoration and detection networks in both the input space and the parameter space. Our analysis shows that restoration networks perform smooth, continuous transformations, while object detectors operate with discontinuous decision boundaries, making them highly sensitive to minor perturbations. This mismatch introduces instability in traditional cascade frameworks, where even imperceptible noise from restoration is amplified during detection, disrupting gradient flow and hindering optimization. To address this, we propose Lipschitz-regularized object detection (LROD), a simple yet effective framework that integrates image restoration directly into the detector's feature learning, harmonizing the Lipschitz continuity of both tasks during training. We implement this framework as Lipschitz-regularized YOLO (LR-YOLO), extending seamlessly to existing YOLO detectors. Extensive experiments on haze and low-light benchmarks demonstrate that LR-YOLO consistently improves detection stability, optimization smoothness, and overall accuracy.

Despite its widespread adoption, Adam's advantage over Stochastic Gradient Descent (SGD) lacks a comprehensive theoretical explanation. This paper investigates Adam's sensitivity to rotations of the parameter space. We observe that Adam's performance in training transformers degrades under random rotations of the parameter space, indicating a crucial sensitivity to the choice of basis in practice. This reveals that conventional rotation-invariant assumptions are insufficient to capture Adam's advantages theoretically. To better understand the rotation-dependent properties that benefit Adam, we also identify structured rotations that preserve or even enhance its empirical performance. We then examine the rotation-dependent assumptions in the literature and find that they fall short in explaining Adam's behaviour across various rotation types. In contrast, we verify the orthogonality of the update as a promising indicator of Adam's basis sensitivity, suggesting it may be the key quantity for developing rotation-dependent theoretical frameworks that better explain its empirical success.

Transformer architectures can solve unseen tasks based on input-output pairs in a given prompt due to in-context learning (ICL). Existing theoretical studies on ICL have mainly focused on linear regression tasks, often with i.i.d.

We consider binary classification restricted to a class of continuous piecewise linear functions whose decision boundaries are (possibly nonconvex) starshaped polyhedral sets, supported on a fixed polyhedral simplicial fan. We investigate the expressivity of these function classes and describe the combinatorial and geometric structure of the loss landscape, most prominently the sublevel sets, for two loss-functions: the 0/1-loss (discrete loss) and a log-likelihood loss function. In particular, we give explicit bounds on the VC dimension of this model, and concretely describe the sublevel sets of the discrete loss as chambers in a hyperplane arrangement. For the log-likelihood loss, we give sufficient conditions for the optimum to be unique, and describe the geometry of the optimum when varying the rate parameter of the underlying exponential probability distribution.

Low-Rank Adaptation (LoRA) has become the de facto parameter-efficient finetuning (PEFT) method for large language models (LLMs) by constraining weight updates to low-rank matrices. Recent works such as Tied-LoRA, VeRA, and VBLoRA push efficiency further by introducing additional constraints to reduce the trainable parameter space. In this paper, we show that the parameter space reduction strategies employed by these LoRA variants can be formulated within a unified framework, Uni-LoRA, where the LoRA parameter space, flattened as a highdimensional vector space RD, can be reconstructed through a projection from a subspace Rd, with d D. We demonstrate that the fundamental difference among various LoRA methods lies in the choice of the projection matrix, P RD d. Most existing LoRA variants rely on layer-wise or structure-specific projections that limit cross-layer parameter sharing, thereby compromising parameter efficiency. In light of this, we introduce an efficient and theoretically grounded projection matrix that is isometric, enabling global parameter sharing and reducing computation overhead. Furthermore, under the unified view of Uni-LoRA, this design requires only a single trainable vector to reconstruct LoRA parameters for the entire LLM - making UniLoRA both a unified framework and a "one-vector-only" solution. Extensive experiments on GLUE, mathematical reasoning, and instruction tuning benchmarks demonstrate that Uni-LoRA achieves state-of-the-art parameter efficiency while outperforming or matching prior approaches in predictive performance.

Math reasoning has been one crucial ability of large language models (LLMs), where significant advancements have been achieved in recent years. However, most efforts focus on LLMs by curating high-quality annotation data and intricate training (or inference) paradigms, while the math reasoning performance of multi-modal LLMs (MLLMs) remains lagging behind. Since the MLLM typically consists of an LLM and vision block, we wonder: \textit{Can MLLMs directly absorb math reasoning abilities from off-the-shelf math LLMs without tuning?} Recent model-merging approaches may offer insights into this question. However, they overlook the alignment between the MLLM and LLM, where we find that there is a large gap between their parameter spaces, resulting in lower performance. Our empirical evidence reveals two key factors behind this issue: the identification of crucial reasoning-associated layers in the model and the mitigation of the gaps in parameter space. Based on the empirical insights, we propose \textbf{IP-Merging} that first \textbf{I}dentifies the reasoning-associated parameters in both MLLM and Math LLM, then \textbf{P}rojects them into the subspace of MLLM aiming to maintain the alignment, finally merges parameters in this subspace. IP-Merging is a tuning-free approach since parameters are directly adjusted. Extensive experiments demonstrate that our IP-Merging method can enhance the math reasoning ability of MLLMs directly from Math LLMs without compromising their other capabilities.

Despite its widespread adoption, Adam's advantage over Stochastic Gradient Descent (SGD) lacks a comprehensive theoretical explanation. This paper investigates Adam's sensitivity to rotations of the parameter space. We observe that Adam's performance in training transformers degrades under random rotations of the parameter space, indicating a crucial sensitivity to the choice of basis in practice. This reveals that conventional rotation-invariant assumptions are insufficient to capture Adam's advantages theoretically. To better understand the rotation-dependent properties that benefit Adam, we also identify structured rotations that preserve or even enhance its empirical performance. We then examine the rotation-dependent assumptions in the literature and find that they fall short in explaining Adam's behavior across various rotation types. In contrast, we verify the orthogonality of the update as a promising indicator of Adam's basis sensitivity, suggesting it may be the key quantity for developing rotation-dependent theoretical frameworks that better explain its empirical success.