We consider the problem of estimating the power spectral density of a Vector-valued Autoregressive Moving-Average (VARMA) process. A well-established approach to accomplishing this task consists of formulating a moment problem that selects the spectral density maximizing an entropy-like index while matching certain moments of the underlying process. However, existing methods assume that the moving-average (MA) component is either fixed or scalar, thereby reducing the class of ARMA models over which the optimal spectral density is sought. In this paper, we propose a new moment problem, based on the notion of transportation entropy, for learning the spectral density of a VARMA process without fixing the MA component or restricting it to be scalar.
In open-world recognition, machine learning models not only make predictions on unknown structures ("generalized open-set"), but also continually learn and improve from those predictions over time. Interactive segmentation, despite its access to user feedback, has so far remained confined to the open-set setting. In this work we introduce MEOW, the first framework that turns non-interactive segmentation backbones into interactive open-world learners without updating learnable model parameters. MEOW combines parameter-free transductive reasoning with memory consolidation to accumulate knowledge and thereby improve future performance. Focusing on medical data, we show that MEOW consistently improves its own predictions on classes never seen during training, and transfers zero-shot to unfamiliar domains, including micro-CT data and other imaging modalities.
We study how outliers affect the hard-margin linear support vector machine when the dimension grows while the sample sizes remain fixed. For one-sided contamination, the centered Gram matrix converges to a constant inlier block plus an outlier-diagonal component, and the limiting dual problem has a closed-form solution. All outliers act through one effective distance. This representation yields explicit thresholds for loss of inlier support, margin shrinkage. intercept displacement, and asymptotic misclassification. We then treat contamination in both classes. The limiting intercept is the multiplier of the class-balance constraint, and the two classes pull it in opposite directions. Consequently, at most one class can collapse or be asymptotically misclassified, whereas symmetric contamination leaves the limiting discriminant function unchanged even though the margin shrinks. Seven numerical experiments verify the dual allocation, phase transitions, and discriminant limits.
Stochastic diffusion samplers often require substantially more neural function evaluations (NFEs) than deterministic samplers to achieve comparable sample quality. Although higher-order discretization accelerates deterministic sampling, its benefits for stochastic sampling remains unclear. We establish matching upper and worst-case lower bounds showing that stochastic samplers with higher-order approximations remain first-order convergent, even with exact score functions. Thus, increasing the approximation order does not generally improve the convergence order. Motivated by this limitation, we introduce forward-value discretization, a training-free framework that evaluates the data/noise prediction model at a lookahead estimate constructed from previous predictions. For deterministic sampling, the resulting Forward DPMSolver (F-DPMSolver), improves sample quality at low NFE budgets through its DDIM-based variant, despite remaining first-order convergent. For stochastic sampling, we develop Forward DDPM (F-DDPM), which evaluates the model at a cleaner lookahead point adjusted for subsequent noise injection. This estimate reuses both the previous model prediction and the previously injected noise. Both variants admit first-order convergence guarantees with one model evaluation per iteration. Across CIFAR-10, ImageNet, FFHQ, and LSUN, F-DDPM consistently achieves lower FID than the evaluated stochastic baselines at matched NFE budgets. With 1.5 times as many NFEs as deterministic baselines, it achieves competitive sample quality, narrowing the practical efficiency gap between stochastic and deterministic sampling. Code is available at https://github.com/Na-Li66/F-DDPM.
Compositional data have been analysed in the literature from three main points of view: directly on the simplex, in log-ratio coordinates, and as directional data on the unit sphere. In this paper, we construct a random forest framework for compositional responses that accommodates all three. Random forests are ensembles of decision trees, and the split criterion determines how each tree grows. We use the log-likelihood gain as the split criterion and consider distributions defined on the different spaces in which compositions can be represented. The Dirichlet likelihood has been studied in the literature, but its covariance is fixed by its mean up to a single scalar, and its maximum likelihood estimator has no closed form. To obtain more flexible criteria, we use two well-known transformations. First, the square-root transformation maps compositions onto the unit sphere, where existing directional distributions require numerical estimation. We introduce a tangent-normal (TN) distribution, which shares its parameters with the Kent distribution, and construct a split criterion whose gain is computed from additive sufficient statistics: the shape parameters are maximised out in closed form, up to an exponentially small truncation constant, so that no parameter is estimated iteratively. Second, the log-ratio transformation maps compositions to Euclidean space, where the multivariate logistic-normal distribution has a flexible covariance and a closed-form maximum log-likelihood, also based on additive sufficient statistics. Both criteria respond to a shift of the mean, measured relative to the covariance, and to a change in the covariance itself. In simulations, the most accurate forest depended on the covariance structure of the composition. The TN forest was the most accurate in two of the three designs.
We present TorchCurves, a PyTorch library that treats vector-valued parametric curves as learnable primitives. Its central operation maps batches of scalar inputs to points on curves represented in B-spline or Legendre polynomial bases, with differentiation through both inputs and coefficients. Curve coefficients can be optimized jointly with the surrounding model or produced by another network. The library provides trainable modules, reusable basis modules, and a functional interface, supporting applications ranging from continuous feature embeddings to Kolmogorov-Arnold networks and functional regression. Its implementation combines classical evaluation algorithms with vectorized PyTorch operations, emphasizing readability and maintainability without custom hardware kernels. Timing experiments compare curve evaluation against NumPy and SciPy and KAN training against EfficientKAN across a range of problem sizes. The results illustrate where this generality remains computationally practical and where overhead is substantial. Library is available on GitHub at https://github.com/alexshtf/torchcurves.
Policy optimization is central to language model post-training, yet principled guidance remains limited. We revisit its theoretical foundations to identify algorithms that learn near-optimal policies using total samples and gradient steps governed by the base model's response coverage, a measure of how readily it generates optimal responses. We call such algorithms coverage-preserving. This guarantee captures the goal of amplifying capabilities accessible through sampling from the base model. (1) Coverage collapse. REINFORCE, the basis for PPO and GRPO, can drive the probability of optimal responses to near zero, even as reward improves. This coverage collapse stalls optimization, and convergence can require exponentially many steps despite good initial coverage. The failure arises even with exact gradients and persists under KL regularization to the base model. (2) Reference resets. We show that repeatedly resetting the KL reference to the most recent policy mitigates coverage collapse. We formalize this intervention in Approximate Policy Mirror Descent (APMD), a meta-algorithm that approximately solves a sequence of KL-regularized RL subproblems. APMD enables stronger regularization, making each subproblem easier to solve, while reference resets allow convergence to global optimality. (3) Coverage-preserving algorithms. For softmax linear policies over pretrained features, we use APMD to establish the first global convergence guarantees showing that widely used families of post-training algorithms are coverage-preserving. These include stylized PPO and GRPO variants with explicit KL regularization and reference resets, and squared-loss algorithms that implement them implicitly, including those used to train the Kimi models. These results clarify how regularization, clipping, and objective choice affect the sampling and optimization costs of post-training.
AUC and the Brier score are standard summaries of a binary risk score, yet they can rank models differently. We introduce a weighted AUC, AUC$_π$, which weights each pair of observations by how differently a decision maker with probability of acting $π$ treats them. We show that all three are built on weighted averages of one curve, the Youden index $J(t) = \mathrm{TPR}(t) - \mathrm{FPR}(t)$ across cutoffs $t$, and that the averages differ in the weights they put on the cutoffs. Each weights a cutoff by how much something rises across it: for AUC, the distribution of all scores; for the Brier score, the true probability $\mathbb{E}[Y \mid S]$; for AUC$_π$, the decision maker's probability of acting. Up to normalization, AUC$_π$ is the Brier skill score of the ideal forecast with its weight moved to the cutoffs the decision maker cares about. We also show that weighting pairs by $|π(s) - π(s')|$ is, up to scale, the only pair weight fixed in advance that makes a pair-weighted AUC an average of the Youden curve for every score distribution. As a ranking measure, AUC weights every model the same way. Hand (2009) shows that it can also be read as an average over cost thresholds, and in that reading each model weights the thresholds differently. The numerator of AUC$_π$, which we call the policy Youden index, is the expected share of class 1 a fixed policy acts on minus the expected share of class 0. It judges every model by the same policy and so avoids this; the full ratio AUC$_π$ does not.
Cross-validation estimates predictive performance by repeatedly splitting the data into training and validation sets, fitting the model on the first and evaluating it on the second. Every observation influences the estimate in two roles, through its own evaluation when it is held out and through the models it helps fit when it trains. The random splits determine its weight in each role. From this representation, we prove an exact finite-sample identity for the variance of the cross-validated estimate. For a given splitting scheme with a common training set size, the variance is a function of these weights, with constants of the model, the metric, and the population. It covers metrics averaged over observations, such as mean squared error, and over case--control pairs, such as the AUC. The identity shows which schemes to use. The variance decomposes into sampling variance, which depends only on the training size, and variance due to the random fold splitting, which arises when the weights change from one draw of the splits to the next. K-fold cross-validation, whose weights are equal across observations by design, is therefore variance optimal at first order; Monte Carlo and bootstrap cross-validation are not. The first-order variance of K-fold does not depend on K, and under reasonable conditions on the model leave-one-out has the smallest variance of any K-fold scheme. For the AUC the same holds for leave-pair-out, which evaluates every case--control pair. Finally, we show the cross-validated estimate is asymptotically normal with the identity's full variance. Together, these results provide a general theoretical foundation for inference on cross-validated performance and establish principles for selecting variance-optimal cross-validation designs.
Mixtures of logistic regressions provide a flexible framework for modeling binary responses, extending standard logistic regression to accommodate unobserved heterogeneity while retaining an interpretable regression structure. In this paper, we study such models with an unspecified mixing distribution. We analyze a sieve nonparametric maximum likelihood estimator (NPMLE) and derive nonasymptotic bounds on the estimation error of the resulting mixture regression function under both fixed and random designs. Our results reveal an explicit tradeoff between statistical estimation error and the approximation error induced by the sieve radius, from which we derive convergence rates under several tail conditions on the mixing distribution. For computation, we propose a finite-particle algorithm that approximates the sieve NPMLE by optimizing the locations and weights of a finitely supported mixing distribution. The method offers numerical advantages over existing algorithms and comes with rigorous convergence guarantees. For any fixed dataset, we establish global convergence of the likelihood to the restricted NPMLE value when the number of particles exceeds a finite threshold. When the true mixing distribution is finitely supported, we further establish uniform global convergence guarantees and derive error bounds that separate algorithmic and statistical errors. Our analysis also yields a denoising bound for the mixing distribution and an identifiability result that complements existing results in the literature.
Learning feature interactions is essential for accurate prediction and interpretable modeling, yet a high interaction score may reflect fitting noise rather than a reproducible pattern. Ranking methods score feature pairs but leave the cutoff to the user; testing methods set the cutoff but rely on repeated model refits or a model of the joint feature distribution. We present LENS, a model-agnostic framework for selecting feature interactions through independent replication. LENS trains two models with the same learning algorithm on disjoint data and accesses them only through predictions. For each feature pair, it removes main effects from the joint-effect surface to isolate an interaction pattern, ranks pairs by its strength, and tests whether the second model reproduces it at a meaningful strength. A single multiple-testing correction determines which pairs to retain, using only two model fits regardless of the number of features. We prove that a replication ratio alone can retain arbitrarily weak patterns shared by both fits, whereas a strength floor restores selection consistency, and we characterize when normalization in Friedman's $H^2$ reverses interaction rankings. On 16 synthetic designs, LENS matches leading model-specific ranking methods (e.g., TreeSHAP) and recovers 91% of material interactions, against 45% for the knockoff-based DIAMOND, without selecting false pairs; with 1,225 candidate pairs and no interactions, it makes substantially fewer false discoveries. On large-scale French motor insurance data, it identifies interactions documented in the actuarial literature and recovers 90% of the deviance gain obtained using all pairs. On two credit datasets, LENS selects few pairs while maintaining or slightly improving predictive performance over the additive baseline.
Transformer feed-forward blocks combine several design choices: squared activations, gating, and tied versus untied parameter sharing, whose separate contributions to expressivity remain poorly understood. We introduce CP-MLPs, a tensor framework in which each multiplicative unit contributes a rank-1 interaction with three roles: detecting an input feature, reading a value, and writing to the output. Tied gated blocks such as ReLU$^2$ force the detector and value roles to share the same direction; untied gated blocks such as ReGLU and SwiGLU let them differ. We show that this distinction yields exact, degree-wise tensor-rank separations: untied detector-value units are strictly more width-efficient for indefinite quadratic interactions and role-asymmetric monomials, while tied units are structurally matched to symmetric pure-power targets. The width advantage of untying therefore depends on the target structure. Experiments show loss plateaus below the exact-representation width, but these plateaus alone do not establish a positive approximation gap: arbitrarily accurate approximation is governed by border rank. In trained gated MLPs, untied blocks retain an advantage in selected comparisons at matched input-projection budget. Code is available at https://github.com/rarefin/CP-MLPs.
Under i.i.d. uniform sampling from a closed Riemannian manifold embedded in Euclidean space, we derive first-order asymptotic expansions for the eigenvalues and eigenvectors of symmetrically normalized graph Laplacians constructed via Gaussian kernels and provide concentration inequalities for the remainder terms. As a direct consequence, we establish the corresponding weak convergence results. Unlike most related works, our analysis does not impose a multiplicity-one assumption on the eigenvalues, thereby extending the applicability of the framework to general manifolds with potentially nontrivial eigenvalue multiplicities.
Scientific design often requires balancing competing objectives that are expensive to evaluate through experiments or simulations. Multi-objective Bayesian optimization (MOBO) addresses this challenge by using probabilistic models to guide the selection of promising designs. In many applications, tunable inputs map to observable intermediate responses, which are then combined through known functions to compute the final objectives. Prior work on Bayesian optimization of composite functions shows that learning from these intermediate responses, which we call Composite modeling, can substantially improve sampling efficiency in single-objective optimization, compared to methods that condition only on black-box observations of the objective function. The benefits of Composite modeling in MOBO, however, remain understudied. In this work, we study the benefits of Composite modeling in MOBO and investigate the factors driving variation in its gains. We first establish single-objective conditions under which intermediate observations provide no additional predictive information beyond objective values, where composite and black box methods use the same warped Gaussian process prior. Controlled numerical experiments with versions of four MOBO methods across eight synthetic and application-inspired scientific-design benchmarks examine how these conditions relate to optimization gains in the multi-objective setting, where the black box approach models the objective functions with a standard multi-output Gaussian process. Both investigations support the claim that composite modeling tends to help more when the function mapping intermediate responses to final objectives is non-bijective and non-affine.
BF16 is widely used to pretrain large language models because it has the same dynamic range as FP32 and converges similarly to FP32 training. However, in reinforcement learning with verifiable rewards, where learning rates are typically low, we find that standard mixed-precision BF16 training converges more slowly than FP16 training at the same learning rate. We attribute this gap to weight sticking: when FP32 shadow weights are quantized to BF16 using round-to-nearest, small updates often leave the BF16 weights used for rollout generation and training unchanged. Consequently, several optimizer steps may be needed before updates to the shadow weights are reflected in the BF16 weights. Raising the BF16 learning rate reduces weight sticking and matches FP16's convergence speed, but it also increases the KL divergence from the reference model. We instead apply stochastic rounding when quantizing the FP32 shadow weights to BF16. This approach matches FP16's convergence speed across different models and datasets while reducing the KL divergence relative to BF16 training with a larger learning rate. Unlike FP16, it keeps BF16's wider dynamic range and needs no loss scaling.
Simulation optimization typically returns a single estimated optimal point. When decisions also depend on criteria evaluated independently of simulation, such as resource cost, the decision maker instead needs the set of designs whose expected performance lies within a tolerance of the optimum, often choosing the tolerance only after seeing the tradeoffs. We study how to identify these near-optimal regions on a continuous design space, simultaneously for all tolerances up to a prescribed maximum $T$, with region estimation error at most $\eta$. We show that region identification is not a by-product of optimization. Specifically, when $\alpha>\nu$, any procedure, even one given the optimal design in advance, needs at least $\Omega\bigl(\eta^{-(2+d/\nu)}\min\{1,KT^{d/\alpha}\}\bigr)$ simulation evaluations, where $d$ is the dimension, $\nu$ the smoothness, and $(K,\alpha)$ the covering parameters that describe the geometry of the near-optimal designs. We propose SASORI, which returns an estimated optimal point and the regions for all tolerances from the same evaluations, matching this lower bound up to logarithmic factors. SASORI also achieves the optimal tradeoff between region estimation error and cumulative regret, and reduces to a minimax-optimal simulation optimization procedure at zero tolerance. Numerical experiments illustrate the effectiveness of our proposed method.
Reliable inference for rankings of large language models based on human pairwise comparisons must preserve the privacy of individual preference records. To meet this need, we develop methods for score and rank inference under the Bradley-Terry model with edge differential privacy. For an objective-perturbed estimator, a new leave-one-out analysis establishes finite-sample bounds on Euclidean and entrywise estimation errors. For sufficiently dense comparison graphs, the error bounds extend to smaller privacy budgets, including regimes in which its privacy term dominates its statistical term. A uniform asymptotic expansion yields asymptotically valid confidence intervals for individual scores and pairwise score differences. To improve finite-sample performance, we account for both regularization bias and privacy-induced randomness by combining a plug-in ridge-bias correction with Gaussian-Laplace calibration. We further establish a bootstrap procedure for simultaneous rank inference, providing asymptotically valid confidence sets for individual ranks and tests of top-$K$ membership. When the comparison probability is public and known, all proposed inferential procedures require no additional privacy budget. We evaluate finite-sample performance through simulations and apply the proposed methods to rank large language models for creative writing.
We study the regression of a scalar response $Y$ on a Euclidean predictor $\mathbf{X} \in \mathbb{R}^p$ and a random predictor object $\mathcal{Z}$ in a general metric space under arbitrary structural dependence. We introduce a joint sufficient dimension reduction $(\mathbf{B}^\top\mathbf{X}, \mathcal{G}_Z)$ constructed via a sequential conditional independence framework, which differs fundamentally from the ordinary central subspace of $Y \mid \mathbf{X}$. Methodologically, $\mathcal{Z}$ is embedded into a reproducing kernel Hilbert space, and a regularized kernel-response envelope model extracts a finite-dimensional reduction of the feature map. Partial adaptive estimation then recovers the Euclidean reduction conditional on the first-stage coordinates. We establish root-$n$ consistency rates for the first-stage envelope subspace under a shrinking regularizer, alongside large-sample consistency for the second-stage Euclidean subspace estimator. Simulations and two real-data applications illustrate the utility and flexibility of the proposed method.
We study the excess risk of constant-stepsize online stochastic gradient descent (SGD) with tail-averaging for high-dimensional linear regression. Building on the instance-dependent spectral framework of Zou et al. (2021), we establish matching initialization-explicit upper and lower bounds on the total excess risk of tail-averaged SGD in terms of the population covariance spectrum $\mathbf{H}$ and initialization $\mathbf{w}_0-\mathbf{w}^*$ whenever a trajectory spectral feedback condition holds. Existing analyses exhibit a fundamental gap in the effective variance: the upper bound of Zou et al. (2021) retains an undamped contribution from the initial error in the head eigenspace $\mathbf{I}_{0:k^\dagger}$, whereas their lower bound omits the variance induced by the head initialization altogether. We show that both the existing upper and lower bounds are loose: the single-hop fast-mode variance contribution from the head initialization is exponentially damped during the burn-in phase, whereas fourth-moment energy scattered from the head initialization into slow spectral modes during the early burn-in steps survives undamped into the averaging window. We prove that our spectral feedback condition is satisfied in two concrete constant-stepsize regimes: (i) for arbitrary covariance spectra $\mathbf{H}$ and any finite horizon, under an explicit horizon-dependent stepsize chosen prior to execution; and (ii) for polynomially decaying spectra $\lambda_i\asymp i^{-a}$ ($a>1$), under a single horizon-independent positive stepsize. Finally, we complement these positive results with explicit one- and two-dimensional separation constructions proving an exponential-versus-polynomial gap in the lower bound of Zou et al. (2021) and showing why fourth-moment assumptions alone cannot determine the asymptotic exponential decay rate without additional structural conditions.
We compute the VC-dimension of metric ball classifiers in the model spaces of Riemannian geometry (Euclidean, spherical, hyperbolic) and present upper and lower bounds on the VC-dimension of metric balls in several other situations, including general normed vector spaces (potentially infinite VC-dimension), vector spaces with $\ell^p$-norms (VC-dimension grows at least linearly in the dimension, faster than Euclidean for $p=1$), flat tori, and general Riemannian manifolds (arbitrarily large VC-dimension after a conformal change). In several situations where the VC-dimension is large or even infinite, we prove generalization bounds under additional assumptions on the data distribution using covering arguments. Both in the VC-dimension and covering setting, we consider data on low-dimensional subsets of high-dimensional Euclidean spaces. As an application, we consider coverage-based algorithms in active learning.