Objective functions based on Hellinger distance yield robust and efficient estimators of model parameters. Motivated by privacy and regulatory requirements encountered in contemporary applications, we derive in this paper \emph{private minimum Hellinger distance estimators}. The estimators satisfy a new privacy constraint, namely, Hellinger differential privacy, while retaining the robustness and efficiency properties. We demonstrate that Hellinger differential privacy shares several features of standard differential privacy while allowing for sharper inference. Additionally, for computational purposes, we also develop Hellinger differentially private gradient descent and Newton-Raphson algorithms. We illustrate the behavior of our estimators in finite samples using numerical experiments and verify that they retain robustness properties under gross-error contamination.
Parameter estimation and classification in binary logistic regression deteriorate severely when the number of minority class observations ($n_1$) is smaller than the feature dimension ($p$). Under such rank-deficient rarity ($n_1 < p$), maximum likelihood estimators suffer from quasi-complete separation, penalized likelihood methods exhibit severe intercept bias, and convex minority oversampling techniques such as SMOTE collapse into an $(n_1 - 1)$-dimensional affine subspace. In this paper, we propose a novel \textit{Self-Calibrated Zero-Complement} estimator that shifts the paradigm from minority interpolation to majority fringe extrapolation. Grounded in Stein's Lemma, we prove that the conditional expectation shift of the abundant majority class ($Y=0$) induced by logistic truncation uniquely identifies the true hyperplane normal $\boldsymbolβ$ after Mahalanobis whitening. We further establish that extrapolating synthetic minority instances outward from the high-quantile density fringe of the majority distribution along the conjugate direction preserves full-rank covariance regularity ($\mathrm{rank} = p$). To accommodate both semi-supervised environments (with an unlabeled background mean) and strictly supervised subsamples, we introduce a hybrid estimator incorporating the intrinsic third-moment skewness of the majority class alongside a location-shift-invariant zero-boundary intercept calibration. Extensive Monte Carlo simulations and real-world benchmark applications (Wisconsin Diagnostic Breast Cancer and Banknote Authentication) under extreme rarity ($n_1 \in \{3, 5\}$) demonstrate that our approach prevents parameter divergence, recovers the oracle hyperplane with high fidelity, and substantially outperforms Firth's regression and SMOTE in F1-score and estimation stability.
Zhengkai Pan, Peter Potaptchik, Wenxi Yao, Michael S. Albergo, Jakiw Pidstrigach
Recent one-step generative models accelerate sampling by learning deterministic flow maps of the underlying dynamics. These methods rely on learning from ordinary differential equations, leaving open how to define an exact distillation procedure for stochastic dynamics. We introduce the Itô map, an any-step stochastic flow map that takes an intermediate state and Brownian path and predicts future states in a single pass. The Itô map formulation yields novel estimators for inference-time control by providing cheap, differentiable access to posterior samples. Empirically, Itô maps produce diverse, conditionally valid endpoint samples from fixed intermediate states and support strong steering performance on synthetic and image-generation benchmarks. These results establish any-step SDE integration as a useful primitive for posterior sampling and stochastic control.
Robust subspace estimation is fundamental to many machine learning and data analysis tasks. Iteratively Reweighted Least Squares (IRLS) is an elegant and empirically effective approach to this problem, yet its theoretical properties remain poorly understood. This paper establishes that, under deterministic conditions, a variant of IRLS with dynamic smoothing regularization converges linearly to the underlying subspace from any initialization. We extend these guarantees to affine subspace estimation, a setting that lacks prior recovery theory. Additionally, we illustrate the practical benefits of IRLS through an application to low-dimensional neural network training. Our results provide the first global convergence guarantees for IRLS in robust subspace recovery and, more broadly, for nonconvex IRLS on a Riemannian manifold.
Anindya De, Elchanan Mossel, Shivam Nadimpalli, Joe Neeman
We give a proper membership-query algorithm for agnostically learning halfspaces under the uniform distribution on $\{\pm1\}^n$. Given query access to an arbitrary function $f:\{\pm1\}^n \to \{\pm1\}$, our algorithm runs in time $\mathrm{poly}(n)+2^{\mathrm{poly}(1/\varepsilon)}$ and outputs a halfspace with error at most $\mathrm{OPT}+\varepsilon$, where $\mathrm{OPT}$ is the error of the best-fitting halfspace for $f$. Previously, even with membership queries, the best known running time was $n^{\mathrm{poly}(1/\varepsilon)}$, achieved by the sample-based algorithm of Kalai, Klivans, Mansour, and Servedio (FOCS 2005). Our algorithm is the first agnostic halfspace learner under the uniform distribution on $\{\pm1\}^n$ whose running time is polynomial in $n$ with an exponent independent of $\varepsilon$. Combining our algorithm with a boosting argument yields a proper agnostic learner using only random examples. It achieves the same $\mathrm{OPT}+\varepsilon$ guarantee, with running time matching that of Kalai et al.\ up to an additive term depending only on $\varepsilon$. To our knowledge, this is the first sample-based proper agnostic learner in the Boolean setting with running time $2^{o(n)}$ for every fixed $\varepsilon>0$.
Learning to stabilize an unknown continuous-time stochastic linear system is a central problem in control theory and reinforcement learning. This paper numerically studies data-driven identification of such systems with possibly unstable dynamics for the purpose of stabilization. We present a method that learns a stabilizing gain from observed trajectories, and examine how the failure probability, namely the probability that the learned gain does not stabilize the system, depends on the amount of data, dimension, noise intensity, excitation, and controllability. Numerical experiments on an unstable batch-reactor benchmark show that the squared identification error decays inversely with the aggregate observation time, that is, the combined length of all trajectories, while the failure probability decays exponentially in the same quantity. Hence, despite instability, a single long trajectory can achieve similar stabilization performance to that of many short trajectories of the same total length. We also observe \emph{signal saturation}: stronger excitation improves stabilization only logarithmically with the excitation strength. Finally, we show experimentally that accurate identification of the full dynamics is \emph{not} necessary for successful stabilization.
Conformal Bayes combines Bayesian posterior predictive scores with conformal calibration. Under continuous label shift, both the score and calibration weights depend on an unknown response density ratio. Existing methods typically estimate a single exponential tilt from pseudo-labels or predictive samples and plug it into calibration. We propose Joint Tilt-Sensitivity Conformal Bayes (JTS-CB), which performs sensitivity analysis over a prespecified family of tilts. Each tilt defines a coupled score-weight pair. The split procedure, JTS-SCB, solves joint min-max calibration by taking the largest tilt-specific threshold from weighted intercept-only quantile regression. JTS-Union retains individual thresholds, yielding prediction sets contained in those of JTS-SCB. Both use calibration-only weights by design, without a general coverage guarantee. We also study a candidate-weighted exact counterpart. Any nonzero linear exponential tilt makes its prediction set unbounded, whereas compact negative-quadratic families yield bounded exact unions under Gaussian scores. For the practical procedures, a localized correction with fallback gives coverage at least nominal minus a target tail-mass bound when the family contains the true tilt, but can substantially increase width. Experiments on controlled shifts of three regression datasets and on synthetic data show that plug-in predictive sampling can approach oracle coverage when the shift is well identified, but remains vulnerable to predictive bias and unstable tilt estimation. Sensitivity analysis can mitigate these failures in the tested settings, at the cost of wider prediction sets. Coupling scores and weights can yield narrower sets with higher coverage than varying weights alone, although this advantage is not uniform across shifts. Excluding the true tilt from the sensitivity family can still cause undercoverage.
Jesto Peter, Timothy M. Hansen, Hossein Moradi Rekabdarkolaee
Accurate forecasting and temporal downscaling of solar irradiance is critical for reliable renewable energy integration and grid management. Traditional methods often struggle to capture rapid irradiance variability. In this work, we develop a Transformer-based architecture for simultaneous forecasting and temporal downscaling. The model takes input data at arbitrary resolutions and generates outputs at any resolutions represented within the training data, leveraging both global and local contextual information. To improve training efficiency, a genetic algorithm-based optimization framework is implemented to identify suitable model hyperparameters, while reducing the computational cost of exhaustive grid search. The effectiveness of the proposed model is shown by analyzing real data in Dallas, Texas. The proposed model achieved root mean square error (RMSE) values of 28.785, 29.544, 30.157, 32.132~W/m\(^2\) for 15-, 30-, 60-minute, and irregularly spaced global input and producing forecast at 5-minutes resolution, respectively. Furthermore, this model produced reliable uncertainty estimates, exhibiting wider prediction intervals for high-resolution inputs during volatile solar conditions, whereas coarser-resolution inputs result in narrower intervals and significantly lower training time and computational cost. Our approach provides a scalable and efficient framework for multi-resolution solar irradiance forecasting and temporal downscaling, suitable for practical energy forecasting and grid management applications.
This paper presents a near-quadratic-time algorithm that correctly matches almost all nodes of two correlated Erdős-Rényi graphs without any incorrect matches. Each graph has average degree $λ$ and its edges are subsampled from a common parent Erdős-Rényi graph with $n$ nodes, with subsampling probability $s$. The nodes of one graph are then relabeled by an unknown permutation. The algorithm is built on a test for correlated versus independent Galton-Watson trees with approximated likelihood ratios. For every constant $s>\sqrt{C_{\mathrm{Otter}}}$, where $C_{\mathrm{Otter}}\approx0.338$ is Otter's tree-counting constant, and every $λ=ω(1)$ with $λ=(\log n)^{O(1)}$, the algorithm outputs a partial matching of size $n-o(n)$ that contains no incorrect matches with high probability. If, in addition, $λs\ge(1+ε)\log n$ for a constant $ε>0$, a seeded completion step recovers the permutation exactly. In both cases, the time complexity of the algorithm is $n^{2+o(1)}$ with high probability, whereas previous polynomial-time algorithms with the same guarantees either have a time-complexity bound whose exponent goes to $\infty$ as $s$ approaches $\sqrt{C_{\mathrm{Otter}}}$ from above, or apply only when $s$ is close to $1$.
An invertible graph aggregation preserves the complete feature table, but can increase the size of a classifier that reads one node at a time. For every even $N\ge4$, we construct a labeled tree with $2N$ vertices and maximum degree three. A shared one-hidden-layer rectified linear unit (ReLU) classifier needs one hidden unit on the raw scalar features and exactly $N$ units after one lazy neighborhood mean. All aggregate values are distinct, and the mixing matrix has condition number at most 8. We determine the complete error curve for heads with a free affine term. The width separation persists under scalar perturbations. Under a specified bounded uniform-noise model with one nuisance coordinate, every one-unit aggregate head has expected error at least $1/12$, while a fixed $N$-unit head has zero error. Controlled fits compare trained heads with these explicit feasible scores. On Roman-empire and Tolokers, aggregation improves fitted prediction, and retaining raw features helps at the largest tested parameter budget. Training-only diagnostics include conditional output-layer optimality bounds for eight fixed runs.
We consider classification in high-dimensional feature spaces where only a vanishing fraction of features are useful and the number of classes may be large. While this rare-and-weak useful-feature regime is well understood in the binary setting, its multiclass version raises new questions: a feature can be useful for class discrimination without separating any specific pair of classes. We focus on a rare feature diversity model, in which most features are useless, but a small fraction promote discrimination among classes in a way that can be detected by a per-feature diversity test, such as one-way ANOVA or the Kruskal-Wallis test. For this setting, we propose Diversity Pursuit Higher Criticism Thresholding (DP-HCT), a feature-screening procedure that applies a higher-criticism threshold to the collection of p-values obtained from per-feature diversity tests. We justify DP-HCT under a Gaussian multiclass Rare-and-Weak Useful Feature model with class-mean contrasts of fixed magnitude in random directions, extending the binary model of Donoho and Jin (2008). When the number of classes is fixed, our model recovers the binary phase transition. When the number of classes grows logarithmically with the number of features, however, we derive a new and strictly harder phase transition, driven by the growing degrees of freedom and their effect on the useful-feature tail rate. We show that whenever classification beyond chance is possible, DP-HCT with ANOVA, followed by a simple classifier based on similarity to class means, achieves asymptotically perfect accuracy. Experiments on real multiclass classification benchmarks demonstrate substantial improvements, and numerical simulations confirm that the derived phase-transition curve accurately predicts the finite-sample behavior of DP-HCT in the growing-class regime.
Recent progress has rapidly advanced our understanding of the mechanisms underlying in-context learning in modern attention-based neural networks. However, existing results focus exclusively on unimodal data; in contrast, the theoretical underpinnings of in-context learning for multi-modal data remain poorly understood. We introduce a mathematically tractable framework for studying multi-modal learning and explore when transformer-like architectures can recover Bayes-optimal performance in-context. To model multi-modal problems, we assume the observed data arises from a latent factor model. Our first result comprises a negative take on expressibility: we prove that single-layer self-attention models fail to recover the Bayes-optimal predictor uniformly over the task distribution. To address this limitation, we introduce a novel, linearized cross-attention mechanism, which we study in the regime where both the number of cross-attention layers and the context length are large. We show that this cross-attention mechanism is provably Bayes optimal when optimized using gradient flow. Our results underscore the benefits of depth for in-context learning and establish the provable utility of cross-attention for multi-modal distributions.
Gaussian process (GP) regression is a popular surrogate modeling tool for computer simulations in engineering and scientific domains. However, it often struggles with high computational costs and low prediction accuracy when the simulation involves too many input variables. For some simulation models, the outputs may only be significantly influenced by a small subset of the input variables, referred to as the ``active variables''. We propose an optimal kernel learning approach to identify these active variables, thereby overcoming GP model limitations and enhancing system understanding. Our method approximates the original GP model's covariance function through a convex combination of kernel functions, each utilizing low-dimensional subsets of input variables. Inspired by the Fedorov-Wynn algorithm from optimal design literature, we develop an optimal kernel learning algorithm to determine this approximation. We incorporate the effect heredity principle, a concept borrowed from the field of ``design and analysis of experiments'', to facilitate efficient computation. Through several examples, we demonstrate that the proposed method provides competitive prediction and variable selection, with favorable computational performance in the examples considered. It is an effective solution for interpreting the surrogate GP regression and simplifying the complex underlying system.
Adversarial training attains strong empirical robustness to specific adversarial attacks by training on concrete adversarial perturbations, but it produces neural networks that are not amenable to strong robustness certificates through neural network verification. On the other hand, earlier certified training schemes directly train on bounds from network relaxations to obtain models that are certifiably robust, but display sub-par standard performance. Recent work has shown that state-of-the-art trade-offs between certified robustness and standard performance can be obtained through a family of losses combining adversarial outputs and neural network bounds. Nevertheless, differently from empirical robustness, verifiability still comes at a significant cost in standard performance. In this work, we propose to leverage empirically-robust teachers to improve the performance of certifiably-robust models through knowledge distillation. Using versatile feature-space distillation objectives, we show that distillation from adversarially-trained teachers can significantly improve the performance of state-of-the-art certified training algorithms for ReLU networks on robust computer vision benchmarks.
Camille Touron, Gabriel V. Cardoso, Julyan Arbel, Pedro L. C. Rodrigues
Compositional score-based approaches to simulation-based inference (SBI) approximate the posterior over a shared parameter given $n$ independent observations by aggregating individually learned posterior scores: currently, there are two main propositions of such methods (Geffner et al. (2023), Linhart et al. (2026)). As the resulting composite score does not correspond to the score of any distribution along the forward diffusion path of the true multi-observation posterior, sampling from it via a reverse SDE leads to an irreducible bias. Annealed Langevin dynamics provides a principled alternative: it treats the composite score as the genuine score of a sequence of tractable bridging densities and samples from them in succession. When properly tuned, it could lead to a controllable bias. However, its hyperparameters, namely step sizes, the number of steps per level, and the number of annealing levels, have so far been chosen empirically. We derive Wasserstein bounds for annealed Langevin with approximate scores and translate them into explicit decision rules for these hyperparameters that guarantee a prescribed sampling accuracy, while highlighting different theoretical aspects of each composite score formulation. In the Gaussian setting, we obtain closed-form expressions for all relevant quantities and prove that the bridging densities of Linhart et al. (2026) consistently admit larger step sizes and require fewer total Langevin steps than those of Geffner et al. (2023). Furthermore, we show empirically that the tuning obtained in the Gaussian setting generalizes to more complex problems, thus providing a well-understood and theoretically grounded starting point for practitioners using compositional score-based approaches.
In modern commerce, firms and platforms typically use customer features from different sources to personalize prices: some features are verified using seller records or third-party data, while others are self-reported and may be strategically manipulated by buyers seeking lower prices. In this paper, we study the problem of source-aware dynamic pricing with strategic buyers, where the buyer's valuation is a sparse linear function of her high-dimensional features plus market noise with an unknown distribution. We develop the Source-Aware Localized exploration and calibrated Exploitation (SALE) policy, which personalizes exploration using past learning and verified features, and corrects for strategic manipulation using verified and reported features during exploitation. Our novel personalized design aims to narrow the range of prices explored, reducing unnecessary exploration at excessively high or low prices. To support learning under this design, we derive a new population equation that enables sparse valuation parameter estimation and, in turn, nonparametric estimation of the unknown noise distribution, even when buyer features are dependent over time. By uncovering the intrinsic sparse structure, our analysis establishes estimation guarantees for both learning tasks using the same exploration data. With these guarantees, we show that SALE achieves a sublinear expected regret bound, explicitly quantifying the benefit of using verified features to narrow exploration intervals. Several numerical studies further demonstrate the appealing performance of SALE.
Machine learning models deployed in decision-making pipelines, from vehicle routing to dynamic pricing to clinical decision support, must remain reliable in the face of low-quality training data. We study a predict-then-optimize setting in which an upstream model predicts the cost vector of a downstream linear program, with a large pool of noisy training data supplemented by a small curated set of clean data. We introduce Reduced Cost Influence Functions (RCIFs), which combine influence functions from robust statistics with the geometry of linear optimization to estimate how each training point shifts downstream decisions. Building on this, we propose a reweighting algorithm that uses RCIFs to downweight harmful training points and reduce downstream regret. Experiments on random linear programs and shortest-path problems show that our approach reduces regret, and accompanying theory characterizes properties of RCIFs and identifies regimes favorable to our method relative to baselines.
Language and vision models became data-efficient by pretraining on plentiful unlabelled data; physical field models have such data, over ten million 3D shapes, but geometry pretraining designs its targets rather than deriving them from the physics. Euler-Lagrange geometry pretraining derives them: for attached flow, a shape enters the flow's Euler-Lagrange equation only through coefficients that distance and normal line (the normal up to sign) fix to first order. Their reconstruction error therefore bounds what a representation loses about any field, and fine-tuning on costly simulations need only assemble the global field. We realise it in AREALS, a masked surface autoencoder whose areal tokens carry position, normal and area, pretrained self-supervised and sign-free on 61,822 raw meshes. Because the operating condition enters only through the far field, sensors can supply it: AREALS completes a hidden field in one forward pass without condition labels. It matches the accuracy of the same architecture trained from scratch with half the simulations on DrivAerML and DrivAerNet++ and about a quarter on WindsorML. On DoubleDelta wings, 20 sensors give $3.9\times$ and $4.0\times$ lower field and force error than GABI, a Bayesian inversion method with a learned prior, and beyond the training angles pretraining halves the error.
We study finite-horizon reinforcement learning with a safety constraint on the probability of reaching a failure set. In contrast to the well-studied safe reinforcement learning literature, we consider problems where the probability of reaching the failure set is vanishingly small, but the consequences of reaching the failure set are catastrophic. Direct simulation and estimation of the optimal policy is naturally ineffective when the failure probability is extremely small. Moreover, on a real system the controller can only choose actions, and this is not enough to make failures frequent. In contrast to the rare event simulation literature, our goal is not to estimate the probability of the rare event occurrence, but to upper bound it such that the failure constraint is satisfied. Thus, instead of sampling failures, we use the large deviations principle. We define a damage variable that on exceeding a certain threshold indicates failure. We study the cumulant generating function (CGF) as its positive root provides an upper bound on the failure probability via Cramer's theorem. We extend this approach to Markov decision processes with hitting times through state-action CGFs, and give finite-horizon and state-dependent versions of the bound. The state-action CGFs do not depend on the policy and can be estimated from regular transitions, so the bound can be checked for any policy a priori. This gives a policy improvement algorithm in which every policy satisfies the failure constraint with high probability.
Training a classifier means letting a validation metric decide which model is kept and when training stops. A natural requirement on that metric is superiority: every correctly classified forecast should score better than every incorrectly classified one, however close the underlying probabilities are. Neither the Brier score nor the logarithmic score satisfies this on its own. We show that superiority is governed by the separation between two endpoints -- the supremum of the score on correct predictions and the infimum on incorrect predictions -- and derive both in closed form for each score. Their difference is the infimal penalty, the greatest lower bound on the penalty needed to restore superiority. Prior work restores superiority with a penalty calibrated from the first endpoint alone; for the Brier base, our infimal penalty is smaller than that published penalty by a factor ranging from 4.00 at three classes to 2.11 at twenty. We further embed the published and infimal penalties in a one-parameter family and show that, under a single-valued tie convention, every member of the family is strictly proper. In a pre-specified two-dataset case study, mean selected-model macro-F1 rises with the penalty in the reported ten-class regimes, though not distinguishably from noise once seed/selection-mode nesting is accounted for; a fifth regime at forty-seven classes does not reproduce even this descriptive pattern. The intended no-superiority baseline is algebraically a rescaling of the family under test, so the experiment cannot attribute the observed pattern specifically to superiority. When superiority itself is the objective, any penalty strictly above the infimal value suffices; choosing a larger penalty is a separate calibration trade-off rather than a requirement for the guarantee.
Offline reinforcement learning is fundamentally limited by the coverage of the available data, while collecting additional trajectories through online exploration may be costly, unsafe, or infeasible. We study whether this limitation can be overcome without directly interacting with the underlying environment. Motivated by practical settings such as collaborator-mediated simulation, where a collaborator can provide information complementary to the offline data, we introduce query-based collaborative reinforcement learning to study how to acquire this information efficiently. The learner starts with a fixed offline dataset and adaptively queries state-time pairs; the collaborator samples an action according to a fixed, possibly suboptimal policy, and the simulator returns the corresponding reward and next state. For finite-horizon tabular problems, we derive an instance-dependent information-theoretic lower bound on the additional queries required to identify an $ε$-optimal policy and develop a local quadratic approximation. Guided by this characterization, we develop an adaptive query-allocation algorithm whose query complexity nearly matches the local benchmark under suitable conditions. Experiments on multiple synthetic and semi-synthetic environments demonstrate the empirical efficiency of the proposed method.
We study ordinary kernel ridge regression over full Hölder-Zygmund balls, measuring errors in Sobolev norms and in pointwise and uniform losses for mixed derivatives. For targets of smoothness $s$, every RKHS norm equivalent to $H^{s+d/2}$ gives the optimal $H^r$ risk for $0\le r<s$; integer choices of $r$ include integrated squared derivative losses. Pointwise and uniform guarantees require additional kernel structure. A localized wavelet kernel attains the sharp derivative rates with the same regularization orders as for function estimation, whereas equivalent changes of the RKHS norm can destroy the population bias rates. For the ordinary Gaussian kernel with a shrinking bandwidth, we prove a uniform derivative risk bound with the optimal polynomial rate and an explicit, unoptimized logarithmic factor. The results allow bounded measurable design densities bounded away from zero and conditionally sub-Gaussian noise.
In this paper, we propose a novel approach to tackle the multiple instance regression (MIR) problem. This problem arises when the data is a collection of bags, where each bag is made of multiple instances corresponding to the same unique real-valued label. Our goal is to train a regression model which maps the instances of an unseen bag to its unique label. This MIR setting is common to remote sensing applications where there is high variability in the measurements and low geographical variability in the quantity being estimated. Our approach, in contrast to most competing methods, does not make the assumption that there exists a prime instance responsible for the label in each bag. Instead, we treat each bag as a set (i.e, an unordered sequence) of instances and learn to map each bag to its unique label by using all the instances in each bag. This is done by implementing an order-invariant operation characterized by a particular type of attention mechanism. This method is very flexible as it does not require domain knowledge nor does it make any assumptions about the distribution of the instances within each bag. We test our algorithm on five real world datasets and outperform previous state-of-the-art on three of the datasets. In addition, we augment our feature space by adding the moments of each feature for each bag, as extra features, and show that while the first moments lead to higher accuracy, there is a diminishing return.
Data-driven sequential decision making in marketing and healthcare can benefit from observations collected in related environments, particularly when data in the environment of interest are limited. Transferring such information requires accounting for differences in rewards and transition dynamics across environments. We study knowledge transfer for offline reinforcement learning through a joint estimation framework that directly integrates source information into target value-function learning. We propose Direct Transfer Fitted Q-Iteration (DFQI), which updates the target Q-function while simultaneously borrowing the source information through an L1 penalty that links the target coefficients to a weighted aggregate of source coefficients, with both determined jointly within each Bellman update. Using sieve approximation, we establish statistical guarantees for target estimation and policy learning in stationary discounted environments with common or heterogeneous transition kernels. Empirical studies further demonstrate the effectiveness of the proposed DFQI framework in transferring source information to improve policy learning with limited target data.
We study the implicit bias of Adam and its similarity to sign gradient descent on homogeneous models in binary classification tasks. Perhaps surprisingly, we uncover that these depend on the learning rate schedule. While previous works studied strongly decaying learning rate schedules $η(t)=\mathrm o \left(t^{1/L-1}\right)$, where $L$ is the homogeneity degree, and showed $\ell_\infty$-margin-maximization biases, we focus on the more common complementary regime, including constant and weakly decaying schedules. For a polynomial schedule $t^α$ and homogeneity degree $L$, we show that when $α> \frac 1L - \frac12$, Adam has no trajectory-independent margin-maximization bias for deep linear networks (in the KKT sense): first, there is no non-trivial, locally Lipschitz, convex implicit regularizer in this setting, including any squared norm. Second, on width-$1$ networks, the result holds without requiring convexity, implying that any implicit regularizer for wide networks must be trivial when restricted to a width-$1$ network. Nevertheless, we positively characterize the implicit bias in this regime as maximizing a trajectory-dependent weighted-$\ell_2$ margin for any $L$-homogeneous model. The fundamental mechanism behind our results is convergence of Adam's preconditioner to a trajectory-dependent vector, which also allows showing that the limiting parameter direction depends on initialization. In the intermediate regime $\frac 1L - 1 \leq α\leq \frac1L - \frac12$ we prove an $\ell_\infty$ bias, conditional on gradient directional convergence to a full-support vector. Of independent interest, we establish a loss decay rate for full batch Adam on general homogeneous models, whose leading order depends only on the second-moment parameter.
Tensor factor models provide low-dimensional representations of tensor-valued observations while preserving their multiway structure. When these observations are held by different clients, privacy constraints may prevent them from being pooled for centralized analysis. We study estimation of common loading spaces in Tucker tensor factor models under federated differential privacy, allowing heterogeneous sample sizes, privacy budgets, and factor strengths. The proposed method combines private stability testing, Huber aggregation, and Gaussian perturbation at each client with weighted aggregation at the server. We establish non-asymptotic upper bounds governed by a scaled harmonic mean of the squared local error rates, and prove minimax optimality up to logarithmic factors under strong factors, fixed factor ranks, and fixed noise variance. We further develop a private calibration procedure that selects the local tuning parameters and aggregation weights without knowledge of the noise level or client-specific factor strengths. The resulting adaptive estimator preserves the privacy guarantee and attains the error rate obtained with oracle tuning. We illustrate the finite-sample performance of the proposed estimators through simulation experiments and an application to a Meditation Traits EEG dataset.
In recent years, Rectified flow (RF) has gained considerable popularity largely due to its generation efficiency and state-of-the-art performance. In this paper, we investigate how well RF automatically adapts to the intrinsic low dimensionality of the support of the target distribution to accelerate sampling. We show that, using a carefully designed \blue{end-point adapted} time-discretization scheme and with sufficiently accurate drift estimates, the RF sampler enjoys an iteration complexity of order $O(k/\varepsilon)$ (up to log factors), where $\varepsilon$ is the precision in total variation distance and $k$ is the intrinsic dimension of the target distribution. In addition, we show that the denoising diffusion probabilistic model (DDPM) procedure is equivalent to a stochastic version of RF by establishing a novel connection between these processes and stochastic localization. Building on this connection, we further design a stochastic RF sampler that also adapts to the low-dimensionality of the target distribution under mild requirements on the accuracy of the drift estimates, and also with a specific time schedule. We illustrate the efficacy of newly designed time-discretization schedules with simulations on the synthetic data and text-to-image (T2I) data experiments.
We establish complexity lower bounds for stochastic first-order algorithms in nonconvex--concave minimax optimization, allowing algorithms to use variance reduction. Our main contribution is a lower bound for a zero-respecting algorithm class that permits variance reduction, extending beyond the algorithmic restrictions imposed by some existing lower bounds. We consider objectives with an $L$-Lipschitz continuous joint gradient, a compact convex dual domain of Euclidean radius at most $D_Y$, and a primal value function, defined by maximizing the objective over the dual variable, with initial suboptimality at most $Δ$. The target accuracy $\varepsilon$ is measured by the gradient norm of the Moreau envelope of the constrained primal value function with parameter $1/(2L)$. Under an unbiased stochastic first-order oracle with variance at most $σ^2$ and mean-squared smoothness, we prove the lower bound $Ω\!\left(L^2D_YΔ\varepsilon^{-3}+L^3D_Y^2Δσ^2\varepsilon^{-6}\right)$. This result quantifies the dependence on accuracy, dual-domain radius, and oracle noise even when variance reduction is allowed. We also establish complementary lower bounds for nonconvex--strongly-concave minimax optimization. With dual strong-concavity parameter $μ>0$ and condition number $κ:=L/μ$, we obtain $Ω\!\left(LΔ\sqrtκ\,\varepsilon^{-2}+LΔκσ^2\varepsilon^{-4}\right)$ under the bounded-variance oracle model. Under the additional mean-squared smoothness condition with constant $\bar L$, we obtain $Ω\!\left(LΔ\sqrtκ\,\varepsilon^{-2}+Δ\bar Lσκ^{3/2}\varepsilon^{-3}\right)$. Together, these results identify complexity barriers across the concave and strongly concave regimes, with the main nonconvex--concave bound remaining valid for algorithms that use variance reduction.
For a risk-averse finite-horizon Markov Decision Process, we introduce a special class of Markov coherent risk measures, called mini-batch measures. We also define the class of multipattern risk-averse problems that reduce to linear MDPs in the risk-neutral case. We use both concepts in a feature-based $Q$-learning method with multipattern $Q$-factor approximation and we prove a high-probability regret bound of $\widetilde{\mathcal{O}}\big(d^{3/2}HN^{H-1}\sqrt{K}\big)$, where $d$ is the number of features, $H$ is the horizon, $N\ge2$ is the mini-batch size, $K$ is the number of episodes, and $\widetilde{\mathcal{O}}$ hides logarithmic factors. We also propose an economical version of the $Q$-learning method that streamlines the policy evaluation (backward) step. The theoretical results are illustrated on a stochastic assignment problem and a short-horizon multi-armed bandit problem.
Convex regression provides a flexible nonparametric estimation framework while incorporating appropriate shape information. However, its standard least-squares formulation has $O(n^2)$ supporting-hyperplane constraints creating a computational bottleneck as the sample size $n$ increases. We study a $K$-nearest-neighbor (KNN) localized relaxation for convex regression where we for each sample, we only maintain the $K$ nearest hyperplane constraints. Under mild regularity conditions, we establish statistical guarantees for our modified estimator in every fixed dimension $d\ge2$. In particular, for $d>4$, the population risk of our modified problem can be shown to have equivalent convergence rates as the original least square convex regression estimator of $\widetilde O(n^{-2/d})$ using neighborhoods sizes (i.e. $K=K_n$) of order $\widetilde O(n^{2(d-1)/(2d-1)})$ for general convex functions and $\widetilde O(n^{(d+1)/(d+2)})$ for strongly convex functions. To corroborate our theory, we implement iterative optimization methods to compare the prediction accuracy of the KNN relaxation with those of the full estimator across synthetic convex benchmarks. Our experiments demonstrate that KNN prediction risk approaches the full-estimator prediction risk at neighborhood sizes matching the theoretical scale.