We address the challenge of certifying the performance of a federated learning model on an unseen target network using only measurements from the source network that trained the model. Specifically, consider a source network "A" with $K$ clients, each holding private, non-IID datasets drawn from heterogeneous distributions, modeled as samples from a broader meta-distribution $\mu$. Our goal is to provide certified guarantees for the model’s performance on a different, unseen network "B", governed by an unknown meta-distribution $\mu'$, assuming the deviation between $\mu$ and $\mu'$ is bounded—either in Wasserstein distance or an $f$-divergence. We derive worst-case uniform guarantees for both the model’s average loss and its risk CDF, the latter corresponding to a novel, adversarially robust version of the Dvoretzky–Kiefer–Wolfowitz (DKW) inequality. In addition, we show how the vanilla DKW bound enables principled certification of the model's true performance on unseen clients within the same (source) network. Our bounds are efficiently computable, asymptotically minimax optimal, and preserve clients' privacy. We also establish non-asymptotic generalization bounds that converge to zero as $K$ grows and the minimum per-client sample size exceeds $\mathcal{O}(\log K)$. Empirical evaluations confirm the practical utility of our bounds across real-world tasks. The project code is available at: github.com/samin-mehdizadeh/Robust-Evaluation-DKW

We consider the problem of optimal transport between two high-dimensional distributions μ,ν in ℝⁿ from a new algorithmic perspective, in which we are given a sample x ∼ μ and we have to find a close y ∼ ν while running in poly(n) time, where n is the size/dimension of x,y. In other words, we are interested in making the running time bounded in dimension of the spaces rather than bounded in the total size of the representations of the two distributions. Our main result is a general algorithmic transport result between any product distribution μ and an arbitrary distribution ν of total cost Δ + δ under 𝓁_p^p cost; here Δ is the cost of the so-called Knothe–Rosenblatt transport from μ to ν, while δ is a computational error that goes to zero for larger running time in the transport algorithm. For this result, we need ν to be "sequentially samplable" with a "bounded average sampling cost" which is a novel but natural notion of independent interest. In addition, we prove the following. - We prove an algorithmic version of the celebrated Talagrand’s inequality for transporting the standard Gaussian distribution Φⁿ to an arbitrary ν under the Euclidean-squared cost. When ν is Φⁿ conditioned on a set S of measure ε, we show how to implement the needed sequential sampler for ν in expected time poly(n/ε), using membership oracle access to S. Hence, we obtain an algorithmic transport that maps Φⁿ to Φⁿ|S in time poly(n/ε) and expected Euclidean-squared distance O(log 1/ε), which is optimal for a general set S of measure ε. - As corollary, we find the first computational concentration (Etesami et al. SODA 2020) result for the Gaussian measure under the Euclidean distance with a dimension-independent transportation cost, resolving a question of Etesami et al. More precisely, for any set S of Gaussian measure ε, we map most of Φⁿ samples to S with Euclidean distance O(√{log 1/ε}) in time poly(n/ε).

Multiple Instance Learning (MIL) is a standard weakly supervised approach for Whole Slide Image (WSI) classification, where performance hinges on both feature representation and MIL pooling strategies. Recent research has predominantly focused on Transformer-based architectures adapted for WSIs. However, we argue that this trend faces a fundamental limitation: data scarcity. In typical settings, Transformer models yield only marginal gains without access to large-scale datasets—resources that are virtually inaccessible to all but a few well-funded research labs. Motivated by this, we revisit simple, non-attention MIL with unsupervised slide features and analyze temperature-$\beta$-controlled log-sum-exp (LSE) pooling. For slides partitioned into $N$ patches, we theoretically show that LSE has a smooth transition at a critical $\beta_{\mathrm{crit}}=\mathcal{O}(\log N)$ threshold, interpolating between mean-like aggregation (stable, better generalization but less sensitive) and max-like aggregation (more sensitive but looser generalization bounds). Grounded in this analysis, we introduce Maxsoft—a novel MIL pooling function that enables flexible control over this trade-off, allowing adaptation to specific tasks and datasets. To further tackle real-world deployment challenges such as specimen heterogeneity, we propose PerPatch augmentation—a simple yet effective technique that enhances model robustness. Empirically, Maxsoft achieves state-of-the-art performance in low-data regimes across four major benchmarks (CAMELYON16, CAMELYON17, TCGA-Lung, and SICAP-MIL), often matching or surpassing large-scale foundation models. When combined with PerPatch augmentation, this performance is further improved through increased robustness. Code is available at \href{https://github.com/jafarinia/maxsoft}{\texttt{https://github.com/jafarinia/maxsoft}}

We propose a novel framework for incorporating unlabeled data into semi-supervised classification problems, where scenarios involving the minimization of either i) adversarially robust or ii) non-robust loss functions have been considered. Notably, we allow the unlabeled samples to deviate slightly (in total variation sense) from the in-domain distribution. The core idea behind our framework is to combine Distributionally Robust Optimization (DRO) with self-supervised training. As a result, we also leverage efficient polynomial-time algorithms for the training stage. From a theoretical standpoint, we apply our framework on the classification problem of a mixture of two Gaussians in $\mathbb{R}^d$, where in addition to the $m$ independent and labeled samples from the true distribution, a set of $n$ (usually with $n\gg m$) out of domain and unlabeled samples are gievn as well. Using only the labeled data, it is known that the generalization error can be bounded by $\propto\left(d/m\right)^{1/2}$. However, using our method on both isotropic and non-isotropic Gaussian mixture models, one can derive a new set of analytically explicit and non-asymptotic bounds which show substantial improvement on the generalization error compared ERM. Our results underscore two significant insights: 1) out-of-domain samples, even when unlabeled, can be harnessed to narrow the generalization gap, provided that the true data distribution adheres to a form of the "cluster assumption", and 2) the semi-supervised learning paradigm can be regarded as a special case of our framework when there are no distributional shifts. We validate our claims through experiments conducted on a variety of synthetic and real-world datasets.

The aim of this paper is to address the challenge of gradual domain adaptation within a class of manifold-constrained data distributions. In particular, we consider a sequence of $T\ge2$ data distributions $P_1,\ldots,P_T$ undergoing a gradual shift, where each pair of consecutive measures $P_i,P_{i+1}$ are close to each other in Wasserstein distance. We have a supervised dataset of size $n$ sampled from $P_0$, while for the subsequent distributions in the sequence, only unlabeled i.i.d. samples are available. Moreover, we assume that all distributions exhibit a known favorable attribute, such as (but not limited to) having intra-class soft/hard margins. In this context, we propose a methodology rooted in Distributionally Robust Optimization (DRO) with an adaptive Wasserstein radius. We theoretically show that this method guarantees the classification error across all $P_i$s can be suitably bounded. Our bounds rely on a newly introduced {\it {compatibility}} measure, which fully characterizes the error propagation dynamics along the sequence. Specifically, for inadequately constrained distributions, the error can exponentially escalate as we progress through the gradual shifts. Conversely, for appropriately constrained distributions, the error can be demonstrated to be linear or even entirely eradicated. We have substantiated our theoretical findings through several experimental results.

Deep neural networks excel at learning the training data, but often provide incorrect and confident predictions when evaluated on slightly different test examples. This includes distribution shifts, outliers, and adversarial examples. To address these issues, we propose \manifoldmixup{}, a simple regularizer that encourages neural networks to predict less confidently on interpolations of hidden representations. \manifoldmixup{} leverages semantic interpolations as additional training signal, obtaining neural networks with smoother decision boundaries at multiple levels of representation. As a result, neural networks trained with \manifoldmixup{} learn flatter class-representations, that is, with fewer directions of variance. We prove theory on why this flattening happens under ideal conditions, validate it empirically on practical situations, and connect it to the previous works on information theory and generalization. In spite of incurring no significant computation and being implemented in a few lines of code, \manifoldmixup{} improves strong baselines in supervised learning, robustness to single-step adversarial attacks, and test log-likelihood.

What is the role of unlabeled data in an inference problem, when the presumed underlying distribution is adversarially perturbed? To provide a concrete answer to this question, this paper unifies two major learning frameworks: Semi-Supervised Learning (SSL) and Distributionally Robust Learning (DRL). We develop a generalization theory for our framework based on a number of novel complexity measures, such as an adversarial extension of Rademacher complexity and its semi-supervised analogue. Moreover, our analysis is able to quantify the role of unlabeled data in the generalization under a more general condition compared to the existing theoretical works in SSL. Based on our framework, we also present a hybrid of DRL and EM algorithms that has a guaranteed convergence rate. When implemented with deep neural networks, our method shows a comparable performance to those of the state-of-the-art on a number of real-world benchmark datasets.

Association mapping of genetic diseases has attracted extensive research interest during the recent years. However, most of the methodologies introduced so far suffer from spurious inference of the associated sites due to population inhomogeneities. In this paper, we introduce a statistical framework to compensate for this shortcoming by equipping the current methodologies with a state-of-the-art clustering algorithm being widely used in population genetics applications. The proposed framework jointly infers the disease-associated factors and the hidden population structures. In this regard, a Markov Chain-Monte Carlo (MCMC) procedure has been employed to assess the posterior probability distribution of the model parameters. We have implemented our proposed framework on a software package whose performance is extensively evaluated on a number of synthetic datasets, and compared to some of the well-known existing methods such as STRUCTURE. It has been shown that in extreme scenarios, up to $10-15$10-15 percent of improvement in the inference accuracy is achieved with a moderate increase in computational complexity.

Nonlinear dimensionality reduction methods have demonstrated top-notch performance in many pattern recognition and image classification tasks. Despite their popularity, they suffer from highly expensive time and memory requirements, which render them inapplicable to large-scale datasets. To leverage such cases we propose a new method called "Path-Based Isomap". Similar to Isomap, we exploit geodesic paths to find the low-dimensional embedding. However, instead of preserving pairwise geodesic distances, the low-dimensional embedding is computed via a path-mapping algorithm. Due to the much fewer number of paths compared to number of data points, a significant improvement in time and memory complexity with a comparable performance is achieved. The method demonstrates state-of-the-art performance on well-known synthetic and real-world datasets, as well as in the presence of noise.