The success of Graph Neural Networks (GNNs) leverages the homophily principle, where connected nodes share similar features and labels. However, this assumption breaks down in heterophilic graphs, where same-class nodes are often distributed across distant neighborhoods rather than immediate connections. Recent attempts expand the receptive field through multi-hop aggregation schemes that explicitly preserve intermediate representations from each hop distance. While effective at capturing heterophilic patterns, these methods require separate weight matrices per hop and feature concatenation, causing parameters to scale linearly with hop count. This leads to high computational complexity and GPU memory consumption. We propose Gated Multi-hop Message Passing (GAMMA), where nodes assess how relevant the aggregated information is from their k-hop neighbors. This assessment occurs through multiple refinement steps where the node compares each hop's embedding with its current representation, allowing it to focus on the most informative hops. During the forward pass, GAMMA finds the optimal mix of multi-hop information local to each node using a single feature vector without needing separate representations for each hop, thereby maintaining dimensionality comparable to single hop GNNs. In addition, we propose a weight sharing scheme that leverages a unified transformation for aggregated features from multiple hops so the global heterophilic patterns specific to each hop are learned during training.

Thompson sampling (TS) is a powerful and widely used strategy for sequential decision-making, with applications ranging from Bayesian optimization to reinforcement learning (RL). Despite its success, the theoretical foundations of TS remain limited, particularly in settings with complex temporal structure such as RL. We address this gap by establishing no-regret guarantees for TS using models with Gaussian marginal distributions. Specifically, we consider TS in episodic RL with joint Gaussian process (GP) priors over rewards and transitions. We prove a regret bound of $\mathcal{\tilde{O}}(\sqrt{KH\Gamma(KH)})$ over $K$ episodes of horizon $H$, where $\Gamma(\cdot)$ captures the complexity of the GP model. Our analysis addresses several challenges, including the non-Gaussian nature of value functions and the recursive structure of Bellman updates, and extends classical tools such as the elliptical potential lemma to multi-output settings. This work advances the understanding of TS in RL and highlights how structural assumptions and model uncertainty shape its performance in finite-horizon Markov Decision Processes.

We study a two-institution stable matching model in which candidates from two distinct groups are evaluated using partially correlated signals that are group-biased. This extends prior work (which assumes institutions evaluate candidates in an identical manner) to a more realistic setting in which institutions rely on overlapping, but independently processed, criteria. These evaluations could consist of a variety of informative tools such as standardized tests, shared recommendation systems, or AI-based assessments with local noise. Two key parameters govern evaluations: the bias parameter $\beta \in (0,1]$, which models systematic disadvantage faced by one group, and the correlation parameter $\gamma \in [0,1]$, which captures the alignment between institutional rankings. We study the representation ratio $\mathcal{R}(\beta, \gamma)$, i.e., the ratio of disadvantaged to advantaged candidates selected by the matching process in this setting.

Dynamic pricing algorithms typically assume continuous price variables, which may not reflect real-world scenarios where prices are often discrete. This paper demonstrates that leveraging discrete price information within a semi-parametric model can substantially improve performance, depending on the size of the support set of the price variable relative to the time horizon. Specifically, we propose a novel semi-parametric contextual dynamic pricing algorithm, namely BayesCoxCP, based on a Bayesian approach to the Cox proportional hazards model. Our theoretical analysis establishes high-probability regret bounds that adapt to the sparsity level $\gamma$, proving that our algorithm achieves a regret upper bound of $\widetilde{O}(T^{(1+\gamma)/2}+\sqrt{dT})$ for $\gamma < 1/3$ and $\widetilde{O}(T^{2/3}+\sqrt{dT})$ for $\gamma \geq 1/3$, where $\gamma$ represents the sparsity of the price grid relative to the time horizon $T$. Through numerical experiments, we demonstrate that our proposed algorithm significantly outperforms an existing method, particularly in scenarios with sparse discrete price points.

In this work, we study $\gamma$-discounted infinite-horizon tabular Markov decision processes (MDPs) and introduce a framework called dynamic policy gradient (DynPG).

The success of Graph Neural Networks (GNNs) leverages the homophily principle, where connected nodes share similar features and labels. However, this assumption breaks down in heterophilic graphs, where same-class nodes are often distributed across distant neighborhoods rather than immediate connections. Recent attempts expand the receptive field through multi-hop aggregation schemes that explicitly preserve intermediate representations from each hop distance. While effective at capturing heterophilic patterns, these methods require separate weight matrices per hop and feature concatenation, causing parameters to scale linearly with hop count. This leads to high computational complexity and GPU memory consumption. We propose Gated Multi-hop Message Passing (GAMMA), where nodes assess how relevant the aggregated information is from their k-hop neighbors. This assessment occurs through multiple refinement steps where the node compares each hop's embedding with its current representation, allowing it to focus on the most informative hops. During the forward pass, GAMMA finds the optimal mix of multi-hop information local to each node using a single feature vector without needing separate representations for each hop, thereby maintaining dimensionality comparable to single hop GNNs. In addition, we propose a weight sharing scheme that leverages a unified transformation for aggregated features from multiple hops so the global heterophilic patterns specific to each hop are learned during training.

We study online learning of feedforward neural networks with the sign activation function that implement functions from the unit ball in $\mathbb{R}^d$ to a finite label set $\mathcal{Y} = \{1, \ldots, Y \}$. First, we characterize a margin condition that is sufficient and in some cases necessary for online learnability of a neural network: Every neuron in the first hidden layer classifies all instances with some margin $\gamma$ bounded away from zero. Quantitatively, we prove that for any net, the optimal mistake bound is at most approximately $\mathtt{TS}(d,\gamma)$, which is the $(d,\gamma)$-totally-separable-packing number, a more restricted variation of the standard $(d,\gamma)$-packing number. We complement this result by constructing a net on which any learner makes $\mathtt{TS}(d,\gamma)$ many mistakes. We also give a quantitative lower bound of approximately $\mathtt{TS}(d,\gamma) \geq \max\{1/(\gamma \sqrt{d})^d, d\}$ when $\gamma \geq 1/2$, implying that for some nets and input sequences every learner will err for $\exp(d)$ many times, and that a dimension-free mistake bound is almost always impossible.

We introduce Exponential Family Discriminant Analysis (EFDA), a unified generative framework that extends classical Linear Discriminant Analysis (LDA) beyond the Gaussian setting to any member of the exponential family. Under the assumption that each class-conditional density belongs to a common exponential family, EFDA derives closed-form maximum-likelihood estimators for all natural parameters and yields a decision rule that is linear in the sufficient statistic, recovering LDA as a special case and capturing nonlinear decision boundaries in the original feature space. We prove that EFDA is asymptotically calibrated and statistically efficient under correct specification, and we generalise it to $K \geq 2$ classes and multivariate data. Through extensive simulation across five exponential-family distributions (Weibull, Gamma, Exponential, Poisson, Negative Binomial), EFDA matches the classification accuracy of LDA, QDA, and logistic regression while reducing Expected Calibration Error (ECE) by $2$-$6\times$, a gap that is structural: it persists for all $n$ and across all class-imbalance levels, because misspecified models remain asymptotically miscalibrated. We further prove and empirically confirm that EFDA's log-odds estimator approaches the Cramér-Rao bound under correct specification, and is the only estimator in our comparison whose mean squared error converges to zero. Complete derivations are provided for nine distributions. Finally, we formally verify all four theoretical propositions in Lean 4, using Aristotle (Harmonic) and OpenGauss (Math, Inc.) as proof generators, with all outputs independently machine-checked by AXLE (Axiom).

We provide faster randomized algorithms for computing an $\epsilon$-optimal policy in a discounted Markov decision process with $A_{\text{tot}}$-state-action pairs, bounded rewards, and discount factor $\gamma$.