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 $\gamma$.

We study convex resource allocation problems with $m$ hard constraints under $(\varepsilon,\delta)$-joint differential privacy (Joint-DP or JDP) in an offline setting. To approximately solve the problem, we propose a generic algorithm called Noisy Dual Mirror Descent. The algorithm applies noisy Mirror Descent to a dual problem from relaxing the hard constraints for private shadow prices, and then uses the shadow prices to coordinate allocations in the primal problem.

We consider a constrained Markov Decision Problem (CMDP) where the goal of an agent is to maximize the expected discounted sum of rewards over an infinite horizon while ensuring that the expected discounted sum of costs exceeds a certain threshold. Building on the idea of momentum-based acceleration, we develop the Primal-Dual Accelerated Natural Policy Gradient (PD-ANPG) algorithm that ensures an $\epsilon$ global optimality gap and $\epsilon$ constraint violation with $\tilde{\mathcal{O}}((1-\gamma)^{-7}\epsilon^{-2})$ sample complexity for general parameterized policies where $\gamma$ denotes the discount factor. This improves the state-of-the-art sample complexity in general parameterized CMDPs by a factor of $\mathcal{O}((1-\gamma)^{-1}\epsilon^{-2})$ and achieves the theoretical lower bound in $\epsilon^{-1}$.

We study the problem of PAC learning $\gamma$-margin halfspaces with Massart noise. We propose a simple proper learning algorithm, the Perspectron, that has sample complexity $\widetilde{O}((\epsilon\gamma)^{-2})$ and achieves classification error at most $\eta+\epsilon$ where $\eta$ is the Massart noise rate. Prior works (DGT19, CKMY20) came with worse sample complexity guarantees (in both $\epsilon$ and $\gamma$) or could only handle random classification noise (DDKWZ23,KITBMV23)--- a much milder noise assumption. We also show that our results extend to the more challenging setting of learning generalized linear models with a known link function under Massart noise, achieving a similar sample complexity to the halfspace case. This significantly improves upon the prior state-of-the-art in this setting due to CKMY20, who introduced this model.

We consider the problem of Federated Q-learning, where $M$ agents aim to collaboratively learn the optimal Q-function of an unknown infinite horizon Markov Decision Process with finite state and action spaces. We investigate the trade-off between sample and communication complexity for the widely used class of intermittent communication algorithms. We first establish the converse result, where we show that any Federated Q-learning that offers a linear speedup with respect to number of agents in sample complexity needs to incur a communication cost of at least $\Omega(\frac{1}{1-\gamma})$, where $\gamma$ is the discount factor. We also propose a new Federated Q-learning algorithm, called Fed-DVR-Q, which is the first Federated Q-learning algorithm to simultaneously achieve order-optimal sample and communication complexities. Thus, together these results provide a complete characterization of the sample-communication complexity trade-off in Federated Q-learning.

Offline safe reinforcement learning (RL) aims to find an optimal policy using a pre-collected dataset when data collection is impractical or risky. We propose a novel linear programming (LP) based primal-dual algorithm for convex MDPs that incorporates ``uncertainty'' parameters to improve data efficiency while requiring only partial data coverage assumption. Our theoretical results achieve a sample complexity of $\mathcal{O}(1/(1-\gamma)\sqrt{n})$ under general function approximation, improving the current state-of-the-art by a factor of $1/(1-\gamma)$, where $n$ is the number of data samples in an offline dataset, and $\gamma$ is the discount factor. The numerical experiments validate our theoretical findings, demonstrating the practical efficacy of our approach in achieving improved safety and learning efficiency in safe offline settings.

We study the sample complexity of learning an $\varepsilon$-optimal policy in an average-reward Markov decision process (MDP) under a generative model. For weakly communicating MDPs, we establish the complexity bound $\widetilde{O}\left(SA\frac{\mathsf{H}}{\varepsilon^2} \right)$, where $\mathsf{H}$ is the span of the bias function of the optimal policy and $SA$ is the cardinality of the state-action space. Our result is the first that is minimax optimal (up to log factors) in all parameters $S,A,\mathsf{H}$, and $\varepsilon$, improving on existing work that either assumes uniformly bounded mixing times for all policies or has suboptimal dependence on the parameters. We also initiate the study of sample complexity in general (multichain) average-reward MDPs.

Transformers have shown impressive capabilities across various tasks, but their performance on compositional problems remains a topic of debate. In this work, we investigate the mechanisms of how transformers behave on unseen compositional tasks. We discover that the parameter initialization scale plays a critical role in determining whether the model learns inferential (reasoning-based) solutions, which capture the underlying compositional primitives, or symmetric (memory-based) solutions, which simply memorize mappings without understanding the compositional structure. By analyzing the information flow and vector representations within the model, we reveal the distinct mechanisms underlying these solution types. We further find that inferential (reasoning-based) solutions exhibit low complexity bias, which we hypothesize is a key factor enabling them to learn individual mappings for single anchors.