Taylor Swift marries Travis Kelce in'moving' ceremony where the bride wore Dior and walked down the aisle to one of her own songs... before partying the night away at Madison Square Garden Mississippi teen's'bad decision' cost him his life just days after graduation, mom says, as mourning family demands answers Lena Dunham leaves Taylor Swift wedding guests GASPING with shockingly rude dinner speech after taking microphone... as world famous celeb is dramatically turned away: Insiders leak outrageous MSG gossip Blake Lively's fury after Taylor Swift left her off wedding guest list as sources say it's the final straw Taylor Swift's celebrity wedding guests FLEE lavish MSG reception early... as Travis Kelce extravaganza runs into the early hours The WORST dressed guests at Taylor Swift and Travis Kelce's lavish wedding Does this photo capture Travis Kelce's last dose of Dutch courage before marriage? After all the wild lengths Taylor went to hide wedding... one image seems all too human Probe into fiery Missouri plane crash that killed 11 skydivers and pilot takes shocking twist... as investigators reveal head-scratching findings Trump takes swipe at Iranian leaders during America 250 speech revealing he gave them'a week off' for Ayatollah funeral Meghan's Taylor Swift wedding humiliation: KENNEDY's Montecito mole tells all as Prince William rubs salt in the wound! Bruised Tom Selleck feasts on McDonald's just after gym visit as fears grow about actor's grim new look All the BEST dressed guests at Taylor Swift and Travis Kelce's extravagant wedding Taylor Swift's wedding officiant is Adam Sandler! Twisted family secrets of postal worker mom, 35, slaughtered on delivery route just six months after husband's shock death: Horrifying new details of her final moments emerge The end of the America we know? Victoria Beckham risks Brooklyn's fury as she extends another olive branch in anniversary post for David - after'fuming' son said he'wished they'd stop posting about him' Taylor Swift's '40-page prenup': How $2BILLION in assets divide up... and the one major concession Travis is predicted to have written in as special clause Iconic Las Vegas casinos' years-long infestation with bed bugs exposed...as unearthed records reveal YEARS of disgusting stays that have sent mortified guests fleeing Elon Musk's next target could be your smartphone as SpaceX schemes to take on America's biggest mobile phone companies TV icon, 84, shares rare throwback photos ahead of Fourth of July holiday... can you guess who it is?

Recent work has established a fundamental trilemma between Byzantine robustness, local differential privacy (LDP), and optimization error in distributed learning. We show that this trilemma does not universally extend to generalization error, but instead depends critically on the privacy regime. Specifically, in the high-noise regime (strong privacy), we prove that increasing privacy reduces the generalization error, i.e., there is no tension between robustness and privacy. In the low-noise regime (weaker privacy), however, the tension between robustness and privacy reappears and increasing privacy indeed degrades generalization. Our theory explains this surprising non-monotonic behavior of the generalization error via matching lower and upper bounds on the algorithmic stability of Byzantine-robust distributed learning under LDP constraints. We corroborate and further analyze these theoretical findings with empirical evaluations.

The aggregation with exponential weights (AEW) estimator is not fully understood in the basic setting of model selection aggregation with squared loss. In particular, whether it is minimax-rate optimal in expectation for large enough fixed temperatures and under random design has been an open problem since its introduction, which was explicitly posed by Lecué and Mendelson (2013). In this paper, we settle this problem by showing that \emph{without} requiring a Bernstein-type assumption, the AEW indeed achieves the excess risk $T \log (M) / (n+1)$ in expectation, whenever the temperature $T$ satisfies $(L^2/T)\exp(B/T)\leq μ/2$. Here, the number of dictionary elements is $M$, the estimator has observed $n$ i.i.d. samples from any distribution, and the loss is assumed to be bounded by $B$, $L$-Lipschitz continuous and $μ$-strongly convex. For squared loss, we show that $T\geq 4 b^2$ suffices when the predictions and labels are $[0,b]$-valued. Because AEW is known to be suboptimal in expectation for temperatures below some constant, this shows that AEW has a sharp phase transition when the temperature is large enough but constant, as conjectured by Lecué and Mendelson.

We study uniform-in-time propagation-of-chaos for continuous-time Stein Variational Gradient Descent (SVGD). Classical finite-time propagation-of-chaos estimates for mean-field systems typically deteriorate rapidly with time and therefore do not directly explain the long-time relation between the finite-particle system and its mean-field limit. We obtain two complementary classes of uniform-in-time propagation-of-chaos results. For broad distributional metrics, we introduce a cutoff strategy which combines finite-time propagation-of-chaos estimates up to an $N$-dependent horizon with independent quantitative long-time convergence estimates for the finite-particle and mean-field SVGD flows. This yields uniform-in-averaging-time propagation-of-chaos bounds in Langevin kernel Stein discrepancy, Wasserstein-1 distance, and Wasserstein-2 distance, with logarithmic or iterated-logarithmic rates depending on the metric, target and kernel class. We also develop a finite-dimensional theory for matrix-valued finite-rank kernels. For Gaussian targets with bilinear kernels, the SVGD dynamics close exactly on first and second moments, yielding genuine uniform-in-physical-time parametric propagation-of-chaos rates in finite-dimensional Stein-feature metrics. We then prove a conjugacy principle showing that these feature-level estimates transfer to conjugate target-kernel pairs under orientation-preserving diffeomorphisms, thereby extending the theory to broad classes of nonlinear, including multimodal, targets. Together, these results highlight the contrast between generic distributional metrics, for which our general approach yields logarithmic rates, and closed finite-dimensional Stein observables, for which parametric $N^{-1/2}$ propagation-of-chaos rates persist uniformly in time.

Machine learning algorithms and their training datasets have grown substantially in both size and complexity over the past decade. This increased model complexity has made it challenging to interpret and predict their behavior in unobserved scenarios. Hence, many applications that involve societal decisions still rely on simple, interpretable models like linear regression, often after feature engineering. Examples of such applications include predicting national housing prices, estimating wages across industries, forecasting loan amounts across banks, predicting life insurance premiums across groups, and projecting energy consumption across communities [CGKMN24]. A shared safety and sometimes legal concern across the above applications is the potential for wildly different model qualities for different distributions, i.e., outputting a notably worse model for some source data distributions [Dat14; BS16; HPS16; VVB18; SBFVV19; BHJKR21; CGNSG23; Cho16; KLMR18; ADW19; CGKMN24; SVWZ24].

Transformer-based large models have demonstrated remarkable generalization abilities across different tasks by leveraging a context-aware attention module for in-context learning. With richer context, transformers adapt more effectively to the current use case without any parameter updates. However, the quadratic computational and memory complexity with respect to context length significantly slows data processing in softmax transformers. Linear transformers were proposed to address this issue by reducing the complexity to linear dependence on context length, but the design and understanding of the feature mapping in linear attention, from a theoretical viewpoint, remain unclear. In this paper, we investigate the approximation and generalization abilities of linear transformers under a two-staged sampling process from domain generalization. We show that linear transformers perform in-context learning as learning a mapping from context distributions to response functions. A dimension-independent convergence rate is obtained for our generalization analysis, which also exhibits the tradeoff between the regularities of data distributions and latent features. Guided by our theoretical framework, we propose a new perspective on activation and loss design for linearizing pretrained softmax large language models.

We study policy optimization for online episodic tabular Markov decision processes with unknown transition kernels, aiming for best-of-both-worlds guarantees together with data-dependent regret bounds. Recent work (Dann et al., 2023; Li et al., 2026) has shown that policy optimization can adapt to both adversarial and stochastic losses with first-order, second-order, and path-length bounds, but only under known transitions, leaving open whether such data-dependent guarantees are achievable by policy optimization when the transition kernel is unknown. We resolve this by developing a new algorithm based on optimistic follow-the-regularized-leader that attains these guarantees under unknown transitions. The key ingredient is a new design of optimistic $Q$-function estimators together with a data-dependent transition bonus that controls estimator bias through the loss-prediction error. Our analysis further identifies an unavoidable transition-dependent complexity term that captures the intrinsic cost of estimating the transition kernel. As a result, we obtain first-order, second-order, and path-length bounds with the transition-dependent complexity term while simultaneously achieving gap-dependent $\mathrm{polylog}(T)$ regret in the stochastic regime.

Abstractitations, recent work explores online RLHF that iterates between generating on-policy responses and collecting preferences [Lee et al., 2024, Park et al., 2022]. Among online The Self-Improving Alignment (SAIL) algorithmapproaches, SAIL reduces a bilevel alignment formulation addresses distribution shift by reducing a bilevelto a computationally efficient single-level surrogate and formulation of the problem to an efficient, single-reports strong empirical gains [Ding et al., 2024]. Empirically, SAIL has demonstratedisting online pipelines are largely heuristic and do not anastrong performance on this task. However, a for-lytically control the distributional shift induced by iterative mal analysis of its convergence properties has beendata collection [Chakraborty et al., 2024, Shen et al., 2024], lacking. We identify a key theoretical challenge: which has been linked to suboptimal performance in practice the standard SAIL objective function is not guar- [Sharma et al., 2024]. To address this limita-A growing line of work argues that the coupling between tion, we propose a regularized objective, SAILreward learning and policy updates is fundamentally bilevel and should be modeled as such [Chakraborty et al., 2024].RevKL, which incorporates a reverse KullbackAs a follow-up, Ding et al. [2024] reduces the bilevel align-Leibler (KL) divergence penalty to improve the optimization landscape. Our central theoretical con-ment objective to a tractable single-level surrogate and retribution is to prove that this regularized objectiveports strong empirical gains, yet it lacks formal convergence satisfies the Polyak-Lojasiewicz (PL) conditionguarantees. Related theoretical analyses in bilevel/RLHFstyle problems exist [e.g., Yang et al., 2025, Chakrabortywithin a bounded parameter space. We establish et al., 2024, Gaur et al., 2025], yet they either focus onglobal convergence guarantees, achieving a nearlinear sample complexity.

Stochastic Gradient Descent ($\textsf{SGD}$) is one of the most classical optimization algorithms with favorable theoretical guarantees, yet the practical implementation of $\textsf{SGD}$ differs subtly from its well-known form and is often referred to as Shuffling Stochastic Gradient Descent ($\textsf{Shuffling SGD}$). A particularly popular strategy in $\textsf{Shuffling SGD}$ is Random Reshuffling ($\textsf{RR}$), which has achieved great empirical success across numerous experiments. Despite its strong performance, $\textsf{RR}$ has long been considered a heuristic due to a lack of theoretical support. Over the last decade, people have finally established provable convergence rates for $\textsf{RR}$, thus justifying its observed superiority. However, for smooth convex optimization, two clouds over the convergence theory of $\textsf{RR}$ remain to this day. More precisely, according to the current theory, $\textsf{Shuffling SGD}$ under $\textsf{RR}$ converges only when the stepsize is smaller than a threshold proportional to $1/n$, where $n$ is the number of summands in the objective (or the number of data points). Consequently, the optimally tuned theoretical rate of $\textsf{Shuffling SGD}$ under $\textsf{RR}$ is strictly worse than that of $\textsf{SGD}$ when the number of epochs is smaller than another threshold proportional to $n$. These two restrictions heavily limit the applicability of existing theories and leave a critical mismatch with practice. In this work, for the first time, we prove that $\textsf{RR}$ dominates $\textsf{SGD}$ in smooth convex optimization under any reasonable stepsize after any finite number of epochs, thereby addressing a longstanding open question.

Neural networks are known to be susceptible to over-reliance on spurious correlations. However, the precise mechanism by which models exploit shortcut features is not fully understood, and algorithms to mitigate this behavior rely on as yet unjustified assumptions about the learned representations. In this work, we provide the first end-to-end theoretical characterization of spurious feature learning for two-layer ReLU neural networks trained by online minibatch SGD on the logistic loss. We consider data drawn from the high-dimensional Boolean hypercube with a quadratic signal function (namely XOR) and a linear spurious correlation. We show that SGD learns the spurious feature first, and exponentially fast. Moreover, the optimization dynamics couple the spurious and signal features, with a stronger spurious component inhibiting signal feature learning. Our analysis reveals precise phase transitions in the learning dynamics. In the first phase, alignment between the signs of the spurious feature and second-layer weight drives rapid growth of the spurious feature. In the second phase, large majority group margin slows learning and the signal feature remains suppressed. When the spurious correlation is maximally strong, we show theoretically that the spurious feature dominates even at the sample complexity threshold where XOR would be learned in isolation (i.e., if the spurious feature was absent). In contrast, when the correlation strength is constant, we provide preliminary empirical evidence that the model can eventually learn the XOR signal, although the spurious feature is not forgotten.