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We consider the problem of validating whether a neural posterior estimate $q(\theta \mid x)$ is an accurate approximation to the true, unknown true posterior $p(\theta \mid x)$. Existing methods for evaluating the quality of an NPE estimate are largely derived from classifier-based tests or divergence measures, but these suffer from several practical drawbacks. As an alternative, we introduce the *Conditional Localization Test* (**CoLT**), a principled method designed to detect discrepancies between $p(\theta \mid x)$ and $q(\theta \mid x)$ across the full range of conditioning inputs. Rather than relying on exhaustive comparisons or density estimation at every $x$, CoLT learns a localization function that adaptively selects points $\theta_l(x)$ where the neural posterior $q$ deviates most strongly from the true posterior $p$ for that $x$. This approach is particularly advantageous in typical simulation-based inference settings, where only a single draw $\theta \sim p(\theta \mid x)$ from the true posterior is observed for each conditioning input, but where the neural posterior $q(\theta \mid x)$ can be sampled an arbitrary number of times. Our theoretical results establish necessary and sufficient conditions for assessing distributional equality across all $x$, offering both rigorous guarantees and practical scalability. Empirically, we demonstrate that CoLT not only performs better than existing methods at comparing $p$ and $q$, but also pinpoints regions of significant divergence, providing actionable insights for model refinement.

In this work, westudy algorithms for learning Gaussian mixtures subject todifferential privacy[32], which has become thede facto standard for individual privacy in statistical analysis of sensitive data. Intuitively, differential privacy guarantees that the output of the algorithm does not depend significantly on any one individual's data, which in this case means any one sample.

Table 1 compares 2 sufficienttorecovertime. Thisiscalled "spherical andmargin.

After prediction, the true labelyt is revealed and the lossℓ(yt,ˆyt)is incurred.

LetT be the time horizon andPT be the path-length that essentially reflects the non-stationarity of environments, the state-of-the-art dynamicregretis O( p T(1+PT)).

In online multiclass classification a learner has to repeatedly predict the label that corresponds to a feature vector. Algorithms in this setting have a wide range of applications ranging from predicting the outcomes ofsport matches torecommender systems.

Inparticular, dynamic regret [Zinkevich,2003;Zhang et al.,2018a]and adaptiveregret [Hazan and Seshadhri, 2009; Daniely et al., 2015] are proposed as two principled metrics to guide the algorithm design. Theunknowncomparators orunknown intervals bring considerable uncertainty to online optimization.

The sharpest known high probability generalization bounds for uniformly stable algorithms (Feldman, Vondrak, NeurIPS 2018, COLT, 2019), (Bousquet, Klochkov, Zhivotovskiy, COLT, 2020) contain a generally inevitable sampling error term of order $\Theta(1/\sqrt{n})$. When applied to excess risk bounds, this leads to suboptimal results in several standard stochastic convex optimization problems. We show that if the so-called Bernstein condition is satisfied, the term $\Theta(1/\sqrt{n})$ can be avoided, and high probability excess risk bounds of order up to $O(1/n)$ are possible via uniform stability. Using this result, we show a high probability excess risk bound with the rate $O(\log n/n)$ for strongly convex and Lipschitz losses valid for \emph{any} empirical risk minimization method.