We study dynamic regret in online convex optimization, where the objective is to achieve low cumulative loss relative to an arbitrary benchmark sequence. By observing that competing with an arbitrary sequence of comparators u1,...,uT in W Rd can be reframed as competing with a fixed comparator function u: [1,T] W, we cast dynamic regret minimization as a static regret problem in a function space. By carefully constructing a suitable function space in the form of a Reproducing Kernel Hilbert Space (RKHS), our reduction enables us to recover the optimal RT(u1,...,uT) = O( pP t ut ut 1 T) dynamic regret guarantee in the setting of linear losses, and yields new scale-free and directionallyadaptive dynamic regret guarantees. Moreover, unlike prior dynamic-to-static reductions--which are valid only for linear losses--our reduction holds for any sequence of losses, allowing us to recover O u 2H +deff(λ)lnT bounds when the losses have meaningful curvature, where deff(λ)is a measure of complexity of the RKHS. Despite working in an infinite-dimensional space, the resulting reduction leads to algorithms that are computable in practice, due to the reproducing property of RKHSs.

Existing visual token pruning methods target prompt alignment and visual preservation with static strategies, overlooking the varying relative importance of these objectives across tasks, which leads to inconsistent performance. To address this, we derive the first closed-form error bound for visual token pruning based on the Hausdorff distance, uniformly characterizing the contributions of both objectives. Moreover, leveraging ϵ-covering theory, we reveal an intrinsic trade-off between these objectives and quantify their optimal attainment levels under a fixed budget. To practically handle this trade-off, we propose Multi-Objective Balanced Covering (MoB), which reformulates visual token pruning as a bi-objective covering problem. In this framework, the attainment trade-off reduces to budget allocation via greedy radius trading. MoB offers a provable performance bound and linear scalability with respect to the number of input visual tokens, enabling adaptation to challenging pruning scenarios. Extensive experiments show that MoB preserves 96.4% of performance for LLaVA-1.5-7B using only 11.1% of the original visual tokens and accelerates LLaVA-Next-7B by 1.3-1.5 with negligible performance loss. Additionally, evaluations on Qwen2-VL and Video-LLaVA confirm that MoB integrates seamlessly into advanced MLLMs and diverse vision-language tasks. The code is available at https://github.com/YChenL/MoB.

We study the generalization of ridge-regularized nonlinear least-squares models via on-average algorithmic stability, deriving error bounds for local minimizers in terms of a data-dependent effective dimension that reflects the geometry of the gradient model at the trained parameters, through the empirical Jacobian Gram matrix and a residual-curvature term. In the linear case, where the curvature term vanishes, this recovers the classical effective dimension of the Jacobian kernel covariance, but evaluated at the trained model rather than at initialization as is typical in neural tangent kernel analyses. We further bound this effective dimension via covering complexity of the gradient features, leading to guarantees that depend on learned geometry rather than parameter count. In particular, for manifold-supported data and piecewise Lipschitz Jacobians, the bounds scale with intrinsic dimension, while for one-hidden-layer ReLU networks, the mechanism can be made explicit through counts of activation-stable regions. Experiments on synthetic manifolds, clustered distributions, and benchmark datasets illustrate trained-Jacobian compression, the tightness of the residual-curvature linearization, and agreement between the stability bound and observed generalization gaps. A key feature of our bounds is the simplicity of their derivation, which follows from first principles using the Brascamp-Lieb inequality under strongly log-concave noise.

Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict using the Fisher Information Matrix to quantify the effective degrees of freedom ($d_{eff}$) in a physics-constrained model. Unlike the classical $d_{eff}$ which measures how many parameter directions are informed by data against a statistical prior, our $d_{eff}$ measures the dimension of the parameter directions unconstrained by the differential operator. For operators with finite-dimensional kernel, we show that $d_{eff}$ converges to the kernel dimension exactly, independent of network width, depth, or activation function, recasting it from a fit diagnostic into a structural invariant of the underlying continuous operator. For operators with infinite-dimensional kernel, $d_{eff}$ instead measures the network's finite-dimensional representational bandwidth for that kernel rather than recovering an integer invariant. Importantly, $d_{eff}$ also serves as an a priori structural diagnostic. Driving $d_{eff}$ of a well-posed problem to zero certifies that the physics and boundary constraints have absorbed the network's free directions. Building on this characterization, we introduce subspace projection strategies for boundary adaptation. Rather than retraining from scratch, we project parameter updates into the null space of the pre-trained physics operator so that new boundary conditions are satisfied without disturbing the learned physics. Gradient-based fine-tuning can match or exceed this but needs more wall-clock time and tuning, whereas subspace projection delivers near-equivalent quality in seconds to minutes. We validate on linear and nonlinear operators, demonstrating accurate adaptation to initial and boundary shifts and unencountered constraint types.

Existing guarantees for misspecified kernelized bandit optimization pay for misspecification through kernel complexity: in generic offline bounds, the misspecification level $\varepsilon$ is multiplied by $\sqrt{d_\mathrm{eff}}$, where $d_\mathrm{eff}$ is the kernel effective dimension, while in online regret bounds, the corresponding penalty is $\sqrt{γ_n}\,n\varepsilon$, where $γ_n$ is the maximum information gain after $n$ rounds of interaction. In this work, we show that, for a large class of kernels, the misspecification amplification can be reduced to logarithmic or polylogarithmic growth. In the offline setting, we first prove high-probability simple-regret bounds whose misspecification term is governed by a spectral Lebesgue constant. This yields logarithmic amplification for one-dimensional monotone spectra and polylogarithmic amplification for multivariate Fourier-diagonal product kernels. In the online setting, we modify a domain-splitting algorithm and prove a cumulative regret bound of $\widetilde{\mathcal O}(\sqrt{γ_n n}+n\varepsilon)$ under mild localized eigendecay assumptions, removing the extra $\sqrt{γ_n}$ factor from the misspecification term. The common principle is localization: spectral localization controls the Lebesgue constant of the offline approximation operator, while domain splitting implements the spatial analogue of this mechanism in the online setting, preventing local misspecification errors from being amplified globally.

In second-order optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration. Randomized sketching has emerged as a powerful technique for constructing estimates of the Hessian which can be used to perform approximate Newton steps. This involves multiplication by a random sketching matrix, which introduces a trade-off between the computational cost of sketching and the convergence rate of the optimization algorithm. A theoretically desirable but practically much too expensive choice is to use a dense Gaussian sketching matrix, which produces unbiased estimates of the exact Newton step and which offers strong problem-independent convergence guarantees. We show that the Gaussian sketching matrix can be drastically sparsified, significantly reducing the computational cost of sketching, without substantially affecting its convergence properties. This approach, called Newton-LESS, is based on a recently introduced sketching technique: LEverage Score Sparsified (LESS) embeddings. We prove that Newton-LESS enjoys nearly the same problem-independent local convergence rate as Gaussian embeddings, not just up to constant factors but even down to lower order terms, for a large class of optimization tasks. In particular, this leads to a new state-of-the-art convergence result for an iterative least squares solver. Finally, we extend LESS embeddings to include uniformly sparsified random sign matrices which can be implemented efficiently and which perform well in numerical experiments.

We develop a geometric framework that links objective accuracy to structural recovery in prototype-based clustering. The analysis is algorithm-agnostic and applies to a broad class of admissible loss functions. We define a clustering condition number that compares within-cluster scale to the minimum loss increase required to move a point across a cluster boundary. When this quantity is small, any solution with a small suboptimality gap must also have a small misclassification error relative to a benchmark partition. The framework also clarifies a fundamental trade-off between robustness and sensitivity to cluster imbalance, leading to sharp phase transitions for exact recovery under different objectives. The guarantees are deterministic and non-asymptotic, and they separate the role of algorithmic accuracy from the intrinsic geometric difficulty of the instance. We further show that errors concentrate near cluster boundaries and that sufficiently deep cluster cores are recovered exactly under strengthened local margins. Together, these results provide a geometric principle for interpreting low objective values as reliable evidence of meaningful clustering structure.

We are interested in a framework of online learning with kernels for lowdimensional, but large-scale and potentially adversarial datasets. We study the computational and theoretical performance of online variations of kernel Ridge regression.

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We further show that optimistic posterior sampling can control this Hellinger distance, when we measure model error via data likelihood. This technique allows us to design and analyze unified posterior sampling algorithms with state-of-the-art sample complexity guarantees for many model-based RL settings.