We consider online model selection with decentralized data over M clients, and study the necessity of collaboration among clients. Previous work proposed various federated algorithms without demonstrating their necessity, while we answer the question from a novel perspective of computational constraints. We prove lower bounds on the regret, and propose a federated algorithm and analyze the upper bound. Our results show (i) collaboration is unnecessary in the absence of computational constraints on clients; (ii) collaboration is necessary if the computational cost on each client is limited to o(K), where K is the number of candidate hypothesis spaces. We clarify the unnecessary nature of collaboration in previous federated algorithms for distributed online multi-kernel learning, and improve the regret bounds at a smaller computational and communication cost.

We introduce the problem of model selection for contextual bandits, where a learner must adapt to the complexity of the optimal policy while balancing exploration and exploitation. Our main result is a new model selection guarantee for linear contextual bandits. We work in the stochastic realizable setting with a sequence of nested linear policy classes of dimension d_1 d_2 \ldots, where the m \star -th class contains the optimal policy, and we design an algorithm that achieves \tilde{O}l(T {2/3}d {1/3}_{m \star}) regret with no prior knowledge of the optimal dimension d_{m \star} . This is the first model selection result for contextual bandits with non-vacuous regret for all values of d_{m \star}, and to the best of our knowledge is the first positive result of this type for any online learning setting with partial information. The core of the algorithm is a new estimator for the gap in the best loss achievable by two linear policy classes, which we show admits a convergence rate faster than the rate required to learn the parameters for either class.

The vast majority of statistical theory on binary classification characterizes performance in terms of accuracy. However, accuracy is known in many cases to poorly reflect the practical consequences of classification error, most famously in imbalanced binary classification, where data are dominated by samples from one of two classes. The first part of this paper derives a novel generalization of the Bayes-optimal classifier from accuracy to any performance metric computed from the confusion matrix. Specifically, this result (a) demonstrates that stochastic classifiers sometimes outperform the best possible deterministic classifier and (b) removes an empirically unverifiable absolute continuity assumption that is poorly understood but pervades existing results. We then demonstrate how to use this generalized Bayes classifier to obtain regret bounds in terms of the error of estimating regression functions under uniform loss. Finally, we use these results to develop some of the first finite-sample statistical guarantees specific to imbalanced binary classification.

In latent diffusion models (LDMs), denoising diffusion process efficiently takes place on latent space whose dimension is lower than that of pixel space. Decoder is typically used to transform the representation in latent space to that in pixel space. While a decoder is assumed to have an encoder as an accurate inverse, exact encoder-decoder pair rarely exists in practice even though applications often require precise inversion of decoder. In other words, encoder is not the left-inverse but the right-inverse of the decoder; decoder inversion seeks the left-inverse. Prior works for decoder inversion in LDMs employed gradient descent inspired by inversions of generative adversarial networks. However, gradient-based methods require larger GPU memory and longer computation time for larger latent space.

We study theoretical properties of a broad class of regularized algorithms with vector-valued output. These spectral algorithms include kernel ridge regression, kernel principal component regression and various implementations of gradient descent. First, we rigorously confirm the so-called saturation effect for ridge regression with vector-valued output by deriving a novel lower bound on learning rates; this bound is shown to be suboptimal when the smoothness of the regression function exceeds a certain level.Second, we present an upper bound on the finite sample risk for general vector-valued spectral algorithms, applicable to both well-specified and misspecified scenarios (where the true regression function lies outside of the hypothesis space), and show that this bound is minimax optimal in various regimes. All of our results explicitly allow the case of infinite-dimensional output variables, proving consistency of recent practical applications.

Transformers have achieved great success in recent years. Interestingly, transformers have shown particularly strong in-context learning capability -- even without fine-tuning, they are still able to solve unseen tasks well purely based on task-specific prompts. In this paper, we study the capability of one-layer transformers in learning the one-nearest neighbor prediction rule. Under a theoretical framework where the prompt contains a sequence of labeled training data and unlabeled test data, we show that, although the loss function is nonconvex, when trained with gradient descent, a single softmax attention layer can successfully learn to behave like a one-nearest neighbor classifier. Our result gives a concrete example on how transformers can be trained to implement nonparametric machine learning algorithms, and sheds light on the role of softmax attention in transformer models.

Implicit bias describes the phenomenon where optimization-based training algorithms, without explicit regularization, show a preference for simple estimators even when more complex estimators have equal objective values. Multiple works have developed the theory of implicit bias for binary classification under the assumption that the loss satisfies an exponential tail property. However, there is a noticeable gap in analysis for multiclass classification, with only a handful of results which themselves are restricted to the cross-entropy loss. In this work, we employ the framework of Permutation Equivariant and Relative Margin-based (PERM) losses [Wang and Scott, 2024] to introduce a multiclass extension of the exponential tail property. This class of losses includes not only cross-entropy but also other losses. Using this framework, we extend the implicit bias result of Soudry et al. [2018] to multiclass classification.

Large Language Models (LLMs) and other large foundation models have achieved impressive results, but their size exacerbates existing resource consumption and latency challenges. In particular, the large-scale deployment of these models is hindered by the significant resource requirements during inference. In this paper, we study two approaches for mitigating these challenges: employing a cache to store previous queries and learning a model selector to choose from an ensemble of models for query processing.Theoretically, we provide an optimal algorithm for jointly optimizing both approaches to reduce the inference cost in both offline and online tabular settings. By combining a caching algorithm, namely Greedy Dual Size with Frequency (GDSF) or Least Expected Cost (LEC), with a model selector, we achieve optimal rates in both offline and online settings. Empirically, simulations show that our caching and model selection algorithm greatly improves over the baselines, with up to 50\times improvement over the baseline when the ratio between the maximum cost and minimum cost is 100 .

While 3D Gaussian Splatting has recently become popular for neural rendering, current methods rely on carefully engineered cloning and splitting strategies for placing Gaussians, which does not always generalize and may lead to poor-quality renderings. For many real-world scenes this leads to their heavy dependence on good initializations. In this work, we rethink the set of 3D Gaussians as a random sample drawn from an underlying probability distribution describing the physical representation of the scene--in other words, Markov Chain Monte Carlo (MCMC) samples. Under this view, we show that the 3D Gaussian updates can be converted as Stochastic Gradient Langevin Dynamics (SGLD) update by simply introducing noise. We then rewrite the densification and pruning strategies in 3D Gaussian Splatting as simply a deterministic state transition of MCMC samples, removing these heuristics from the framework.

Dimensionality reduction is a classical technique widely used for data analysis. One foundational instantiation is Principal Component Analysis (PCA), which minimizes the average reconstruction error. In this paper, we introduce the multi-criteria dimensionality reduction problem where we are given multiple objectives that need to be optimized simultaneously. As an application, our model captures several fairness criteria for dimensionality reduction such as the Fair-PCA problem introduced by Samadi et al. [NeurIPS18] and the Nash Social Welfare (NSW) problem. In the Fair-PCA problem, the input data is divided into k groups, and the goal is to find a single d-dimensional representation for all groups for which the maximum reconstruction error of any one group is minimized.