We present HUME, a simple model-agnostic framework for inferring human labeling of a given dataset without any external supervision. The key insight behind our approach is that classes defined by many human labelings are linearly separable regardless of the representation space used to represent a dataset. HUME utilizes this insight to guide the search over all possible labelings of a dataset to discover an underlying human labeling. We show that the proposed optimization objective is strikingly well-correlated with the ground truth labeling of the dataset. In effect, we only train linear classifiers on top of pretrained representations that remain fixed during training, making our framework compatible with any large pretrained and self-supervised model. Despite its simplicity, HUME outperforms a supervised linear classifier on top of self-supervised representations on the STL-10 dataset by a large margin and achieves comparable performance on the CIFAR-10 dataset. Compared to the existing unsupervised baselines, HUME achieves state-of-the-art performance on four benchmark image classification datasets including the large-scale ImageNet-1000 dataset. Altogether, our work provides a fundamentally new view to tackle unsupervised learning by searching for consistent labelings between different representation spaces.

The performance of reproducing kernel Hilbert space-based methods is known to be sensitive to the choice of the reproducing kernel. Choosing an adequate reproducing kernel can be challenging and computationally demanding, especially in data-rich tasks without prior information about the solution domain. In this paper, we propose a learning scheme that scalably combines several single kernel-based online methods to reduce the kernel-selection bias. The proposed learning scheme applies to any task formulated as a regularized empirical risk minimization convex problem. More specifically, our learning scheme is based on a multi-kernel learning formulation that can be applied to widen any single-kernel solution space, thus increasing the possibility of finding higher-performance solutions. In addition, it is parallelizable, allowing for the distribution of the computational load across different computing units. We show experimentally that the proposed learning scheme outperforms the combined single-kernel online methods separately in terms of the cumulative regularized least squares cost metric.

Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in uncertainty quantification where the function is the solution map of a parametric or stochastic differential equation (DE). Yet, sparse polynomial approximation lacks a complete theory. On the one hand, there is a well-developed theory of best $s$-term polynomial approximation, which asserts exponential or algebraic rates of convergence for holomorphic functions. On the other, there are increasingly mature methods such as (weighted) $\ell^1$-minimization for computing such approximations. While the sample complexity of these methods has been analyzed with compressed sensing, whether they achieve best $s$-term approximation rates is not fully understood. Furthermore, these methods are not algorithms per se, as they involve exact minimizers of nonlinear optimization problems. This paper closes these gaps. Specifically, we consider the following question: are there robust, efficient algorithms for computing approximations to finite- or infinite-dimensional, holomorphic and Hilbert-valued functions from limited samples that achieve best $s$-term rates? We answer this affirmatively by introducing algorithms and theoretical guarantees that assert exponential or algebraic rates of convergence, along with robustness to sampling, algorithmic, and physical discretization errors. We tackle both scalar- and Hilbert-valued functions, this being key to parametric or stochastic DEs. Our results involve significant developments of existing techniques, including a novel restarted primal-dual iteration for solving weighted $\ell^1$-minimization problems in Hilbert spaces. Our theory is supplemented by numerical experiments demonstrating the efficacy of these algorithms.

Tuning hyperparameters, such as the stepsize, presents a major challenge of training machine learning models. To address this challenge, numerous adaptive optimization algorithms have been developed that achieve near-optimal complexities, even when stepsizes are independent of problem-specific parameters, provided that the loss function is $L$-smooth. However, as the assumption is relaxed to the more realistic $(L_0, L_1)$-smoothness, all existing convergence results still necessitate tuning of the stepsize. In this study, we demonstrate that Normalized Stochastic Gradient Descent with Momentum (NSGD-M) can achieve a (nearly) rate-optimal complexity without prior knowledge of any problem parameter, though this comes at the cost of introducing an exponential term dependent on $L_1$ in the complexity. We further establish that this exponential term is inevitable to such schemes by introducing a theoretical framework of lower bounds tailored explicitly for parameter-agnostic algorithms. Interestingly, in deterministic settings, the exponential factor can be neutralized by employing Gradient Descent with a Backtracking Line Search. To the best of our knowledge, these findings represent the first parameter-agnostic convergence results under the generalized smoothness condition. Our empirical experiments further confirm our theoretical insights.

Deep Neural Networks and Reinforcement Learning methods have empirically shown great promise in tackling challenging combinatorial problems. In those methods a deep neural network is used as a solution generator which is then trained by gradient-based methods (e.g., policy gradient) to successively obtain better solution distributions. In this work we introduce a novel theoretical framework for analyzing the effectiveness of such methods. We ask whether there exist generative models that (i) are expressive enough to generate approximately optimal solutions; (ii) have a tractable, i.e, polynomial in the size of the input, number of parameters; (iii) their optimization landscape is benign in the sense that it does not contain sub-optimal stationary points. Our main contribution is a positive answer to this question. Our result holds for a broad class of combinatorial problems including Max- and Min-Cut, Max-$k$-CSP, Maximum-Weight-Bipartite-Matching, and the Traveling Salesman Problem. As a byproduct of our analysis we introduce a novel regularization process over vanilla gradient descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.

We introduce a differentiable clustering method based on stochastic perturbations of minimum-weight spanning forests. This allows us to include clustering in end-to-end trainable pipelines, with efficient gradients. We show that our method performs well even in difficult settings, such as data sets with high noise and challenging geometries. We also formulate an ad hoc loss to efficiently learn from partial clustering data using this operation. We demonstrate its performance on several data sets for supervised and semi-supervised tasks.

Modern policy optimization methods in reinforcement learning, such as TRPO and PPO, owe their success to the use of parameterized policies. However, while theoretical guarantees have been established for this class of algorithms, especially in the tabular setting, the use of general parameterization schemes remains mostly unjustified. In this work, we introduce a novel framework for policy optimization based on mirror descent that naturally accommodates general parameterizations. The policy class induced by our scheme recovers known classes, e.g., softmax, and generates new ones depending on the choice of mirror map. Using our framework, we obtain the first result that guarantees linear convergence for a policy-gradient-based method involving general parameterization. To demonstrate the ability of our framework to accommodate general parameterization schemes, we provide its sample complexity when using shallow neural networks, show that it represents an improvement upon the previous best results, and empirically validate the effectiveness of our theoretical claims on classic control tasks.

Text-to-image diffusion models show great potential in synthesizing a large variety of concepts in new compositions and scenarios. However, the latent space of initial seeds is still not well understood and its structure was shown to impact the generation of various concepts. Specifically, simple operations like interpolation and finding the centroid of a set of seeds perform poorly when using standard Euclidean or spherical metrics in the latent space. This paper makes the observation that, in current training procedures, diffusion models observed inputs with a narrow range of norm values. This has strong implications for methods that rely on seed manipulation for image generation, with applications to few-shot and long-tail learning tasks. To address this issue, we propose a novel method for interpolating between two seeds and demonstrate that it defines a new non-Euclidean metric that takes into account a norm-based prior on seeds. We describe a simple yet efficient algorithm for approximating this interpolation procedure and use it to further define centroids in the latent seed space. We show that our new interpolation and centroid techniques significantly enhance the generation of rare concept images. This further leads to state-of-the-art performance on few-shot and long-tail benchmarks, improving prior approaches in terms of generation speed, image quality, and semantic content.

In this paper, we introduce faster first-order primal-dual algorithms for minimizing a convex function subject to strongly convex function constraints. Before our work, the best complexity bound was $\mathcal{O}(1/{\varepsilon})$, and it remains unclear how to improve this result by leveraging the strong convexity assumption. We address this issue by developing novel techniques to progressively estimate the strong convexity of the Lagrangian function. Our approach yields an improved complexity of $\mathcal{O}(1/\sqrt{\varepsilon})$, matching the complexity lower bound for strongly-convex-concave saddle point optimization. We show the superior performance of our methods in sparsity-inducing constrained optimization, notably Google's personalized PageRank problem. Furthermore, we show that a restarted version of the proposed methods can effectively identify the sparsity pattern of the optimal solution within a finite number of steps, a result that appears to have independent significance.

A primary goal in strategic classification is to learn decision rules which are robust to strategic input manipulation. Earlier works assume that strategic responses are known; while some recent works address the important challenge of unknown responses, they exclusively study sequential settings which allow multiple model deployments over time. But there are many domains$\unicode{x2014}$particularly in public policy, a common motivating use-case$\unicode{x2014}$where multiple deployments are unrealistic, or where even a single bad round is undesirable. To address this gap, we initiate the study of strategic classification under unknown responses in the one-shot setting, which requires committing to a single classifier once. Focusing on the users' cost function as the source of uncertainty, we begin by proving that for a broad class of costs, even a small mis-estimation of the true cost can entail arbitrarily low accuracy in the worst case. In light of this, we frame the one-shot task as a minimax problem, with the goal of identifying the classifier with the smallest worst-case risk over an uncertainty set of possible costs. Our main contribution is efficient algorithms for both the full-batch and stochastic settings, which we prove converge (offline) to the minimax optimal solution at the dimension-independent rate of $\tilde{\mathcal{O}}(T^{-\frac{1}{2}})$. Our analysis reveals important structure stemming from the strategic nature of user responses, particularly the importance of dual norm regularization with respect to the cost function.