This paper investigates non-stationary online learning using the metric of interval regret, which requires an online algorithm to perform well over every time interval. We propose the first online learning algorithm that achieves an interval regret bound scaling with gradient variation, a fundamental measure of the cumulative change in online function gradients, which relates to various problem-dependent quantities and is closely connected to stochastic optimization and other problems. Our method employs a simple and efficient two-layer online ensemble structure that achieves strong theoretical guarantees. Specifically, it enjoys a regret bound that simultaneously adapts to various problem-dependent quantities while also preserving the minimax-optimal rate in the worst case. Moreover, recognizing the challenge of hyperparameter tuning, we introduce a Lipschitz- and smoothness-agnostic variant that automatically adapts to these potentially unknown constants. This is primarily enabled by a novel Lipschitz-adaptive meta algorithm, which may be of independent interest. Beyond interval regret, our method also yields broader implications: it provides versatile bounds for interval dynamic regret, a stronger measure that competes with changing comparators over any interval, and yields the first piecewise characterization for stochastic extended adversarial optimization. Theoretical findings are validated by experiments.

Independent Component Analysis (ICA) was introduced in the 1980's as a model

Representation learning based on multi-task pretraining has become a powerful approach in many domains. In particular, task-aware representation learning aims to learn an optimal representation for a specific target task by sampling data from a set of source tasks, while task-agnostic representation learning seeks to learn a universal representation for a class of tasks. In this paper, we propose a general and versatile algorithmic and theoretic framework for \emph{active representation learning}, where the learner optimally chooses which source tasks to sample from. This framework, along with a tractable meta algorithm, allows most arbitrary target and source task spaces (from discrete to continuous), covers both task-aware and task-agnostic settings, and is compatible with deep representation learning practices. We provide several instantiations under this framework, from bilinear and feature-based nonlinear to general nonlinear cases. In the bilinear case, by leveraging the non-uniform spectrum of the task representation and the calibrated source-target relevance, we prove that the sample complexity to achieve $\varepsilon$-excess risk on target scales with $(k^*)^2 ||v^*||_2^2 \varepsilon^{-2}$ where $k^*$ is the effective dimension of the target and $||v^*||_2^2 \in (0,1]$ represents the connection between source and target space. Compared to the passive one, this can save up to $\frac{1}{d_W}$ of sample complexity, where $d_W$ is the task space dimension. Finally, we demonstrate different instantiations of our meta algorithm in synthetic datasets and robotics problems, from pendulum simulations to real-world drone flight datasets. On average, our algorithms outperform baselines by 20%-70%.

Universal online learning aims to achieve optimal regret guarantees without requiring prior knowledge of the curvature of online functions. Existing methods have established minimax-optimal regret bounds for universal online learning, where a single algorithm can simultaneously attain $\mathcal{O}(\sqrt{T})$ regret for convex functions, $\mathcal{O}(d \log T)$ for exp-concave functions, and $\mathcal{O}(\log T)$ for strongly convex functions, where $T$ is the number of rounds and $d$ is the dimension of the feasible domain. However, these methods still lack problem-dependent adaptivity. In particular, no universal method provides regret bounds that scale with the gradient variation $V_T$, a key quantity that plays a crucial role in applications such as stochastic optimization and fast-rate convergence in games. In this work, we introduce UniGrad, a novel approach that achieves both universality and adaptivity, with two distinct realizations: UniGrad.Correct and UniGrad.Bregman. Both methods achieve universal regret guarantees that adapt to gradient variation, simultaneously attaining $\mathcal{O}(\log V_T)$ regret for strongly convex functions and $\mathcal{O}(d \log V_T)$ regret for exp-concave functions. For convex functions, the regret bounds differ: UniGrad.Correct achieves an $\mathcal{O}(\sqrt{V_T \log V_T})$ bound while preserving the RVU property that is crucial for fast convergence in online games, whereas UniGrad.Bregman achieves the optimal $\mathcal{O}(\sqrt{V_T})$ regret bound through a novel design. Both methods employ a meta algorithm with $\mathcal{O}(\log T)$ base learners, which naturally requires $\mathcal{O}(\log T)$ gradient queries per round. To enhance computational efficiency, we introduce UniGrad++, which retains the regret while reducing the gradient query to just $1$ per round via surrogate optimization. We further provide various implications.

Independent Component Analysis (ICA) was introduced in the 1980's as a model

We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call noisy power method. Our result characterizes the convergence behavior of the algorithm when a large amount noise is introduced after each matrix-vector multiplication. The noisy power method can be seen as a meta-algorithm that has recently found a number of important applications in a broad range of machine learning problems including alternating minimization for matrix completion, streaming principal component analysis (PCA), and privacy-preserving spectral analysis.

Representation learning based on multi-task pretraining has become a powerful approach in many domains. In particular, task-aware representation learning aims to learn an optimal representation for a specific target task by sampling data from a set of source tasks, while task-agnostic representation learning seeks to learn a universal representation for a class of tasks. In this paper, we propose a general and versatile algorithmic and theoretic framework for \emph{active representation learning}, where the learner optimally chooses which source tasks to sample from. This framework, along with a tractable meta algorithm, allows most arbitrary target and source task spaces (from discrete to continuous), covers both task-aware and task-agnostic settings, and is compatible with deep representation learning practices. We provide several instantiations under this framework, from bilinear and feature-based nonlinear to general nonlinear cases.

Representation learning based on multi-task pretraining has become a powerful approach in many domains. In particular, task-aware representation learning aims to learn an optimal representation for a specific target task by sampling data from a set of source tasks, while task-agnostic representation learning seeks to learn a universal representation for a class of tasks. In this paper, we propose a general and versatile algorithmic and theoretic framework for \emph{active representation learning}, where the learner optimally chooses which source tasks to sample from. This framework, along with a tractable meta algorithm, allows most arbitrary target and source task spaces (from discrete to continuous), covers both task-aware and task-agnostic settings, and is compatible with deep representation learning practices. We provide several instantiations under this framework, from bilinear and feature-based nonlinear to general nonlinear cases.

Modern power systems integrate renewable distributed energy resources (DERs) as an environment-friendly enhancement to meet the ever-increasing demands. However, the inherent unreliability of renewable energy renders developing DER management algorithms imperative. We study the energy-sharing problem in a system consisting of several DERs. Each agent harvests and distributes renewable energy in its neighborhood to optimize the network's performance while minimizing energy waste. We model this problem as a bandit convex optimization problem with constraints that correspond to each node's limitations for energy production. We propose distributed decision-making policies to solve the formulated problem, where we utilize the notion of dynamic regret as the performance metric. We also include an adjustment strategy in our developed algorithm to reduce the constraint violations. Besides, we design a policy that deals with the non-stationary environment. Theoretical analysis shows the effectiveness of our proposed algorithm. Numerical experiments using a real-world dataset show superior performance of our proposal compared to state-of-the-art methods.

In this paper, we consider the sequential decision problem where the goal is to minimize the general dynamic regret on a complete Riemannian manifold. The task of offline optimization on such a domain, also known as a geodesic metric space, has recently received significant attention. The online setting has received significantly less attention, and it has remained an open question whether the body of results that hold in the Euclidean setting can be transplanted into the land of Riemannian manifolds where new challenges (e.g., curvature) come into play. In this paper, we show how to get optimistic regret bound on manifolds with non-positive curvature whenever improper learning is allowed and propose an array of adaptive no-regret algorithms. To the best of our knowledge, this is the first work that considers general dynamic regret and develops "optimistic" online learning algorithms which can be employed on geodesic metric spaces.