In this work, we study offline convex optimization with smooth objectives, where the classical Nesterov's Accelerated Gradient (**NAG**) method achieves the optimal accelerated convergence. Extensive research has aimed to understand **NAG** from various perspectives, and a recent line of work approaches this from the viewpoint of online learning and online-to-batch conversion, emphasizing the role of *optimistic online algorithms* for acceleration. In this work, we contribute to this perspective by proposing novel *optimistic online-to-batch conversions* that incorporate optimism theoretically into the analysis, thereby significantly simplifying the online algorithm design while preserving the optimal convergence rates. Specifically, we demonstrate the effectiveness of our conversions through the following results: *(i)* when combined with simple online gradient descent, our optimistic conversion achieves the optimal accelerated convergence; *(ii)* our conversion also applies to strongly convex objectives, and by leveraging both optimistic online-to-batch conversion and optimistic online algorithms, we achieve the optimal accelerated convergence rate for strongly convex and smooth objectives, for the first time through the lens of online-to-batch conversion; *(iii)* our optimistic conversion can achieve universality to smoothness~---~applicable to both smooth and non-smooth objectives without requiring knowledge of the smoothness coefficient~---~and remains efficient as non-universal methods by using only one gradient query in each iteration. Finally, we highlight the effectiveness of our optimistic online-to-batch conversions by a precise correspondence with **NAG**.

Smoothness is known to be crucial for acceleration in offline optimization, and for gradient-variation regret minimization in online learning. Interestingly, these two problems are actually closely connected --- accelerated optimization can be understood through the lens of gradient-variation online learning. In this paper, we investigate online learning with *Hölder* functions, a general class encompassing both smooth and non-smooth (Lipschitz) functions, and explore its implications for offline optimization. For (strongly) convex online functions, we design the corresponding gradient-variation online learning algorithm whose regret smoothly interpolates between the optimal guarantees in smooth and non-smooth regimes. Notably, our algorithms do not require prior knowledge of the Hölder smoothness parameter, exhibiting strong adaptivity over existing methods. Through online-to-batch conversion, this gradient-variation online adaptivity yields an optimal universal method for stochastic convex optimization under Hölder smoothness. However, achieving universality in offline strongly convex optimization is more challenging. We address this by integrating online adaptivity with a detection-based guess-and-check procedure, which, for the first time, yields a universal offline method that achieves accelerated convergence in the smooth regime while maintaining near-optimal convergence in the non-smooth one.

We investigate online Linear Quadratic Regulator (LQR) with bandit feedback and semi-adversarial disturbances. Previous works assume costs with *homogeneous* curvatures (i.e., with a uniform strong convexity lower bound), which can be hard to satisfy in many real scenarios and prohibits adapting to true curvatures for better performance. In this paper, we initiate the study of bandit LQR control with *heterogeneous* cost curvatures, aiming to strengthen the algorithm's adaptivity. To achieve this, we reduce the problem to bandit convex optimization with memory via a ``with-history'' reduction to avoid hard-to-control truncation errors. Then we provide a novel analysis for an important *stability* term that appeared in both regret and memory, using *Newton decrement* developed in interior-point methods. The analysis enables us to guarantee memory-related terms introduced in the reduction and also provide a simplified analysis for handling heterogeneous curvatures in bandit convex optimization. Finally, we achieve interpolated guarantees that can not only recover existing bounds for convex and quadratic costs but also attain new implications for cases of corrupted and decaying quadraticity.

We investigate the problem of universal online learning with gradient-variation regret. Universal online learning aims to achieve regret guarantees without prior knowledge of the curvature of the online functions. Moreover, we study the problem-dependent gradient-variation regret as it plays a crucial role in bridging stochastic and adversarial optimization as well as game theory. In this work, we design a universal approach with the *optimal* gradient-variation regret simultaneously for strongly convex, exp-concave, and convex functions, thus addressing an open problem highlighted by [Yan et al. [2023]](https://openreview.net/forum?id=AA1xrgAP5z). Our approach is *simple* since it is algorithmically efficient-to-implement with a two-layer online ensemble structure and only $1$ gradient query per round, and theoretically easy-to-analyze with a novel and alternative analysis to the gradient-variation regret. Concretely, previous works on gradient variations require controlling the algorithmic stability, which is challenging and leads to sub-optimal regret and less efficient algorithm design. Our analysis overcomes this issue by using a Bregman divergence negative term from linearization and a useful smoothness property.

Multi-player online games depict the interaction of multiple players with each other over time. Strongly monotone games are of particular interest since they have benign properties and also relate to many classic games that have applications in real life. Existing works mainly focus on the time-invariant case with provable guarantees established. However, the research of the more general *time-varying* games in changing environments is underexplored and the best-known result cannot match the guarantees in the time-invariant case. In this work, we present a new decentralized online algorithm for time-varying strongly monotone games, which greatly improves existing results and obtains fast rates, matching the best time-invariant guarantee without knowing the environmental non-stationarity. Furthermore, to achieve faster rates, we generalize the RVU property with smoothness and establish a series of problem-dependent bounds that also match the best time-invariant one. To realize all those results, we make a comprehensive use of the techniques in non-stationary and universal online learning.

In this paper, we propose an online convex optimization approach with two different levels of adaptivity. On a higher level, our approach is agnostic to the unknown types and curvatures of the online functions, while at a lower level, it can exploit the unknown niceness of the environments and attain problem-dependent guarantees. Specifically, we obtain $\mathcal{O}(\log V_T)$, $\mathcal{O}(d \log V_T)$ and $\hat{\mathcal{O}}(\sqrt{V_T})$ regret bounds for strongly convex, exp-concave and convex loss functions, respectively, where $d$ is the dimension, $V_T$ denotes problem-dependent gradient variations and the $\hat{\mathcal{O}}(\cdot)$-notation omits $\log V_T$ factors. Our result not only safeguards the worst-case guarantees but also directly implies the small-loss bounds in analysis. Moreover, when applied to adversarial/stochastic convex optimization and game theory problems, our result enhances the existing universal guarantees. Our approach is based on a multi-layer online ensemble framework incorporating novel ingredients, including a carefully designed optimism for unifying diverse function types and cascaded corrections for algorithmic stability. Notably, despite its multi-layer structure, our algorithm necessitates only one gradient query per round, making it favorable when the gradient evaluation is time-consuming. This is facilitated by a novel regret decomposition equipped with carefully designed surrogate losses.

Online control with non-stochastic disturbances and adversarially chosen convex cost functions, referred to as online non-stochastic control, has recently attracted increasing attention. We study online non-stochastic control with partial feedback, where learners can only access partially observed states and partially informed (bandit) costs. The problem setting arises naturally in real-world decision-making applications and strictly generalizes exceptional cases studied disparately by previous works. We propose the first online algorithm for this problem, with an $\tilde{O}(T^{3/4})$ regret competing with the best policy in hindsight, where $T$ denotes the time horizon and the $\tilde{O}(\cdot)$-notation omits the poly-logarithmic factors in $T$. To further enhance the algorithms' robustness to changing environments, we then design a novel method with a two-layer structure to optimize the dynamic regret, a more challenging measure that competes with time-varying policies. Our method is based on the online ensemble framework by treating the controller above as the base learner. On top of that, we design two different meta-combiners to simultaneously handle the unknown variation of environments and the memory issue arising from the online control. We prove that the two resulting algorithms enjoy $\tilde{O}(T^{3/4}(1+P_T)^{1/2})$ and $\tilde{O}(T^{3/4}(1+P_T)^{1/4}+T^{5/6})$ dynamic regret respectively, where $P_T$ measures the environmental non-stationarity. Our results are further extended to unknown transition matrices. Finally, empirical studies in both synthetic linear and simulated nonlinear tasks validate our method's effectiveness, thus supporting the theoretical findings.

We study the problem of Online Convex Optimization (OCO) with memory, which allows loss functions to depend on past decisions and thus captures temporal effects of learning problems. In this paper, we introduce dynamic policy regret as the performance measure to design algorithms robust to non-stationary environments, which competes algorithms' decisions with a sequence of changing comparators. We propose a novel algorithm for OCO with memory that provably enjoys an optimal dynamic policy regret in terms of time horizon, non-stationarity measure, and memory length. The key technical challenge is how to control the switching cost, the cumulative movements of player's decisions, which is neatly addressed by a novel switching-cost-aware online ensemble approach equipped with a new meta-base decomposition of dynamic policy regret and a careful design of meta-learner and base-learner that explicitly regularizes the switching cost. The results are further applied to tackle non-stationarity in online non-stochastic control (Agarwal et al., 2019), i.e., controlling a linear dynamical system with adversarial disturbance and convex cost functions. We derive a novel gradient-based controller with dynamic policy regret guarantees, which is the first controller provably competitive to a sequence of changing policies for online non-stochastic control.

Feature evolvable learning has been widely studied in recent years where old features will vanish and new features will emerge when learning with streams. Conventional methods usually assume that a label will be revealed after prediction at each time step. However, in practice, this assumption may not hold whereas no label will be given at most time steps. A good solution is to leverage the technique of manifold regularization to utilize the previous similar data to assist the refinement of the online model. Nevertheless, this approach needs to store all previous data which is impossible in learning with streams that arrive sequentially in large volume. Thus we need a buffer to store part of them. Considering that different devices may have different storage budgets, the learning approaches should be flexible subject to the storage budget limit. In this paper, we propose a new setting: Storage-Fit Feature-Evolvable streaming Learning (SF2EL) which incorporates the issue of rarely-provided labels into feature evolution. Our framework is able to fit its behavior for different storage budgets when learning with feature evolvable streams with unlabeled data. Besides, both theoretical and empirical results validate that our approach can preserve the merit of the original feature evolvable learning i.e., can always track the best baseline and thus perform well at any time step.

The learnware paradigm attempts to change the current style of machine learning deployment, i.e., user builds her own machine learning application almost from scratch, to a style where the previous efforts of other users can be reused, given a publicly available pool of machine learning models constructed by previous users for various tasks. Each learnware is a high-quality pre-trained model associated with its specification. Although there are many models in the learnware market, only a few, even none, may be potentially helpful for the current job. Therefore, how to identify and deploy useful models becomes one of the main concerns, which particularly matters when the user’s job involves certain unseen parts not covered by the current learnware market. It becomes more challenging because, due to the privacy consideration, the raw data used for training models in the learnware market are inaccessible. In this paper, we develop a novel scheme that works can effectively reuse the learnwares even when the user’s job involves unseen parts. Despite the raw training data are inaccessible, our approach can provably identify samples from the unseen parts while assigning the rest to proper models in the market for predicting under a certain condition. Empirical studies also validate the efficacy of our approach.