Bandit Convex Optimization is a fundamental class of sequential decision-making problems, where the learner selects actions from a continuous domain and observes a loss (but not its gradient) at only one point per round. We study this problem in non-stationary environments, and aim to minimize the regret under three standard measures of non-stationarity: the number of switches $S$ in the comparator sequence, the total variation $\Delta$ of the loss functions, and the path-length $P$ of the comparator sequence. We propose a polynomial-time algorithm, Tilted Exponentially Weighted Average with Sleeping Experts (TEWA-SE), which adapts the sleeping experts framework from online convex optimization to the bandit setting. For strongly convex losses, we prove that TEWA-SE is minimax-optimal with respect to known $S$ and $\Delta$ by establishing matching upper and lower bounds. By equipping TEWA-SE with the Bandit-over-Bandit framework, we extend our analysis to environments with unknown non-stationarity measures. For general convex losses, we introduce a second algorithm, clipped Exploration by Optimization (cExO), based on exponential weights over a discretized action space. While not polynomial-time computable, this method achieves minimax-optimal regret with respect to known $S$ and $\Delta$, and improves on the best existing bounds with respect to $P$.

We consider Constrained Online Convex Optimization (COCO) with adversarially chosen constraints. At each round, the learner chooses an action before observing the loss and constraint function for that round. The goal is to achieve small static regret against the best point satisfying all constraints while also controlling cumulative constraint violation ($\mathsf{CCV}$). For strongly convex losses, state-of-the-art algorithms achieve $O(\log T)$ regret and $O(\sqrt{T \log T})$ $\mathsf{CCV}.$ The corresponding best-known bounds for convex losses is $O(\sqrt{T})$ regret and $O(\sqrt{T} \log T)$ $\mathsf{CCV}$. In this paper, we give a simple projection-based algorithm that simultaneously achieves $O(\log T)$ regret and $O(\log T)$ $\mathsf{CCV}$ for strongly-convex losses, yielding an exponential improvement in the $\mathsf{CCV}$. For the convex losses, our algorithm improves the $\mathsf{CCV}$ to $O(\sqrt{T})$ while maintaining the optimal $O(\sqrt{T})$ regret. The key to our improvement is a recent geometric result for self-contracted curves, which may be of independent interest.

Gradient Boosting Decision Trees (GBDTs) dominate tabular machine learning, with modern implementations like XGBoost, LightGBM, and CatBoost being based on Newton boosting: a second-order descent step in the space of decision trees. Despite its empirical success, the global convergence of Newton boosting is poorly understood compared to first-order boosting. In this paper, we introduce Restricted Newton Descent, which studies convex optimization with Newton's method on Hilbert spaces with inexact iterates, based on the concepts of cosine angle and weak gradient edge. Within this framework, we recover Newton boosting with GBDTs and classical finite-dimensional theory as special cases. We first prove that vanilla Newton boosting achieves a linear rate of convergence for smooth, strongly convex losses that satisfy a Hessian-dominance condition. To handle general convex losses with Lipschitz Hessians, we extend a recent gradient regularized Newton scheme to the restricted weak learner setting. This scheme minimally modifies the classical algorithm by introducing an adaptive $\ell_2$-regularization term proportional to the square root of the gradient norm at each iteration. We establish a $\mathcal{O}(\frac{1}{k^2})$ rate for this scheme, thereby obtaining a globally convergent second-order GBDT algorithm with a rate matching that of first-order boosting with Nesterov momentum. In numerical experiments, we show that our scheme converges while vanilla Newton boosting may diverge.

Neural Information Processing Systems http://nips.cc/

How the regret scales in a finite time horizonT with problem parameters is a central theme in studies of different algorithms.

The online Frank-Wolfe (OFW) method has gained much popularity for online convex optimization due to its projection-free property. Previous studies show that OFW can attain an $O(T^{3/4})$ regret bound for convex losses and an $O(T^{2/3})$ regret bound for strongly convex losses. However, they assume that each gradient queried by OFW is revealed immediately, which may not hold in practice and limits the application of OFW. To address this limitation, we propose a delayed variant of OFW, which allows gradients to be delayed by arbitrary rounds. The main idea is to perform an update similar to OFW after receiving any delayed gradient, and play the latest decision for each round. Despite its simplicity, we prove that our delayed variant of OFW is able to achieve an $O(T^{3/4}+dT^{1/4})$ regret bound for convex losses and an $O(T^{2/3}+d\log T)$ regret bound for strongly convex losses, where $d$ is the maximum delay. This is quite surprising since under a relatively large amount of delay (e.g., $d=O(\sqrt{T})$ for convex losses and $d=O(T^{2/3}/\log T)$ for strongly convex losses), the delayed variant of OFW enjoys the same regret bound as that of the original OFW.