We study adversarial online learning with hidden-convex losses, i.e., nonconvex losses that become convex after a nonlinear reparameterization. Ghai, Lu and Hazan (2022) proved that, under geometric and smoothness assumptions, online gradient descent (OGD) on such nonconvex losses approximately simulates online mirror descent (OMD) on the underlying convex losses with a suitable regularizer, yielding $\mathcal{O}(T^{2/3})$ regret. They left open whether the optimal $Θ(\sqrt{T})$ regret from online convex optimization can be recovered in this hidden-convex setting. We answer this question affirmatively. More specifically, via a sharper discrete-time algorithmic equivalence argument, we prove that OGD achieves $\mathcal{O}(\sqrt{T})$ regret under the same assumptions, matching the optimal worst-case rate for adversarial online convex optimization. We also address another open question of Ghai, Lu and Hazan (2022) by clarifying the geometry required for this algorithmic equivalence. We replace the diagonal-Jacobian sufficient condition with a necessary-and-sufficient Hessian compatibility condition, thereby expanding the class of admissible reparameterizations. We complement our tight regret bound with a lower bound showing that the Hessian compatibility assumption is essential for OGD; when it fails, we construct a smooth reparameterization and an adversarial sequence of hidden-convex losses for which OGD suffers $Ω(T)$ regret. Finally, we extend our analysis to one-point bandit feedback and prove a $\mathcal{O}(T^{3/4})$ expected regret bound for bandit OGD with spherical smoothing, matching its classical rate on convex losses.

Asian N. Cristianini, An based.

Since the switching costs introduce coupling across all stages, multi-step-ahead (long-term) predictions areincorporated toimprovethe online performance.

LetT be the time horizon andPT be the path-length that essentially reflects the non-stationarity of environments, the state-of-the-art dynamicregretis O( p T(1+PT)).

Adversarial online linear optimization (OLO) is essentially about making performance tradeoffs with respect to the unknown difficulty of the adversary. In the setting of one-dimensional fixed-time OLO on a bounded domain, it has been observed since Cover (1966) that achievable tradeoffs are governed by probabilistic inequalities, and these descriptive results can be converted into algorithms via dynamic programming, which, however, is not computationally efficient. We address this limitation by showing that Stein's method, a classical framework underlying the proofs of probabilistic limit theorems, can be operationalized as computationally efficient OLO algorithms. The associated regret and total loss upper bounds are "additively sharp", meaning that they surpass the conventional big-O optimality and match normal-approximation-based lower bounds by additive lower order terms. Our construction is inspired by the remarkably clean proof of a Wasserstein martingale central limit theorem (CLT) due to Röllin (2018). Several concrete benefits can be obtained from this general technique. First, with the same computational complexity, the proposed algorithm improves upon the total loss upper bounds of online gradient descent (OGD) and multiplicative weight update (MWU). Second, our algorithm can realize a continuum of optimal two-point tradeoffs between the total loss and the maximum regret over comparators, improving upon prior works in parameter-free online learning. Third, by allowing the adversary to randomize on an unbounded support, we achieve sharp in-expectation performance guarantees for OLO with noisy feedback.

A common assumption in machine learning is that the training set and test set are drawn from the same distribution [25]. However, this assumption often does not hold in practice when models are deployed in the real world [3, 28]. One common type of distribution shift is label shift, where the conditional distribution p(x|y) is fixed but the label distribution p(y) changes over time.

Orthogonal Gradient Descent (OGD) has emerged as a powerful method for continual learning. However, its Euclidean projections do not leverage the underlying information-geometric structure of the problem, which can lead to suboptimal convergence in learning tasks. To address this, we propose incorporating the natural gradient into OGD and present \textbf{ONG (Orthogonal Natural Gradient Descent)}. ONG preconditions each new task-specific gradient with an efficient EKFAC approximation of the inverse Fisher information matrix, yielding updates that follow the steepest descent direction under a Riemannian metric. To preserve performance on previously learned tasks, ONG projects these natural gradients onto the orthogonal complement of prior tasks' natural gradients. We provide an initial theoretical justification for this procedure, introduce the Orthogonal Natural Gradient Descent (ONG) algorithm, and present preliminary results on the Permuted and Rotated MNIST benchmarks. Our preliminary results, however, indicate that a naive combination of natural gradients and orthogonal projections has potential issues. This finding has motivated continued future work focused on robustly reconciling these geometric perspectives to develop a continual learning method, establishing a more rigorous theoretical foundation with formal convergence guarantees, and extending empirical validation to large-scale continual learning benchmarks.

We consider online convex optimization with time-varying stage costs and additional switching costs.

In Online Convex Optimization (OCO), when the stochastic gradient has a finite variance, many algorithms provably work and guarantee a sublinear regret. However, limited results are known if the gradient estimate has a heavy tail, i.e., the stochastic gradient only admits a finite $\mathsf{p}$-th central moment for some $\mathsf{p}\in\left(1,2\right]$. Motivated by it, this work examines different old algorithms for OCO (e.g., Online Gradient Descent) in the more challenging heavy-tailed setting. Under the standard bounded domain assumption, we establish new regrets for these classical methods without any algorithmic modification. Remarkably, these regret bounds are fully optimal in all parameters (can be achieved even without knowing $\mathsf{p}$), suggesting that OCO with heavy tails can be solved effectively without any extra operation (e.g., gradient clipping). Our new results have several applications. A particularly interesting one is the first provable convergence result for nonsmooth nonconvex optimization under heavy-tailed noise without gradient clipping. Furthermore, we explore broader settings (e.g., smooth OCO) and extend our ideas to optimistic algorithms to handle different cases simultaneously.