Quantum state tomography, the task of learning an unknown quantum state, is a fundamental problem in quantum information. In standard settings, the complexity of this problem depends significantly on the type of quantum state that one is trying to learn, with pure states being substantially easier to learn than general mixed states. A natural question is whether this separation holds for any quantum state learning setting. In this work, we consider the online learning framework and prove the surprising result that learning pure states in this setting is as hard as learning mixed states. More specifically, we show that both classes share almost the same sequential fat-shattering dimension, leading to identical regret scaling under the $L_1$-loss. We also generalize previous results on full quantum state tomography in the online setting to learning only partially the density matrix, using smooth analysis.

Abstract--We present an iterative approach for planning and controlling motions of underactuated robots with uncertain dynamics. At its core, there is a learning process which estimates the perturbations induced by the model uncertainty on the active and passive degrees of freedom. The generic iteration of the algorithm makes use of the learned data in both the planning phase, which is based on optimization, and the control phase, where partial feedback linearization of the active dofs is performed on the model updated on-line. The performance of the proposed approach is shown by comparative simulations and experiments on a Pendubot executing various types of swing-up maneuvers. Very few iterations are typically needed to generate dynamically feasible trajectories and the tracking control that guarantees their accurate execution, even in the presence of large model uncertainties.

The problem of dynamic pricing examines strategies of setting and adjusting prices in response to varying customer behaviors and market conditions. The mainstream of existing works on dynamic pricing, including Kleinberg and Leighton (2003); Broder and Rusmevichientong (2012); Cohen et al. (2020); Wang et al. (2021b), focuses on the estimation of demand curves while putting aside the decisions on the supply side. Another series of literature, including Besbes and Zeevi (2009); Chen et al. (2019, 2021a); Keskin et al. (2022), takes supply and inventories into account. However, these works simplify the supply cost as uniform and static, underestimating the difficulty of allocating products through sophisticated supply chains among multiple parties such as factories, warehouses, and retailers. On the other hand, the problem of resource allocation - to serve different demand classes with various types of resources - presents a complex challenge within the field of operations research. Analogous to online dynamic pricing, the recent proliferation of e-platforms has magnified the importance of developing online allocation algorithms that efficiently manage supply and demand on the fly while maximizing cumulative utilities.

First, the main class of losses that the paper introduces, that of relative Lipschitz continuity (Def. In particular, given that the losses are (RLC) then one can recover relative Lipschitz continuity via a direct combination of convexity and Cauchy-Schwartz inequality. Moreover, conversely every relative Lipschitz continuous loss can be seen as (RLC) if one chooses the respective Riemannian metric accordingly; this becomes even more evident for the example that the paper presents, if f(x) x {2} for x\in R, then one can straightforwardly choose the Riemannian metric in such a manner that the respective dual norm would be \ v\ _{x,\ast} v /x and (RLC) follows. That said, this weakens significantly the contributions concerning FTRL and the like, since in Antonakopoulos et. On the other hand, concerning the most intriguing part that of establishing logarithmic regret for the case where the loss functions are in addition relatively strongly convex, there is no obvious way to establish any relevant examples that satisfy simultaneously relative Lipschitz continuity and relative strong convexity, besides of course the euclidean ones.

This paper treats the problem of online convex optimization without Lipschitz continuity of the loss functions. The authors consider a variant of Lipschitz continuity called "relative Lipschitz continuity": this notion is originally due to Lu (2019) and involves a Bregman divergence instead of the standard norm in comparing nearby points. In this context, the authors prove the following results: - Under only relative Lipschitz continuity: an O(sqrt{T}) regret bound for follow-the-regularized-leader (FTRL) and a "stabilized" variant of the online mirror descent (OMD) algorithm. These results are similar to standard bounds in the literature for Lipschitz continuous / strongly convex functions. The extension to *relative* Lipschitz continuous / strongly convex functions was welcomed by the reviewers, but two major issues were identified: 1. An earlier ICLR paper by Antonakopoulos et al. (2020) already provides O(\sqrt{T}) bounds for FTRL and OMD under a closely related "Riemannian Lipschitz continuity" condition.

Post-rebuttal: I am satisfied with the rebuttal. I am interested to know more about "One reason why such rates are common in online learning is the connection of the sequential Rademacher complexity with uniform convergence of martingale difference sequences in the corresponding Banach space (see [2] for details)." If the paper is accepted and space is allowed, I suggest to elaborate on this more thoroughly.

One of the main issues of this paper is clarity, however. We trust that you will work very hard to improve clarity for the final submission, as suggested in the reviews.

Decentralized learning networks aim to synthesize a single network inference from a set of raw inferences provided by multiple participants. To determine the combined inference, these networks must adopt a mapping from historical participant performance to weights, and to appropriately incentivize contributions they must adopt a mapping from performance to fair rewards. Despite the increased prevalence of decentralized learning networks, there exists no systematic study that performs a calibration of the associated free parameters. Here we present an optimization framework for key parameters governing decentralized online learning in supervised regression and classification problems. These parameters include the slope of the mapping between historical performance and participant weight, the timeframe for performance evaluation, and the slope of the mapping between performance and rewards. These parameters are optimized using a suite of numerical experiments that mimic the design of the Allora Network, but have been extended to handle classification tasks in addition to regression tasks. This setup enables a comparative analysis of parameter tuning and network performance optimization (loss minimization) across both problem types. We demonstrate how the optimal performance-weight mapping, performance timeframe, and performance-reward mapping vary with network composition and problem type. Our findings provide valuable insights for the optimization of decentralized learning protocols, and we discuss how these results can be generalized to optimize any inference synthesis-based, decentralized AI network.

This paper considers the implicit update algorithm for online learning (a.k.a. It is shown that the algorithm achieves a regret bound that is adapted to the variability of the sequence of loss functions. This holds even without the smoothness of the loss. I believe this is a firm contribution to the fields of online learning and stochastic optimization. Firstly, Implicit updates are known to have practical advantages, but their theoretical understanding has been limited to the fact that they enjoy the same worst-case regret guarantees as their explicit counterparts. This is one of a very few works (if not the first one) which shows a nontrivial advantages of the implicit methods and thus makes a significant progress in better understanding of their behavior.

The paper concerns a new online learning problem subject to the constraint of individual fairness. It provides a framework that reduces online classification in the considered model to standard online classification, obtaining an algorithm with sublinear regret both in terms of accuracy and fairness, as well as strong generalization bounds in the i.i.d. All the reviewers liked the paper and the proposed metric-free approach. The appreciated an interesting problem formulation and a clean reduction technique to a known online learning problem. The paper received very high uniform scores of 8 from each reviewer. The reviewers found some issues with the presentation, and I hope the authors will address them in the final version of the manuscript.