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1. Faster online calibration without randomization: interval forecasts and the power of two choices

   Gupta, Chirag; Ramdas, Aaditya (July 2022, 35th Annual Conference on Learning Theory)

   We study the problem of making calibrated probabilistic forecasts for a binary sequence generated by an adversarial nature. Following the seminal paper of Foster and Vohra (1998), nature is often modeled as an adaptive adversary who sees all activity of the forecaster except the randomization that the forecaster may deploy. A number of papers have proposed randomized forecasting strategies that achieve an ϵ-calibration error rate of O(1/sqrt T), which we prove is tight in general. On the other hand, it is well known that it is not possible to be calibrated without randomization, or if nature also sees the forecaster's randomization; in both cases the calibration error could be Ω(1). Inspired by the equally seminal works on the "power of two choices" and imprecise probability theory, we study a small variant of the standard online calibration problem. The adversary gives the forecaster the option of making two nearby probabilistic forecasts, or equivalently an interval forecast of small width, and the endpoint closest to the revealed outcome is used to judge calibration. This power of two choices, or imprecise forecast, accords the forecaster with significant power -- we show that a faster ϵ-calibration rate of O(1/T) can be achieved even without deploying any randomization. 

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2. Stronger Calibration Lower Bounds via Sidestepping

   Qiao, Mingda; Valiant, Gregory (January 2021, Proceedings of the Annual ACM Symposium on Theory of Computing)

   null (Ed.)

   We consider an online binary prediction setting where a forecaster observes a sequence of T bits one by one. Before each bit is revealed, the forecaster predicts the probability that the bit is 1. The forecaster is called well-calibrated if for each p in [0,1], among the n_p bits for which the forecaster predicts probability p, the actual number of ones, m_p, is indeed equal to p*n_p. The calibration error, defined as \sum_p |m_p - p n_p|, quantifies the extent to which the forecaster deviates from being well-calibrated. It has long been known that an O(T^(2/3)) calibration error is achievable even when the bits are chosen adversarially, and possibly based on the previous predictions. However, little is known on the lower bound side, except an sqrt(T) bound that follows from the trivial example of independent fair coin flips. In this paper, we prove an T^(0.528) bound on the calibration error, which is the first bound above the trivial sqrt(T) lowerbound for this setting. The technical contributions of our work include two lower bound techniques, early stopping and sidestepping, which circumvent the obstacles that have previously hindered strong calibration lower bounds. We also propose an abstraction of the prediction setting, termed the Sign-Preservation game, which may be of independent interest. This game has a much smaller state space than the full prediction setting and allows simpler analyses. The T^0.528 lower bound follows from a general reduction theorem that translates lower bounds on the game value of Sign-Preservation into lower bounds on the calibration error. 

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3. An analysis of constraint-relaxation in PDE-based inverse problems

   https://doi.org/10.1088/1361-6420/adab89  

   van_Leeuwen, Tristan; Yang, Yunan (January 2025, Inverse Problems)

   Abstract Many inverse problems are naturally formulated as a PDE-constrained optimization problem. These non-linear, large-scale, constrained optimization problems know many challenges, of which the inherent non-linearity of the problem is an important one. In this paper, we focus on a relaxed formulation of the PDE-constrained optimization problem and provide analysis for its properties including convexity under certain assumptions. Starting from an infinite-dimensional formulation of the inverse problem with discrete data, we propose a general framework for the analysis and discretisation of such problems. The relaxed formulation of the PDE-constrained optimization problem is shown to reduce to a weighted non-linear least-squares problem. The weight matrix turns out to be the Gram matrix of solutions of the PDE, and in some cases be estimated directly from the measurements. The latter observation points to a potential way to unify recently proposed data-driven reduced-order models for inverse problems with PDE-constrained optimization. We provide a number of representative case studies and numerical examples to illustrate our findings. 

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4. Parameter-Free Online Convex Optimization with Sub-Exponential Noise

   Jun, Kwang-Sung; Orabona, Francesco (June 2019, Proceedings of the Thirty-Second Conference on Learning Theory)

   Abstract We consider the problem of unconstrained online convex optimization (OCO) with sub- exponential noise, a strictly more general problem than the standard OCO. In this setting, the learner receives a subgradient of the loss functions corrupted by sub-exponential noise and strives to achieve optimal regret guarantee, without knowledge of the competitor norm, i.e., in a parameter-free way. Recently, Cutkosky and Boahen (COLT 2017) proved that, given unbounded subgradients, it is impossible to guarantee a sublinear regret due to an exponential penalty. This paper shows that it is possible to go around the lower bound by allowing the observed subgradients to be unbounded via stochastic noise. However, the presence of unbounded noise in unconstrained OCO is challenging; existing algorithms do not provide near-optimal regret bounds or fail to have a guarantee. So, we design a novel parameter-free OCO algorithm for Banach space, which we call BANCO, via a reduction to betting on noisy coins. We show that BANCO achieves the optimal regret rate in our problem. Finally, we show the application of our results to obtain a parameter-free locally private stochastic subgradient descent algorithm, and the connection to the law of iterated logarithms. 

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5. Distributed No-Regret Learning for Multi-Stage Systems with End-to-End Bandit Feedback

   https://doi.org/10.1145/3641512.3686369  

   Hou, I-Hong (October 2024, ACM)

   This paper studies multi-stage systems with end-to-end bandit feedback. In such systems, each job needs to go through multiple stages, each managed by a different agent, before generating an outcome. Each agent can only control its own action and learn the final outcome of the job. It has neither knowledge nor control on actions taken by agents in the next stage. The goal of this paper is to develop distributed online learning algorithms that achieve sublinear regret in adversarial environments. The setting of this paper significantly expands the traditional multi-armed bandit problem, which considers only one agent and one stage. In addition to the exploration-exploitation dilemma in the traditional multi-armed bandit problem, we show that the consideration of multiple stages introduces a third component, education, where an agent needs to choose its actions to facilitate the learning of agents in the next stage. To solve this newly introduced exploration-exploitation-education trilemma, we propose a simple distributed online learning algorithm, ϵ-EXP3. We theoretically prove that the ϵ-EXP3 algorithm is a no-regret policy that achieves sublinear regret. Simulation results show that the ϵ-EXP3 algorithm significantly outperforms existing no-regret online learning algorithms for the traditional multi-armed bandit problem. 

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