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1. Second Order Path Variationals in Non-Stationary Online Learning

   Baby, Dheeraj and; Wang, Yu-Xiang (April 2023, Proceedings of Machine Learning Research)

   Ruiz, Francisco and (Ed.)

   We consider the problem of universal dynamic regret minimization under exp-concave and smooth losses. We show that appropriately designed Strongly Adaptive algorithms achieve a dynamic regret of $$\tilde O(d^2 n^{1/5} [\mathcal{TV}_1(w_{1:n})]^{2/5} \vee d^2)$$, where $$n$$ is the time horizon and $$\mathcal{TV}_1(w_{1:n})$$ a path variational based on second order differences of the comparator sequence. Such a path variational naturally encodes comparator sequences that are piece-wise linear – a powerful family that tracks a variety of non-stationarity patterns in practice (Kim et al., 2009). The aforementioned dynamic regret is shown to be optimal modulo dimension dependencies and poly-logarithmic factors of $$n$$. To the best of our knowledge, this path variational has not been studied in the non-stochastic online learning literature before. Our proof techniques rely on analysing the KKT conditions of the offline oracle and requires several non-trivial generalizations of the ideas in Baby and Wang (2021) where the latter work only implies an $$\tilde{O}(n^{1/3})$$ regret for the current problem. 

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2. Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond

   Baby, Dheeraj; Wang, Yu-Xiang (April 2022, Proceedings of Machine Learning Research)

   We study the framework of universal dynamic regret minimization with strongly convex losses. We answer an open problem in Baby and Wang 2021 by showing that in a proper learning setup, Strongly Adaptive algorithms can achieve the near optimal dynamic regret of 𝑂̃ (𝑑1/3𝑛1/3TV[𝑢1:𝑛]2/3∨𝑑) against any comparator sequence 𝑢1,…,𝑢𝑛 simultaneously, where 𝑛 is the time horizon and TV[𝑢1:𝑛] is the Total Variation of comparator. These results are facilitated by exploiting a number of new structures imposed by the KKT conditions that were not considered in Baby and Wang 2021 which also lead to other improvements over their results such as: (a) handling non-smooth losses and (b) improving the dimension dependence on regret. Further, we also derive near optimal dynamic regret rates for the special case of proper online learning with exp-concave losses and an 𝐿∞ constrained decision set. 

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3. Constrained Variational Policy Optimization for Safe Reinforcement Learning

   Liu, Zuxin; Cen, Zhepeng; Isenbaev, Vladislav; Liu, Wei; Wu, Steven; Li, Bo; Zhao, Ding (January 2022, Proceedings of the 39th International Conference on Machine Learning)

   afe reinforcement learning (RL) aims to learn policies that satisfy certain constraints before deploying them to safety-critical applications. Previous primal-dual style approaches suffer from instability issues and lack optimality guarantees. This paper overcomes the issues from the perspective of probabilistic inference. We introduce a novel Expectation-Maximization approach to naturally incorporate constraints during the policy learning: 1) a provable optimal non-parametric variational distribution could be computed in closed form after a convex optimization (E-step); 2) the policy parameter is improved within the trust region based on the optimal variational distribution (M-step). The proposed algorithm decomposes the safe RL problem into a convex optimization phase and a supervised learning phase, which yields a more stable training performance. A wide range of experiments on continuous robotic tasks shows that the proposed method achieves significantly better constraint satisfaction performance and better sample efficiency than baselines. The code is available at https://github.com/liuzuxin/cvpo-safe-rl. 

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4. The SMART approach to instance-optimal online learning

   Banerjee, Siddhartha; Bhatt; Alankrita; Yu, Christina_Lee (July 2024, Proceedings of Machine Learning Research)

   We devise an online learning algorithm -- titled Switching via Monotone Adapted Regret Traces (SMART) -- that adapts to the data and achieves regret that is instance optimal, i.e., simultaneously competitive on every input sequence compared to the performance of the follow-the-leader (FTL) policy and the worst case guarantee of any other input policy. We show that the regret of the SMART policy on any input sequence is within a multiplicative factor e/(e−1)≈1.58 of the smaller of: 1) the regret obtained by FTL on the sequence, and 2) the upper bound on regret guaranteed by the given worst-case policy. This implies a strictly stronger guarantee than typical `best-of-both-worlds' bounds as the guarantee holds for every input sequence regardless of how it is generated. SMART is simple to implement as it begins by playing FTL and switches at most once during the time horizon to the worst-case algorithm. Our approach and results follow from an operational reduction of instance optimal online learning to competitive analysis for the ski-rental problem. We complement our competitive ratio upper bounds with a fundamental lower bound showing that over all input sequences, no algorithm can get better than a 1.43-fraction of the minimum regret achieved by FTL and the minimax-optimal policy. We also present a modification of SMART that combines FTL with a ``small-loss" algorithm to achieve instance optimality between the regret of FTL and the small loss regret bound. 

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5. Dynamic Regret of Policy Optimization in Non-Stationary Environments

   Fei, Yingjie; Yang, Zhuoran; Wang, Zhaoran; Xie, Qiaomin (December 2020, Advances in neural information processing systems)

   null (Ed.)

   We consider reinforcement learning (RL) in episodic MDPs with adversarial full-information reward feedback and unknown fixed transition kernels. We propose two model-free policy optimization algorithms, POWER and POWER++, and establish guarantees for their dynamic regret. Compared with the classical notion of static regret, dynamic regret is a stronger notion as it explicitly accounts for the non-stationarity of environments. The dynamic regret attained by the proposed algorithms interpolates between different regimes of non-stationarity, and moreover satisfies a notion of adaptive (near-)optimality, in the sense that it matches the (near-)optimal static regret under slow-changing environments. The dynamic regret bound features two components, one arising from exploration, which deals with the uncertainty of transition kernels, and the other arising from adaptation, which deals with non-stationary environments. Specifically, we show that POWER++ improves over POWER on the second component of the dynamic regret by actively adapting to non-stationarity through prediction. To the best of our knowledge, our work is the first dynamic regret analysis of model-free RL algorithms in non-stationary environments. 

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