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Online paging is a fundamental problem in the field of online algorithms, in which one maintains a cache of $k$ slots as requests for fetching pages arrive online. In the weighted variant of this problem, each page has its own fetching cost; a substantial line of work on this problem culminated in an (optimal) $O(\log k)$-competitive randomized algorithm, due to Bansal, Buchbinder and Naor (FOCS'07). Existing work for weighted paging assumes that page weights are known in advance, which is not always the case in practice. For example, in multi-level caching architectures, the expected cost of fetching a memory block is a function of its probability of being in a mid-level cache rather than the main memory. This complex property cannot be predicted in advance; over time, however, one may glean information about page weights through sampling their fetching cost multiple times. We present the first algorithm for online weighted paging that does not know page weights in advance, but rather learns from weight samples. In terms of techniques, this requires providing (integral) samples to a fractional solver, requiring a delicate interface between this solver and the randomized rounding scheme; we believe that our work can inspire online algorithms to other problems that involve cost sampling.

Recent studies have shown that large language models' (LLMs) mathematical problem-solving capabilities can be enhanced by integrating external tools, such as code interpreters, and employing multi-turn Chain-of-Thought (CoT) reasoning. While current methods focus on synthetic data generation and Supervised Fine-Tuning (SFT), this paper studies the complementary direct preference learning approach to further improve model performance. However, existing direct preference learning algorithms are originally designed for the single-turn chat task, and do not fully address the complexities of multi-turn reasoning and external tool integration required for tool-integrated mathematical reasoning tasks. To fill in this gap, we introduce a multi-turn direct preference learning framework, tailored for this context, that leverages feedback from code interpreters and optimizes trajectory-level preferences. This framework includes multi-turn DPO and multi-turn KTO as specific implementations. The effectiveness of our framework is validated through training of various language models using an augmented prompt set from the GSM8K and MATH datasets. Our results demonstrate substantial improvements: a supervised fine-tuned Gemma-1.1-it-7B model's performance increased from 77.5% to 83.9% on GSM8K and from 46.1% to 51.2% on MATH. Similarly, a Gemma-2-it-9B model improved from 84.1% to 86.3% on GSM8K and from 51.0% to 54.5% on MATH., Comment: A multi-turn direct preference learning framework for tool-integrated reasoning tasks

Policy Optimization (PO) methods are among the most popular Reinforcement Learning (RL) algorithms in practice. Recently, Sherman et al. [2023a] proposed a PO-based algorithm with rate-optimal regret guarantees under the linear Markov Decision Process (MDP) model. However, their algorithm relies on a costly pure exploration warm-up phase that is hard to implement in practice. This paper eliminates this undesired warm-up phase, replacing it with a simple and efficient contraction mechanism. Our PO algorithm achieves rate-optimal regret with improved dependence on the other parameters of the problem (horizon and function approximation dimension) in two fundamental settings: adversarial losses with full-information feedback and stochastic losses with bandit feedback.

Reinforcement Learning from Human Feedback (RLHF) has become the standard approach for aligning Large Language Models (LLMs) with human preferences, allowing LLMs to demonstrate remarkable abilities in various tasks. Existing methods work by emulating the preferences at the single decision (turn) level, limiting their capabilities in settings that require planning or multi-turn interactions to achieve a long-term goal. In this paper, we address this issue by developing novel methods for Reinforcement Learning (RL) from preference feedback between two full multi-turn conversations. In the tabular setting, we present a novel mirror-descent-based policy optimization algorithm for the general multi-turn preference-based RL problem, and prove its convergence to Nash equilibrium. To evaluate performance, we create a new environment, Education Dialogue, where a teacher agent guides a student in learning a random topic, and show that a deep RL variant of our algorithm outperforms RLHF baselines. Finally, we show that in an environment with explicit rewards, our algorithm recovers the same performance as a reward-based RL baseline, despite relying solely on a weaker preference signal.

In many real-world applications, it is hard to provide a reward signal in each step of a Reinforcement Learning (RL) process and more natural to give feedback when an episode ends. To this end, we study the recently proposed model of RL with Aggregate Bandit Feedback (RL-ABF), where the agent only observes the sum of rewards at the end of an episode instead of each reward individually. Prior work studied RL-ABF only in tabular settings, where the number of states is assumed to be small. In this paper, we extend ABF to linear function approximation and develop two efficient algorithms with near-optimal regret guarantees: a value-based optimistic algorithm built on a new randomization technique with a Q-functions ensemble, and a policy optimization algorithm that uses a novel hedging scheme over the ensemble.

PhysioZoo is a collaborative platform designed for the analysis of continuous physiological time series. The platform currently comprises four modules, each consisting of a library, a user interface, and a set of tutorials: (1) PhysioZoo HRV, dedicated to studying heart rate variability (HRV) in humans and other mammals; (2) PhysioZoo SPO2, which focuses on the analysis of digital oximetry biomarkers (OBM) using continuous oximetry (SpO2) measurements from humans; (3) PhysioZoo ECG, dedicated to the analysis of electrocardiogram (ECG) time series; (4) PhysioZoo PPG, designed to study photoplethysmography (PPG) time series. In this proceeding, we introduce the PhysioZoo platform as an open resource for digital physiological biomarkers engineering, facilitating streamlined analysis and data visualization of physiological time series while ensuring the reproducibility of published experiments. We welcome researchers to contribute new libraries for the analysis of various physiological time series, such as electroencephalography, blood pressure, and phonocardiography. You can access the resource at physiozoo.com. We encourage researchers to explore and utilize this platform to advance their studies in the field of continuous physiological time-series analysis., Comment: 4 pages, 2 figure, 50th Computing in Cardiology conference in Atlanta, Georgia, USA on 1st - 4th October 2023

Quantile regression (QR) is a statistical tool for distribution-free estimation of conditional quantiles of a target variable given explanatory features. QR is limited by the assumption that the target distribution is univariate and defined on an Euclidean domain. Although the notion of quantiles was recently extended to multi-variate distributions, QR for multi-variate distributions on manifolds remains underexplored, even though many important applications inherently involve data distributed on, e.g., spheres (climate and geological phenomena), and tori (dihedral angles in proteins). By leveraging optimal transport theory and c-concave functions, we meaningfully define conditional vector quantile functions of high-dimensional variables on manifolds (M-CVQFs). Our approach allows for quantile estimation, regression, and computation of conditional confidence sets and likelihoods. We demonstrate the approach's efficacy and provide insights regarding the meaning of non-Euclidean quantiles through synthetic and real data experiments.

We derive a new analysis of Follow The Regularized Leader (FTRL) for online learning with delayed bandit feedback. By separating the cost of delayed feedback from that of bandit feedback, our analysis allows us to obtain new results in three important settings. On the one hand, we derive the first optimal (up to logarithmic factors) regret bounds for combinatorial semi-bandits with delay and adversarial Markov decision processes with delay (and known transition functions). On the other hand, we use our analysis to derive an efficient algorithm for linear bandits with delay achieving near-optimal regret bounds. Our novel regret decomposition shows that FTRL remains stable across multiple rounds under mild assumptions on the Hessian of the regularizer.

Policy Optimization (PO) is one of the most popular methods in Reinforcement Learning (RL). Thus, theoretical guarantees for PO algorithms have become especially important to the RL community. In this paper, we study PO in adversarial MDPs with a challenge that arises in almost every real-world application -- \textit{delayed bandit feedback}. We give the first near-optimal regret bounds for PO in tabular MDPs, and may even surpass state-of-the-art (which uses less efficient methods). Our novel Delay-Adapted PO (DAPO) is easy to implement and to generalize, allowing us to extend our algorithm to: (i) infinite state space under the assumption of linear $Q$-function, proving the first regret bounds for delayed feedback with function approximation. (ii) deep RL, demonstrating its effectiveness in experiments on MuJoCo domains., Comment: ICML 2023

Quantile regression (QR) is a powerful tool for estimating one or more conditional quantiles of a target variable $\mathrm{Y}$ given explanatory features $\boldsymbol{\mathrm{X}}$. A limitation of QR is that it is only defined for scalar target variables, due to the formulation of its objective function, and since the notion of quantiles has no standard definition for multivariate distributions. Recently, vector quantile regression (VQR) was proposed as an extension of QR for vector-valued target variables, thanks to a meaningful generalization of the notion of quantiles to multivariate distributions via optimal transport. Despite its elegance, VQR is arguably not applicable in practice due to several limitations: (i) it assumes a linear model for the quantiles of the target $\boldsymbol{\mathrm{Y}}$ given the features $\boldsymbol{\mathrm{X}}$; (ii) its exact formulation is intractable even for modestly-sized problems in terms of target dimensions, number of regressed quantile levels, or number of features, and its relaxed dual formulation may violate the monotonicity of the estimated quantiles; (iii) no fast or scalable solvers for VQR currently exist. In this work we fully address these limitations, namely: (i) We extend VQR to the non-linear case, showing substantial improvement over linear VQR; (ii) We propose {vector monotone rearrangement}, a method which ensures the quantile functions estimated by VQR are monotone functions; (iii) We provide fast, GPU-accelerated solvers for linear and nonlinear VQR which maintain a fixed memory footprint, and demonstrate that they scale to millions of samples and thousands of quantile levels; (iv) We release an optimized python package of our solvers as to widespread the use of VQR in real-world applications., Comment: 35 pages, 15 figures, code: https://github.com/vistalab-technion/vqr

Policy optimization is among the most popular and successful reinforcement learning algorithms, and there is increasing interest in understanding its theoretical guarantees. In this work, we initiate the study of policy optimization for the stochastic shortest path (SSP) problem, a goal-oriented reinforcement learning model that strictly generalizes the finite-horizon model and better captures many applications. We consider a wide range of settings, including stochastic and adversarial environments under full information or bandit feedback, and propose a policy optimization algorithm for each setting that makes use of novel correction terms and/or variants of dilated bonuses (Luo et al., 2021). For most settings, our algorithm is shown to achieve a near-optimal regret bound. One key technical contribution of this work is a new approximation scheme to tackle SSP problems that we call \textit{stacked discounted approximation} and use in all our proposed algorithms. Unlike the finite-horizon approximation that is heavily used in recent SSP algorithms, our new approximation enables us to learn a near-stationary policy with only logarithmic changes during an episode and could lead to an exponential improvement in space complexity.

The standard assumption in reinforcement learning (RL) is that agents observe feedback for their actions immediately. However, in practice feedback is often observed in delay. This paper studies online learning in episodic Markov decision process (MDP) with unknown transitions, adversarially changing costs, and unrestricted delayed bandit feedback. More precisely, the feedback for the agent in episode $k$ is revealed only in the end of episode $k + d^k$, where the delay $d^k$ can be changing over episodes and chosen by an oblivious adversary. We present the first algorithms that achieve near-optimal $\sqrt{K + D}$ regret, where $K$ is the number of episodes and $D = \sum_{k=1}^K d^k$ is the total delay, significantly improving upon the best known regret bound of $(K + D)^{2/3}$., Comment: NeurIPS 2022

We study cooperative online learning in stochastic and adversarial Markov decision process (MDP). That is, in each episode, $m$ agents interact with an MDP simultaneously and share information in order to minimize their individual regret. We consider environments with two types of randomness: \emph{fresh} -- where each agent's trajectory is sampled i.i.d, and \emph{non-fresh} -- where the realization is shared by all agents (but each agent's trajectory is also affected by its own actions). More precisely, with non-fresh randomness the realization of every cost and transition is fixed at the start of each episode, and agents that take the same action in the same state at the same time observe the same cost and next state. We thoroughly analyze all relevant settings, highlight the challenges and differences between the models, and prove nearly-matching regret lower and upper bounds. To our knowledge, we are the first to consider cooperative reinforcement learning (RL) with either non-fresh randomness or in adversarial MDPs.

Some of the most powerful reinforcement learning frameworks use planning for action selection. Interestingly, their planning horizon is either fixed or determined arbitrarily by the state visitation history. Here, we expand beyond the naive fixed horizon and propose a theoretically justified strategy for adaptive selection of the planning horizon as a function of the state-dependent value estimate. We propose two variants for lookahead selection and analyze the trade-off between iteration count and computational complexity per iteration. We then devise a corresponding deep Q-network algorithm with an adaptive tree search horizon. We separate the value estimation per depth to compensate for the off-policy discrepancy between depths. Lastly, we demonstrate the efficacy of our adaptive lookahead method in a maze environment and Atari.

We study the Stochastic Shortest Path (SSP) problem in which an agent has to reach a goal state in minimum total expected cost. In the learning formulation of the problem, the agent has no prior knowledge about the costs and dynamics of the model. She repeatedly interacts with the model for $K$ episodes, and has to minimize her regret. In this work we show that the minimax regret for this setting is $\widetilde O(\sqrt{ (B_\star^2 + B_\star) |S| |A| K})$ where $B_\star$ is a bound on the expected cost of the optimal policy from any state, $S$ is the state space, and $A$ is the action space. This matches the $\Omega (\sqrt{ B_\star^2 |S| |A| K})$ lower bound of Rosenberg et al. [2020] for $B_\star \ge 1$, and improves their regret bound by a factor of $\sqrt{|S|}$. For $B_\star < 1$ we prove a matching lower bound of $\Omega (\sqrt{ B_\star |S| |A| K})$. Our algorithm is based on a novel reduction from SSP to finite-horizon MDPs. To that end, we provide an algorithm for the finite-horizon setting whose leading term in the regret depends polynomially on the expected cost of the optimal policy and only logarithmically on the horizon.

Reinforcement learning typically assumes that agents observe feedback for their actions immediately, but in many real-world applications (like recommendation systems) feedback is observed in delay. This paper studies online learning in episodic Markov decision processes (MDPs) with unknown transitions, adversarially changing costs and unrestricted delayed feedback. That is, the costs and trajectory of episode $k$ are revealed to the learner only in the end of episode $k + d^k$, where the delays $d^k$ are neither identical nor bounded, and are chosen by an oblivious adversary. We present novel algorithms based on policy optimization that achieve near-optimal high-probability regret of $\sqrt{K + D}$ under full-information feedback, where $K$ is the number of episodes and $D = \sum_{k} d^k$ is the total delay. Under bandit feedback, we prove similar $\sqrt{K + D}$ regret assuming the costs are stochastic, and $(K + D)^{2/3}$ regret in the general case. We are the first to consider regret minimization in the important setting of MDPs with delayed feedback., Comment: AAAI 2022

We study regret minimization in non-episodic factored Markov decision processes (FMDPs), where all existing algorithms make the strong assumption that the factored structure of the FMDP is known to the learner in advance. In this paper, we provide the first algorithm that learns the structure of the FMDP while minimizing the regret. Our algorithm is based on the optimism in face of uncertainty principle, combined with a simple statistical method for structure learning, and can be implemented efficiently given oracle-access to an FMDP planner. Moreover, we give a variant of our algorithm that remains efficient even when the oracle is limited to non-factored actions, which is the case with almost all existing approximate planners. Finally, we leverage our techniques to prove a novel lower bound for the known structure case, closing the gap to the regret bound of Chen et al. [2021]., Comment: NeurIPS 2021

Stochastic shortest path (SSP) is a well-known problem in planning and control, in which an agent has to reach a goal state in minimum total expected cost. In this paper we present the adversarial SSP model that also accounts for adversarial changes in the costs over time, while the underlying transition function remains unchanged. Formally, an agent interacts with an SSP environment for $K$ episodes, the cost function changes arbitrarily between episodes, and the transitions are unknown to the agent. We develop the first algorithms for adversarial SSPs and prove high probability regret bounds of $\widetilde O (\sqrt{K})$ assuming all costs are strictly positive, and $\widetilde O (K^{3/4})$ in the general case. We are the first to consider this natural setting of adversarial SSP and obtain sub-linear regret for it.

Stochastic shortest path (SSP) is a well-known problem in planning and control, in which an agent has to reach a goal state in minimum total expected cost. In the learning formulation of the problem, the agent is unaware of the environment dynamics (i.e., the transition function) and has to repeatedly play for a given number of episodes while reasoning about the problem's optimal solution. Unlike other well-studied models in reinforcement learning (RL), the length of an episode is not predetermined (or bounded) and is influenced by the agent's actions. Recently, Tarbouriech et al. (2019) studied this problem in the context of regret minimization and provided an algorithm whose regret bound is inversely proportional to the square root of the minimum instantaneous cost. In this work we remove this dependence on the minimum cost---we give an algorithm that guarantees a regret bound of $\widetilde{O}(B_\star |S| \sqrt{|A| K})$, where $B_\star$ is an upper bound on the expected cost of the optimal policy, $S$ is the set of states, $A$ is the set of actions and $K$ is the number of episodes. We additionally show that any learning algorithm must have at least $\Omega(B_\star \sqrt{|S| |A| K})$ regret in the worst case.

Policy optimization methods are one of the most widely used classes of Reinforcement Learning (RL) algorithms. Yet, so far, such methods have been mostly analyzed from an optimization perspective, without addressing the problem of exploration, or by making strong assumptions on the interaction with the environment. In this paper we consider model-based RL in the tabular finite-horizon MDP setting with unknown transitions and bandit feedback. For this setting, we propose an optimistic trust region policy optimization (TRPO) algorithm for which we establish $\tilde O(\sqrt{S^2 A H^4 K})$ regret for stochastic rewards. Furthermore, we prove $\tilde O( \sqrt{ S^2 A H^4 } K^{2/3} ) $ regret for adversarial rewards. Interestingly, this result matches previous bounds derived for the bandit feedback case, yet with known transitions. To the best of our knowledge, the two results are the first sub-linear regret bounds obtained for policy optimization algorithms with unknown transitions and bandit feedback., Comment: Accepted to ICML 2020

We consider online learning in episodic loop-free Markov decision processes (MDPs), where the loss function can change arbitrarily between episodes, and the transition function is not known to the learner. We show $\tilde{O}(L|X|\sqrt{|A|T})$ regret bound, where $T$ is the number of episodes, $X$ is the state space, $A$ is the action space, and $L$ is the length of each episode. Our online algorithm is implemented using entropic regularization methodology, which allows to extend the original adversarial MDP model to handle convex performance criteria (different ways to aggregate the losses of a single episode) , as well as improve previous regret bounds.