State-free Reinforcement Learning
Authors: Mingyu Chen
Aldo Pacchiano
Xuezhou Zhang
Venue: NeurIPS 2024 poster (2024)
Keywords: Reinforcement Learning
Parameter-free
Online Learning
Primary area: reinforcement_learning
TL;DR: We propose and study the problem of state-free RL, i.e. algorithms that do not require the state space as input.

Abstract: In this work, we study the \textit{state-free RL} problem, where the algorithm does not have the states information before interacting with the environment. Specifically, denote the reachable state set by $\mathcal{S}^\Pi := \{ s|\max_{\pi\in \Pi}q^{P, \pi}(s)>0 \}$, we design an algorithm which requires no information on the state space $S$ while having a regret that is completely independent of $\mathcal{S}$ and only depend on $\mathcal{S}^\Pi$. We view this as a concrete first step towards \textit{parameter-free RL}, with the goal of designing RL algorithms that require no hyper-parameter tuning.

State-free Reinforcement Learning
Mingyu Chen
Aldo Pacchiano
Xuezhou Zhang Boston University xuezhouz@bu edu
Boston University
Boston University
mingyuc@bu. edu
Broad Institute of MIT and Harvard pacchian@bur edu
Abstract
In this work, study the state-free RL problem, where the algorithm does not have the states information before interacting with the environment. Specifically, denote the reachable state set by S := {s] max gfx((=" (s) > 0). we design an algorithm which requires no information on the state space while having a regret that is completely independent of S and only depend 8 We view this as a concrete first step towards parameter-free RL, with the goal of designing RL algorithms that require hyper-parameter tuning.
Introduction
Reinforcement learning (RL) studies the problem where an agent interacts with an unknown envi- romment optimize cumulative rewards/losses [Sutton and Barto 2018). While the nature of the environment is in principle hidden from the agent, many existing algorithms [Azar et al. 2017 Jin
al. 2018 Zanette and Brunskill 2019 Zhang al., 2020 202T implicitly require prior knowledge of parameters of the environment, such as the size of the state space, action space, time horizon and so on. Such parameters play a crucial role in these algorithms, as they are used in the construction of variable initializations, exploration bonuses, confidence sets, etc. However, in most real-world problems, these parameters are not known a priori, resulting in the need for the system designer to perform hyper-parameter tuning in a black-box fashion, which is known to be extremely costly in RL compared their supervised learning counterparts [Pacchiano al., 2020). In supervised learning algorithms, selecting among M hyper-parameters only degrades the sample complexity by factor of O(log(M)) In contrast, in RL problems it will incur a O (VVM) multiplier the regret, making hyper-parameter tuning prohibitively expensive. This is one of the major roadblocks to broader applicability of RL to real-world scenarios.
Motivated by the above observation, we propose and advocate for the study of parameter-free reinforcement learning, i.e. the design of RL. algorithms that have no or as few hyper-pue aeeer-prameterss as possible, with the eventual goal of eliminating the need for heavy As concrete first step, in this paper, we focus on the problem of state-free RL in tabular MDPs. In particular, we will show that there exist state-free RL algorithms which do not require the state space S as input parameter to the algorithm, nor do their regret scale with the innate state space size (S]. In particular, we design a black-box reduction framework called State-Free Reinforcement Learning (SFRL.. Given any existing RI algorithm for stochastic or adversarial MDPs, this framework can transform it into a state-free RL algorithm through a black-bol box reduction. We also show that the same framework can be adapted to induce action-free free and horizon- free algorithms,, the three of which now makes tabular MDP algorithm completely parameter-fite free, i.c. it requires no input parameters whatsoever, and their regret bound automatically adapt to the intrinsic complexity of the problem.
tuning in practice.
The rest of the paper organized as follows. Following the discussion of related works and problem formulation, we start by discussing the technical challenges of state-free learning and why existing algorithmic and analysis framework are not able to achieve (Section 4). Built upon these insights, we propose an intuitive black-box reduction framework SF-RL, that transforms any RL algorithm
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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into a frre RL algorithm, albeit incurring a multiplicative cost to the regret (Section 5), Further improvements are then made to eliminate the additional cost through novel confidence interval design, which can be of independent interest (Section 6).
2 Related Works
Parameter-free algorithms:
Acknowledgedly. parameter-fie free learning is not a new concept and has been studied extensively in the optimization and online learning community. Parameter-free algorithms refer to algorithms that do not require the learner to specify certain hyperparameters
in advance. These algorithms are appealing in both theory and practice, considering that tuning algorithmic parameters is challenging task [Bottou] 2012 Schaul et al. 2013]. The types of nyperparameters "set free" varies depending on the specific problem. For example, for online learning and bandit problems, the hyperparameters are considered as the scale bound of the losses De Rooij et al., 2014| Orabona and Pál 2018 Duchi et al. 2011 Chen and Zhang 2023), J. or the range of the decision set OOabona and Pal. 2016 Cutkosky and Orabona [018 Zhang et al. 2022
van der Hoeven et al.
2020]; for neural network optimization, the hyperparameters can be the learning
rate of the optimizer
Defazio and Mishchenko 2023 Carmon and Hinder [202 Iygi et al. 2023
Cutkosky et al. 2024 Khaled and Jin [024]: for model selection, the hyperparameters are the choice
of the hypothesis class
[Foster et al.) 2017 2019].
Surprisingly, the reinforcement learning (RL) community has overlooked the concept of parameter- free learning almost entirely. To the best of our knowledge, the only related work is from Chen and Zhang [224], where the authors proposed an algorithm that adapts to the scale of the losses in the setting of adversarial MDPs. In this work we focus on the problem of developing parameter-free free RL algorithms where the parameter to be focused on are those related to the environment transition, particularly the state space. Almost all RL algorithms assume knowledge of the state-space, For example, existing based reinforcement learning algorithms [Azar et al., 2017 Jin et al.] 2018 Zanette and Brunskill 2019 /Zhng et al. 2020 202 make use of the state space size to construct the UCB bonus. When the state space is unknown, it is unclear whether these algorithms can still build valid UCB bonus that ensures optimism and achieve bounded regrets.
Instance-dependens algorithms: iistance-dependeen learning is a closely related concept to parameter-free learning. Instance-dependent algorithms dynamically adjust to the input data they find, and achieve regret that not only scaling with the number of iterations T, but also adapt certain "measures of hardness" of the environment. Such algorithms perform better than the worst-case regret if the environment is "benign". In reinforcement learning, the most common "measures of hardness" considered in the community are Variance [Zanette and Brunskill 2019 Zhou et al. 2023 Zhang
2023 Zhao et al., [023] and Gap (Simchowitz and Jamieson] 2019 Xu et al. 2021 Dann et al.
al.
2021 Jonsson et al.) 2020 Wagenmaker et al. [202]] Tirinzoni et al. 2021]. both related to algorithms provide regret bounds that scale with the underlying conditional variance of the Q* function. Sap-dependentt algorithms provide regret bounds of order Ö(lo T ggp a)) where the gap notion is defined as the difference of the optimal value function and the Q" q'-fucction at a sub-optimal action V*(s) Q* a) (Dann et al. [222T] Additionally, some studies consider problems similar to ours, that is, how to adapt to the "measure of hardness" of the state space. Given an initial state space, Fruit et al. [2018) proposes an algorithm that adapts to the size of the reachable state space, resulting in improved performance when the initial state space is vacuous.
the reward of the environment. Specifically, surrance-dependee
space,(Friit
The difference between instance-depender algorithm and parameter-free algorithm is subtle. Both family of algorithms have the capability to adapt to the input data, allowing them to sequentially tune the hyperparameters and ultimately converge to the optimal hyperparameters inherent in the data, Consequently, when the number of iterations becomes sufficiently large, both instance-dependent algorithms and parameter-free free algorithms tend to provide the same theoretical guarantees. However, this does not mean that the two types of algorithms are the same. The most significant difference is that ristance-dependee state-space adaptability as an example. Let N represent the true number of states. An instance- dependent algorithm must be provided with an initial value M > N. If this value is invalid, i.e., M < N, the algorithm will fail function properly. Moreover, the regret of deeendetde algorithms is typically related to the initial input, even though this dependency may fade away
algorithms require appropriate hyper-paraneters initialization. Taking
2

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as the number of iterations increases. This is also why we cannot simply set M to infinity in an instance-dependent algorithm and call parameter- that is, the regret an instance-dependen algorithm always includes some burn-in terms that scale with M. As M goes to infinity, the burn-in term eventually dominates. this sense, parameter-free learning strictly harder problem than instance-dependen learning.
Problem Formulation
This paper focuses on the episodic MDP setting with finite horizon, unknown transition, and bandit feedback. A MDP is defined by tuple M = A, H, P), where S= (1....,,5 S} denotes the state space, AA = [l....,,,, AA} denotes the action space, and H denotes the planning horizon. P S x A x S + [0, 1] is unknown transition function where P(s'|s. a) the probability of reaching state after taking action a state s. For every t E [T'. we define the loss function. In stochastic MDPs, the loss function & is drawn from a line-independent distribution. In adversarial MDPs, the loss function & is determined by the adversary, which can depend the player's actions before t. The learning proceeds in episodes. In each episode t, the learner starts from state 81 and decides a stochastic policy T. II: S A [0. 1] with (a)ss being the probability of taking action a in state Afterwards, the learner executes the policy in the MDP for H steps and observes a state-action-loss trajectory (81, ( (11 a..,..., 8H, am,, (sn, an)) before reaching the end state 8 With slight abuse of notation, we assume (T)) = E [E l((sn, PP) π.. The performance is measured by the regret, which is defined by
Markov Decision Process:
4 8 A + [0, 1]
R(T)= = Σ ((+))
min xEll
(T))
Without loss of generality, we consider a layered-structuree MDP: the state space is partitioned into H 2 horizons .o...., SH+ such that S S Sn. S, Sy for every + j. So { and SH+1 = {sH+1}.
Occupancy measure: Given the transition function P and a policy the occupancy measure 9:S x A [0, induced by P and æ is defined as
-, = a (s 0) Σ Ps
Using occupancy measures, the MDP problem can be interpreted in way that makes it similar to Multi-armed Bandit (MAB) because for any policy n, the loss can be expressed as
(<TT) = Σ Σ d Pe (1)
sE[S] dE(A)
Using this formula the regret can be written as ((T) = ET Pi Po
State-free RL: We say state s E S is reachable to policy set II if there exists a policy E II such that g Pix (() >> O. We further define SlI = {a E S] maxzen g P:x (s) > 0} to represent all the reachable states to IT in S. The formal definition state-free algorithm is proposed below.
Definition 3.1. (State-free algorithm): We say RI. algorithm is state-free if given any policy set 11, the regret bound for the algorithm can be adaptive to | and independent to (S), without any knowledge of the state space priori.
At first glance, designing ttate-free algorithms appears straightforward: if the learner had access to the transition P, it can compute max P. (s) for every state unreachable states, thereby reducing the state space S to S" Through this reduction, any existing MDP algorithm can be made state-free. However, such method is infeasible since P is always unknown in practice. Without the knowledge of P, it becomes challenging even impossible to determine whether a state is reachable or not. the following section, we elaborate on the technical challenges of the problem for both stochastic and adversarial loss settings.
and then remove all the

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4 Technical challenges
In this section, we explain the technical challenges for the state-free learning. Specifically, we consider weakened setup. We assume for moment that the algorithm has access to the state space S but not the reachable space S It is clear that this setup is weaker than the state-free definition, as in the state-free setting, the information about S is also unknown.
We start with the stochastic setting, where the loss function & is sampled by a inme-independent distribution for all & E [T]. As the most prominent setting in RL research, numerous works have since been devoted to improving the regret guarantee and the analysis framework [Brafman and
Tennenholtz 2003 Kakade 2003 Jaksch et al. 2010 Azar et al. 2017 Jin et al., 2018
2018]/amm Dann et al.
2017 2017|/Zaette and Brunskill 2019 Bai et al. 2019 Zhang et al. 2020
202
Ménard et al. 2021
Li et al. 2021] Domingues et al., [222]. Surprisingly, although existing works have not mentioned
the state-free concept explicitly, we find that some algorithms can almost achieve state-free learning
without algorithmic modifications. In particular, we have
Proposition 4.1. For stochastic MDPs, UCBVI (Azar et al. 2017] is weakly state-free algorithm, that is, with only the knowledge of S, the regret guarantee of UCBVI adaptive to |S and independent to [S| except in the logarithmic terms.
Proposition 4. offers some positive insights into existing algorithms. The source of the log- dependence on S] is straightfforward: the analysis of RL algorithms needs to ensure that con- centration incqualities hold for all states with probability least ò. At this point, since the events among the states are independent from each other, the algorithms have to take a union bound across the state space make concentration holds simultaneously in all states, which implies that the confidence level needs to be divided by S). This leads to regret guarantee that scale with log((S)).
Remark 4.2. In Appendix A we propose a simple technique to get rid of the log-dependence [S] under the UCBVI frumework. The key is to allocate the confudence for each visited (s, a) pairs sequentially, instead of applying uniform confidence allocation across all states in (S). We further show that such a method removes the need of S information in the algorithm design. Based on this method, it is suffices conclude that (a modified version of) UCB CBBVI is a state-free algorithm.
We now turn our attention to the adversarial setting. In adversarial MDPs, the loss is determined by the adversary and can be depend on previous actions. Adversarial MDPs have been studied extensively in recent years Jin et al. [2019] Dai et al. (2022). [ee et al. [2020), [uo al. [202T]. Given the positive results for stochastic MDPs, one might hope that existing adversarial MDP algorithms can naturally achieve a state-free regret guarantees. Unfortunately, this not the case. In particular, we have the following observation.
Observation 4.3. (Informal) In adversarial MDPs, using the existing algorithms and analysis framework, the regret guarantee cannot escape polynomial-leve dependence on (S].
Here we briefly explain Observation 4.3 In all prior works on adversarial MDPs, the analysis relics on bounding the gap between the approximation transition function P and the true one i.c. EeESTXA ()) P(-|s, a)1l In this case, for any state s' E S, regardless of whether s' is reachable not, the estimation error P((s'ss,) a) P(s'ls (s; a)l may remain non-zero for all (s, a) pairs. Consequently, the larger the ()) the greater the inaccuracy of the transition estimation. At this point, one may wonder if the learner can directly set P(s'|s, a) 0 for all unvisited s' E S, so that [P(s'|s,, a) P(s|s, a) is always zero when s' is unreachable. However, since the learner does not have the knowledge of P. it is impossible to determine whether state is unreachable, even the learner has never visited the state before. In this regard, if s' is actually reachable, the transition estimator will become invalid. Such dilemma constitutes the main challenge of the problem.
The above offers some ligh-level intuitions into the complexity of designing state-free algorithms. In the next section, we introduce our new algorithms for state-free RI. that operate without prior knowledge of S.

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State space S
Pruned Space S
Figure 1: An illustration of the mapping between the state space S and the pruned space S-,, The left side represents the original state space S, where grey nodes denote the states in S- and red nodes denote the others. The right side is the corresponding pruned space S-, where blue nodes denote the auxiliary states {s} [e(H)- Given the structure, for any trajectory in space (purple arrows), we can find a dual trajectory (yellow and green arrows) in the pruned space.
Black-bol bbxx reduction for State-free RL
In this section, outline the main contribution of the paper. To generalize our results further, we denote the e-reachable state space as Su {s E S] max (s) > e). By definition s" S Our algorithm SF-RL is illustrated in Algorithm| The algorithm maintains a pruned state space, denoted by S which includes all the identified e-reachable states and H additional auxiliary states. Throughout
A(A) i...., T, SF-RL first obtains the policy x; III S from a black-box adversarial MDP algorithm, namely ALG, which operates on S Then, by playing an arbitrary action on states not in S compatible with 1I, it extends the pruned policy के to Tee E II, which is defined over the domain S, and then receives the trajectory o, after playing TT. Given the trajectory, if there exists a new state & S that can be confirmed to be e-reachable, the algorithm will update S and restart the ALG subroutine. Otherwise, SF-RL pretends that the trajectory was produced only by interacting with S and sends the pruned trajectory o, back to ALG.
The key novelty of the algorithm lies in the design of the pruned space S and trajectory of. For every [H] the auxiliary state s represents the collection of states S, S,S These behave "absorbing" states, coalescing all the transitions states in S Given the pruned space S' and ot, we can build the pruned trajectory of { o/,, ( a'), [e Specifically, of can be split into two parts based on the horizon that first encounters the state not in S-, i..., For the state-action actte-actio--ooss pairs before the split horizon, we set o, to be the same as in og. Otherwise, we let the states to be the corresponding auxiliary states, the actions to be n which represents an arbitrary action, and the loss to be zero. An illustration of the design is provided in Figure
arg max, (S1A E S+).
In order to analyze the performance our black-box SF-RL algorithm we assume the input ALG comes equipped with regret bound,
Assumption 5.1. (Regret guarantee for black-box algorithm ALG): With probability 6, for all K O, the regret guarantee for ALG followln K epochs of interaction with MDP M (S. P) bounded by
R (L < reg (18), H, (H|S)AA(//8) VK,
H,,
where reg([S], [A,, H) is coefficient that depends polynomially on |S) and log( H SKAA /5). Moreover, we assume that the coefficient reg is non-decreasing as [S], |AI. H. K, 1/8 increase.
This definition works for most algorithms in both stochastic and adversarial environments, e.g.. for stochastic MDPs, by setting ALG as UCBVI Azar et al. [2017), the coefficient can be set as O(H ((HH/SIAA) log( ()l[(S) the coefficient can be set as (HHS)) /[A| log(S[ K/()) reg(.) is the regret coefficient function for input algorithm ALG, the regret bound for Algorithm satisfies
for adversarial MDPs. by setting ALG as UOB UOB-REPS Jin et al. [2019]
5

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Theorem 5.2. With probability 1 - 6, the state-free algorithm SF-RL achieves regret bound
(T) S O reg (1St1.) + H, A), H, log (H(S4))1AA)1))) ( /
As shown in Theorem 5.2 the regret bound consists of two terms. The first term is /SSUl.c times the regret of the black-box algorithm ALG over I" iterations, while the second term can be considered as the regret incurred by the barely reachable states we have disregarded. The trade-off between these two terms is reasonable because it is impossible to discard states that not e-reachable without incurring any cost. By setting to the unknown state size is I Additionally,
0, Theorem [5.2 immediately provides a regret bound adaptive remark that SF does not require any prior knowledge about the state space in the algorithm design, which means that SF-RL is state-free by design. Below we discuss the main steps in establishing the above result.
Proof Highlight: We first define p1 S A S - [0. on the pruned space S-, Specifically, for every h E [ff]. we V(s,a,s)) S ( Ax Sitt1 { 1 v(s,a,8') E Sy \ \ A x (8
be the underlying transition function
p- (5'1s, a) P(s|s,a), P (811,a)= 1
Σ
P(st)s,a),
V(s,a, 8') E $t x A x Sh Similarly, we define ef st A + [0, 1] to be the loss function on the pruned space S-, which satisfies that
p+ = 166' = st+ti:
a)) s & heEmn v(s,a) E S x A otherwise
4. (s,0) 110 0,
Note that the tuple M = (S+ H, is a well-defined MDP. In what follows we use the subscript to represent the estimators of the objects above at the beginning of epoch t, e.g., S P.l. The key lemma of the proof is the following.
Lemma 5.3. suffices to consider of, which is the pruned trajectory corresponding to ot, as an instance by executing policy मं on the pruned space S- with transition function Pt and loss lz.
Lemma 5.3 reveals how the black-box algorithm ALG can work. By Assumption 5.1 in order make the regret independent to (S]. we let ALG perform on the pruned MDP M instead of M. However, since M is actually a "virtual" MDP, we cannot account for ALG's interaction with it. To ensure that ALG can be updated correctly, in Lemma 5.3 we show the pruned trajectory of can be viewed trajectory from executing policy n. on M and f,', Denote the optimal in-hindsight policy as T, = arg min Et= (Ee. TT) and let प. be the corresponding policy on the pruned space, we start by the regret decomposition below.
T
T
T
R(
Σ
la
tth ptt a
(et +
Pire RF,,,
Σ
lq Pt
a .4
t=1
2
Here, term represents ALG's regret and term 2 the difference between S and S-,
corresponds to the sum of the error incurred by
Bounding Let intervals Ii,..., Im be a partition of [T]. such that P.t (m) for E I.
We can rewrite the regret as
M
Σ
(a
Pul (mmiite
EET.
For brevity we consider |S*| < T.. Detailed regret is provided in the appendix.
"Note that the optimal choice of e should not be general, e.g., by setting ALG as UCBVI and e = 1/T, SF-RL achieves regret O (H) // /IA/T HI" Such regret will be much smaller than the regret under when [I.t/ |.

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Since Em=1 8/2m² 8, using Lemma 5.3 and Assumption 5.1 1 can be bounded below with probability at least 1 - d.
reg (19tm)) [A], H1 log (2m (2m² H (/m v VIzml.
Now continue the proof by bounding M and (m)) As in SF-RL, state S will be added in the pruned space if it satisfies Σ₁=1 1 {2)/2 (og(2H²(2/0) 1/2 > ct. Such design ensures that all states added in S- are at least e-reachable, which formalized the following lemma. Lemma 5,4. With probability - 6. for every state & E S, it will be added in S only if the state is c-reachable, ie., max p (s)
By Lemma 5.4 suffices to say that (s(mm) S | + H for all m E [M] and M S 18 as the states not in S cannot be added in S This result also implies that ALG can be restarted at most |8tt.e] times, thus M S < + 1. Given the above, we can finally bound by
M
S reg (15
+ H, [A], log (2H() + /))))
Σ
Im
m==1
reg (1S + H, |A|,, H, log (2H(1S ( V
VISTI.ITT.
Bounding 2 The proof relies on the following lemma.
Lemma 5.5. Given the primed space Sth and the corresponding transition P any policy there is
0 S (q pe. iq p .et) +
SHHH Σ g Pat SEST
{s)
Using Lemma 5.5 we immediately have
2
Ee)
Σ 2mll
Pa (())
H
(s)l((s
t=
#ES
It then suffices to bound the right hand side of the inequality. Denote by X₁ = -Eses Lrl)siss }. By definition, we have X, E [0, H and E[X,(F,-1] Exes EsEsqP" (s)1,{s & st). Using Lemma B.1 in the appendix, with probability
8, we have
T
2 SH
L(s) SS 1 2H log
SES t=1
As in SF-RL, a state has been visited 2ct + 21ogg 22)) times, the state will be added in S which means 11, & 5 will be for the rest j > t. This implies that Σ 1,(s ø S/} is at most 2c7 + 2 2 (2 /8) + 2 for all E S. Moreover, if a state is not reachable by any policy in 11, always have Σ q ( (s) & S 0. Therefore,
conclude that 2 S + 2H2 S log (2H2222/)) + S + 2 log Finally, realizing that we have conditioned on the events stated in Assumption 5.1 Lemma 5.4|and and Lemma C.I which happens with probability at least - 36. By combining
and and rescaling 6, we complete the proof. Remark 5.6. Interestingly, the SF-RL framework can be extended to build horizon-free and action- free algorithm, that is, algorithms that do not require the horizon length (when the horizon length is variable) and action space as input parameters. Specifically, given S+, we denote H by the maximum horizon corresponding to the states S- {st !H 1h-1 which represents the max- imum horizon index among identified e-reachable states. We further denote A by the actions 1
corresponding to S- \. (sx Ih=1- When S is updated, we let ALG restart with typer-parameters (S (SL H H 1-H-). where Sh and Pil represent the states and transitions within the first H horizons of S and p.. By using the sub-trajectory of of within the first H horizons as the trajectory input of ALC, it suffices to note that Lemma 5.3 still holds, thereby the black-box reduction also works. With such extension, SF- RL requires no exper-parameter from the environment and can be regarded as completely parameter-free.
7

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Algorithm Black-box Reduction for State-free RL (SF-RL)
1: Input: action space A, horizon H, black-box algorithm ALG, confidence 6, pessimism level
2: for t= 1 to T do
3: Receive policy n. :8! (AA) from ALG
" (-18); E
4: Derive N. : (AA) such that (T(-|s) x (-1s), otherwise
5: Play policy ne, receive trajectory 0, = (8h, 0/ 0h, (s( un) hE(H]
6:
if 3s E or, s.t., & S -j= 1, [s)/2 - log 22H2 /8) 12 1/2 at then
7:
Update S
U is E S: E {s) log 22 /2 2211 d}
(1s) [1(+ss) Se 1 (hE(m)
8:
Update policy set III x (1s) E {a }.
otherwise
9:
4 Restart ALG with state space - action space A, policy set 11= and confidence
2/5+T²
else
11:
Derive the pruned trajectory of = { ohi C a/, hnEm such that
8/
Shs S/
$1ch E S-
pi aj {{ Oh,
81th E otherwise
14(5%, d/,) = { (sh:: an);
81sh E SL otherwise
otherwise
12:
Send the pruned trajectory of to ALG
end if
14: end for
Improved regret bound for State-free RL
In the previous section, introduce black-box framework SF-RL that transforms any existing RI. algorithm into a state-free RL algorithm. However, the regret guarantee for SF-RL is suboptimal: compared to ALG itself, SF-RL incurs an V(Sll.[ multiplicative term to the regret bound. This is mainly because RF-RL needs to restart the black-box algorithm ALG whenever S updates. Such restarting strategy inevitably leads to the loss of the learned MDP model. For this reason, and in order to achieve optimal regret rates, we need to design state-free algorithms that do not lose model information. In this section, we introduce a novel approach that enables SF-RL retain previous transition information after restarting ALG. We illustrate that such method improves the regret guarantee of SF-RL by [lu.. term for adversarial MDPs when combined with a specific choice of ALG. This bound matches the best known regret bound for adversarial MDPs given known state space.
In existing adversarial MDP algorithms, the model information is captured within the confidence set of transition functions. Take Jin et al. [2019] as example. For epoch 2 1, let Ne(s, a) and Ms(s')s [s, a) be the total number of visits pair (s, a) and (s, a, s') before epoch The confidence set of Jin et al. [2019] is defined as (s,a) - P:(5'ls, a)) S e=(''1s, V(s,a,8)) E Sn x A S4+1, Vh
P. = { P:
where P.(s' a) = Mi(s' ss, a)/ max(1, (s, a)) the empirical transition function for epoch and (s'( a) is the confidence width defined
Pi(s, a) In
471S1JA
14
[TTIS|LA)
(
21 21
N,
3 max as) -
As in Lemma 2 of Jinee et al. [2019], by empirical Bernstein inequality and union bound, one can establish that P P, for all i 0 with probability at least 5. -8.
Intuitively, such a construction of confidence set tends be overly conservative for our state-free setup. On the one hand, it requires taking union bound over all (s, a, s') S x A x S pairs, resulting in an inevitable log-dependence on (S]. On the other hand, even if state unreachable, the confidence width (x(s') ls,s, a) is not zero for all (s, a). Furthermore, since -F-RL RL operates within the pruned space

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S' it is necessary to construct the confidence set on S instead of S. Given by these observations, we propose new construction of the confidence sets. For every s S, we denote t(s) by the epoch index when the algorithm first accesses to state s. If a state has not been visited, we define ((s) = oo. Without of loss generality, we denote by t(s, s') = max (ax the index when both 8 and s' are reached. Denote P (s']s, a) = (s") s, a) - ( (s, a)) max(1, NN a) - Nv. a)} be the partial empirical transition function corresponding to epochs (t' 1, 1. We further define S." as the states visited before I. For every / E [7], we build P,, such that
P- P (s' a)) Is''sss)
als,a,s)) E Sh \ (st) A Sh+1 \ (8 vh a(s,a, 5') E ( x A S/+1 Vh
P.-
(s = 1
where (1661::0))==(((1)0)))))) defined by
[b'10, (1]1,,)) a) and (5'1,, a) are two confidence intervals
(s'1s, a) (0)(1sss))===(((0)))))) [8('ss, = { [0. (s [)]] t(s') 2 t(s) +
[0.
else
pi(o.s0) (8'18, () log (1/5(8, (8,0,,,0,00 80))
20 log (1,((,,,,8))))
(s)ss,, - max (N.(s, Ne a) 1,11 max {Ni(s,a) No 1,,1)
(5''ss = 0| [ | 24 (t/8(s,a)) (N 1,1]
Here, (s's [ss and (s'ss, a) are two Bernstein-type confidence intervals. Let us explain the high-level ideas of the design. First, to avoid wasting confidence on unreachable states, we initialize the confidence level of I, (5'|s, a) only if both and s' are visited. The probability parameter 6(s'|s, a) is Fi(s,s') -eeauurable because it only depends on the data before epoch t(s, s'). Thus, in order to avoid correlation, we can only use the data collected after epoch f(8, 8') + 1 to construct the confidence interval Z (s'ls, a). This leads problem: when t(s') is much greater than t(s). we drop too much data that could be used to estimate P(s (s, a), resulting in ( (s'1s, a) being loose compared the existing confidence interval designed in Jin et al. 2019 To address this issue, we introduce the second confidence interval I (s'|s a). The logic behind the estimator (s's''s, a) is that t(s') >> t(s) when the probability P(s'|s, a) is very small and therefore (N a) 1 can be used to certify an upper bound to P(s'|s, a). The confidence level of (s')s, a) can only be determined after epoch t(s',, whereas the confidence interval I22 (s'ls, a) is constructed based on data between (()) and ((s'). By combining Z, (s'1s, a) and (5 a). such construction makes use of all the data after ((s) to ensure tight confidence interval. Considering that ((s) is the first time reached, we essentially lose only one data point, which acceptable. By carefully designing the confidence level 8(s, o, s') and 6(s, a), can show that
Lemma 6.1. Let i(s) 5 be the index of state s sorted by the arriving time. By setting 8(s, a) = T(("" A and 6(8, a, 8') a(1 +i()")) with probability at least 1 5, there is P,t P for all E (T).
Lemma shows that such construction of the confidence set is valid. Based on the new confidence set, we show that the regret bound of SF-RL can be improved by taking Pr- as an additional input to the black-box algorithm ALG. We summarize the result as follows.
Theorem 6.2. (Informal) By initializing ALG as UOB-REPA Jin et al. (2019) and laking Pt as an additional imput to ALG every epoch, with probability - 6, the state-free algorithm SF- RL achieves regret bound
R(T) SO H [HSST
LA
log
SAAATT
+
which matches the best existing result of non- state-fre algorithms for adversarial MDPs.

---

7 Conclusion
This paper initiates the study of state-free RL, where the algorithm does not require the information of state space as hyper-paramete input. Our framework SF-RL allows us to transform any existing RL algorithm into a state-free RL algorithm through a black-box reduction. Future work includes extending the framework SF-RL from the tabular setting to the setting with function approximation.
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12

---

A Omitted details for Section 2
A.1 Details for Proposition 4.1
In this subsection, illustrate that UCBVI is actually a weakly state-free algorithm. As in Azar et al. [017]. UCBVI algorithm consists two parts: value iteration and policy execution. In every epoch, policy execution executes policy that is greedy the current Q values and adds newly encountered trajectory to the dataset. Then, value iteration uses this dataset to update the Q values of state-action pairs. Specifically, value iteration proceeds from the horizons H,..., For every (8,a) a) E SA A, the update can be expressed as
Q(s, a) = min a) (P())s,)), v()) + b(8, })
where P(-|s, a) is the empirical transitions estimation, V(-) m Q(:, the corresponding V value, and b(s, a) is the exploration bonus, which is defined by
| 5
b(s,a)
b(s,a)=cHI) = cHI. - a)(s.a)' where I. log
Throughout the algorithm, can note that the knowledge of S is only applied in the design of the exploration bonus. Specifically, the exploration bonus is designed to ensure that the following event holds for all epochs with probability at least - 8 by Hoeffding inequality and union bound.
< = ((P((sss a())s,a), V*OA b(s, a), V(s,a) E ESx
(I)
When the reachable space S is known, by substituting S with S in advance, as in Theorem 2 of Azar et al. (2017), the regret of UCBVI is well bounded by O(H / [[A[ [og(Toog((a)) || When S is unknown, we have to utilize S to design the bonus, resulting in the exploration bonus being amplified by a factor of / //og((()) [[og]) In this case, by optimism lemma, the regret guarantee increases by at most the same factor. Such result suggests that UCBVI is weakly state-free.
A.2 Details for Remark 4.2
Based on Proposition 4. here we show how escape the log-dependence on |S] under the UCBVI framework. The idea is simple: when constructing the exploration bonus, instead of allocating confidence | ||||T T to every state-action-epoch pair uniformly, we initialize the confidence level for states based on their arriving time. Let i(s) be the index of state sorted by the arriving time. The exploration bonus set by
((sa)= -HH max max (s 1,13 where L = log 2[((S)
5
S, allocate confidence 8/2 8/2 to its corresponding state-action satte-action-epoc pair. In this regard, the confidence allocated to state s is bounded by {s) and the total confidence is bounded by Σ 8/21² 5. Specifically, to avoid the correlation between the confidence level and the subsequent confidence sequence, we initialize the confidence sequence of (P(-Is, a) P(- (s, a). V*()) at epoch ((s) 1, where ((s) is the epoch index that state 8 is visited for the first time. This is because the confidence level / (/) [|A[ can be determined when the algorithm first reaches state s. In this way, we loose one data point for constructing the confidence sequence. This is why the count of visits is max{NN a) - 1, 1} rather than (s, a). Additionally, by the definition of b(s, a). it suffices note that b(s, a) 2 H. for the epochs before t(s). which implies that the bonus ensures optimism for all epochs before t(s) with probability Combining with the above, suffices to note that with probability the bonus b(s. a) ensures optimism for all epochs. By union bound, we can conclude that our proposed exploration bonuses also make the event in ① hold with probability 1 - 8. Given that i(s) < [ for all visited states, the new exploration bonus is of the same order as the bonus where s"' is known. In this way, we make the regret bound be completely independent |S], and turn UCBVI to fully state-fres free algorithm.
Broadly speaking, for every visited state
8/ (
13

---

A.3 Details for Proposition 4.3
In this subsection, we demonstrate that, within the current analysis framework for adversarial MDPs, eliminating the polynomial dependence on [S] is impossible. To the best of our knowledge, in order to handle the unknown transitions, all existing works share the same idea that maintain a confidence set P. of transition functions for E [7]. The confidence set can be denoted by
P. 2 (P: PP( a() S fe(s"|s,a)) a), s) ESXAXS).
where c,(6'18, a) is the confidence width and ((s'(s,a) is the empirical estimation of the true transition P(s'|s, a). Specifically, cs(s'|s, a) set based on empirical Bernstein inequality, i...
a()')s,a)= = ( log(SAT/5) Ne(s,a)
P(s)ss
log(SAT/6) N1(s,a)
Y(s,a, e(s,,,,') Sx
where (r a) denotes the counter of state-action pair as. This setting ensures that / P, for all & E [T] with probability at least 6. Given the confidence set, the algorithm selects P. P, as the approximation of P, and chooses policy based on the approximation transition. For clear understanding, we decompose the regret into two terms, i.e.,
T Σ 6 PeTT q Pat
T
R(T) = Σ la Pixe q Pix" 2e)
Σ
(g
Pm
P.. l
REGRET
ERROR
Here, the first term REGRET represents the regret of the algorithm with the approximation transition. In some senses, bounding REGRET can be reduced to a bandit problem. In every round t, the algorithm chooses an occupancy measure d: E (P,,, IT) and corresponding (Pr, Te) E (Pt, IT), then obtain a partial observation of the loss &. For the second term ERROR, corresponds to the error using P. to approximate P. Considering the adversarial environment, there exists a "worst enough" loss sequence f,...., by such that
ERROR (SA)ESUXA
Σ
la (s -g (
Thus, bounding ERROR is essentially equates bounding the right hand side of the above. At this point, one needs to demonstrate that the confidence set shrinks in the correct rate over time, so that the sum of the gap between g n. and g... can be well bounded.
Having provided sufficient background, we now explain why existing methods fail to achieve state- free regret bounds. First, since the confidence set requires to work for all (s. a, s') pairs, we have to take a union bound on the "good event" for all (s, a, s') E A x S. This essentially cause (<(s')s (s, a) to be log-dependent on |S), More important, to bound [ (s, a) g gfo (s,a)), a)), existing methods mainly follow the proof of Lemma in Jin et al. [2019] that is, for every (s, a) E SlI A
and n. E II, there exists P. E Pe such that
h()-1
(s, g
P.Te (s,
Σ
Σ 8h.0
c((sutlls, Pr an (s,
By the definition of confidence width, have (e ox) > O(I/N((sh: an)) for all (s), ah 9h+1) pairs. Furthermore, if state 8h-11 is unvisited, the algorithm has no information for the transitions after the state. Assuming that S₁, * S for all [H]. there will always exist a "worst enough" P. E Pe such that the probability of reaching s via 8441 with policy ni is 1, iee. (s(8844)) = 1. In this case, we have
Σ q Prite (s, g Pa (s,a)
14

---

h(8)-1
Σ
Pite (sh, an) aEA
Σ
gfe (B,
Ne(sn,
h=
h())-1
Σ
(sn an) {8h+1 is unreachable)
Ne(sn, an)
11
h(s)-1
Σ
Pite (sh, an) -15++11)
E
h(*) A==
(|St+1l
Sll Sttll
Based the analysis, it suffices to that ERROR is least of order O(Eme[mn "h
(1S D)), which polynomially depends on |S when |S < [S].
Such result suggests that current analysis cannot derive state-free regret bound for adversarial MDP.
B Omitted proof of Section
B.1 Proof of Lemma 5.3
Fix S-, we consider an pruned trajectory of derived by SF-RL such that
of = {s,, a1 (1),
ts ah, fe(sh, an). sh+1, at 0, ,,....,.77 s7 a+,0].
By definition, it suffices to note that $1sh E S- and and sh+1 S+, In the following, we complete the proof by demonstrating that the likelihood of obtaining of using algorithm SF-RL is the same to the likelihood of obtaining of by executing _.! on P and f,
We first analyze the likelihood of obtaining of by SF-RL. Let's first review the algorithm. Initially, SF-RL executes policy ni and obtains the trajectory ot, which is the underlying transition / and loss 4, Then, using S-, SF-RL RL degrades o, to the pruned trajectory of In order to generate of defined above, og needs to satisfy 1). at horizon to h, the state-action pairs are (s1, a1) to (sh, an) respectively. 2). at horizon h + the visited state cannot belong to S-. Therefore, the likelihood can be denoted by
h-1
h-1
(1-
1
P(o, SF-RL) = Te(an 8t)
II k=0
n (ax 8k) II
P(sktal ax)
Σ
P(s
A=
sTES Matten)
Likelibood visiting {sz.ax 1i
Likelihood of visiting states not in S at borizon h++
Now we study the likelihood of obtaining of by executing प. on p- and f By definition, for every s E S, there E, (s, a) = Ls a) if E St, and E (s, a) = otherwise. Using the observation, we can rewrite the loss (sk, a&) by &t (sk, 0x) for E 1..... ... h, and rewrite the rest zero loss by &t (st at). Therefore, suffices focus on the likelihood of obtaining state-action pairs of or. In this regard, we have
B(o IP+ xx))))
h-11
h-1
H
A-1 x (a ((a+)s) II p+ (ot lotriottaa
11 K (axlak) II p+ (ex41lse,
ot (onlan) (ot
II
I kmo
x (ax)es)
II km0
p-t (e2+1)
n (onlsn))p- ( as)
πI Ke(ae 2x) II P(enssla).
sc(onlsn)P (ohanlen.an)
km
II (s)) II
(e 1-
Σ
p(o
15

---

where the first equality is because IT az (s*1))) 1 (s 1 ll a²) by definition. The second and third equalities are by the definition of p1 and nt. that is, ar (ax|s2) (1k, ag) and P- (sx+1 Sk, az) P(sk+1|sM ox) for k O,..., h, The last equality is by the definition of P-l ($ (s,, ax). Using the above, it suffices to show that P(o, [SF-RL) P(o, 1pt,et x,) thereby we complete the proof.
B.2 Proof of Lemma 5.4
The proof is mainly based on the following lemma.
Lemma B.1. Let for 0 be filtration and (X₂ 2 0) be a sequence of non-negative random variables with B[Xx][F-11] = Pt with Pt being Fi-1 mees-alsuaable. Given confidence level being Fe-measurable, there is
1 8
P
3n > 1
Σ
X,22
Σ
Pj + log
|=++1
j=1+1
T1 Σ
3 t,
Pj 2
Σ 5=(+1
Xj + log
S 5.
8
j=f+1
Let i(s) be the index of state sorted by the arriving time. Recall t(s) is the epoch when the algorithm first accesses to state s. Apparently, i(s) is F Using Lemma B.1 and union bound, we immediately have
1,s}
log (2g(2)())3/8)
P Vs 8, vn t(s),
Σ
P (s)
2 21-8 8.
)=((a)+1
f=((a)+1
Assuming the above holds for true. By SF-RL, state s will be added S only if Σ [s]//22 log(2i) (2((())))/))))) //////////22 2 - 1/2 2 el. Since there are at most Hn states that can be visited before epoch L we have i(s) S Ht((s) < Hn. Moreover, it is obvious that E 15(8] E1 15(8) + by
the definition of t(s). Thus, we have
n
1,(5} log (og(2is)²//))
n max 9 P.T ((T(s) 2 #ell
Σ
(s)
Σ
2
f=1(+)+
j=((s)+1
1s(s}
1og(2H?222/0
2 en,
2
2
2
j=1
which implies that state is e-reachable. This completes the proof.
B.3 Proof of Lemma
To prove Lemma
5.5
the key observation is that
H
H 4 (sn. (n E S
(
pt
E
(sh, ap - =
Σ
where the first equality is by occupancy measure and the second equality is by Lemma 5.3 observation, we have
Using the
Pr Li) (g p
p- ,
=
Lahn, PN)
E
(shn PP
E
(shh on PPTT
E
E(sh, an)1 E
1=1
16

---

thus we have (0"t,,0 10tt ( 2 0 and
"0 li) (q ph (et) S HE [13 sn 8 HPPa
S Σ
o (s) $ s s
aE8u
which completes the proof.
C Proof of Section 4
C.1 Proof of Lemma 6.1
We first fix horizon h. For every (s,a, E Sh A Sk+1, considering that 8(8, a, s') is meeauurlll by empirical Bernstein inequality and union bound, we immediately have a)) I2 (s'1s, a) for all 2 5(s,5") + 1 with probability 1 - 8(s, a, s'). By the definition of Pl, there is P (s) = P(s'ls, a) once s, E S Furthermore, the confidence sequence of ( s, a) is initialized once both states & and s' have been visited, which is potentially as soon as epoch ((s,5') s') + Given EE a(s,a,s) 8/2, it suffices say the following event
E = ( E (s'ss [T), (s. S { x A x Sitt1 \ ( h}
holds true with probability at least 1 - 8/2.
It now suffices to focus on the second confidence interval (s's's,, a). For every (s,a) E Sv. A, define
S² s E S Σ 1, =
r=((s)
be the states such that the snte-action-state pair (s, a, s') is unvisited before epoch t. Notice that S..0 is -e-r-measurable and B[1,[S, |s, ss, confidence 5(s, a), by empirical Bernstein inequality and union bound, it suffices to claim that
F P(S. |s, a). Given Fets)= measurable
P(S ((S""ls,a)La (,,) Σ +=)++
1,(87 1s,
(=t(+)-1
Σ L157 1,, (115%%,,, (s, 7=1(a)+1
2f 8(s,a)
+
14 log
3(s,a)
2t
Vt 2 f(s) + 1
3
with probability at least - 8(s,
By the definition of St it suffices to note that E - 4, 15, S | Using the above, we can note that
2t ((s,a))
14 log
2r 6(8,0)
2 log
3
t
2|S + 24 log
8(s, a)
with probability at least 8(s,
Consider state E S such that t(s') > t(s) 1. By definition, it suffices to note that s' E S3.a for
all t(s) +1 S t(s'), which implies that P(s'(s,)) S P(Staals, a) for all t(s) + 1 < T 5 ((s'). In this case, we finally have
((d)) 21=1(0)41 P(S (s,
2| ((
+ 24 log
*(x")
8(s,0)
P(s'(s,a)
AA vs':t(s') 2 t(ss +1
E 1,(8,0]
max (ax((N 1,
17

---

with probability at least 1 - ö(s, a). Given E EBES a) S 6/2 and P (s'ss,a) = P(s'|s,)) it suffices to say that the following event
2 (s'lss E ( a. a): [T], (s, S \ { x Ax Sitt1 \ ( 1. h}
holds true with probability least 8/2. By a union bound of events &1 and Ez. we complete the proof.
C.2 Proof of Theorem 6.2
The proof of Theorem 6.2 is technical but mostly follows the same ideas of that for Lemma Di al. [2019]. First, as in Jin et al. [2019). we illustrate that the confidence set Pf is tight enough, i.e., the difference between the true transition function and any transition function from the confidence set can be well bounded.
Lemma C.I. Under the event of Lemma 6.1 for all epoch i E [T], all P7t E Pt. all = O,... 1-11 and (8, 8') E Sh \ (8% x A x Sh+1 (.h+1), we have
(s1ss)) (111s))))) so (
P(e')s,a) log max a))
sil LAIT
SS log
max (N,
Lemma C.2. (Refined regret guarantee for Theorem in Jin et al. (2019)) With probability 8, for all K >> O, with confidence sets Pi..... .K. the regret guarantee for epochs interaction with MDP M (S, H)P) and loss sequence ,,,..., lx is bounded by
following K
1 S ( HIISIAAIJ (HI))AIIKKoo(( log((S) Al|K/6) +
K DEMD
(x - - a a)) [ea(s. AEI
Σ
where ( ]EE[K] is a collection of policies and (P²) (P])sss is a collection of transition functions selected by pessimism, i.e,, for all E S and [K].
PK
(t. P. (2, [(x(s.))).
PEP REA
Recall the proof sketch in Section ALG's regret and the error incurred by the difference between S and S respectively. By the proof of there is S (T) It suffices to focus on
we decompose the regret I(T)) into 1 and 2 which represent
Theorem 5.2
1 (..B] By Lemma
we have
HISL ||A||
I.m log( |1| I.n 1/8)
Σ LESt
pt (8,0) 9 at at (s. le! (o.a))
m=1
q Pfivi (1, - get pt me (aaa))
so m|s // ||]TToo(((5..0)1111177
Σ
where P. E P.- for all t E [T] and s E Sr It suffices to focus on the second term. For every E St 1 he a E A, let h(s)) be the index of horizon to which s belongs. According to the proof of Lemma in et al. 2019 (specifically their Eq. (15)), we have
(f (8,0) -t " (s.a)
Σ
Σ
P. (em+ alem Bm)) p.t (Am+1lem ( gfttert Pt (e, em+1)
am 65 Pm Pm Ais
115 S8 4m
where Stmm represents the states 8t horizon m. Intuitively, in order to continue the proof, should apply Lemma C.I
C.I and bound (sm=+1mmmm am. - P.t. (sm+1l 19,, am)l by E (am+1 Smn am). However, when S.z sy or Sm+1 s the confidence width ez (Sm+al |mm am) is not well defined. To address this, we show that the terms related states 8 or a can disregarded. We prove case by case, i.c.,
18

---

1. (8m S m) By the definition of P. and P, we always have
Pr (5m+1|$m am) = P. (sm+1 8m: am) 1fsm+1 sm+1k
which implies that |P/ (8m+1 18m; (m)) (f( (8m+11 18m, am)) = 0.
2. (sm o in Sm+1 = S s By the definition of P, after visiting state sn++i the probabil- ity of visiting state s + ((o)) is zero. This means that gf (s, aalsm1)mm=)))) =0,
3. (8m * sm 81 # m)+1): By Lemma C.1
bound [PP (8m++1 (8m++1/mm am)
(9 am) by (sm+1 (9m: mm).
Using the above, it suffices show that
q pl (s.a) - - od (s.a)
M+)-1
Σ
Σ
(em44 amm:am]ff (smm nt (om, am ai ( alom+1)
ES Math ham A. 1m ES MM m
k(a)
Σ
(em 12 118m amly Pite (em, am " (e, 8m
am 3m68/ (m M Jaa C. 165 tmm Mo Me
where the second inequality is by Lemma 5.5 i...
q g क (sm, am) = (q Ptt 1 {8mm aml) S Po {8 am)) q (m aen
By the exact analysis, also have
q Pf (1)) -9 ind (e, 11
A(d)-
$ E
Σ hamtl ee Mea hi / h41 68
(1o P ad (shi (om " (ora wber)
M
4 qi (a|s)
Σ
Σ hathag ed es tn
65 1sh ahda otti (sh, sm+s))
, E
M
< <x((ass)
Σ
Σ
46 2o (th a' lmm41)
Ammi 11 E Mo hag 1
To simplify notation, in the following, & (Sh+1 18h, an). We further denote W by all the state-action-state pairs Si, \ {s A S (8/41) 3 and wille by all the reachable inte-action-state pairs S/, x A x h+1 Using the above two inequalities, we have
the shorthands wn (sn, any an 8h41 1) and ef (wn))
Pris (s,a) - -.t (s)
M()-= Σ
Σ
PM (sm, emgg Ft mit (e, alsm+1)
m=0
Win
h(t)-1
h(a)
Σ
Σ
(w (ttm))attoe Pot oo (om, and me(als)
Σ
Σ
e (e a' (em41)
m=0
Wm
Amm+ 1 wh
M())11
Σ
Σ
(w PiTs Piri (Am am Pote (e, (em+1)
m=i
Wan W
h(a)-1
h(a)-1
Σ E.
(w )q Po (xmm am))
me(als) Σ
Σ c(e)) " af (sma))
The last inequality is based on Lemma 5.4 i... W.,. with probability
Wille for all t [T] and h e [H] Following the proof in Jii al. [2019], we can take the sum over states E St \ Hsthhenn }RELH and actions E A.
Σ .Cs APD a
PE (0 - .rttt (8.0)
19

---

h
Σ
Σ
Σ
(wml
am)s Pitt alsm+1)
1m] rE8TlIC m=0
Ww m the
h
h(a)11
Σ
Σ
Σ
Σ
(wmma Pirt (smian amd Te(als)
Σ
Σ
1=1
1-1
$
(wim Jg
Pi (Ams Rmm)
1m H
m m0
Wm 2 Wm
T
Σ
Σ
Σ
E
Σ Σ
(0(m)) Pirte (Am, amd (il
me(a a)
sestlie
Mo I. Amm+1 "A eew.. TI
$
Σ
Σ
Σ
(w Piti (Am. am))
kilk mi
+
Σ
1
Σ Σ
Σ
4f (wmyy ( (am, am )ei (w//39 Pi Po (mm+1)
05mchekeH
Mm W.ll. h-m ew
1-1
$
Σ
Σ
Σ
o (wim) 10" PiTe Pite (sms amm)
kC ma0 E
+18
E
Σ uLew
(1mml Pire (smm am Je] (10f (e ofd (mm-
Here, the technical part of the proof has completed. It can be noted that the form of the right hand side of the above is exactly the same as the corresponding formula in the proof Jf et al. [2019] Lee et al. [2020]. except that we have reduced the state space from S to Sll.e. Since S is meamaaauralle, the concentration inequalities used in previous works still hold in our proof. Furthermore, compared to the confidence width defined in Jin et [2019), our cj (sm+1 Sm. Qm) has only increased by burn-in term of order O( max No (5m; am )m))) This ensures that the term in the final regret bound of order Ö(VT) will not depend on | polynomially. For the completeness of the proof, with the help of Lemma C.6 in Lee et al. 20200 we can derive a regret bound with the dependence on all parameters explicit, i.e.,
E
Σ
pP (s,a) - gPll and (s.
|S")) + "s"" log] log?
+ Iss" log
Summing up to the first term and 2 we finally upper bound R(T) by
|S"I
(AIT)
+ iss" log (IS" + +cHIs"(T))
is" [i "IA log
5
D Proof of Auxiliary Lemmas and Corollaries
D.1 Proof of Lemma B.1
Without loss of generality, we assume the confidence level 8 is Fo -e-muauuablle. We first note that
E [[exp(Xx - 2P.) E 1 + (X, 2P:) + (X, -1 = E - P X? $ 1,
E. 1 E (P. - 2Xx) (P. (x E
P. x2 <
where the first inequality is due to exp(x) < 1 + x² for x e [-,
Denote by Yn = exp( exp(En (X, 2P))) and Z,, exp( (P. 2X,)). it suffices to note that both Yi...., Yr and Z,,... Zr are neg-nigee supermartingales. By Ville's inequality, we immediately have
1 8
1 8 5
PP 3m 0,y 2
<0 IP
3m > 0, 2m
20

---

After taking log on both Y, and Zn we get
n
P 3n 0, Σ (X, 2P;) 2 log (
S P
3n 0, Σ
(P. 2x,) 2 log 8 sd
which completes the proof.
D.2 Proof of Lemma C.I
Given E [T] and h [H] and (s, a') E St (s
Ax Ax Sh+ M 1). Notice that t 2 t(s) and
I 2 U(s') by definition. Under the event of Lemma 6.1 suffices to prove [Z] (5'(6)))) a) a)T (s' |s, a) s (s'11,, a). We discuss case by case.
1. ((ss) > t(s) and (s(s,a) > (8, In this case, there is (s( = (1, a) Ne(s, 4)/2. Thus we have
pltoaa (s'ss,a)
( pels.a" (s'ls, log ((s)01s1))
log
1 3(s.0,0))
(s((ss))
max (Nc(8, Nete 1,,1}
max { (N((s, a)
Ne(r, (8, 1,1}
Pltee"' (s'ls,,) log Jlaine
20 M[raa)))
max (N. (),
1,13
max (Ne(.,
N1 1,111 1,1]
pitd' (s' 8, log ((s,a.6')
40 3(5,0,5
< {P(s||s.))+8) max (Ne(s, 2,
max (N)(s, 2, 1
Viewing this as a quadratic inequality v plllo (s'1s,
(s' a). we have
\VPollsaa++ + max (Ni(8, 2 13 V ( (s11s,))
log (((e(a."))
This leads to
((s')s,a) log M log M(5,(5a)
I (8
max (Ni(s,
max { (Ne(s,
2. (1(8)) > ((8) and a) < 2N (s, a)): By the definition of (7'8'(s,, a), we have
(s('sss 2|
26 log
|S | log T(s.a)
M(saa
SO
max - 1, 1,
mmxx(N(ss,a),11
3. ((ss) < (())): When l(s') < ((s), we have (s (s, a) 1, thereby (s, a) is also on the same order of Ni(s, a). Using similar proof as the first case, suffices to obtain
P(s' a) log 8(s.a.a)) max (Ni(s, a),
log b(s.a,o7)
max (N)
Using the above complete the proof.
21

---

D.3
Details for Corollary C.2
As in Jin et al. 2019 we decompose the regret into four terms
R UOB-REPS (K)
Pank q PRITE
(x)
(xc
h=1
K PuiTe g p.:ta
K
Px)
( p: Ex 1x)
+
k=1
Here, P& Pk an approximation transition selected from the confidence set Pk. &g is the loss
estimator, which is defined by
(x)
()(s,a)= ax(s,a) + 1x((s,a) is visited),
maxper, seP, q (s, a) is an upper bound of where y is an adaptive exploration rate and ( a) = max the probability of visiting state-action pair (s, a) with confidence set Pk.
Now show how to refine the regret bound proposed in Theorem 3 [Jin et al. 2019]. For the first term, we immediately have
K
K Σ
of faite (x) S
l0 Pite -oee-gPeee (l
1=1
k=1 dES
Pite gft
1= sES
by the definition of PE. For the second term, there is
K
K la
K PuiTa Blbal - Ex).
Σ
Pu lu Ex)
PooT bu - E(Lu])
Since Rote, Ex) < H for sure, applying Azuma's inequality we have Ek Re [1[Pal - Eea s H / 22 log (1/5). It suffices to focus on Σk= (q PriTT ,ex -E(a)).
Pe (8, (1
9 (s,
PT lu Elbaa)
Σ
ux(s,a)+ +7
1=1
1= sES,
K
TT (s,
Uh (8) lu (x( (ux
( a) - - (s, a))
1=1 sEA
- - (s,(ss))(((((,,,)+++)1)))
1=1 sE
Notice that
ux(8,a) max & P.Te (. = (aass)) max q P. (s).
PEPa
PEP.
Therefore, denote by P= arg max pep, PEP, q P.TM (s), we have
(x) (s, < Σ Ig (s, - (s
GEAA
acA
<
1g th (s.u) (s.
a
by the definition of Pg.
22

---

For the third and fourth terms, the proof completely follow the proof of Lemma 12 and 14 in Jin et al. 2019 i.e.,
HIn((S]AAI)
7HIn(H/6)
a PRink
(()) 00
//S||AAKK
and
K
Hin([S]
Pm 4. (x) 0O
7
By setting 77 =
F1 SIA (8) [
we finally upper bound uo
nOOB-REPS/P (K) by
( HISTAAI log
|
l (8.0) - -f (s, (o)l)))(((ssaa))
k= aEE
Notice that the proof above requires tuning the learning and exploration rate in terms of remove the restriction, a standard method is to let learning rate and exploration rate be adaptive, i.e.
7): H Log (|//)) Using the adaptive rates and taking union bound over all k, we can A|k
obtain the results Corollary C.2
23

---

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Reviews:
[{"review_id": "pyx9i7wgpS", "reviewer": "Reviewer_sD2S", "summary": "The paper studies the problem of online reinforcement learning in the tabular setting when no prior knowledge about the size of the state space is available. Unlike existing algorithms, that usually require the size S as input parameter, the proposed algorithm is fully adaptive and its final performance scales with the set of reachable states.", "questions": "See weaknesses.", "limitations": "See above.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The algorithmic solution is quite elegant since it can be applied to any \"basic\" RL algorithm with regret guarantees.\n* The final result achieves the desired removal of the dependency on S, which is replaced by the size of the reachable states.\n* The result holds for both stochastic and adversarial settings and it can be extended to removing the dependency on the horizon H as well.", "weaknesses": "* I would encourage the authors to provide a clean comparison of the final bounds in the stochastic setting with the best available bounds. In particular, I'm wondering whether the restart leads to extra log terms.\n* Related the previous point, I suggest the authors to make explicit the bounds for simple doubling trick strategies, so as to have a point of comparison.\n* What is exactly the role of epsilon? It looks like it can be directly set to 0 and everything works the same.\n\nAdditional references of interest\n\n* “Layered State Discovery for Incremental Autonomous Exploration” https://arxiv.org/pdf/2302.03789 This paper extends the seminal work of Lim and Auer on “Autonomous exploration” where the state space is possibly infinite (countable). In this paper, the authors managed to resolve an issue in the original paper and removed any dependency on the total number of states, where making the bound completely adaptive to the set of reachable states. Given the similarity between finite horizon and bounded distance exploration, I wonder whether there is any connection to draw between these two works. My impression is that there is quite a strong resemblance between the concept of pruned states and L-reachable states. The main technical difference is that in autonomous exploration the agent needs to explicitly restart to avoid getting “lost” in long episodes, whereas in finite horizon, the reset naturally happens each H steps.\n* “Near Optimal Exploration-Exploitation in Non-Communicating Markov Decision Processes” https://arxiv.org/abs/1807.02373 This paper considers the case where the state space is somewhat “misspecified” (i.e., the set of available states is actually larger than the set of reachable states). In this case, the authors still require knowing the possibly large set of available states, so I’m referring to this paper more for completeness than for strict comparison.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the instructive feedback! Below we address some of the questions raised by the reviewer.\n\n**Q1**: I would encourage the authors to provide a clean comparison of the final bounds in the stochastic setting with the best available bounds. In particular, I'm wondering whether the restart leads to extra log terms.\n\n**R1**: \nFor the stochastic setting, our results suggest that existing algorithm UCBVI actually achieves weakly state-free learning, where its regret is only dependent on $|\\mathcal{S}|$ on the log terms (Proposition 4.1). We also propose a straightforward technique, which effectively eliminates the logarithmic dependence on $|\\mathcal{S}|$ under UCBVI framework (Proposition 4.2). These two propositions imply that there is no need to use our designed algorithm SF-RL for the stochastic setting. \n\nFurthermore, in inhomogeneous finite-horizon MDP, it seems that UCBVI has achieved asymptotically optimal regret. In this regard, Proposition 4.2 suggests that UCBVI (with small modifications) can also achieve state-free asymptotically optimal regret, which matches the best available bound. Besides, there is no restart in UCBVI, thus there is no extra log terms. \n\n**Q2**: Related the previous point, I suggest the authors to make explicit the bounds for simple doubling trick strategies, so as to have a point of comparison.\n\n**R2**:\nIf the reviewer are referring to apply doubling trick on the state space, we will encounter some issues in the analysis. In Theorem 6.2, when a state $s$ is placed into $\\mathcal{S}$, we initialize its corresponding high probability event ($\\delta(s)$). In this case, we need to know exactly which state s refers to (i.e., the index of $s$). However, under doubling trick strategies, the states generated each time we double are void, which makes it impossible to establish the corresponding high probability events.\n\n**Q3**:\nWhat is exactly the role of epsilon? It looks like it can be directly set to $0$ and everything works the same.\n\n**R3**:\nGood question! $\\epsilon$ controls the tradeoff between the regret of ALG and the error incurred by the barely reachable states we discard.\n    Specifically, when $|\\mathcal{S}^\\Pi|< \\infty$, our results suggest that the optimal choice of $\\epsilon$ should be $o(1/\\sqrt{T})$ instead of $0$.\n    For example, by setting $\\epsilon=1/T$, Theorem 6.2 achieves regret bound $\\tilde{\\mathcal{O}}(H|\\mathcal{S}^{\\Pi, 1/T}|\\sqrt{|\\mathcal{A}|T}+H|\\mathcal{S}^{\\Pi}|)$.\n    When $|\\mathcal{S}^{\\Pi, 1/T}|\\ll |\\mathcal{S}^{\\Pi}|$, this regret will be much smaller than the regret under $\\epsilon = 0$.\n\n\n**Q4**:\nComparison of ``Layered State Discovery for Incremental Autonomous Exploration''.\n\n**R4**:\n    The objective of [1] is fundamentally different to the objective of our work (even if we transform the infinite horizon setting to our finite horizon setting).\n    In [1], the objective is to find **all** the incrementally L-controllable states. \n    If we transform this to our finite-horizon setting, the objective should be to find all the incrementally $\\epsilon$-reachable states.\n    However, in our work, the objective is to minimize the regret.\n    In this regard, our algorithm does not require to find all $\\epsilon$-reachable states.\n    Specifically, as described in Algorithm 1, if an $\\epsilon$-reachable state has not been reached $\\epsilon t$ times, it will not be included in $\\mathcal{S}_t^\\bot$.\n    This implies that in our work the pruned space $\\mathcal{S}_t^\\bot$ may **never** be the same to $\\mathcal{S}^{\\epsilon, \\Pi}$, even if $t$ goes to infinite.\n\n**Q5**: Comparison of ``Near Optimal Exploration-Exploitation in Non-Communicating Markov Decision Processes''.\n\n**R5**: The problem studied in [2] is similar to the one considered in our work, i.e., how to achieve regret adaptive to $|\\mathcal{S}^\\Pi|$ instead of $|\\mathcal{S}|$.\nHowever, [2] still requires knowing the possibly large set of available states $\\mathcal{S}$. \nSpecifically, as in its Theorem 1, the regret bound still has a term polynomial to the size of the large set of available states $|\\mathcal{S}|$.\nIn this regard, when $|\\mathcal{S}|$ is large enough or infinite, its regret bound will become vacuous.\nSuch a result aligns perfectly with our Observation 4.3: if the information of $\\mathcal{S}$ is used as the input for the algorithm, using existing analysis framework, it is difficult to prevent the regret guarantee from a polynomial-level dependence on $|\\mathcal{S}|$.\n\nThanks for these additional references! We will add these comparisons in the camera-ready version.\n\n\n**References**:\n\n[1]: Layered State Discovery for Incremental Autonomous Exploration https://arxiv.org/pdf/2302.03789.\n\n[2]: Near Optimal Exploration-Exploitation in Non-Communicating Markov Decision Processes https://arxiv.org/abs/1807.02373", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RxBM20zfwE", "reviewer": "Reviewer_8pEE", "summary": "This paper proposes a kind of parameter-free reinforcement learning where the algorithm does not need to have the information about states before interacting with the environment. To achieve this, the authors design a black-box reduction framework which can transform any existing RL algorithm for stochastic or adversarial MDPs into a state-free RL algorithm. The paper focuses on establishing the theoretical regret bound of such a black-box approach.", "questions": "See above", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper proposes an interesting problem and a black-box framework for transforming any existing RL algorithm into a state-free algorithm.  The theoretical analysis seems sound and the technique may be of interest for the theoretical community.", "weaknesses": "While I understand that the focus of this paper is on theories, it could still be informative to include some toy experiments. For example, how would a non state-free algorithm behave if given \"wrong\" or insufficient knowledge/estimates of the state space, and how would the corresponding state-free algorithm (via the reduction) behaves.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the comment! We would like to respectfully ask the reviewer to reassess our paper in light of the rebuttal. We believe the reviewer's concern was peripheral to our main contributions and would like to receive a fair assessment of our work. \n\n**Q1**: \nWhile I understand that the focus of this paper is on theories, it could still be informative to include some toy experiments. For example, how would a non state-free algorithm behave if given \"wrong\" or insufficient knowledge/estimates of the state space, and how would the corresponding state-free algorithm (via the reduction) behaves.\n\n**R1**:\n This is a theoretical work, so there are no simulation experiments. Specifically, we are uncertain how existing non-state-free algorithms would perform in our state-free setting, making it challenging to design experiments.\n    For example, consider a non-state-free algorithm running on $\\mathcal{S}'$. During some rounds, it executes a policy  $\\pi$ and obtains a trajectory that includes a state $ s \\notin \\mathcal{S}' $. In this case, it is unclear how does the algorithm update its policy.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CbXG3eSO7x", "reviewer": "Reviewer_zatZ", "summary": "The paper proposes a black-box approach to make any no-regret algorithm to a state-free algorithm. The topic this paper works on is a very important topic. The theoretical results in this paper are significant, if correct. That said, the algorithm description is hard to follow, and hence, I could not verify if the results are correct or not.", "questions": "I listed my questions in Weakness section.", "limitations": "The paper is mostly theoretical, and it has no potential negative societal impact.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 1, "contribution": 4, "strengths": "Introduction and Related Work are both well-written and useful to understand what the issue in current RL theory is. The topic this paper works on is a very important topic. The theoretical results in this paper are significant, if correct.", "weaknesses": "That said, the algorithm proposed in this paper is not easy to understand. I tried to understand it by reading its description multiple times, but still I do not really understand it. It is because there are some undefined symbols and ambiguous definitions. For example\n- What is $π^\\perp$?\n- What do \"compatible with $\\Pi$\" and \"it extends the pruned policy $\\pi^\\perp_t$ to $\\pi_t$\" in Line 185-187 mean?\n- What is the initial set of $\\mathcal{S}^\\perp$?\n- What are inputs and outputs of $\\mathrm{ALG}$?\n- Figure 1 is not really helpful to understand how the pruned space, dual trajectory, etc are obtained.\n\n**Minor Comment**\n- A paper by [Fruit et al. (2018)](https://arxiv.org/abs/1807.02373) seems related in that it also find unreachable states. I think it should be cited.\n\n# After rebuttal review\nI read the paper again and now understand the algorithm. The contribution of the paper is significant, and I highly recommend acceptance to the AC.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the comment! We believe there are some misunderstandings, and have clarified them below. We hope the reviewer could reevaluate our paper, and are very happy to respond more questions during the reviewer-author discussion period.\n\n**Q1**:     What is $\\pi^\\bot$?\n\n**R1**:     $\\pi^\\bot$ represents a fixed default policy defined on state space $\\mathcal{S}$. It can be understood as ``select an action arbitrarily\" (Line 4 of Algorithm 1).\n\n**Q2**:     What does ``by playing an arbitrary action on states not in $\\mathcal{S}^\\bot$ compatible with $\\Pi$,  it extends the pruned policy $\\pi_t^\\bot$ to $\\pi_t$'' mean?\n\n**R2**: Note that $\\pi_t^\\bot$ is a function with domain $\\mathcal{S}^\\bot$, where an available policy $\\pi_t$ needs to a function with domain $\\mathcal{S}$.\nIn this regard, given $\\pi_t^\\bot$, we need to ``lift'' the function domain from $\\mathcal{S}^\\bot$ to $\\mathcal{S}$.\nThis is achieved by playing an arbitrary action on states not in $\\mathcal{S}^\\bot$ compatible with $\\Pi$.\n\n**Q3**: What is the initial set of $\\mathcal{S}^\\bot$?  \n\n**R3**:         As we explained in Lines 182-184, $\\mathcal{S}^\\bot$ includes all the identified $\\epsilon$-reachable states and $H$ additional auxiliary states.\n        Therefore, at the beginning of the game, since there is no identified $\\epsilon$-reachable states, the initial set of $\\mathcal{S}^\\bot$ is $H$ additional auxiliary states.\n\n**Q4**:     What are inputs and outputs of ALG?\n\n**R4**:         As we explained in Lines 9-12 of Algorithm 1, the initial input of ALG is a state space, an action space, a policy set, and a confidence coefficient.\n        At each round, ALG outputs a policy that operates within the given state and action space. \n        Upon executing the policy, a trajectory is generated, which then serves as the new input for ALG.\n        Such an interaction process is generally applicable to all existing online MDP algorithms.\n\n\n**Q5**: Figure 1 is not really helpful to understand how the pruned space, dual trajectory, etc are obtained.\n\n**R5**:\nWe are disappointed to hear our figure wasn't helpful, we would love to hear the reviewer's ideas for improvement during the discussion period. We really want to understand what was confusing about this picture because we are committed to improve our paper's presentation to enhance clarity for the camera ready version.\n\n\n\n**Q6**: Comparison of ``Near Optimal Exploration-Exploitation in Non-Communicating Markov Decision Processes''.\n\n**R6**:\nThanks for the additional reference! We will add this in the camera-ready version.\n\nThe problem studied in [1] is similar to the one considered in our work, i.e., how to achieve regret adaptive to $|\\mathcal{S}^\\Pi|$ instead of $|\\mathcal{S}|$.\nHowever, [1] still requires knowing the possibly large set of available states $\\mathcal{S}$. \nSpecifically, as in its Theorem 1, the regret bound still has a term polynomial to the size of the large set of available states $|\\mathcal{S}|$.\nIn this regard, when $|\\mathcal{S}|$ is large enough or infinite, its regret bound will become vacuous.\nSuch a result aligns perfectly with our Observation 4.3: if the information of $\\mathcal{S}$ is used as the input for the algorithm, using existing analysis framework, it is difficult to prevent the regret guarantee from a polynomial-level dependence on $|\\mathcal{S}|$.\n\n\n\n**References**:\n\n[1]: Near Optimal Exploration-Exploitation in Non-Communicating Markov Decision Processes https://arxiv.org/abs/1807.02373", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper resolves a very specific issue of a large part of tabular RL algorithms: the need of knowing the state space in advance as well as how that dependency translates into suboptimal regret bounds. The authors first identified a neat "trick" to remove completely the dependency and knowledge of S from the bounds of UCBVI in stochastic MDPs and replace it by the set of states that are actually reachable. While this relies on a relatively basic change, this result is already interesting in its own. The major contribution is then to have a general scheme to turn regret-minimization algorithms into state-free algorithms for both stochastic and adversarial settings at the same time. 

There is general consensus about the result and novelty of the paper, while reviewer 8pEE raised concerns about the significance of the result, given the absence of empirical evaluation and comparison to non-reset-free algorithms. An additional concern, in my opinion, is that the current paper is limited to the tabular case, whereas most of the theoretical literature on exploration-exploitation has now moved to linear MDPs and non-linear approximation in general. In fact, the knowledge of the state space is not needed in non-tabular MDPs, although it is arguably replaced by the knowledge of the ranges of the linear parameters and the realizability assumption. This makes the paper somewhat borderline. Nonetheless, my recommendation is towards acceptance, given the novelty of the result and the resolution of an open question in the community. 

I would still strongly encourage the authors to improve the current submission along the following axes:
* Make more explicit that the impact of the proposed algorithm is mostly in the adversarial setting, whereas the result in the stochastic setting is more "for the sake of generality", since a similar result can be already obtained with a minor change to UCBVI.
* Instantiate the general bounds to specific cases in order to make it easier for the reader to compare to existing non-state-free bounds.
* Clarify the role of epsilon as discussed in the rebuttal.
* Include the clarifications provided to reviewer zatZ during the rebuttal.
Decision: Accept (poster)