Warm-up Free Policy Optimization: Improved Regret in Linear Markov Decision Processes
Authors: Asaf Cassel
Aviv Rosenberg
Venue: NeurIPS 2024 poster (2024)
Keywords: policy optimization
reinforcement learning theory
regret
Markov Decision Process
linear MDP
Primary area: reinforcement_learning

Abstract: 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.

up Free Policy Optimization: Improved Regret in Linear Markov Decision Processes
Asaf Cassel Tel Aviv University acassel tau. ac il
Aviv Rosenberg Google Research avivros@google.c com
Abstract
Policy Optimization (PO) methods are among the most popular Reinforcement Learning (RL) algorithms in practice. Rehemaly 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 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 im- proved 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.
(Sherman
Introduction
Policy Optimization (PO) is a widely used method in Reinforcement Learning (RL) that achieved tremendous empirical success, with applications ranging from robotics and computer games (Schul
man et al., 2015 2017 Mnih et al. [2055] |Harnoja et al.) 2018 to Large Language Models (LLMs;
Stiennon et al. 2020 Ouyang al. [2022). Theoretical work policy optimization algorithms
initially considered tabular Markov Decision Processes (MDPs; Even-Dar et al. 22009) Neu et al.
(0010b) Shani et
2020 Luo et al. [202]), where the number of states is assumed to be finite and
small. In recent years the theory was generalized to infinite state spaces under function approximation.
specifically under linear function approximation in the linear MDP model [Luo et al. 2021 Dai et al.
2023 Sherman et al. 2023b a
Liu et al. 2023).
Recently, Sherman al. 2023a presented the first policy optimization algorithm that achieves rate-
optimal regret in linear MDPs, i.e., regret bound of (ppoly H, d) K)), where the number of interaction episodes, H is the horizon, and is the dimension of the linear function approximation. However, their algorithm requires a pure exploration warm-up phase to obtain an initial estimate of the transition dynamics. To that end, they utilize the algorithm of /agenmaker al.) (002b) for reward-Fie free exploration which is not based on the policy optimization paradigm. Moreover, although this algorithm computationally efficient, relies on intricate estimation techniques that are hard to implement in practice and unlikely to generalize beyond linear function approximation (see discussion in section
In this paper, we propose a novel contraction mechanism to avoid this costly warm-up phase. Both our contraction mechanism and the warm-up phase serve similar purpose ensuring that the Q-value estimates are bounded and yield "simple" policies. But, unlike the warm-up, our method is integrated directly into the PO algorithm, implemented using simple conditional truncation of the Q-estimates, and only contributes lower-order term to the final regret bound. Moreover, our approach is much
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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more efficient in practice since it does not rely on any reward-free methods, which explore the state space uniformly without taking the reward into account.
Based on this contraction mechanism, we build a new policy optimization algorithm that is simpler, more computationally efficient, easier to implement, and most importantly, improves upon the best-known regret bounds for policy optimization in linear MDPs. Our regret bound holds in two fundamental settings:
1. Adversarial losses with full-information feedback, where the loss function changes arbitrarily between episodes and is revealed to the agent entirely at the end of each episode.
2. Stochastic losses with bandit feedback, where the loss function in each episode is sampled i.i.d from some unknown fixed distribution and the agent only observes instantaneous losses in the state-action pairs that she visits.
In these settings, the best-known regret bound (by Sherman et al. [202bb] ) was Ō(VHT Our algorithm, Contracted Features Policy Optimization (CFPO), achieves Ol v d3 K') regret, yielding a v VH&d improvement over any algorithm for the adversarial setting and matching the value iteration based approach of Jin et al. [2020b] in the stochastic setting. We conjecture that this is the best regret we can hope for without more sophisticated variance reduction techniques [Azar et al. 2017 Zanette and Brunskill 2019 He et al. 2023 Zhang et al. 2024), that have not yet been applied to PO algorithms even in the tabular setting Ignoring logarithmic factors, the regret of CFPO leaves a gap of only v vHd from the ((V K") lower bound for linear MDPs [Zhou et al. 2021a). Finally, our
analysis relies on novel regret decomposition that uses notion of contracted (sub) MDP and may be of separate interest (see section 5
1.1 Related work
Policy optimization in tabular MDPs. The regret analysis of PO methods in tabular MDPs was introduced by EvennDar et al. [2009], which considered the case of known transitions and adversarial losses under full-information feedback. Neu et al. 2210a bJ extended their algorithms to adversarial
losses under bandit feedback. Then, Shani et al. [2020] presented the first PO algorithms for the
case of unknown transitions (for both stochastic and adversarial losses), and finally [uo et al. [021]
devised a PO algorithm with rate-optimal regret for the challenging case of unknown transitions with adversarial losses under bandit feedback. Since then, PO was studied in more challenging cases, e.g.,
delayed feedback [ancewicki et al. 2022 2023
and bestt-ofboot-worllee [Dann et al. 2023].
Other regret mininization methods in tabular MDPs. An alternative popular method for regret minimization tabular MDPs with adversarial losses is O-REPS [Zimin and Neul 2013] Rosenberg Jin al.) [2020a], which optimizes over the global state-action occupancy measures instead of locally over the policies in each state. However, this method is hard to implement
and Mansour 2019a b
in practice and does not generalize the function approximation setting (without restrictive assump-
tions). For stochastic losses. optimistic methods based on Value Iteration (VI: J Jaksch et al. [2010].
Azar al. 2017]. | Zanette and Brunskill (2019) and Q-learning [Jin et al. 2018 Zhang al. 2020
are known to guarantee optimal regret, which has not been established yet for adversarial losses.
Policy optimization in linear MDPs. While Sherman et al. [2023a] established rate-optimal regret for PO methods in linear MDPs with stochastic losses, most of the recent research focused on the case of adversarial losses with bandit feedback [Luo
2021 Neu and Olkhovskayal 2021 Dai
et al. [2233, (Sherman et al.) 2023b Kong et al. 2023 Liu et al. 2023 Zhong and Zhang 2023
where rate-optimality has not been achieved yet.
Other regret minimization methods in linear MDPs and other models for function approxima- tion. Unlike O-REPS methods that do not generalize to linear function approximation, value-based methods (operating under the stochastic loss assumption) are also popular in linear MDPs and have been shown to yield optimal regret [Jin et al. 2020a Zanette et al. 2020 Wagenmaker et al. 2022a
Wu et al. 2022 apply variance reduction techniques to get better regret bounds in the tabular setting, but they use L regularization instead of KL-regularization which does not align with practical PO algorithms Schulman et al.
2015 2017].

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Hu et al. 2022 He et al. 2023 Agarwal et al. 2023] Another line of works Ayoub et al.
2020
Modi al. |020] Cai al. 2020 Zhang et al. 202
Zhou al. 202 b He et al. 2022 Zhou
and Gu 2022 study linear mixture MDP which is a different model that is incomparbble with linear
MDP (Zhou et al.
2021b]. Finally, there rich line of works studying statistical properties of RI.
with more general function approximation [Munos] 2005 Jiang et al.) 2017 Dong et al. 2020 Jin
ct al. 2021
Du et al. 2021 but these usually do not admit computationally efficient algorithms.
Problem setup
Episodic Markov Decision Process (MDP). A finite-horizon episodic MDP M is de- fined by a tuple (X,A, T1, {E" 1K=1. T====
H) with X, set of states, A, a set of actions,
E X. an initial state (assumed to be fixed for simplicity), A(X), the transition probabilities, and [K=1-= sequence of loss functions such that ek = ([H)) lh X' A - [0, I]. is horizon dependent immediate loss function for taking action at state x and horizon h of episode k. A single episode of an MDP is sequence (i a fh (= af) N (X x A [0, 11) such that
H. decision horizon,
P P= (Ph (ernn PA : x
Pr[w/+1 x' x x, att - a| Pn (x' xx,a).
For the losses, we consider two settings: stochastic and adversarial. In the stochastic setting, there exists fixed loss function f = (Cx) l, : X' x
[0. such that e is sampled i.i.d from
a distribution whose expected value is defined by l, i.e., E [ech (2, a) x, a] = (n a). In the
adversarial setting, the loss function sequence {eh { 1== is chosen by an adaptive adversary.
Linear MDP. A linear MDP et al. 2020bb satisfies all the properties of the above MDP but has the following additional structural assumptions. There is known feature mapping Φ ; X' x A - R such that Pa(2'' xr a) 6(z, a)* (2²) where thn X + Fd are unknown parameters. Moreover, for all h [.] k [K], there is unknown vector 0% Rd such that, in the adversarial case, fh a) = (ax, a) T% while in the stochastic case, 8f = fl and (,(2,, a) = 6(2, a) 0₁, We make the following normalization assumptions, common throughout the literature:
1. |0(zz a)ll - 1 for all x E X, E A:
2. |0X|l v for all [H] k [K]:
3. I1l (X'II = 1b S vd for all h E [H];
where [x)) is the entry-wise absolute value of (x) e Rd. We follow the standard assumption in the literature that the action space A is finite. In addition, for of mathematical exposition (e.g. Cassel et al. (024)): ). we also assume that the state space X' finite. This allows for simple matrix notation and avoids technical measure theoretic definitions. Importantly, our results completely independent of the state space size | both computationally and in terms of regret. Thus, there is no particular loss of generality.
Policy and value. A stochastic Markov policy π = (TA))e(H) : [M] x X' + A(A) is a mapping from step and state distribution over actions. Such policy induces distribution trajectories = (Ir,, an) (n))=[H)> i.e., sequences of H state-action pairs. For f (X' x A) + R, which maps trajectories
real values, we denote the expectation with respect to under dynamics P and policy π as p [f Similarly, we denote the probability under this distribution by IP P. We denote the class of stochastic Markov policies as 1IM. For any policy 11, horizon h [M] and episode E [K] we define its loss-to-go, as
VT E.p..
Σ T [eh (1te, The, ane] = J W
which is the expected loss if one starts from state x E X' at horizon & of episode and follows policy π onwards. Note that the inner expectation only relevant for stochastic losses as its argument deterministic in the adversarial setup. The performance of a policy in episode k. also known as its
value, is measured by its expected cumulative loss V.R.7 (x1).
3

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Interaction protocol and regret. We consider standard episodic regret minimization setting where an algorithm performs K interactions with an MDP M. For stochastic losses we consider bandit feedback, where the agent observes only the instantaneous losses along its trajectory, while for adversarial losses we consider full-information feedback, where the agent observes the full loss function & in the end of episode k E [K]. Concretely, at the start of each interaction/episodee k E [K], the agent specifies a stochastic Markov policy x² = (TT) (e()) Subsequently, it observes the trajectory 2x sampled from the distribution Pr,+, and, either the individual episode losses (a)) o E [H] the case of bandit feedback, or the entire loss function th in the case of mull-information feedback.
We measure the quality of any algorithm via its regret the difference between the value of the policies x" generated by the algorithm and that of the best policy in hindsight, i.e.,
Regret Σ Vil (I1)
min xEll
Σ v (21)
Σ
v k
(21)
V (x1),
where the best policy in hindsight is denoted by x* (known to be optimal even among the class of stochastic history-dependent policies).
Notation. Throughout the paper = O(x) a E Rd denote the state-action features horizon episode k. In addition, || = VUT Au. hyyee-parameters follow the notations 8g and 7, for some and 8 (0, 1) denotes confidence parameter. Finally, in the context of algorithm, +- signs refer to compute operations whereas = signs define operators, which are evaluated at specific points part of compute operations.
The role of value clipping
Before presenting contraction technique and main results, discuss the role that value clipping plays in regret minimization and its apparent necessity for linear MDPs. As starting point, is important note that, while commonly used [Azar al. 2017 Luo al. 2021], value clipping strictly necessary in tabular MDPs. To demonstrate this, consider a fairly standard optimistic Value Iteration (VI) algorithm that constructs sample-based estimates é, with empirical error estimates Ae, Ap, defines exploration bonuses (Ar + H Ap), and chooses policy #* that optimal in the empirical MDP whose dynamics are and losses are e Then its single-episode regret may be decomposed
V (x1) V/= (z1)
Vi (x1) (6) bias of optimism
- V (a1)
V (21) V/
VTT (x1) (x)) V2" (x1). (iit)
(6t)-FLL) FTL
optimism
where V is the value under the empirical MDP. Now, by definition of f*, we have that (ii) S 0. Now.
let Ae = -e, AP== = P. Using standard value difference lemma (lemmal 14 in appendix B
we have that (i) S b and
(iii) = E pp
Al(zn, an) b(z,, an) + Σ AP(a' Ih on) on) Vnt
(I)
NE(H
x'EX
< E p.
Ay an) + HAp(w,, an) - b(an, an)
NE(M)
where the inequality also used that V,, E [0, H}. The final regret bound is concluded by summing over l [K] and using bound harmonic sums. We note that similar clipping- free method also works for tabular PO (see Cassel et al.] (2024)) )
Moving to Linear MDPs, one might expect similar approach to work. Unfortunately. the standard approach that estimates the dynamics backup operators s/n, E [H] using regularized least-squares presents a significant challenge. This is because, unlike the tabular setting, the resulting estimate

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Pv.. a) o(x, a) T2 (eq. (2) is not guaranteed to yield a valid probability distribution. i.c., there could exist E X, a E A. h [H] such that
( 2, a) = = 1 and/or
min Pa 22, a) < O. #'EX'
P is still a finite signed- signed-measure, measure, which is enough for the first equality in eq. (1) to hold. However, since p could contain negative probability terms, the inequality in eq. (I) does not hold. These negative probabilities also seem make caleulating #* computationally hard. Finally, the & - norm exceeding 1 may cause term (i) to depend on H exponentially. While some of these issues could be mitigated without clipping, we are not aware of method that resolves all simultaneously.
The use of value clipping opens the path for an alternative value decomposition that replaces p in eq. (T) with E,,-- at the cost of also replacing Vi+, with V/+11 We thus need that |V S H for the inequality in eq. 4 to work. This is made possible using a clipping mechanism that decouples the scale of V h+1 from the magnitude of the bonuses 6, which may be much larger when the error estimates De, P are large. This typically achieved by adding max(0,-} to the recursive formula for the value function. A similar clipping approach also works for tabular PO and VI [Azar et al.
2017
Luo al. 202T]. and even for VI in linear MDPs (Jin al. 2020b].
However, this is not the case for PO in linear MDPs where Sherman et al. [203a explain that this type of value clipping leads to prohibitive complexity of the policy and value function classes, and thus sub-optimal regret. Concretely, the complexity of the soft-max policy class roughly corresponds
to the number of parameters required to represent Eke[K] 0% If Q2, a) = o(z, a/'u,; are linear, then the sum remains linear and depends on d parameters (with slightly larger magnitude). If @h a) max (0, de T [wx),, the sum may, in general, have d' parameters thus degrading the regret. [Sherman et al. (2023a] overcome this issue using a warm-up based truncation technique. In what follows, we suggest an alternative solution that uses novel notion of contracted features and has several advantages over their approach (see discussion at the end of section 4
w/
Algorithm and main result
We present Contracted Features Policy Optimization (CFPO; algorithm [ a policy optimization routine for regret minimization in linear MDPs. The algorithm operates in epochs, each beginning when the uncertainty of the dynamics estimation shrinks by a multiplicative factor, expressed by the determinant of the covariance matrices N la E [H] (see line 13 for the definition of A and line for the epoch change condition). At the start of each epoch e, we reset the policy to its initial (uniform) state, and define the contracted features Z.. hh E [H] (line by multiplying the original features with coefficients in the range [o, 1). and thus shrinking their distance the origin. Inspired by ideas from Zanette et al. [2020), these coefficients are chosen inversely proportional to the current uncertainty of the least squares estimators each state-action pair, essentially degenerating the MDP in areas of high uncertainty. Inside an epoch, at episode k. we compute the estimated reward vector 8h (line 14) and estimated dynamics backup operators vix (eq. (2)). Then, we use these 6² and vi to compute our Q-value estimates with the contracted features (eq. (3), and run online mirror descent (OMD) update over them (eq. (5), i.e., run a policy optimization step with respect to the contracted empirical MDP (more this in section 5.1).
We note that the computational complexity of algorithm I is comparable to other algorithms for regret minimization in linear MDPs, such as LSVI-UCB (Jin et al. 2020b). The following is our main result for algorithm (see the full analysis appendix A).
Theorem 1. Suppose that we run CFPO (algorithm 7 with the parameters defined in theorem 9 (in appendix Ay. Then, with probability at least 1 5, we have
Regret O
(v (VH*d HH log( K) log + / HHadK log( loglA)
Discussion. Policy optimization algorithms typically entail running OMD over estimates Q of the state-action value function Q. as in eq. (5). The crux of the algorithm in obtaining such estimates that satisfy optimistic condition similar to eg. ( while also keeping the complexity of the policy
class bounded. As discussed in Sherman et al. 2023a), the latter depends on Exee 2% (eq. 3)
5

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Algorithm Contracted Features PO for linear MDPs
1: input: d, HI, K, A. 8, Sw 3o, To O.
2: initialize: c f- -1 , < I. h. E [H].
3: for episode k 1, ...,, K do
4: if k 1 or 3h [H], (et((A)) det((A then
5:
e e + and lig t k.
6:
9h (z. a) a( - Pw (xx a) +og
lo(2) 1/(1 exp(-2)))
7:
T | = 1/|A| for all h [H] a A.
8:
end if
9:
Play n² and observe losses (a), af neM and trajectory 2 (x ae))
10:
the case of full-information feedback: observe 0h
11:
Define H++ (2) = O for all X.
for h
H. ..... do
A + + +1+22ela ErE( 0% (1(6))
(Ah) ">E{k- - 07 (1), az),
feedback = bandit
14:
0% <
feedback full.
For any
V R r X. A define:
V
= -AA) Σ V
(2)
TE(K-1)
(1,)))) a)
2h (z) 10% 34 khtt 1 (x, a)laa)))-
(x) = -E Eht(aa x) 2))
(4)
aEA
+ x) x (a(a z)exp( (aap--n.h Q (x,
(5)
end for 17: end for
having a low dimensional representation nearly independent of k. Although standard unclipped estimates admit such a representation, they lack other essential properties (see discussion in section On the other hand, the standard clipping method, which restricts the value to [o, H between each backup operation (see, e.g.. Jin et al. (2020b)), does not admit the desired representation.
Sherman et al. 2023a overcame this issue by employing a warm-up phase based on a reward- free pure exploration algorithm by Waagenaaerr et al. (20226) to obtain initial backup operators 20, [H] and subsets X,, S ,, h E [H] such that: (i) for every x, a E X, x AA the bonuses (b in section B), which are proportional to the estimation uncertainty of the value backup estimates,
small (S 1); and (ii) for all policies π e 1lM, the probability reaching any æ, a & Uhe x A
is small (S K-1/2) To ensure that the overall value estimates remain bounded, they truncate (zero out) the Q-value estimate of these nearly unreachable state-action pairs, an operation that allows for a low-dimensional representation of the policies. Nonetheless, their warm-up approach has several drawbacks.
It runs for Ko poly(d, H)VK episodes, contributing the leading term in their regret guarantee;
It relies on first-order regret algorithm by Wagenmaker et al. 2022a that is not PO- based and uses computationally hard variance- aware Catoni estimator for robust mean estimation of the value backups, instead of the standard least-squares estimator. To maintain computational efficiency, they use approximate version of the estimator, losing factor of Vd in the regret;
Still, to the best of our knowledge, even the approximate estimator must be computed using binary search methods, making hard to apply practical methods that typically rely on gradient-based continuous optimization techniques;

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It runs separate algorithms for each horizon h E [H], using only out of H samples during the warm-up phase;
It is not reward-aware, and thus has explore the space uniformly to ensure that the uncertainty is small for all policies, which could be highly prohibitive in practice.
Our feature contraction approach obtains the desired bounded Q-value estimates and low-cempyy policy class without relying order term of poly(d, d ) log K to the regret guarantee, thus improving the overall dependence on and H. Additionally, it uses all samples, is easy implement, and is reward-aware. To understand the benefit of reward-awe -warddaa consider an MDP where at the initial state the agent has two actions, each leading to a distinct MDP. Now, suppose that both MDPs have only single state and action for the first H/2 steps with one MDP incurring a loss of in these steps while the other incurring O loss. Notice that regardless of the last H/2 steps, the 0 loss MDP will outperform the 1 loss MDP. Nonetheless, the reward-free warm-up, which does not observe the losses, will have to fully explore both MDPs. In contrast, our reward-awar approach would quickly stop exploring the inferior MDP, leading to better performance in practice.
dedicated warm-up phase. Crucially, it only contributes lower
Analysis
In this section, we prove the main claims of result. For full details see appendix A We begin by introducing the main technical tool for our contraction mechanism the contracted MDP.
5.1 Contracted (sub) MDP
For any MDP M (K, {
1k=1, Ii H. and contraction coefficients p [H] X A [0, we define a contracted (sub) MDP MDP AX A, x1, { { k= H) where as e (2, (r( a)ek
f (xr a) E [o. 1] are the contracted losses and Px I. a2.a) are the contracted (sub) probability transitions. Notice that the transitions being a sub-probability measure implies that E (x
Pa (2. P (x' r. E [0
< as compared with a probability measure where
this holds with equality. For any Markov policy π II,, let V. (:pp) : X + R, h E [H] be the loss-to-go (or value) functions of the contracted MDP. In particular, these may be defined by the usual backward recursion
via (x; (ewr((la)
BlCh (2x, x, + Σ [ (4
z'EX
with k. #+1 (z; p) = 0 for all x E X. The following result shows that the value of any contracted MDP lower bounds its non-contracted variant.
Lemma 2. For any p:: [H] x X A (o. E M,, [H) E [K]. and x E X we have that
V.k. (I; p) & Vi. (x)).
Proof. The proof follows by backward induction on h E [H + 1]. For the base case h = H - 1, both values are 0 and the claim holds trivially. Now suppose the claim holds for h + 1, then we have that for all x E X
V.k. (x: p) Bawx((l))
E (l a] Σ xex
Pa(x' ( (zaa)
< Bawx(()a)
E (2 a
Pa(x' ( Σ (.. I
Next, for any epoch e [E], consider its contracted linear MDP (line in algorithm whose contraction coefficients are for (f, a (-Bulllorr, a))l(at))- + log K The following result gives an upper bound on the performance gap between the contracted and non-contracted variants. Lemma 3. For any [E] and v E ddd have that (Ø(z), an) The (In, TT < (482110(zn. an)llagg-- + 2k- Tn)
7

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Proof. We have that
(Ø(œn, an) The (Ih, an))" v= oBBulo((z an (1 - log K) 0x as)'v
2( lo an (((a)t)) + K 0 an)
(4 on
" [ 2K ( mn Vl
where the first relation is by the property of the sigmoid 1 oxx) o the second is by simple algebric argument that a quadratic function bounds the sigmoid (lemma 19 in appendix B and the
last relation uses (ct(AA) - det (A)) by line in algorithm
(see lemma 16 in appendix B
We note that the analogous claim in Sherman al.
2023a shows that for all II
Ep. ( an)) 1l ( ax))" v) S Pr(xn & 2n.) max (xx aav u/.
(6)
2,a
where 2/, is an outcome of the reward-fres warmup phase and Pr(#,, & 2x) = K-1/2, Summing this over k E [K] yields a term that scales as VK. In contrast, we use a standard hound on elliptical potentials (lemma 15 in appendix B get that
(o o All 2K S log K.
RE(K]
This implies that the cost of our contraction is significantly lower than the truncation of Shhrraan al. [2023a]. We achieve this reduced cost by using quadratic (rather than linear) bound on the logistic function. The challenge in our approach is that the above bound only holds for the observed trajectories rather than for all policies as in (Sherman et al. [023a]. In what follows, we overcome this challenge using novel regret decomposition.
5.2
Regret bound
For any epoch [E], let Ke be the set of episodes that it contains, and let V.A." (x1; p denote the value its contracted MDP as defined above and in line of algorithm T
We bound the regret as
Regret-
Σ
Vi V.k
(I1) - V/tT (x1)
kE[K]
Σ
Σ Vha*** (x1) -V/e (41:
(lemma
CE(FF KEKe
Σ
ViRTT* (I1) (/(x1) Σ Σ V+(21) V/aT"" (01
kE[K]
<E(F) keK,
Σ kE[K]
Vhr (z1) vy(1))
Rias Crsst optimism
E pike
Σ
Σ
a mn) (ia xn))
E [M
KEEK REA
(+i)-MD regret
Σ Σ E KEK. he[A]
Σ
E pu .r [hi The (oh, an) T Un V
(883
Optimism
where the last relation is by the extended value difference lemma (see Shani et al. (020] and lemma [4] in appendix B This decomposition is very similar to the standard one for PO algorithms, but with the crucial difference that term (iii) depends on the contracted features -ke (.r,. as) instead of the true features ó(I an).. As by-product, the expectation in terms (ii) and (dit) is taken with respect

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to the contracted MDP instead of the true one. The purpose of this modification will be made clear in the proof of optimism (see lemma 4
In what follows, we bound each term deterministically, conditioned on the following "good event":
E1 Vk [K] h [H] 110 Dillat - 3r }:
F2 k E [K]. [M] 11(v/r f S Sp. 1pp112i+1ll S 22H).
(8)
E1 and E2 are error bounds on the loss and dynamies estimation, respectively. the full feedback setting, Fg holds trivially with 8, O. In the bandit setting, it holds with high probability with 8
O( d log K K(/ by well-established bounds for regularized least-squares estimation (Abbasi- Yadkori et al. 201T]. Showing that Es holds with high probability follows similarly to Sherman et al.
2023a again using least-squares arguments but also using the contraction to ensure that 2 are bounded (see sketch at the end of this section and lemma Giin appendix A for full details), specifically ( The proof of theorem is concluded by bounding each of the terms in the regret decomposition. summing over l E [K] and using a standard bound on elliptical potentials (lemma 15 in appendix B). Term (ii) is bounded using a standard Online Mirror Descent (OMD) argument (lemma 7 in appendix
3, = (Hdd
Optimism and its cost. The following lemmas bound terms (itt) and (i), respectively. Lemma 4 (Optimism). Suppose that eqs. and (8) hold, then
(x a) -% (zz, (Bh + tn V S wh [H], [K], E X, A. Proof. We have that
(x,) a) - कh (a, ()p + thn thn vh+1) = Th (xr, a) ( 0n + (0) vhn )
- Boll%% (z, a |1 ((
< (g. + Bp) (I,, (r.,))la++= Bolloh (1r, (1,,))ll alla
$ (0. Bp 30)110% (x. ())((at))) 0,
where the first relation is by definition of Q (eq.
[ in algorithm D. the second relation is by eqs. ?
and (8) together with Cauchy-Schwar the third relation follows since A A and the last one
by our choice 3. = 3, + 3p (see thcorem in appendix A for hyper-parameter choices).
Notice that the standard PO decomposition would have required that we bound the non-contracted expression Ey..* 1Q% (Ir, a) 6(x, (0f + +%
1/4+)) +//,, this argument and that of lemma can be bounded using eq. (6). However, the equivalent argument
In Sherman et al. [2023a] the gap between
for our contraction is lemma 3 which is bounded only for x² and not for any policy π E M-
Lemma 5 (Cost of optimism). Suppose that eqs. and 8 hold, then for every [K]
Vit (x1) - (x1())) s 3(8, + P
|K an) an)lkag)
NEWH
+ 16H E. P
Σ
| an) 1i + 1-
ME(M
Proof. First, by lemma 14 in appendix B
value difference lemma by Shani et al. 2020
yka (x1) (x(1))
(x1) = Ep,
Σ
Hlan, un) (on. T vnn Ph+11)
Q (xn, an)
Now, using lemma 3 with 0h: + who V&a we have that 10(x, a)T < 4H (by eq. (8) and thus
(ában, un) the (Nn,, an))" (On wn Vh+1)) 16H 16H an an))l(aa)= --1

---

We can thus conclude the proof using standard arguments to show that @h (sn (On + wh Vhit1) - (x as)
2h (I ax) 0% -0% 0% + (1h - ( (ht2)) + Bolløh (on owlllat)) (eq. (3)
s (3. Bp) (2 ab) (())) + Bo 1o110% (I an)ll
(Cauchy-Schward eqs. 7 and (
s 33( + 12) (mm, an) (n)l
(dct(A s 2 (e ", ),80 3, + (p)) (a(x) E [0, Vo E RR)
s 3(3r + Bp)1l on (
as desired.
Bounding the Q-values (proof sketch). The following are the main ideas in showing that Eg
(eq. (8) holds with high probability. First, we define appropriate value classes V/, that contain all value functions V₁ of the form in eq. (4) whose underlying Qn function (eq. (3) satisfies ||Q S 2H + 1 - h). Because both the bonus and contraction operator are kept fixed during each epoch, the log covering number of this class is logarithmic in K (similarly to Sherman et al. (2023a]). Thus, we can use standard least squares arguments (lemma (2) to show that with high
probability [1(bn 7 Vilag I 3, for all & E [K], h E [H], V E Vn. The proof is concluded by showing that || Q1 S $Q.A 2(H + 11 - h), and thus 1/h E Vn, which implies that eq. (8) holds. We prove this by backward induction on h E [H++ +
The base case
H + satisfied because, by definition, Q+++1= = Now, suppose the claim
holds for h + 1 and we show it also holds for h. Recalling the definition of Q in eq. 33. we have that (2, a)l 10% (1x a)" (h V h++1) B (ar, a) (l, (att))
S 1 (z, a) (On + (0% 0n) + (g v/n V++ + thn V/ttall + Bolloh Rh (I, a))l(
S 1 1l + प, (a, a) |1 [105 Onllag + 11 - th V + (b)
where the last inequality also used the triangle and Cauchy- inequalities, and that A/t- = A By the induction hypothesis, | 1 S $Q, and thus V Vu4z. Combining with E1 (eq. (7)) and plugging into the above we get that
12) a)l + SQ + (3, + Pp. + 1o0 The (z, a)l 1.
Now, using a technical algebraic argument (lemma 18 we show that
b (ar, ( I max ly- a- Bwy + log K)] S 2,5=² log(eR).
#20
Finally, plugging this into the above and choosing 3.m 2 2(3, + 3p. + 36) log(e we get (1, a)l < 1+ SQ + 2 (3, + pp. + 9,) log(eK) 2 8Q BQ...
concluding the induction.
Conclusions
In this paper we presented a simple and efficient contraction mechanism for policy optimization in linear MDPs, yielding an overall algorithm with improved regret guarantees under both stochastic (bandit feedback) and adversarial (full feedback) losses. We note that, in the stochastic setting, there are value iteration based methods (He al. (0033 that use variance reduction techniques to achieve better regret bounds. We conjecture that such techniques could be applicable to PO, however, this is highly non-trivias and thus left for future research. Finally, regarding practical implementations, we note that our bonuses and contraction technique are computationally feasible, especially compared to the reward-free warmup phase Sherman et al. (2023a). Nonetheless, it remains open whether our techniques could be applied heuristically to drive exploration in practical deep RL methods. In particular, it would be interesting to examine the necessity of the contraction mechanism. These are challenging questions exploration in deep RL that we leave for future rescarch.
10

---

Acknowledgments and Disclosure of Funding
This project has received funding from the European Research Council (ERC) under the European Union's s Horizon 2020 research and innovation program (grant agreement No. 101078075). Views however those of the author(s) only and do necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. This work received additional support from the Israel Science Foundation (ISF, grant number 2549/19), the Len Blavatnik and the Blavatnik Family Foundation, and the Israeli VATAT data science scholarship.
and opinions expressed
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A Analysis
We begin by defining so-called "good event", followed by optimism, cost of optimism, and Policy Optimization cost. We conclude with the proof of theorem 9
Good event. We define the following good event Eg = Ni=1 Es, over which the regret is determin- istically bounded:
E1 Vk [K], [M] 10% 0ill S $
(eq.
Eg & [K], [H] l1(th
V 1l S Bp 12h+t11l0 s }
(eq. (8)
6
E3
E Ep [Yx] $ Σ
2Y (3( 3p) + 432 log
(9)
kE[K]
kE(K]
where Ys Ene(m 3(8, + Bp)llo(zn, an (n)] 2 1 (1 an) (n)))
Lemma (Good event), Consider the parameter setting of theorem then Pr[Eg] 2 1 5-
1f S 1, Sr S K/(32Hd)
Proof in appendix A.1
Policy online mirror descent. regret in each state.
We use standard online mirror descent arguments to bound the local
Lemma 7 (OMD). Suppose that the good event Eg holds (egs. then
g and 9 and set Tl S 1/3q.
log/A
Σ Σ ( x) -5-mp, = x)) KEK, aEA
+
5 Ve 11,h E [4], x E N.
Y)o
kEK,
Proof. Notice that the policy x is reset at the beginning of every epoch. Then, the lemma follows
directly by lemma 13 with vr(a)) = ()), = < æ) and noting that 12% a)l S Bo by eq.
Epoch schedule. The algorithm operates in epochs. At the beginning each epoch, the policy is reset to be uniformly random. We denote the total number of epochs by E, the first episode within epoch e by kic and the set of episodes within epoch by K The following lemma bounds the number of epochs.
Lemma 8. The number of epochs E is bounded by (3/2)dH log(2K).
Proof. Let Tr {c), ch..... be the epochs where the condition dct(Ak) 2 det((A was trig- gered line 4 of algorithm Then we have that
dA de
E Th otherwise.
dct(A >
Unrolling this relation, we get that
dct(A 2/1-1 det I 2 Tal-1
and changing sides, and taking the logarithm get that
|Ta < log, dot (A
< 1 + d logg2 |
K
(det(A) S [|A|1))
K-1
< 1 dlogg
1+
Σ k=1
(triangle inequality)
+ dlgg K
(1 S 1)
< (og 2K.
15

---

We conclude that
E = Th)l
Σ
|h S (3/2)dH log(2K)
kE(M)
Regret bound.
Theorem 9. Suppose that we run algorithm
with parameters
3dH log((K) log
70
BB = Br + Bp. -(33 + Bp) log(eK),
K KB2
where B, = 2/ 2d
8, = 28Hd//oogL0KKHH/1) 2H. Then with probability
at least 5 we incur regret at most
Regret & 264 Kd (og + 8 logH2 log|
12K
+ da log log|AA) log
OO Kd5/Kooollll/11 log( (K//)) V VKdlt log K) log Al)
Proof. First, 32 K/(32Hd) or 1j 2 1/$q then
Regret < KH 32H m log]AA) log(2K),
and the proof is concluded. Otherwise, if B S (3(2Hd) then suppose that the good event E, holds (eqs. 7 ( and (9)). By lemma this holds with probability at least 8. For any epoch E [E]. let Ke the of episodes that contains, and let V.A. ( denote the value of its contracted MDP as defined in section 5.1 and line of of algorithm We bound the regret as
Regret
Σ KE[K]
(I1)
(x1)
Σ
Σ Vili (21) -D (#1:p²e)
(lemma
cE(E) KeK.
Σ Vi (I1) - (1(())) + Σ Σ ()((1)) 1/ (ke)
kE[K]
cE[A] KEK.
E [E
Vi (x ()))
(4) Bins / Cost of optimism
Σ
Σ
E piie
Σ
Σ
ahln 2n) at(a zn))
E [E AE(M)
LKEK REA
(1)) regret
Σ E
E Pie , (0t and Ty (2 T ((h whn V/h 1
CE(F) KEK.
hE(m]
Oppi)-O
where the last relation by the extended value difference lemma (see Shani et (020 and lemma 14 in appendix B).
16

---

For term
(i), we use lemma as follows.
(t) - Σ
E
Σ AE[H]
3(3, + Bp)ll((zn, aw)l(as)) + 860g)
+ +8H0Q
kE[K]
+ +0H 20H 3Q log 6
s Σ
6(8 + Bp)ll((ah (alloaa ( + 16.9 1 (at))
kE[R]
ME(M]
(eq.
2
4(B, + (pp) 2 32)
6
< <6 22 + (200lk))333333333 Hog(K)))oo))e)oooo
20 0 log
(emma 15
8
12K
< 6(8, + H + 32Hd3q log
By lemmas and
(with our choice of 17.) we have
log A
(ii) <
E Ept. ,TT
+ To
Σ
B
S 4H BQ v KdH log(2K) log Al.
110
hE[H] [
RE K.
By lemma
(iit) < 0. Putting all bounds together, we get that
12K
Regret <633, + H +33G3 log
+HB / KdH 12g() log
12K
s 2644 K Kd 1og + 8y KdH (2K log 64H log
= =0 log logg + KdH log( A).
A.1 Proofs of good event
We begin by defining function classes and properties necessary for the uniform convergence arguments over the value functions. We then proceed to define a proxy good event, whose high probability occurrence straightforward prove. We then show that the proxy event implies the desired good event.
Value and policy classes.
We define the following class of restricted Q-functions:
iCCs, Cw. Cq)
= 10, E Ca). Cw, 1Q0 2) s Co}.
where Q(zz, a; B, A) [wT p(x, a) - Bl10(1, a)laa
+ log K). Next, we de-
fine the following class of soft-max policies:
IICC,, T(- (:2) Q @(Ca, Cw. Cw.
where (a 1x; Q)
exp(Q(z,a)) Ed'ed (xp(Q())a'))" Finally, define the following class of restricted value
functions:
DCC,, Co) = [(::T, Q) T C, Cq), Ō(Cg.Cu. Ca)}.
(10)
where Vx:; T., Q) EnsA л(a 1x)((z a). The following lemma provides the bound the covering number the value function class defined above.
Lemma 10. For any e, Cw > O, Cp, Co 2 we have
log Nc (V(Cp, JoggN(D(Cs
Cwe Chr Ca)) Gel log(1 + 4 CV192K3CCCBBB) KC,, + Vdl/e),
where No is the covering number of a class in supremun distance.
17

---

Proof. We begin by showing that the class of Q function is Lipschitz in its parameters. For ease of
notation, denote y = 0(z, a). Then
(a a; B, w. A) llal| a0-Bw log K)
(o(-) E [0, 11, |lll S 1, 1 (o() [0. Myvl 1)
V a; B, w. A) 11y 0(-Buullvlla log K)ll s
[Q 3, W, A) Q(2, a; 3, w. A')
$ / allll|a 11y1lar1 ( + log K)
+ Bllvll A- (0 1llllll11 + log K) - 0 (o-llllllll I1ylllll + log K)1
(A') - B11AA 1/2 (A) wull -Bw 1 -
(1-11, (+)) I-Lipschitz, a E [0, 1))
S 2/3 - 11)
(llil S 1, = I,3 2 1)
S V2K33 I|A A
(lemma 17
A, = (2K)-!I)
S V2K33 II - 1l
(H S 11-11p)
We thus have that for any such
Q(xr, az B, A)
Q(x, a; B², w', ,A)
< QQ(z, az B, w. A) - Q(2, a: w. w,,al + aQ(x,
a; B', A) B', Q(x, a: 8¹, 8''''. w A)]
+ 1Q(x, a; B¹, w A) Q(z a; B w/ A'')
- |3 - 8 + || w'll 22K B A AlIF
z /3(lw - w/1 + |B 3² (V2K 332 AT((²)
I max(3, V6K } / (1lw - w/1l + 13 - ,/1² + 1|A - - = max V6K [1(3, w, A) - (3",, w/,,"', ,')I
where (B, w. A) vectorization of the parameters. Assuming that C. 2 1, we conclude that ÕlCs, Cw, Cq) is -Lipschtzz in supremum norm, i.e.,
1 w, A) ('.,,,,'', w', Al S V6KC33.l1(3, W. A) (8', w', ,')ll- Next, notice that our policy class [[C,,, K) is soft-max over the functions thus fitting Lemma 12 of Sherman et al. 2023a We conclude that the policy class is Cau in f1 norm, Le.,
W. (3', w/ A')ll- Now, let V,V' DCC Cve, CQ) and 0 (B1, w1, A1, B2, 122 A2), 00 (3), w/, M1, w/z A2) +d(++) be their respective parameters. We have that for all x E X
|| 1; w. A) T( x;; 1x;3', w/, AA]I S
[i n., Q) V(a: n/, Q')l < [V( n, Q) V( 23) + [V n, Q') - V( n Q'1
(62)
For the first term
Σ T(a (z)( 82,
u2, A2) - Qz, a; 02 w/y N2))
aEA
Σ x(a x))
Q( a5 w2, A2) Q(x, az 82 wby No)
(triangle inequality)
aEA
(3g, Ab)l- ( is v V6KC Lipsschizz Cauchy-Schwarc ($2, W2 VVKCC2&
For the second term
(i
(( x'aa < Colla(- x) x(- 12)1 A1 - (3), (..w({,)))) Ai)l
18

---

where the first transition used that [| Cq for all Q E ê(Cs. Cw. CO) and the second used the Lipschitz property of the policy class shown above. Combining the terms and assuming that Co 2 1 we get that
[V T. Q) V(a 2() C A1) - (8i, A1)l
VV6KKCGCC30,l W2, A2) - (32, w/ Ab) V1922K
implying that (C3, Car: Cq) is Cg Lipschit in supremum norm. Finally, notice that
|||| < [8:| + [82] + ||0 ||w2||
||A2|l S 2KKC 2KC 2Vd, |||
and applying lemma 24 concludes the proof.
Proxy good event. We define proxy good event Eg = E1 n Eg n Eg where
Eg = = [K], h [H). V e (B + Bp, 28 BQN+1) l1 Vil s 3z3
(11)
where 3Q HH + h), h E HH + 1]. Then we have the following result.
Lemma 11 (Proxy good event). Consider the parameter setting of lemma Then Pr Pr[E,] 2 8.
Proof. First, by lemma 21 and our choice of parameters, Ei (eq. (7) holds with probability least 1 1-8/3 6/3. Next, applying lemmas and 22] we get that with probability at least 1 5-/3 simultaneously for all & E [K], E [M], V V3 + 3p. 28 (q,++1)
11 V
I 480 V
+ + 12d2 21gp://02006070022222220----/-11 KC + 1))
< 480 dlog(2k)) + 2lg - 12d log(1 + 2K / (K/2222))))((//( 23 + 1))
< 480 (log((2))=2oog66//))))) - 12d log(7K9/2)
< 4,Qd 12lod//21og(10000 12log(10K5HH)5)
(ogHd/log(10K3
Ap
implying Ey (eq. Finally, notice that || 1 S 1, thus < Y, < (33(3,+ Using lemma 20 Bernstein-type ttype inequality for martingales, conclude that E3 (eq. (9) holds with probability at least 8/3.
The good event. The following results show that the proxy good event implies the good event.
Lemma 12. Suppose that Eg holds. Tf [1((B, + 28),28(k3) for all [M] then Q 1(3 + 3,, 28gK, Bq.t), ) V E D(3, + 3p. 28 8q,1) for all E [H + 1].
Proof. We show that the claim holds by backward induction on E [H + 1].
Base case = H + 1: Since V4= = it is also implied that Q = O. Because B, w = 0 E Y(3, + 3p, 2,, Bo 0) we have that Q) E 2(8, 3p. 23g K, .Q 0), and
similarly V (B, + 8p, 28 8Q 0).
19

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Induction step: Now, suppose the claim holds for h + 1 and we show it also holds for h. We have that
(2, (2x,a))
10 (x, a)" w Bolloh (z, (
- 10% (z, a)"a"a))) (On + (0) 0n) + (3%) 2/n) W+i + s'n 1441 + Bollø% (I, a)llaato) S 1 + 11 (r, a))l )>)(())))-1 [10% 0n. IlAy + 11 - th Vitillls + Bbb
(triangle inequality, Cauchy-Schwarz, A = A/
BQN (3, + pp.h + 16)110% (xx, (at))
(induction hypothesis, eqs.
and definition)
(3, + Ap., + 3b) maxly 920
a--Bwy log K)]
(
2 2log(eK) (3r + 3p,h + 3b)
< + .Q.++1
(lemma 18
Bw
(3w 2 2(, + 8mn + Bb) log(eK))
8q.h.
Additionally, 9o = 3r + Bp (A 1 I, | I 1 E EvErK] 10h S 2K,, thus (A = (2K)-11, and
|| = 10 + / || S K + B0K & 23 = Cw.
We conclude that Q 2(3, + 3p 28 82 Since म E [1((33, + 8p). 23 we also conclude that V//t E VB3, Bp 28 K, B) proving the induction step and finishing the proof.
Lemma (restatement of ( Consider the parameter setting of theorem nf S 1, B S K/(32Hld) then Pr[E,] 2
Proof. Suppose that Eg holds. By lemma 11 this occurs with probability at least 1 - 8. We show that Eg implies Eg. thus concluding the proof. Notice that
k-1
(a/x) & exp Σ
O% (c.
k-1
mi -no ka)lld(z, I)
expp o(- 1 + logK).
á(z,a)T
Σ
k'=he
We show by induction on k E Ke that म [B, + 3p). 23 for all h E [H]. For the base
case, k = ke, nr are uniform, corresponding to w. B = 0 [T( [3, + (p), K²). Now, suppose the claim holds for all k' < k. Then by lemma 12 have that Q E 2 Sp, 28QK, (Q,). for all k' < and h E [H]. This implies that 1 nw || S K2 for all E [H. thus Th E II(K(B, + 3p), K2) for all h [H] concluding the induction step.
Now, since ThK E [1( + B,), 2 for all h E [K], E [H], we can apply lemma 12 to get that Q% Ô(3, Pp, 2,9 K, R),, VA D(B + Sp. 28Q K, (Q.a) for all k E [K], h H Using Eg (eq. (11) we conclude that Es (eq. 88 holds, thus concluding the proof.
20

---

B
Technical tools
B.1 Online Mirror Descent
We begin with a standard regret bound for entropy regularized online mirror descent (hedge). See (Sherman et al. 2023a Lemma 25).
Lemma 13. Let yı, ..... Ur IR be any sequence of vectors, and nj > such that my)(a) -1 for all E (T).e e [A]. Then if xa E Aa is given by (1(a) 1/A Va, and for t 2 1:
E my)())
1141(a)= = E (((d')
then,
T
log A
va(a)(z((a) x<a)) -
+ 1 Σ f= [E(A]
Σ
(a)a
A
"1
[A
B.2
Value difference lemma
We use the following extended value difference lemma by Shani et al. 2020
We note that the lemma holds unchanged even for MDP-like structures where the transition kernel P is a sub-stochastic transition kernel, i.e., one with non-negative values that sum to at most one (instead of exactly one).
Lemma 14 (Extended Value difference Lemma 1 in Shani et al. [020]) , Let M be a (sub) MDP. T, 1l be two policies, Qn. X' A + R, E [M] be arbitrary function, and Vn, defined as Vn(x)= = EREA (a x x))n (2, (2. a). Then
IR be
( - V((21) Ep
Σ NE|H] aEA
Σ
a) 2x) #(a znl)
+ Ep
Σ hE(H]
Ln( (n) + Σ P(æ' Ihe Ini Vnti -i
FEX
We note that, in the context of linear MDP (x(2n,a)) + Evex | In, b(nn, T + the Vn+1).
B.3 Algebraic lemmas
Next, is a well-known bound on harmonic sums [see, e.g., Cohen et al. 2019 Lemma 13J. This is used show that the optimistic and true losses are close the realized predictions.
Lemma 15. Let E dd be sequence such that ||ee|| < X. and define Vn Al Σt=l 242g. Then
T
T T
Σ I S
S / 2T'd log(T + 1).
Next, need the following well-known matrix inequality. Lemma (Cohen et [2019) Lemma 27). IfN M then for any vector v
|| det. N
||0113r
dct M
Next, we need bound on the Lipschitz constant of the spectral norm of a square-root matrix.
Lemma 17. For any X > 0 and matrices A' E Idxdd satisfying A, 2 Al we have that
1/2 A 1//
<
2 |A
AlI.
21

---

Proof. Let je be an eigenvalue of A 1/2 / with eigenvector E R. Then we have that
1 (A- = |2 (A 1/2 A2/2)A + + ( A2/2)xl
llaaa (A +A"
Next, notice that |x - Aa)=| - ||
| ||A - Al and xT (A + A z 2//[2=]/. We thus
therefore change sides to get that
|e| - 2V ||A AlII,
and since we can take μ = -1 the proof is concluded.
Finally, we need the following bounds on the logistic function.
Lemma 18. For any K 2 B > 0 we have that
2 l))(e
max(y a(- By log K)] S y20
Proof. First, y S (2 / then using (1/) E [0, 1] we have that
y'o(-By' log K) S 3/ S (2 log K)/3,
as desired. Now, if y' 2 log K then
y
y'a(-By' + log K) S {3/o(-By//2) 1 Byy'
By'/2
B
where the first inequality also used that oyy) is increasing and the last inequality used that 1 +c" for all y 2 O.
Lemma 19. For any K 2 1, 2 0 we have that a(= log K) S 2(= + K-1).
Proof. Recall the logistic function oz)) 1/(1 + = and define the function g(z) o(z
log K) (z + K K-12222 We show that g(z) < for all 0. First, notice that
K-1 g(0) o0- log K) K (K -1-1i-K- 0.
Next, recall that o' (z) = 0 (z)o and thus
(2) g((2) = a( - log K)o(-2 + log K) - 2(2 + K K-1/2).
Examining 0 we further have that
g' (0) lo- Koo K) 2K 1/2
-K 1K 1 2K- 1/2
K- S 2[(K + - < O,
where the last inequalities used K 2 Now, we have that
g"(=) = a(2 - log K)a(- + log K)² - (z - log K))a log K) - 2,0
where the inequality is since (2) E [0, 1] for all 2 E R. Since g g'( < 0 and gz) < 0 for
all 2 0, we conclude that g(z) < 0 for all 2 2 0. The proof is concluded using the AM-GM inequality.
B.4 Concentration bounds
We give the following Bernstein type tail bound (see e.g., [Rosenberg et al. 2020 Lemma D.4].
Lemma 20. Let (Xthss be sequence of random variables with expectation adapted to a filtration
Fe. Suppose that 0 < X₃ < 1 almost surely. Then with probability at least 1 - ô
T
T
E[X, [F <
Σ
X, + log 3
1=1
22

---

We state the well- self normalized error bounds for regularized least squares estimation of the rewards and dynamics (see e.g., Abbasi- Yadkon al. [201T]. Din al. (2020b))
Lemma 21 (reward error bound), Let 0% be as in line of algorithm 1- for all k 2 1, E [M]
With probability at least
||0, Alll $ 2 2d 2KH/8).
Lemma 22 (dynamics error uniform convergence). Let Ji R E be the linear operator defined in eq. ② inside algorithm 7 For all h { let V/. C E be set of mappings V X R such that V loo S 3 and 3 z 1. With probability at least 1 8, for all [H], V Va+1 and k 2
11(v% -i---/)/1 VIl S 43 / + + 2og
where e & BVd/2K, - Nh. and Nu,e is the e coverring number of Vi, with respect to the supremum distance.
B.5 Covering numbers
The following results are (mostly) standard bounds on the covering number of several function classes
Lemma 23. For any > 0, the e-covering of the Euclidean ball in R.d with radius R 2 0 is upper bounded by (1 + 2R/c)a.
Lemma 24. Let V (V(::) ||||| S W} denote class of functions V X - R. Suppose that any V V L-Lipschitz with respect to and supremum distance, i.e.,
[V(;;1)) (+: & T| 02ll, |10:11, || & W.
Let Ne be the c-ooering number of V with respect to the supremum distance. Then
log.N. s dlog(1 + 2W L/e)
Proof. Let O,yr. be an (e/L)-covering of the Euclidean ball in Rd with radius W. Define V. (V(:0) 0 1./L]. By lemma 23 we have that log V.l d'log(1 + 2/c)... We show that V. is an e-cover Ve, thus concluding the proof. Let E and be its associated parameter. Let 0' E O./2 be the point in the cover nearest 9 and V E V its associated function. Then we have that
- - V'lla (IV(:)) v(:6))l s L| l L(eLL)
23

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Reviews:
[{"review_id": "TsY6sSdiRE", "reviewer": "Reviewer_JkeU", "summary": "This paper studies online learning for linear MDPs with stochastic and adversarial full-information losses. The authors propose a new contraction mechanism, avoiding the costly initial exploration phase in previous papers and achieving a better regret bound.", "questions": "Currently, I do not have questions.", "limitations": "Yes.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper studies an important problem with improved regret bounds.\n2. The new algorithm gets rid of a costly initial exploration phase in previous papers using new techniques, which may give more insights for future works in both theory and practice.", "weaknesses": "I do not see obvious weakness. For writing suggestions, I feel it is beneficial to discuss more about the difficulty of bounding covering numbers when applying PO for linear MDP. Although [Zhong and Zhang 2023] and [Sherman et al. 2023a] gave comprehensive discussions, it is good to discuss more in this paper to ensure the readers have a whole picture (e.g. discuss more why clipping Q-functions does not lead to good covering numbers for policy class).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for the positive comments. We agree that reiterating the explanations on the effect of clipping on the covering number will make the paper more self-contained and we will include it in the final version of the paper.", "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": "5MKGm39HIl", "reviewer": "Reviewer_x3ta", "summary": "This paper presents a new policy optimization algorithm called Contracted Features Policy Optimization (CFPO) for reinforcement learning in linear Markov Decision Processes (MDPs). The key contribution is eliminating the need for a costly warm-up phase used in previous state-of-the-art methods, while achieving improved regret bounds.", "questions": "na", "limitations": "The authors should include a more detailed discussion of the study's limitations in the conclusion part.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper addresses a significant issue in reinforcement learning theory, improving on recent findings for policy optimization in linear MDPs.\n\nThe proposed CFPO algorithm eliminates the need for a separate warm-up phase, offering a substantial practical advantage over previous methods.\n\nThe regret bounds show improvement compared to prior work, achieving $O(\\sqrt{H^4 d^3 K})$ regret, which represents a $\\sqrt{H^3 d}$ enhancement.\n\nThe analysis introduces novel techniques, such as the contracted MDP concept, which may have broader applications in the field.", "weaknesses": "While the regret bound in this paper shows a better dependency on K compared to previous works, it has a worse dependency on d and H. Although K is the most critical factor, it might be more appropriate to discuss the improvement in regret when K exceeds a certain threshold.\n\nThe paper is purely theoretical, lacking experimental results to validate the practical performance of the proposed algorithm. More discussion on practical implications and potential applications would be beneficial.\n\n\nDespite the improvement, there remains a gap between the upper and lower bounds for this problem. Further discussion on potential approaches to address this gap would be valuable. For example, can previous variance reduction techniques be applied to this method?\n\n The paper could be strengthened by adding a conclusion section to summarize the key findings", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the time and effort put into the review. The following addresses the points made in your review:\n1. Regarding dependence on $K,d,H$: Are you referring to the additive regret term that is logarithmic in $K$? As you mentioned, most works assume that $K$ is the dominant factor and thus this term is omitted. For our algorithm it becomes a low order term when $K \\ge \\sqrt{H^4 d^3}$. Notice that the regret bound of $\\sqrt{H^4 d^3 K}$ is trivial, i.e., $ \\ge KH$, when $K \\le \\sqrt{H^2 d^3}$ and thus the additional term is non-trivial only between these two values. The regret bound of Sherman et al. (2023) is non-trivial only for $K \\ge \\sqrt{H^5 d^4}$ and thus our regret bound is equivalent or better for all values of $K,d,H$. Overall, you bring up a subtle point. If you think this will improve the paper then we are willing to explain this in the final version.\n2. Discussion: We will add a discussion about potential applications, limitations, and the gap from the lower bound. We conjecture that it is possible to use variance reduction methods for PO. However, this seems quite complicated and thus left for future research.", "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": "h8F4WbPkJ0", "reviewer": "Reviewer_Gemc", "summary": "This paper equips the rare-switching mechanism with a novel feature shrinkage technique to achieve efficient policy optimization (PO) for linear MDPs with adversarial losses or bandit feedback. By shrinking features in directions of high uncertainty, the authors show that the proposed algorithm has its regret optimal in terms of $K$, the number of episodes, up to a logarithmic factor. Compared with prior work on PO for adversarial linear MDPs, the proposed algorithm does not invoke sophisticated black boxes or incur any $\\text{poly}(d, H)$ burn-in cost and enjoys a regret upper bound with lower dependence on the horizon $H$ and the ambient dimension $d$.", "questions": "1. Any reference on the proposition mentioned on Line 134-135 that the Markov $\\pi^*$ in hindsight is optimal among all policies?", "limitations": "not applicable", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The proposed feature shrinkage technique, which enables the estimated state-action value to be bounded without resort to truncation techniques, is simple and does not incur any additional statistical or computational overhead up to constant factors.\n2. The authors notice that the extended value difference lemma [1] is applicable (for the proposed contracted subMDP) even if the transition kernel is sub-stochastic, which is a key observation that might be of independent interest and useful for the reinforcement learning (RL) community.\n\n\n[1] Shani, L., Efroni, Y., Rosenberg, A., & Mannor, S. (2020, November). Optimistic policy optimization with bandit feedback. In International Conference on Machine Learning (pp. 8604-8613). PMLR.", "weaknesses": "## Weaknesses\n\n1. The authors should make it clear in the main text whether the analysis or the final regret bound depends on $|\\mathcal{X}| < \\infty$ in any significant way. As far as I can tell, the regret bound does not rely on the finite $\\mathcal{X}$ assumption, but the authors should further clarify this point.\n2. Line 266: [1] does not utilize any feature shrinkage technique, so it is confusing and not adequate to mention \"the corresponding claim\" and use the $\\bar{\\phi}^{k_e}$ notation in equation (6) without further explanation.\n3. Line 49: Though the results in this paper is significant, it is way too assertive to say that no simpler algorithm with lower regret exists for adversarial linear MDPs. The authors should at least elucidate their conjecture about the optimality of the proposed algorithm in a more appropriate manner.\n4. Minor issue: Technically speaking, the two $\\sum_{h}$ signs between Line 276 and 277 should not be put outside the expectation in that your expectation is taken with respect to the randomness of a trajectory.\n5. Between Line 151 and 152: the $\\Delta P(x_h, a_h)$ part seems to be a typo\n6. The key connection between the proposed technique, especially Lemma 18, and the boundedness of the estimated state-action value function (i.e., the arguments between Line 549 and 550) is not explained in any way in the main text. (BTW, in the proof of Lemma 18, it seems that $K\\geq 1$ should be changed to $K\\geq \\mathrm{e}$)\n7. Minor typo on Line 265: $\\succ$ -> $\\leq$\n8. Minor typo on Line 579: $\\hat{V}$ -> $\\hat{V}_h$\n\n[1] Sherman, U., Cohen, A., Koren, T., & Mansour, Y. (2023). Rate-optimal policy optimization for linear markov decision processes. arXiv preprint arXiv:2308.14642.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the thorough review and helpful comments, we will incorporate them in our revision. The following responds to your individual points:\n1. Finite $\\mathcal{X}$: You are correct, the regret does not depend on the assumption that $\\mathcal{X}$ is finite. In short, the assumption is purely technical as it helps avoid measure theoretic notations and allows for a cleaner and more approachable analysis. The explanation for this was deferred to an existing paper but we agree that it should be included in our paper for the sake of completeness.\n2. L266: Thank you for pointing this out, we will revise this explanation in the final version. Our intention was to say that the reward free warm-up in [1] gives a guarantee of the form in eq.(6) but where $\\bar{\\phi}_h^{k_e}(x,a)$ is replaced with $\\phi(x,a) \\mathbb{1}(x \\in \\mathcal{Z}_h)$. The overall implication is that, through the lens of contracted sub-MDPs, the reward free warm-up would give an overly conservative contraction that incurs additional cost compared to our approach.\n3. L49: The purpose of this phrase was to say that while our rates are not minimax, improving them further likely requires more delicate algorithmic techniques that are not well-understood for PO even in tabular MDPs. In particular, we are not aware of any method besides the mentioned variance reduction technique that achieves better rates. We will soften the phrasing of the final version such that it is more clear that this is a conjecture rather than a proven fact.\n4. L276-277: The expectation is indeed over trajectories. However, because their length is fixed, we can take the $\\sum_h$ outside due to the linearity of the expectation. As you pointed out, the arguments work either way.\n5. L151-152 $\\Delta P$: We overloaded notation here (perhaps excessively). We’ll disambiguate it in the final version.\n6. Key connection… boundedness of $\\hat{Q}$: We omitted this due to space constraints. We’ll include an explanation in the final version. Thanks for pointing out that $K \\ge e$ in Lemma 18.\n7. Minor typos (points 7-8): Thanks for finding these! We’ll fix them for the final version.\n8. Markov $\\pi^\\star$ is optimal even among history dependent policies: There are probably several sources for this claim. For example, Reinforcement Learning:  Foundations (p.56)  by Shie Mannor, Yishay Mansour, and Aviv Tamar.", "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": "5mfbNmpKoO", "reviewer": "Reviewer_gZJ8", "summary": "This paper improves the Policy Optimization methods for learning MDP by eliminating the undesired warm-up phase and replacing it with a simple and efficient contraction mechanism. For linear MDP, it is shown that the proposed Policy Optimization algorithm achieves regret with improved dependence on problem parameters (the horizon and function approximation dimension) under the settings of adversarial losses with full-information feedback and stochastic losses with bandit feedback. The contraction mechanism serves the purpose of ensuring the Q-value estimates are bounded and yield simple policies in the sense of efficient computation and easy implementation. The regret bound improves upon the best known ones for policy optimization.", "questions": "Please see Weaknesses.", "limitations": "Yes.", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 4, "contribution": 3, "strengths": "**Significance**: \n\n**1.** This paper improved the regret bound for policy optimization algorithms for linear MDPs. Specifically, the CFPO algorithm achieves a $\\sqrt{H^3 d}$ improvement over the best known regret for adversarial setting, and matches the performance of the value-iteration based algorithm. \n\n**2** The policy optimization algorithm is computationally more efficient and easier to implement. This is because the algorithm uses all samples and is reward-aware (thus no waste of information). \n\n**Clarity**: This paper is clearly written.\n\n**1.** All necessary related work is properly discussed in my opinion.\n\n**2.** The algorithm is introduced with a clear explanation. In section 4, a detailed walk-through of the CFPO algorithm is given with explicit items comparing the new algorithm with existing one, making it clear why the new one enjoys improved performance. A simple example is given (bottom of page 6) to demonstrate why reward-awareness is beneficial in PO.", "weaknesses": "There is no major technical flaw detected in this paper. \n\nWeakness:\n\nThere is no experimental result to corroborate the theoretical results. Though PO algorithms are a bit more complicated to implement than the value-iteration algorithms in my opinion, for linear MDP, a numerical simulation or simple synthetic experiment should not seem too difficult.\n\nAnd I think experimental results is especially important for this work, since it claims the deviced algorithm is easier to implement compared to existing PO algorithms. Therefore, experimental results verifying the correctness of the improved regret and the easier implementation than existing PO baselines are anticipated. For example, for the improved regret, it would be good to see from experiment that the dependence on the problem parameter is indeed improved. A very simple synthetic environment should be enough.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the time and effort put into the review.\n\nRegarding synthetic experiments, we are not aware of existing benchmarks that are specific to linear MDPs rather than tabular ones. However, in tabular MDPs the contraction is not necessary and thus we do not expect to see improvement. We agree that coming up with interesting experimental setups for linear MDPs would be valuable to the community but leave this to future research. Please note that most papers on regret minimization in MDPs do not include experiments.", "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 introduces the CFPO algorithm for linear MDPs, improving upon prior work by removing the warm-up phase and achieving better regret bounds. Reviewers praised its theoretical contributions, particularly the novel contraction mechanism, and its potential impact on reinforcement learning. However, they raised concerns about the lack of experimental validation and suggested clarifying some technical dependencies on problem parameters.

In rebuttal phase, the authors clarified the points raised, addressed minor presentation issues, and acknowledged the need for future experiments. Due to the paper’s strong theoretical results, reviewers were overall positive after the rebuttal. I recommend acceptance of this paper as a poster presentation.
Decision: Accept (poster)