Warm-up Free Policy Optimization: Improved Regret in Linear Markov Decision Processes

Authors: asaf  cassel
aviv  rosenberg
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-07 23:24:29

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.

Main paper: 
 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 [Schulman et al., 2015 , 2017 , Mnih et al., 2015 , Haarnoja et al., 2018] to Large Language Models (LLMs; Stiennon et al. [2020] , Ouyang et al. [2022] ). Theoretical work on policy optimization algorithms initially considered tabular Markov Decision Processes (MDPs; Even-Dar et al. [2009] , Neu et al. [2010b] , Shani et al. [2020] , Luo et al. [2021] ), 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 et al. [2023a] presented the first policy optimization algorithm that achieves rateoptimal regret in linear MDPs, i.e., a regret bound of O(poly(H, d) √ K), where K is the number of interaction episodes, H is the horizon, and d 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 Wagenmaker et al. [2022b] for reward-free exploration which is not based on the policy optimization paradigm. Moreover, although this algorithm is computationally efficient, it relies on intricate estimation techniques that are hard to implement in practice and unlikely to generalize beyond linear function approximation (see discussion in section 4).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 a 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 a simple conditional truncation of the Q-estimates, and only contributes a lower-order term to the final regret bound. Moreover, our approach is much 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. [2023b] ) was O( √ H 7 d 4 K). Our algorithm, Contracted Features Policy Optimization (CFPO), achieves O( √ H 4 d 3 K) regret, yielding a √ H 3 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. 1 Ignoring logarithmic factors, the regret of CFPO leaves a gap of onlyHd from the Ω( √ H 3 d 2 K) lower bound for linear MDPs [Zhou et al., 2021a] . Finally, our analysis relies on a novel regret decomposition that uses a notion of contracted (sub) MDP and may be of separate interest (see section 5).Policy optimization in tabular MDPs. The regret analysis of PO methods in tabular MDPs was introduced by Even-Dar et al. [2009] , which considered the case of known transitions and adversarial losses under full-information feedback. Neu et al. [2010a,b] 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 Luo et al. [2021] 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 [Lancewicki et al., 2022 [Lancewicki et al., , 2023] ] and best-of-both-worlds [Dann et al., 2023] .Other regret minimization methods in tabular MDPs. An alternative popular method for regret minimization in tabular MDPs with adversarial losses is O-REPS [Zimin and Neu, 2013 , Rosenberg and Mansour, 2019a ,b, Jin et 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 in practice and does not generalize to the function approximation setting (without restrictive assumptions). For stochastic losses, optimistic methods based on Value Iteration (VI; Jaksch et al. [2010] , Azar et al. [2017] , Zanette and Brunskill [2019] ) and Q-learning [Jin et al., 2018 , Zhang et 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 et al., 2021 , Neu and Olkhovskaya, 2021 , Dai et al., 2023 , 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 approximation. 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 , Hu et al., 2022 , He et al., 2023 , Agarwal et al., 2023] . Another line of works [Ayoub et al., 2020 , Modi et al., 2020 , Cai et al., 2020 , Zhang et al., 2021 , Zhou et al., 2021a ,b, He et al., 2022 , Zhou and Gu, 2022] study linear mixture MDP which is a different model that is incomparable with linear MDP [Zhou et al., 2021b] . Finally, there is a rich line of works studying statistical properties of RL with more general function approximation [Munos, 2005 , Jiang et al., 2017 , Dong et al., 2020 , Jin et 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 defined by a tuple (X , A, x 1 , { k } K k=1 , P, H) with X , a set of states, A, a set of actions, H, decision horizon, x 1 ∈ X , an initial state (assumed to be fixed for simplicity), P = (P h ) h∈[H] , P h : X × A → ∆(X ), the transition probabilities, and{ k } K k=1 , sequence of loss functions such that k = ( k h ) h∈[H] , k h : X × A → [0, 1], is a horizon dependent immediate loss function for taking action a at state x and horizon h of episode k. A single episode k of an MDP is a sequence(x k h , a k h , k h (x k h , a k h )) h∈[H] ∈ (X × A × [0, 1]) H such that Pr[x k h+1 = x | x k h = x, a k h = a] = P h (x | x, a).For the losses, we consider two settings: stochastic and adversarial. In the stochastic setting, there exists a fixed loss function= ( h ) h∈[H] , h : X × A → [0, 1] such that k is sampled i.i.d from a distribution whose expected value is defined by , i.e., E k h (x, a) | x, a = h (x, a).In the adversarial setting, the loss function sequence { k } K k=1 is chosen by an adaptive adversary.Linear MDP. A linear MDP Jin et al. [2020b] satisfies all the properties of the above MDP but has the following additional structural assumptions. There is a known feature mapping φ :X × A → R d such that P h (x | x, a) = φ(x, a) T ψ h (x ) where ψ h : X → R d are unknown parameters. Moreover, for all h ∈ [H], k ∈ [K], there is an unknown vector θ k h ∈ R d such that, in the adversarial case, k h (x, a) = φ(x, a) T θ k h , while in the stochastic case, θ k h = θ h and h (x, a) = φ(x, a) T θ h .We make the following normalization assumptions, common throughout the literature:1. φ(x, a) ≤ 1 for all x ∈ X, a ∈ A; 2. θ k h ≤ √ d for all h ∈ [H], k ∈ [K]; 3. |ψ h |(X ) = x∈X |ψ h (x)| ≤ √ d for all h ∈ [H];where |ψ h (x)| is the entry-wise absolute value of ψ h (x) ∈ R d . We follow the standard assumption in the literature that the action space A is finite. In addition, for ease of mathematical exposition (e.g. Cassel et al. [2024] ), we also assume that the state space X is finite. This allows for simple matrix notation and avoids technical measure theoretic definitions. Importantly, our results are completely independent of the state space size |X |, both computationally and in terms of regret. Thus, there is no particular loss of generality.π = (π h ) h∈[H] : [H] × X → ∆(A)is a mapping from a step and a state to a distribution over actions. Such a policy induces a distribution over trajectories ι = (x h , a h ) h∈[H] , i.e., sequences of H state-action pairs. For f : (X × A) H → R, which maps trajectories to real values, we denote the expectation with respect to ι under dynamics P and policy π as E P,π [f (ι)]. Similarly, we denote the probability under this distribution by P P,π [•] .We denote the class of stochastic Markov policies as Π M . For any policy π ∈ Π M , horizon h ∈ [H] and episode k ∈ [K] we define its loss-to-go, asV k,π h (x) = E P,π H h =h E[ k h (x h , a h ) | x h , a h ] x h = x ,which is the expected loss if one starts from state x ∈ X at horizon h of episode k and follows policy π onwards. Note that the inner expectation is only relevant for stochastic losses as its argument is 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 k,π 1 (x 1 ).Interaction protocol and regret. We consider a 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 k in the end of episode k ∈ [K]. Concretely, at the start of each interaction/episode k ∈ [K], the agent specifies a stochastic Markov policy π k = (π k h ) h∈ [H] . Subsequently, it observes the trajectory ι k sampled from the distribution P P,π k , and, either the individual episode lossesk h (x k h , a k h ), h ∈ [H]in the case of bandit feedback, or the entire loss function k in the case of full-information feedback.We measure the quality of any algorithm via its regret -the difference between the value of the policies π k generated by the algorithm and that of the best policy in hindsight, i.e.,Regret = K k=1 V k,π k 1 (x 1 ) -min π∈Π M K k=1 V k,π 1 (x 1 ) = K k=1 V k,π k 1 (x 1 ) -V k,π 1 (x 1 ),where the best policy in hindsight is denoted by π (known to be optimal even among the class of stochastic history-dependent policies).Notation. Throughout the paperφ k h = φ(x k h , a k h ) ∈ R d denote the state-action features at horizon h of episode k. In addition, v A = √ v T Av.Hyper-parameters follow the notations β z and η z for some z, and δ ∈ (0, 1) denotes a confidence parameter. Finally, in the context of an algorithm, ← signs refer to compute operations whereas = signs define operators, which are evaluated at specific points as part of compute operations. 
the role of value clipping: 
 Before presenting our contraction technique and main results, we discuss the role that value clipping plays in regret minimization and its apparent necessity for linear MDPs. As a starting point, it is important to note that, while commonly used [Azar et al., 2017 , Luo et al., 2021] , value clipping is not strictly necessary in tabular MDPs. To demonstrate this, consider a fairly standard optimistic Value Iteration (VI) algorithm that constructs sample-based estimates ˆ , P with empirical error estimates ∆ , ∆ P , defines exploration bonuses b = (∆ + H • ∆ P ), and chooses a policy π that is optimal in the empirical MDP whose dynamics are P and losses are ˆ -b. Then its single-episode regret may be decomposed asV π 1 (x 1 ) -V π 1 (x 1 ) = V π 1 (x 1 ) -V π 1 (x 1 ) (i)-bias / cost of optimism + V π 1 (x 1 ) - V π 1 (x 1 ) (ii)-FTL / ERM + V π 1 (x 1 ) -V π 1 (x 1 ) (iii)-optimism, where V is the value under the empirical MDP. Now, by definition of π , we have that (ii) ≤ 0. Now, let ∆ = ˆ -, ∆P = P -P . Using a standard value difference lemma (lemma 14 in appendix B) we have that (i) b and(iii) = E P ,π   h∈[H] ∆ (x h , a h ) -b(x h , a h ) + x ∈X ∆P (x | x h , a h )V π h+1 (x )   (1) ≤ E P ,π   h∈[H] ∆ (x h , a h ) + H∆ P (x h , a h ) -b(x h , a h )   = 0,where the inequality also used thatV π h ∈ [0, H].The final regret bound is concluded by summing over k ∈ [K] and using a bound on harmonic sums. We note that a similar clipping-free method also works for tabular PO (see Cassel et al. [2024] ).Moving on to Linear MDPs, one might expect a similar approach to work. Unfortunately, the standard approach that estimates the dynamics backup operators ψ h , h ∈ [H] using regularized least-squares presents a significant challenge. This is because, unlike the tabular setting, the resulting estimate 2)) is not guaranteed to yield a valid probability distribution, i.e., there could exist x ∈ X , a ∈ A, h ∈ [H] such that Ph (• | x, a) 1 = c > 1 and/or minPh (• | x, a) = φ(x, a) T ψ h (•) (eq. (x ∈X Ph (x | x, a) < 0. P is still a finite signed-measure, which is enough for the first equality in eq. (1) to hold. However, since E P ,π could contain negative probability terms, the inequality in eq. ( 1) does not hold. These negative probabilities also seem to make calculating π computationally hard. Finally, the 1 -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 a method that resolves all simultaneously.The use of value clipping opens the path for an alternative value decomposition that replaces E P ,π in eq. ( 1) with E P,π at the cost of also replacing V π h+1 with V π h+1 . We thus need that | V π h+1 | H for the inequality in eq. ( 1) to work. This is made possible using a clipping mechanism that decouples the scale of V π h+1 from the magnitude of the bonuses b, which may be much larger when the error estimates ∆ , ∆ P are large. This is 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 et al., 2021] , and even for VI in linear MDPs [Jin et al., 2020b] . However, this is not the case for PO in linear MDPs where Sherman et al. [2023a] 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 k∈ [K] Qk h . If Qk h (x, a) = φ(x, a) T w k h are linear, then the sum remains linear and depends on d parameters (with slightly larger magnitude). If Qk h (x, a) = max{0, φ(x, a) T w k h }, the sum may, in general, have dK 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 a novel notion of contracted features and has several advantages over their approach (see discussion at the end of section 4).Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Justification: The paper does not include experiments.Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. • The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. • The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). 
algorithm and main result: 
 We present Contracted Features Policy Optimization (CFPO; algorithm 1), 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, as expressed by the determinant of the covariance matrices Λ k h , h ∈ [H] (see line 13 for the definition of Λ k h and line 4 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 φke h , h ∈ [H] (line 6) by multiplying the original features with coefficients in the range [0, 1], and thus shrinking their distance to 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 in 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 θ k (line 14) and estimated dynamics backup operators ψ k h (eq. ( 2)). Then, we use these θ k and ψ k h to compute our Q-value estimates with the contracted features (eq. ( 3)), and run an online mirror descent (OMD) update over them (eq. ( 5)), i.e., run a policy optimization step with respect to the contracted empirical MDP (more on this in section 5.1).We note that the computational complexity of algorithm 1 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 1 (see the full analysis appendix A).Theorem 1. Suppose that we run CFPO (algorithm 1) with the parameters defined in theorem 9 (in appendix A). Then, with probability at least 1 -δ, we haveRegret = O H 4 d 3 K log(K) log(KH/δ) + H 5 dK log(K) log|A| .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 is in obtaining such estimates that satisfy an optimistic condition similar to eq. ( 1), while also keeping the complexity of the policy class bounded. As discussed in Sherman et al. [2023a] , the latter depends on k ∈[k] Qk h (eq. ( 3))Algorithm 1 Contracted Features PO for linear MDPs1: input: d, H, K, A, δ, β w , β b , η o > 0. 2: initialize: e ← -1, Λ 1 h ← I, h ∈ [H]. 3: for episode k = 1, 2, . . . , K do 4: if k = 1 or ∃h ∈ [H], det(Λ k h ) ≥ 2 det(Λ ke h ) then 5:e ← e + 1 and k e ← k.6:φke h (x, a) = φ(x, a) • σ -β w φ(x, a) (Λ ke h ) -1 + log K . {σ(z) = 1/(1 + exp(-z))} 7: π k h (a | x) = 1/|A| for all h ∈ [H], a ∈ A, x ∈ X . 8: end if 9:Play π k and observe losses( k h (x k h , a k h )) h∈[H] and trajectory ι k = (x k h , a k h ) h∈[H] .10:In the case of full-information feedback: observe θ k h .11:Define V k H+1 (x) = 0 for all x ∈ X .12:for h = H, . . . , 1 do 13:Λ k+1 h ← I + τ ∈[k] φ τ h (φ τ h ) T .14:θ k h ← (Λ k h ) -1 τ ∈[k-1] φ τ h τ h (x τ h , a τ h ), feedback = bandit θ k h , feedback = f ull.15:For any V : X → R, x ∈ X , a ∈ A define:EQUATIONEQUATIONEQUATIONEQUATIONend 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 3). On the other hand, the standard clipping method, which restricts the value to [0, 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 rewardfree pure exploration algorithm by Wagenmaker et al. [2022b] to obtain initial backup operators ψ 0 h , h ∈ [H] and subsets Xh ⊆ X , h ∈ [H] such that: (i) for every x, a ∈ Xh × A the bonuses (b in section 3), which are proportional to the estimation uncertainty of the value backup estimates, are small (≤ 1); and (ii) for all policies π ∈ Π M , the probability of reaching any x, a / ∈ ∪ h∈[H] Xh × A is small ( 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 K 0 = poly(d, H) √ K episodes, contributing the leading term in their regret guarantee;• It relies on a first-order regret algorithm by Wagenmaker et al. [2022a] that is not PObased and uses a 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 an approximate version of the estimator, losing a factor of √ d in the regret;• Still, to the best of our knowledge, even the approximate estimator must be computed using binary search methods, making it hard to apply in practical methods that typically rely on gradient-based continuous optimization techniques;• It runs separate algorithms for each horizon h ∈ [H], using only 1 out of H samples during the warm-up phase; • It is not reward-aware, and thus has to 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-complexity policy class without relying on a dedicated warm-up phase. Crucially, it only contributes a lower order term of poly(d, H) log K to the regret guarantee, thus improving the overall dependence on d and H. Additionally, it uses all samples, is easy to implement, and is reward-aware. To understand the benefit of reward-awareness, 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 a single state and action for the first H/2 steps with one MDP incurring a loss of 1 in these steps while the other incurring 0 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-aware approach would quickly stop exploring the inferior MDP, leading to better performance in practice. 
analysis: 
 In this section, we prove the main claims of our result. For full details see appendix A. We begin by introducing the main technical tool for our contraction mechanism -the contracted MDP.For any MDP M = (X , A,x 1 , { k } K k=1 , P, H) and contraction coefficients ρ : [H] × X × A → [0, 1] we define a contracted (sub) MDP M(ρ) = (X , A, x 1 , { ¯ k } K k=1 , P , H) where as ¯ k h (x, a) = ρ h (x, a) k h (x, a) ∈ [0, 1] are the contracted losses and Ph (x | x, a) = ρ h (x, a)P h (x | x, a) ∈ [0, 1]are the contracted (sub) probability transitions. Notice that the transitions being a sub-probability measure implies that x ∈X P h (x | x, a) ≤ 1 as compared with a probability measure where this holds with equality. For any Markov policy π ∈ Π M , let V k,π h (•; ρ) : X → R, h ∈ [H] be the loss-to-go (or value) functions of the contracted MDP. In particular, these may be defined by the usual backward recursionV k,π h (x; ρ) = E a∼π(•|x) E[ ¯ k h (x, a) | x, a] + x ∈X Ph (x | x, a) V k,π h+1 (x ; ρ) ,with V k,π H+1 (x; ρ) = 0 for all x ∈ X . The following result shows that the value of any contracted MDP lower bounds its non-contracted variant.Lemma 2. For any ρ :[H] × X × A → [0, 1], π ∈ Π M , h ∈ [H], k ∈ [K], and x ∈ X we have that V k,π h (x; ρ) ≤ V k,π h (x).Proof. The proof follows by backward induction on h ∈ [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 ∈ XV k,π h (x; ρ) = E a∼π(•|x) E[ ¯ k h (x, a) | x, a] + x ∈X Ph (x | x, a) V k,π h+1 (x ; ρ) ≤ E a∼π(•|x) E[ k h (x, a) | x, a] + x ∈X P h (x | x, a)V k,π h+1 (x ) = V k,π h (x).Next, for any epoch e ∈ [E], consider its contracted linear MDP (line 6 in algorithm 1) whosecontraction coefficients are ρ ke h (x, a) = σ -β w φ(x, a) (Λ ke h ) -1 + 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 ∈ [E] and v ∈ R d we have that(φ(x h , a h ) -φke h (x h , a h )) T v ≤ (4β 2 w φ(x h , a h )Proof. We have that(φ(x h , a h ) -φke h (x h , a h )) T v = σ(β w φ(x h , a h ) (Λ ke h ) -1 -log K) • φ(x h , a h ) T v ≤ 2(β 2 w φ(x h , a h ) 2 (Λ ke h ) -1 + K -1 ) φ(x h , a h ) T v ≤ (4β 2 w φ(x h , a h ) 2 (Λ k h ) -1 + 2K -1 ) φ(x h , a h ) T v ,where the first relation is by the property of the sigmoid 1 -σ(x) = σ(-x), the second is by a simple algebric argument that a quadratic function bounds the sigmoid (lemma 19 in appendix B), and the last relation uses det(Λ k h ) ≤ 2 det(Λ ke h ) by line 4 in algorithm 1 (see lemma 16 in appendix B).We note that the analogous claim in Sherman et al. [2023a] shows that for all π ∈ Π MEQUATIONwhere Z h is an outcome of the reward-free warmup phase and Pr(x h / ∈ Z h ) ≈ K -1/2 . Summing this over k ∈ [K] yields a term that scales as √ K.In contrast, we use a standard bound on elliptical potentials (lemma 15 in appendix B) to get thatk∈[K] (4β 2 w φ(x k h , a k h ) 2 (Λ k h ) -1 + 2K -1 ) log K.This implies that the cost of our contraction is significantly lower than the truncation of Sherman et al. [2023a] . We achieve this reduced cost by using a 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. [2023a] . In what follows, we overcome this challenge using a novel regret decomposition.For any epoch e ∈ [E], let K e be the set of episodes that it contains, and let V k,π 1 (x 1 ; ρ ke ) denote the value of its contracted MDP as defined above and in line 6 of algorithm 1. We bound the regret asRegret = k∈[K] V k,π k 1 (x 1 ) -V k,π 1 (x 1 ) ≤ e∈[E] k∈Ke V k,π k 1 (x 1 ) - V k,π 1 (x 1 ; ρ ke ) (lemma 2) = k∈[K] V k,π k 1 (x 1 ) -V k 1 (x 1 ) + e∈[E] k∈Ke V k 1 (x 1 ) - V k,π 1 (x 1 ; ρ ke ) = k∈[K] V k,π k 1 (x 1 ) -V k 1 (x 1 ) (i)-Bias / Cost of optimism + e∈[E] h∈[H] E P ke ,π k∈Ke a∈A Qk h (x h , a)(π k h (a | x h ) -π h (a | x h )) (ii)-OMD regret + e∈[E] k∈Ke h∈[H] E P ke ,π Qk h (x h , a h ) -φke h (x h , a h ) T (θ k h + ψ h V k h+1 ) (iii)-Optimism ,where the last relation is by the extended value difference lemma (see Shani et al. [2020] and lemma 14 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 φke h (x h , a h ) instead of the true features φ(x h , a h ). As a by-product, the expectation in terms (ii) and (iii) is taken with respect 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":EQUATIONEQUATIONE 1 and E 2 are error bounds on the loss and dynamics estimation, respectively. In the full feedback setting, E 1 holds trivially with β r = 0. In the bandit setting, it holds with high probability with β r = O( d log(KH/δ)) by well-established bounds for regularized least-squares estimation [Abbasi-Yadkori et al., 2011] . Showing that E 2 holds with high probability follows similarly to Sherman et al.[2023a], again using least-squares arguments but also using the contraction to ensure that Qk h are bounded (see sketch at the end of this section and lemma 6 in appendix A for full details), specifically β p = O(Hd log(KH/δ)). The proof of theorem 1 is concluded by bounding each of the terms in the regret decomposition, summing over k ∈ [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 A).Optimism and its cost. The following lemmas bound terms (iii) and (i), respectively.Lemma 4 (Optimism). Suppose that eqs. ( 7) and (8) hold, then Notice that the standard PO decomposition would have required that we bound the non-contracted expressionQk h (x, a) -φke h (x, a) T (θ k h + ψ h V k h+1 ) ≤ 0 , ∀h ∈ [H], k ∈ [K], x ∈ X , a ∈ A. Proof. We have that Qk h (x, a) -φke h (x, a) T (θ h + ψ h V k h+1 ) = φke h (x, a) T ( θ k h -θ h + ( ψ k h -ψ h ) V k h+1 ) -β b φke h (x, a) (Λ ke h ) -1 ≤ (β r + β p ) φke h (x, a) Λ k h -1 -β b φke h (x, a) (Λ ke h ) -1 ≤ (β r + β p -β b ) φke h (x, a) (Λ ke h ) -1 = 0,E P,π [ Qk h (x, a) -φ(x, a) T (θ k h + ψ h V k h+1 )].In Sherman et al. [2023a] the gap between this argument and that of lemma 4 can be bounded using eq. ( 6). However, the equivalent argument for our contraction is lemma 3, which is bounded only for π k and not for any policy π ∈ Π M .Lemma 5 (Cost of optimism). Suppose that eqs. (7) and (8) hold, then for every k ∈[K] V k,π k 1 (x 1 ) -V k 1 (x 1 ) ≤ 3(β r + β p )E P,π k   h∈[H] φ(x h , a h ) (Λ k h ) -1   + 16Hβ 2 w E P,π k   h∈[H] φ(x h , a h ) 2 (Λ k h ) -1   + 16H 2 K -1 .Proof. First, by lemma 14 in appendix B, a value difference lemma by Shani et al. [2020] ,V k,π k 1 (x 1 ) -V k 1 (x 1 ) = E P,π k   h∈[H] φ(x h , a h ) T θ h + ψ h V k h+1 -Qk k (x h , a h )   . Now, using lemma 3 with v = θ k h + ψ h V k h+1we have that |φ(x, a) T v| ≤ 4H (by eq. ( 8)) and thus[φ(x h , a h ) -φke h (x h , a h )] T θ h + ψ h V k h+1 ≤ 16Hβ 2 w φ(x h , a h ) 2 (Λ k h ) -1 + 16H 2 K -1 .We can thus conclude the proof using standard arguments to show thatφke h (x h ,a h ) T θ h + ψ h V k h+1 -Qk k (x h , a h ) = φke h (x h , a h ) T θ k h -θ k h + (ψ h -ψ k h ) V k h+1 + β b φke h (x h , a h ) (Λ ke h ) -1(eq. ( 3))≤ (β r + β p ) φke h (x h , a h ) (Λ k h ) -1 + β b φke h (x h , a h ) (Λ ke h ) -1(Cauchy-Schwarz, eqs. ( 7) and ( 8))≤ 3(β r + β p ) φke h (x h , a h ) (Λ k h ) -1 (det(Λ k h ) ≤ 2 det(Λ ke h ), β b = β r + β p ) ≤ 3(β r + β p ) φ(x h , a h ) (Λ k h ) -1 , (σ(x) ∈ [0, 1], ∀x ∈ R)as desired.Bounding the Q-values (proof sketch). The following are the main ideas in showing that E 2 (eq. ( 8)) holds with high probability. First, we define appropriate value classes V h that contain all value functions V h of the form in eq. ( 4) whose underlying Q h function (eq. ( 3)) satisfiesQ h ∞ ≤ 2(H + 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 22) to show that with high probability(ψ h -ψ k h )V Λ k h ≤ β p for all k ∈ [K], h ∈ [H], V ∈ V h . The proof is concluded by showing that Qk h ≤ β Q,h = 2(H + 1 -h),and thus V k h ∈ V h , which implies that eq. ( 8) holds. We prove this by backward induction on h ∈ [H + 1].The base case h = H + 1 is satisfied because, by definition, Qk H+1 = 0. Now, suppose the claim holds for h + 1 and we show it also holds for h. Recalling the definition of Q in eq. ( 3), we have that| Qk h (x, a)| = | φke h (x, a) T ( θ k h + ψ k h V k h+1 ) -β b φke h (x, a) (Λ ke h ) -1 | ≤ | φke h (x, a) T (θ h + ( θ k h -θ h ) + ( ψ k h -ψ h ) V k h+1 + ψ h V k h+1 )| + β b φke h (x, a) (Λ ke h ) -1 ≤ 1 + V k h+1 ∞ + φke h (x, a) (Λ ke h ) -1 θ k h -θ h Λ k h + ( ψ k h -ψ h ) V k h+1 Λ k h + β b ,where the last inequality also used the triangle and Cauchy-Schwarz inequalities, and that Λ ke hΛ k h . By the induction hypothesis, Qk h+1 ∞ , V k h+1 ∞ ≤ β Q,h+1 and thus V k h+1 ∈ V h+1 .Combining with E 1 (eq. ( 7)) and plugging into the above we get that| Qk h (x, a)| ≤ 1 + β Q,h+1 + (β r + β p,h + β b ) φke h (x, a) (Λ ke h ) -1. Now, using a technical algebraic argument (lemma 18), we show that φkeh (x, a) (Λ ke h ) -1 ≤ max y≥0 [y • σ(-β w y + log K)] ≤ 2β -1 w log(eK).Finally, plugging this into the above and choosing β w ≥ 2(β r + β p,h + β b ) log(eK), we get| Qk h (x, a)| ≤ 1 + β Q,h+1 + 2β -1 w (β r + β p,h + β b ) log(eK) ≤ 2 + β Q,h+1 = β Q,h , 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 et al. [2023] ) 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-trivial 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 in 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 on exploration in deep RL that we leave for future research. 
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 Appendix: We begin by defining a 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 E g = 3 i=1 E i , over which the regret is deterministically bounded:E 1 = ∀k ∈ [K], h ∈ [H] : θ k h -θ k h Λ k h ≤ β r ;(eq. ( 7))E 2 = k ∈ [K], h ∈ [H] : (ψ h -ψ k h ) V k h+1 Λ k h ≤ β p , Qk h+1 ∞ ≤ β Q ;(eq. ( 8))EQUATION)whereY k = h∈[H] 3(β r + β p ) φ(x h , a h ) (Λ k h ) -1 + 4β Q β 2 w φ(x h , a h ) 2 (Λ k h ) -1 . Lemma 6 (Good event). Consider the parameter setting of theorem 9. If η o ≤ 1, β 2 w ≤ K/(32Hd) then Pr[E g ] ≥ 1 -δ.Proof in appendix A.1.Policy online mirror descent. We use standard online mirror descent arguments to bound the local regret in each state.Lemma 7 (OMD). Suppose that the good event E g holds (eqs. (7), ( 8) and ( 9)) and setη o ≤ 1/β Q , then k∈Ke a∈A Qk h (x, a)(π h (a | x) -π k h (a | x)) ≤ log|A| η o + η o k∈Ke β 2 Q , ∀e ∈ [E], h ∈ [H], x ∈ X .Proof. Notice that the policy π k is reset at the beginning of every epoch. Then, the lemma follows directly by lemma 13 with y t (a) = -Qk h (x, a), x t (a) = π k h (a | x) and noting that | Qk h (x, a)| ≤ β Q by eq. ( 8).Epoch schedule. The algorithm operates in epochs. At the beginning of 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 k e , and the set of episodes within epoch e by K e . The following lemma bounds the number of epochs.Lemma 8. The number of epochs E is bounded by (3/2)dH log(2K).h , e 2 h , . . .} be the epochs where the condition det(Λ k h ) ≥ 2 det(Λ ke h ) was triggered in line 4 of algorithm 1. Then we have thatdet(Λ ke h ) ≥ 2 det(Λ ke-1 h ) , e ∈ T h det(Λ ke-1 h ) , otherwise.Unrolling this relation, we get thatdet(Λ K h ) ≥ 2 |T h |-1 det I = 2 |T h |-1, and changing sides, and taking the logarithm we get that|T h | ≤ 1 + log 2 det Λ K h ≤ 1 + d log 2 Λ K h (det(A) ≤ A d ) ≤ 1 + d log 2 1 + K-1 k=1 φ k h 2 (triangle inequality) ≤ 1 + d log 2 K ( φ k h ≤ 1) ≤ (3/2)d log 2K.We conclude thatE = | ∪ h∈[H] T h | ≤ h∈[H] |T h | ≤ (3/2)dH log(2K).Theorem 9. Suppose that we run algorithm 1 with parametersη o = 3dH log(2K) log|A| Kβ 2 Q , β b = β r + β p , β w = 4(β r + β p ) log(eK),where β r = 2 2d log(6KH/δ),β p = 28Hd log(10K 5 H/δ), β Q = 2H.Then with probability at least 1 -δ we incur regret at mostRegret ≤ 264 Kd 3 H 4 log(2K) log(10K 5 H/δ) + 8 KdH 5 log(2K) log|A| + 64H 2 d max{β 2 w , log|A|} log 12K δ = O( Kd 3 H 4 log(K) log(KH/δ) + KdH 5 log(K) log|A|). Proof. First, if β 2 w > K/(32Hd) or η ≥ 1/β Q then Regret ≤ KH ≤ 32H 2 d max{β 2 w , log|A|} log(2K),and the proof is concluded. Otherwise, if β 2 w ≤ K/(32Hd) then suppose that the good event E g holds (eqs. ( 7), ( 8) and ( 9)). By lemma 6, this holds with probability at least 1 -δ. For any epoch e ∈ [E], let K e be the set of episodes that it contains, and let V k,π 1 (x 1 ; ρ ke ) denote the value of its contracted MDP as defined in section 5.1 and line 6 of algorithm 1. We bound the regret asRegret = k∈[K] V k,π k 1 (x 1 ) -V k,π 1 (x 1 ) ≤ e∈[E] k∈Ke V k,π k 1 (x 1 ) - V k,π 1 (x 1 ; ρ ke ) (lemma 2) = k∈[K] V k,π k 1 (x 1 ) -V k 1 (x 1 ) + e∈[E] k∈Ke V k 1 (x 1 ) - V k,π 1 (x 1 ; ρ ke ) = k∈[K] V k,π k 1 (x 1 ) -V k 1 (x 1 ) (i)-Bias / Cost of optimism + e∈[E] h∈[H] E P ke ,π k∈Ke a∈A Qk h (x h , a)(π k h (a | x h ) -π h (a | x h )) (ii)-OMD regret + e∈[E] k∈Ke h∈[H] E P ke ,π Qk h (x h , a h ) -φke h (x h , a h ) T (θ k h + ψ h V k h+1 ) (iii)-Optimism ,where the last relation is by the extended value difference lemma (see Shani et al. [2020] and lemma 14 in appendix B).For term (i), we use lemma 5 as follows.(i) ≤ k∈[K] E P,π k   h∈[H] 3(β r + β p ) φ(x h , a h ) (Λ k h ) -1 + 8β Q β 2 w φ(x h , a h ) 2 (Λ k h ) -1   + 8Hβ Q ≤ k∈[K]   h∈[H] 6(β r + β p ) φ(x k h , a k h ) (Λ k h ) -1 + 16β Q β 2 w φ(x k h , a k h ) 2 (Λ k h ) -1   + 20Hβ Q β 2 w log 6 δ(eq. ( 9),β w ≥ 4(β r + β p ) ≥ 32) ≤ 6(β r + β p )H 2Kd log(2K) + 32β Q β 2 w Hd log(2K) + 20Hβ Q β 2 w log 6 δ (lemma 15) ≤ 6(β r + β p )H 2Kd log(2K) + 32Hdβ Q β 2 w log 12K δ .By lemmas 7 and 8 (with our choice of η o ) we have(ii) ≤ h∈[H] e∈[E] E P ke ,π log A η o + η o k∈Ke β 2 Q ≤ 4Hβ Q KdH log(2K) log|A|.By lemma 4 (iii) ≤ 0. Putting all bounds together, we get thatRegret ≤ 6(β r + β p )H 2Kd log(2K) + 32Hdβ Q β 2 w log 12K δ + 4Hβ Q KdH log(2K) log|A| ≤ 264 Kd 3 H 4 log(2K) log(10K 5 H/δ) + 8 KdH 5 log(2K) log|A| + 64H 2 dβ 2 w log 12K δ = O( Kd 3 H 4 log(K) log(KH/δ) + KdH 5 log(K) log|A|).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 is straightforward to 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:Q(C β , C w , C Q ) = Q(•, •; β, w, Λ, Z) | β ∈ [0, C β ], w ≤ C w , (2K) -1 I Λ I, Q(•, •; w, Λ, Z) ∞ ≤ C Q , where Q(x, a; β, w, Λ) = [w T φ(x, a) -β φ(x, a) Λ ] • σ(-β w φ(x, a) Λ + log K).Next, we define the following class of soft-max policies: ,a )) . Finally, we define the following class of restricted value functions:Π(C β , C w ) = π(• | •; Q) | Q ∈ Q(C β , C w , ∞) , where π(a | x; Q) = exp( Q(x,a)) a ∈A exp( Q(xEQUATION) where V (x; π, Q) = a∈A π(a | x) Q(x, a).The following lemma provides the bound on the covering number of the value function class defined above.Lemma 10. For any , C w > 0, C β , C Q ≥ 1, we have log N V(C β , C w , C Q ) ≤ 6d 2 log(1 + 4( √ 192K 3 C Q C β β w )(KC β + KC w + √ d)/ ),where N is the covering number of a class in supremum distance.Proof. We begin by showing that the class of Q function is Lipschitz in its parameters. For ease of notation, denote y = φ(x, a). Then∇ β Q(x, a; β, w, Λ) = y Λ • σ(-β w y Λ + log K) ≤ 1 (σ(•) ∈ [0, 1], y ≤ 1, Λ I) ∇ θ Q(x, a; β, w, Λ) = y • σ(-β w y Λ + log K) ≤ 1 (σ(•) ∈ [0, 1], y ≤ 1) |Q(x, a;β, w, Λ) -Q(x, a; β, w, Λ )| ≤ β| y Λ -y Λ | • σ(-β w y Λ + log K) + β y Λ |σ(-β w y Λ + log K) -σ(-β w y Λ + log K)| ≤ β (Λ 1/2 -(Λ ) 1/2 )y + ββ w y Λ (Λ 1/2 -(Λ ) 1/2 )y ( • , σ(•) 1-Lipschitz, σ ∈ [0, 1]) ≤ 2ββ w Λ 1/2 -(Λ ) 1/2 ( y ≤ 1, Λ I, β w ≥ 1) ≤ √ 2Kββ w Λ -Λ (lemma 17, Λ, Λ (2K) -1 I) ≤ √ 2Kββ w Λ -Λ F . ( • ≤ • F ) We thus have that for any such y |Q(x, a; β, w,Λ) -Q(x, a; β , w , Λ )| ≤ |Q(x, a; β, w, Λ) -Q(x, a; β , w, Λ)| + |Q(x, a; β , w, Λ) -Q(x, a; β , w , Λ)| + |Q(x, a; β , w , Λ) -Q(x, a; β , w , Λ )| ≤ |β -β | + w -w + √ 2Kββ w Λ -Λ F ≤ 3( w -w 2 + |β -β | 2 + ( √ 2Kββ w ) 2 Λ -Λ 2 F ) ≤ max{3, √ 6Kββ w } ( w -w 2 + |β -β | 2 + Λ -Λ 2 F ) = max{3, √ 6Kββ w } (β, w, Λ) -(β , w , Λ ) ,where (β, w, Λ) is a vectorization of the parameters. Assuming that C β ≥ 1, we conclude thatQ(C β , C w , C Q ) is √ 6KC β β w -Lipschitz in supremum norm, i.e., Q(•, •; β, w, Λ) -Q (•, •; β , w , Λ ) ∞ ≤ √ 6KC β β w (β, w, Λ) -(β , w , Λ ) .Next, notice that our policy class Π(C β K, C w K) is a soft-max over the Q functions thus fitting Lemma 12 of Sherman et al. [2023a] . We conclude that the policy class is√ 24K 3 C β β w -Lipschitz, in 1 -norm, i.e., π(• | x; β, w, Λ) -π(• | x; β , w , Λ ) 1 ≤ √ 24K 3 C β β w (β, w, Λ) -(β , w , Λ ) . Now, let V, V ∈ V(C β , C w , C Q ) and θ = (β 1 , w 1 , Λ 1 , β 2 , w 2 , Λ 2 ), θ = (β 1 , w 1 , Λ 1 , β 2 , w 2 , Λ 2 ) ∈R 2(1+d+d 2 ) be their respective parameters. We have that for all x ∈ X |V (x; π, Q) -V (x; π , Q )| ≤ |V (x; π, Q) -V (x; π, Q )| (i) + |V (x; π, Q ) -V (x; π , Q )| (ii).For the first term(i) = a∈A π(a | x)( Q(x, a; β 2 , w 2 , Λ 2 ) -Q(x, a; β 2 , w 2 , Λ 2 )) ≤ a∈A π(a | x) Q(x, a; β 2 , w 2 , Λ 2 ) -Q(x, a; β 2 , w 2 , Λ 2 ) (triangle inequality) ≤ √ 6KC β β w (β 2 , w 2 , Λ 2 ) -(β 2 , w 2 , Λ 2 ) . ( Q is √ 6KC β β w -Lipschitz, Cauchy-Schwarz)For the second term(ii) = a∈A Q (x, a)(π(a | x) -π (a | x)) ≤ C Q π(• | x) -π(• | x) 1 ≤ √ 96K 3 C Q C β β w (β 1 , w 1 , Λ 1 ) -(β 1 , w 1 , Λ 1 ) ,where the first transition used thatQ ∞ ≤ C Q for all Q ∈ Q(C β , C w , C Q )and the second used the Lipschitz property of the policy class shown above. Combining the terms and assuming that C Q ≥ 1 we get that|V (x; π, Q) -V (x; π , Q )| ≤ √ 96K 3 C Q C β β w (β 1 , w 1 , Λ 1 ) -(β 1 , w 1 , Λ 1 ) + √ 96K 3 C Q C β β w (β 2 , w 2 , Λ 2 ) -(β 2 , w 2 , Λ 2 ) ≤ √ 192K 3 C Q C β β w θ -θ , implying that V(C β , C w , C Q ) is √ 192K 3 C Q C β β w -Lipschitz in supremum norm. Finally, notice that θ ≤ |β 1 | + |β 2 | + w 1 + w 2 + Λ 1 F + Λ 2 F ≤ 2KC β + 2KC w + 2 √ d,and applying lemma 24 concludes the proof.Proxy good event. We define a proxy good event Ēg = E 1 ∩ Ē2 ∩ E 3 whereEQUATIONwhereβ Q,h = 2(H + 1 -h), h ∈ [H + 1].Then we have the following result.Lemma 11 (Proxy good event). Consider the parameter setting of lemma 6. Then Pr[ Ēg ] ≥ 1 -δ.Proof. First, by lemma 21 and our choice of parameters, E 1 (eq. ( 7)) holds with probability at least 1-δ/3. Next, applying lemmas 10 and 22, we get that with probability at least 1-δ/3 simultaneously for all k ∈[K], h ∈ [H], V ∈ V(β r + β p , 2β Q K, β Q,h+1 ) (ψ h -ψ k h )V Λ k h ≤ 4β Q,h+1 d log(2K) + 2 log(6H/δ) + 12d 2 log(1 + 8K( √ 192K 3 C β β w )(KC β + KC w + 1)) ≤ 4β Q d log(2K) + 2 log(6H/δ) + 12d 2 log(1 + 2K( √ 192K 3 K/(32Hd))( 1 4 K K/(32Hd) + 2β Q K 2 + 1)) ≤ 4β Q d log(2K) + 2 log(6H/δ) + 12d 2 log(7K 9/2 ) ≤ 4β Q d 12 log(10K 5 H/δ) ≤ 28Hd log(10K 5 H/δ) = β p ,implying Ē2 (eq. ( 11)). Finally, notice thatφ k h (Λ k h ) -1 ≤ 1, thus 0 ≤ Y k ≤ H(3(β r +β p )+4β Q β 2 w ).Using lemma 20, a Bernstein-type inequality for martingales, we conclude that E 3 (eq. ( 9)) holds with probability at least 1 -δ/3. The good event. The following results show that the proxy good event implies the good event.Lemma 12. Suppose that Ēg holds. Ifπ k h ∈ Π(K(β r + β p ), 2β Q K 2 ) for all h ∈ [H] then Qk h ∈ Q(β r + β p , 2β Q K, β Q,h ), V k h ∈ V(β r + β p , 2β Q K, β Q,h ) for all h ∈ [H + 1].Proof. We show that the claim holds by backward induction on h ∈[H + 1]. Base case h = H + 1: Since V k H+1 = 0 it is also implied that Qk H+1 = 0. Because β, w = 0 ∈ Q(β r + β p , 2β Q K, β Q,H+1 = 0) we have that Qk H+1 ∈ Q(β r + β p , 2β Q K, β Q,H+1 = 0), and similarly V k H+1 ∈ V(β r + β p , 2β Q K, β Q,H+1 = 0).Induction step: Now, suppose the claim holds for h + 1 and we show it also holds for h. We have that| Qk h (x, a)| = | φke h (x, a) T w k h -β b φke h (x, a) (Λ ke h ) -1 | ≤ | φke h (x, a) T (θ h + ( θ k h -θ h ) + ( ψ k h -ψ h ) V k,i h+1 + ψ h V k,i h+1 )| + β b φke h (x, a) (Λ ke h ) -1 ≤ 1 + V k,i h+1 ∞ + φke h (x, a) (Λ ke h ) -1 θ k h -θ h Λ k h + ( ψ k h -ψ h ) V k,i h+1 Λ k h + β b (triangle inequality, Cauchy-Schwarz, Λ ke h Λ k h ) ≤ 1 + β Q,h+1 + (β r + β p,h + β b ) φke h (x, a) (Λ ke h ) -1(induction hypothesis, eqs. ( 7) and ( 11))EQUATIONw k h = θ k h + ψ k h V k,i h+1 ≤ K + β Q K ≤ 2β Q K = C w . We conclude that Qk h ∈ Q(β r + β p , 2β Q K, β Q,h ). Since π k h ∈ Π(K(β r + β p ), 2β Q K 2 ), we also conclude that V k h ∈ V(β r + β p , 2β Q K, β Q,h), proving the induction step and finishing the proof. Lemma (restatement of lemma 6). Consider the parameter setting of theorem 9. Ifη o ≤ 1, β 2 w ≤ K/(32Hd) then Pr[E g ] ≥ 1 -δ.Proof. Suppose that Ēg holds. By lemma 11, this occurs with probability at least 1 -δ. We show that Ēg implies E g , thus concluding the proof. Notice thatπ k h (a|x) ∝ exp η k-1 k =ke Qk h (x, a) = exp σ(-β w φ(x, a) (Λ ke h ) -1 + log K) • φ(x, a) T k-1 k =ke ηw k h -ηβ b (k -k e ) φ(x, a) (Λ ke h ) -1.k ∈ K e that π k h ∈ Π(K(β r + β p ), 2β Q K 2 ) for all h ∈ [H]. For the base case, k = k e , π k h are uniform, corresponding to w, β = 0 ∈ Π(K(β r + β p ), 2β Q K 2 ). Now, suppose the claim holds for all k < k. Then by lemma 12 we have that Qkh ∈ Q(β r + β p , 2β Q K, β Q,h ) for all k < k and h ∈ [H]. This implies that k-1 k =ke ηw k h ≤ 2β Q K 2 for all h ∈ [H], thus π k h ∈ Π(K(β r + β p ), 2β Q K 2 ) for all h ∈ [H], concluding the induction step. Now, since π k h ∈ Π(K(β r + β p ), 2β Q K 2 ) for all k ∈ [K], h ∈ [H],we can apply lemma 12 to get that Qkh ∈ Q(β r + β p , 2β Q K, β Q,h ), V k h ∈ V(β r + β p , 2β Q K, β Q,h ) for all k ∈ [K], h ∈ [H].Using Ē2 (eq. ( 11)) we conclude that E 2 (eq. ( 8)) holds, thus concluding the proof.We begin with a standard regret bound for entropy regularized online mirror descent (hedge). See [Sherman et al., 2023a, Lemma 25] .Lemma 13. Let y 1 , . . . , y T ∈ R A be any sequence of vectors, and η > 0 such that ηy t (a) ≥ -1 for all t ∈ [T ], a ∈ [A]. Then if x t ∈ ∆ A is given by x 1 (a) = 1/A ∀a, and for t ≥ 1:x t+1 (a) =x t (a)e -ηyt(a)a ∈[A] x t (a )e -ηyt(a ) , then,max x∈∆ A T t=1 a∈[A] y t (a)(x t (a) -x(a)) ≤ log A η + η T t=1 a∈[A]x t (a)y t (a) 2 .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. [2020] ). Let M be a (sub) MDP, π, π ∈ Π M be two policies, Qh : X × A → R, h ∈ [H] be arbitrary function, and Vh : X → R be defined as Vh (x) = a∈A πh (a | x) Qh (x, a). ThenV π 1 (x 1 ) -V1 (x 1 ) = E P,π   h∈[H] a∈A Qh (x h , a)(π(a | x h ) -π(a | x h ))   + E P,π   h∈[H] h (x h , a h ) + x ∈X P (x | x h , a h ) Vh+1 (x ) -Qh (x h , a h )   .We note that, in the context of linear MDP h (x h , a h ) + x ∈X P (x | x h , a h ) Vh+1 (x ) = φ(x h , a h ) T (θ h + ψ h Vh+1 ).Next, is a well-known bound on harmonic sums [see, e.g., Cohen et al., 2019, Lemma 13] . This is used to show that the optimistic and true losses are close on the realized predictions.Lemma 15. Let z t ∈ R d be a sequence such that z t 2 ≤ λ, and defineV t = λI + t-1 s=1 z s z T s . Then T t=1 z t V -1 t ≤ T T t=1 z t 2 V -1 t ≤ 2T d log(T + 1).Next, we need the following well-known matrix inequality.Lemma 16 (Cohen et al. [2019] , Lemma 27). If N M 0 then for any vector vv 2 N ≤ det N det M v 2 MNext, we need a bound on the Lipschitz constant of the spectral norm of a square-root matrix.Lemma 17. For any λ > 0 and matrices Λ, Λ ∈ R d×d satisfying Λ, Λ λI we have thatΛ 1/2 -Λ 1/2 ≤ 1 2 √ λ Λ -Λ .We state the well-known self normalized error bounds for regularized least squares estimation of the rewards and dynamics (see e.g., Abbasi-Yadkori et al. [2011] , Jin et al. [2020b] ).Lemma 21 (reward error bound). Let θ k h be as in line 14 of algorithm 1. With probability at least1 -δ, for all k ≥ 1, h ∈ [H] θ h -θ k h Λ k h ≤ 2 2d log(2KH/δ).Lemma 22 (dynamics error uniform convergence). Let ψ k h : R X → R d be the linear operator defined in eq. (2) inside algorithm 1. For all h ∈[H], let V h ⊆ R X be a set of mappings V : X → R such that V ∞ ≤ β and β ≥ 1. With probability at least 1 -δ, for all h ∈ [H], V ∈ V h+1 and k ≥ 1 (ψ h -ψ k h )V Λ k h ≤ 4β d log(K + 1) + 2 log(HN /δ),where≤ β √ d/2K, N = h∈[H] N h, ,and N h, is the -covering number of V h with respect to the supremum distance.The following results are (mostly) standard bounds on the covering number of several function classes.Lemma 23. For any > 0, the -covering of the Euclidean ball in R d with radius R ≥ 0 is upper bounded by (1 + 2R/ ) d .Lemma 24. Let V = {V (•; θ) : θ ≤ W } denote a class of functions V : X → R. Suppose that any V ∈ V is L-Lipschitz with respect to θ and supremum distance, i.e.,V (•; θ 1 ) -V (•; θ 2 ) ∞ ≤ L θ 1 -θ 2 , θ 1 , θ 2 ≤ W.Let N be the -covering number of V with respect to the supremum distance. Then ). We show that V is an -cover of V , thus concluding the proof. Let V ∈ V and θ be its associated parameter. Let θ ∈ Θ /L be the point in the cover nearest to θ and V ∈ V its associated function. Then we have thatV (•) -V (•) ∞ = V (•; θ) -V (•; θ ) ∞ ≤ L θ -θ ≤ L( /L) = .NeurIPS Paper Checklist Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.Wu et al. [2022] apply variance reduction techniques to get better regret bounds in the tabular setting, but they use L2-regularization instead of KL-regularization which does not align with practical PO algorithmsSchulman et al. [2015,2017].(Λ k h ) -1 + 2K -1 ) φ(x h , a h ) T v . 


Decision: 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.

Reviews: review: paper_summary: paper_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.
strength_weakness: strength_weakness: 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).
questions: questions: Currently, I do not have questions.
limitations: limitations: Yes.
paper_summary: paper_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.
strength_weakness: strength_weakness: 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.

The 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.


Despite 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?

 The paper could be strengthened by adding a conclusion section to summarize the key findings
questions: questions: na
limitations: limitations: The authors should include a more detailed discussion of the study's limitations in the conclusion part.
paper_summary: paper_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 $d$.
strength_weakness: strength_weakness: ## Weaknesses

1. 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.
2. 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.
3. 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.
4. 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.
5. Between Line 151 and 152: the $\Delta P(x_h, a_h)$ part seems to be a typo
6. 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 $K\geq \mathrm{e}$)
7. Minor typo on Line 265: $\succ$ -> $\leq$
8. Minor typo on Line 579: $\hat{V}$ -> $\hat{V}_h$

[1] Sherman, U., Cohen, A., Koren, T., & Mansour, Y. (2023). Rate-optimal policy optimization for linear markov decision processes. arXiv preprint arXiv:2308.14642.
questions: questions: 1. Any reference on the proposition mentioned on Line 134-135 that the Markov $\pi^*$ in hindsight is optimal among all policies?
limitations: limitations: not applicable
paper_summary: paper_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.
strength_weakness: strength_weakness: There is no major technical flaw detected in this paper. 

Weakness:

There 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.

And 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.
questions: questions: Please see Weaknesses.
limitations: limitations: Yes.

Comments: title: Thanks.
comment: Thanks for your rebuttal. I hope the authors will consider incorporating these clarifications in their revision. I will maintain my positive assessment.
The authors have appropriately answered all questions I asked. I will keep my positive evaluation and recommend this paper for acceptance.
I would like to thank the authors for the rebuttal. I have no additional concerns and thus raised the score.