On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift

Introduction

Policy gradient methods have a long history in the reinforcement learning (RL) literature (Williams, 1992; Sutton et al., 1999; Konda and Tsitsiklis, 2000; Kakade, 2001) and are an attractive class of algorithms as they are applicable to any differentiable policy parameterization; admit easy extensions to function approximation; easily incorporate structured state and action spaces; are easy to implement in a simulation based, model-free manner. Owing to their flexibility and generality, there has also been a flurry of improvements and refinements to make these ideas work robustly with deep neural network based approaches (see e.g. Schulman et al. (2015, 2017)).

Despite the large body of empirical work around these methods, their convergence properties are only established at a relatively coarse level; in particular, the folklore guarantee is that these methods converge to a stationary point of the objective, assuming adequate smoothness properties hold and assuming either exact or unbiased estimates of a gradient can be obtained (with appropriate regularity conditions on the variance). However, this local convergence viewpoint does not address some of the most basic theoretical convergence questions, including: 1) if and how fast they converge to a globally optimal solution (say with a sufficiently rich policy class); 2) how they cope with approximation error due to using a restricted class of parametric policies; or 3) their finite sample behavior. These questions are the focus of this work.

Overall, the results of this work place policy gradient methods under a solid theoretical footing, analogous to the global convergence guarantees of iterative value function based algorithms.

This work focuses on first-order and quasi second-order policy gradient methods which directly work in the space of some parameterized policy class (rather than value-based approaches). We characterize the computational, approximation, and sample size properties of these methods in the context of a discounted Markov Decision Process (MDP). We focus on: 1) tabular policy parameterizations, where there is one parameter per state-action pair so the policy class is complete in that it contains the optimal policy, and 2) function approximation, where we have a restricted class or parametric policies which may not contain the globally optimal policy. Note that policy gradient methods for discrete action MDPs work in the space of stochastic policies, which permits the policy class to be differentiable. We now discuss our contributions in the both of these contexts.

We consider three algorithms: two of which are first order methods, projected gradient ascent (on the simplex) and gradient ascent (with a softmax policy parameterization); and the third algorithm, natural policy gradient ascent, can be viewed as a quasi second-order method (or preconditioned first-order method). Table 1 summarizes our main results in this case: upper bounds on the number of iterations taken by these algorithms to find an $\epsilon$-optimal policy, when we have access to exact policy gradients.

Arguably, the most natural starting point for an analysis of policy gradient methods is to consider directly doing gradient ascent on the policy simplex itself and then to project back onto the simplex if the constraint is violated after a gradient update; we refer to this algorithm as projected gradient ascent on the simplex. Using a notion of gradient domination (Polyak, 1963), our results provably show that any first-order stationary point of the value function results in an approximately optimal policy, under certain regularity assumptions; this allows for a global convergence analysis by directly appealing to standard results in the non-convex optimization literature.

A more practical and commonly used parameterization is the softmax parameterization, where the simplex constraint is explicitly enforced by the exponential parameterization, thus avoiding projections. This work provides the first global convergence guarantees using only first-order gradient information for the widely-used softmax parameterization. Our first result for this parameterization establishes the asymptotic convergence of the policy gradient algorithm; the analysis challenge here is that the optimal policy (which is deterministic) is attained by sending the softmax parameters to infinity.

In order to establish a finite time, convergence rate to optimality for the softmax parameterization, we then consider a log barrier regularizer and provide an iteration complexity bound that is polynomial in all relevant quantities. The use of our log barrier regularizer is critical to avoiding the issue of gradients becomingly vanishingly small at suboptimal near-deterministic policies, an issue of significant practical relevance. The log barrier regularizer can also be viewed as using a relative entropy regularizer; here, we note the general approach of entropy based regularization is common in practice (e.g. see (Williams and Peng, 1991; Mnih et al., 2016; Peters et al., 2010; Abdolmaleki et al., 2018; Ahmed et al., 2019)). One notable distinction, which we discuss later, is that our analysis is for the log barrier regularization rather than the entropy regularization.

For these aforementioned algorithms, our convergence rates depend on the optimization measure having coverage over the state space, as measured by the distribution mismatch coefficient $D_{\infty}$ (see Table 1 caption). In particular, for the convergence rates shown in Table 1 (for the aforementioned algorithms), we assume that the optimization objective is the expected (discounted) cumulative value where the initial state is sampled under some distribution, and $D_{\infty}$ is a measure of the coverage of this initial distribution. Furthermore, we provide a lower bound that shows such a dependence is unavoidable for first-order methods, even when exact gradients are available.

We then consider the Natural Policy Gradient (NPG) algorithm (Kakade, 2001) (also see Bagnell and Schneider (2003); Peters and Schaal (2008)), which can be considered a quasi second-order method due to the use of its particular preconditioner, and provide an iteration complexity to achieve an $\epsilon$-optimal policy that is at most $\frac{2}{(1-\gamma)^{2}\epsilon}$ iterations, improving upon the previous related results of (Even-Dar et al., 2009; Geist et al., 2019) (see Section 2). Note the convergence rate has no dependence on the number of states or the number of actions, nor does it depend on the distribution mismatch coefficient $D_{\infty}$. We provide a simple and concise proof for the convergence rate analysis by extending the approach developed in (Even-Dar et al., 2009), which uses a mirror descent style of analysis (Nemirovsky and Yudin, 1983; Cesa-Bianchi and Lugosi, 2006) and also handles the non-concavity of the policy optimization problem.

This fast and dimension free convergence rate shows how the variable preconditioner in the natural gradient method improves over the standard gradient ascent algorithm. The dimension free aspect of this convergence rate is worth reflecting on, especially given the widespread use of the natural policy gradient algorithm along with variants such as the Trust Region Policy Optimization (TRPO) algorithm (Schulman et al., 2015); our results may help to provide analysis of a more general family of entropy based algorithms (see for example Neu et al. (2017)).

We now summarize our results with regards to policy gradient methods in the setting where we work with a restricted policy class, which may not contain the optimal policy. In this sense, these methods can be viewed as approximate methods. Table 2 provides a summary along with the comparisons to some relevant approximate dynamic programming methods.

Our results are also suggestive that a broader class of incremental algorithms — such as CPI (Kakade and Langford, 2002), PSDP (Bagnell et al., 2004), and MD-MPI Geist et al. (2019) which make small changes to the policy from one iteration to the next — may also permit a sharper analysis, where the dependence of worst-case density ratios can be avoided through an appropriate approximation/estimation decomposition; this is an interesting direction for future work (a point which we return to in Section 7). One significant advantage of NPG is that the explicit parametric policy representation in NPG (and other policy gradient methods) leads to a succinct policy representation in comparison to CPI, PSDP, or related boosting-style methods (Scherrer and Geist, 2014), where the representation complexity of the policy of the latter class of methods grows linearly in the number of iterations (since these methods add one policy to the ensemble per iteration). This representation complexity is likely why the latter class of algorithms are less widely used in practice.

Related Work

We now discuss related work, roughly in the order which reflects our presentation of results in the previous section.

For the direct policy parameterization in the tabular case, we make use of a gradient domination-like property, namely any first-order stationary point of the policy value is approximately optimal up to a distribution mismatch coefficient. A variant of this result also appears in Theorem 2 of Scherrer and Geist (2014), which itself can be viewed as a generalization of the approach in Kakade and Langford (2002). In contrast to CPI (Kakade and Langford, 2002) and the more general boosting-based approach in Scherrer and Geist (2014), we phrase this approach as a Polyak-like gradient domination property (Polyak, 1963) in order to directly allow for the transfer of any advances in non-convex optimization to policy optimization in RL. More broadly, it is worth noting the global convergence of policy gradients for Linear Quadratic Regulators (Fazel et al., 2018) also goes through a similar proof approach of gradient domination.

Empirically, the recent work of Ahmed et al. (2019) studies entropy based regularization and shows the value of regularization in policy optimization, even with exact gradients. This is related to our use of the log barrier regularization.

For our convergence results of the natural policy gradient algorithm in the tabular setting, there are close connections between our results and the works of Even-Dar et al. (2009); Geist et al. (2019). Even-Dar et al. (2009) provides provable online regret guarantees in changing MDPs utilizing experts algorithms (also see Neu et al. (2010); Abbasi-Yadkori et al. (2019a)); as a special case, their MDP Experts Algorithm is equivalent to the natural policy gradient algorithm with the softmax policy parameterization. While the convergence result due to Even-Dar et al. (2009) was not specifically designed for this setting, it is instructive to see what it implies due to the close connections between optimization and regret (Cesa-Bianchi and Lugosi, 2006; Shalev-Shwartz et al., 2012). The Mirror Descent-Modified Policy Iteration (MD-MPI) algorithm (Geist et al., 2019) with negative entropy as the Bregman divergence results is an identical algorithm as NPG for softmax parameterization in the tabular case; Corollary 3 (Geist et al., 2019) applies to our updates, leading to a bound worse by a $1/(1-\gamma)$ factor and also has logarithmic dependence on $|\mathcal{A}|$. Our proof for this case is concise and may be of independent interest. Also worth noting is the Dynamic Policy Programming of Azar et al. (2012), which is an actor-critic algorithm with a softmax parameterization; this algorithm, even though not identical, comes with similar guarantees in terms of its rate (it is weaker in terms of an additional $1/(1-\gamma)$ factor) than the NPG algorithm.

We now turn to function approximation, starting with a discussion of iterative algorithms which make incremental updates in which the next policy is effectively constrained to be close to the previous policy, such as in CPI and PSDP (Bagnell et al., 2004). Here, the work in Scherrer and Geist (2014) show how CPI is part of broader family of boosting-style methods. Also, with regards to PSDP, the work in Scherrer (2014) shows how PSDP actually enjoys an improved iteration complexity over CPI, namely $O(\log 1/\epsilon_{\text{opt}})$ vs. $O(1/\epsilon_{\text{opt}}^{2})$. It is worthwhile to note that both NPG and projected gradient ascent are also both incremental algorithms.

Assuming bounded concentrability coefficient, there are a notable set of provable average case guarantees for the MD-MPI algorithm (Geist et al., 2019) (see also (Azar et al., 2012; Scherrer et al., 2015)), which are stated in terms of various norms of function approximation error. MD-MPI is a class of algorithms for approximate planning under regularized notions of optimality in MDPs. Specifically, Geist et al. (2019) analyze a family of actor-critic style algorithms, where there are both approximate value functions updates and approximate policy updates. As a consequence of utilizing approximate value function updates for the critic, the guarantees of Geist et al. (2019) are stated with dependencies on concentrability coefficients.

When dealing with function approximation, computational and statistical complexities are relevant because they determine the effectiveness of approximate updates with finite samples. With regards to sample complexity, the work in Szepesvári and Munos (2005); Antos et al. (2008) provide finite sample rates (as discussed above), further generalized to actor-critic methods in Azar et al. (2012); Scherrer et al. (2015). In our policy optimization approach, the analysis of both computational and statistical complexities are straightforward, since we can leverage known statistical and computational results from the stochastic approximation literature; in particular, we use the stochastic projected gradient ascent to obtain a simple, linear time method for the critic estimation step in the natural policy gradient algorithm.

In terms of the algorithmic updates for the function approximation setting, our development of NPG bears similarity to the natural actor-critic algorithm Peters and Schaal (2008), for which some asymptotic guarantees under finite concentrability coefficients are obtained in Bhatnagar et al. (2009). While both updates seek to minimize the compatible function approximation error, we perform streaming updates based on stochastic optimization using Monte Carlo estimates for values. In contrast Peters and Schaal (2008) utilize Least Squares Temporal Difference methods (Boyan, 1999) to minimize the loss. As a consequence, their updates additionally make linear approximations to the value functions in order to estimate the advantages; our approach is flexible in allowing for wide family of smoothly differentiable policy classes (including neural policies).

Finally, we remark on some concurrent works. The work of Bhandari and Russo (2019) provides gradient domination-like conditions under which there is (asymptotic) global convergence to the optimal policy. Their results are applicable to the projected gradient ascent algorithm; they are not applicable to gradient ascent with the softmax parameterization (see the discussion in Section 5 herein for the analysis challenges). Bhandari and Russo (2019) also provide global convergence results beyond MDPs. Also, Liu et al. (2019) provide an analysis of the TRPO algorithm (Schulman et al., 2015) with neural network parameterizations, which bears resemblance to our natural policy gradient analysis. In particular, Liu et al. (2019) utilize ideas from both Even-Dar et al. (2009) (with a mirror descent style of analysis) along with Cai et al. (2019) (to handle approximation with neural networks) to provide conditions under which TRPO returns a near optimal policy. Liu et al. (2019) do not explicitly consider the case where the policy class is not complete (i.e when there is approximation). Another related work of Shani et al. (2019) considers the TRPO algorithm and provides theoretical guarantees in the tabular case; their convergence rates with exact updates are $O(1/\sqrt{T})$ for the (unregularized) objective function of interest; they also provide faster rates on a modified (regularized) objective function. They do not consider the case of infinite state spaces and function approximation. The closely related recent papers (Abbasi-Yadkori et al., 2019a, b) also consider closely related algorithms to the Natural Policy Gradient approach studied here, in an infinite horizon, average reward setting. Specifically, the EE-Politex algorithm is closely related to the Q-NPG algorithm which we study in Section 6.2, though our approach is in the discounted setting. We adopt the name Q-NPG to capture its close relationship with the NPG algorithm, with the main difference being the use of function approximation for the $Q$-function instead of advantages. We refer the reader to Section 6.2 (and Remark 6.5) for more discussion of the technical differences between the two works.

Setting

A (finite) Markov Decision Process (MDP) $M=(\mathcal{S},\mathcal{A},P,r,\gamma,\rho)$ is specified by: a finite state space $\mathcal{S}$; a finite action space $\mathcal{A}$; a transition model $P$ where $P(s^{\prime}|s,a)$ is the probability of transitioning into state $s^{\prime}$ upon taking action $a$ in state $s$; a reward function $r:\mathcal{S}\times\mathcal{A}\to$ where $r(s,a)$ is the immediate reward associated with taking action $a$ in state $s$; a discount factor $\gamma\in[0,1)$; a starting state distribution $\rho$ over $\mathcal{S}$.

A deterministic, stationary policy $\pi:\mathcal{S}\to\mathcal{A}$ specifies a decision-making strategy in which the agent chooses actions adaptively based on the current state, i.e., $a_{t}=\pi(s_{t})$. The agent may also choose actions according to a stochastic policy $\pi:\mathcal{S}\to\Delta(\mathcal{A})$ (where $\Delta(\mathcal{A})$ is the probability simplex over $\mathcal{A}$), and, overloading notation, we write $a_{t}\sim\pi(\cdot|s_{t})$.

where the expectation is with respect to the randomness of the trajectory $\tau$ induced by $\pi$ in $M$. Since we assume that $r(s,a)\in$, we have $0\leq V^{\pi}(s)\leq\frac{1}{1-\gamma}$. We overload notation and define $V^{\pi}(\rho)$ as the expected value under the initial state distribution $\rho$, i.e.

The goal of the agent is to find a policy $\pi$ that maximizes the expected value from the initial state, i.e. the optimization problem the agent seeks to solve is:

where the $\max$ is over all policies. The famous theorem of Bellman and Dreyfus (1959) shows there exists a policy $\pi^{\star}$ which simultaneously maximizes $V^{\pi}(s_{0})$, for all states $s_{0}\in\mathcal{S}$.

This work studies ascent methods for the optimization problem:

where $\{\pi_{\theta}|\theta\in\Theta\}$ is some class of parametric (stochastic) policies. We consider a number of different policy classes. The first two are complete in the sense that any stochastic policy can be represented in the class. The final class may be restrictive. These classes are as follows:

Direct parameterization: The policies are parameterized by

where $\theta\in\Delta(\mathcal{A})^{|\mathcal{S}|}$, i.e. $\theta$ is subject to $\theta_{s,a}\geq 0$ and $\sum_{a\in\mathcal{A}}\theta_{s,a}=1$ for all $s\in\mathcal{S}$ and $a\in\mathcal{A}$.

The softmax parameterization is also complete.

Restricted parameterizations: We also study parametric classes $\{\pi_{\theta}|\theta\in\Theta\}$ that may not contain all stochastic policies. In particular, we pay close attention to both log-linear policy classes and neural policy classes (see Section 6). Here, the best we may hope for is an agnostic result where we do as well as the best policy in this class.

While the softmax parameterization is the more natural parametrization among the two complete policy classes, it is also informative to consider the direct parameterization.

It is worth explicitly noting that $V^{\pi_{\theta}}(s)$ is non-concave in $\theta$ for both the direct and the softmax parameterizations, so the standard tools of convex optimization are not applicable. For completeness, we formalize this as follows (with a proof in Appendix A, along with an example in Figure 2):

There is an MDP $M$ (described in Figure 2) such that the optimization problem $V^{\pi_{\theta}}(s)$ is not concave for both the direct and softmax parameterizations.

In order to introduce these methods, it is useful to define the discounted state visitation distribution $d_{s_{0}}^{\pi}$ of a policy $\pi$ as:

where $\Pr^{\pi}(s_{t}=s|s_{0})$ is the state visitation probability that $s_{t}=s$, after we execute $\pi$ starting at state $s_{0}$. Again, we overload notation and write:

where $d_{\rho}^{\pi}$ is the discounted state visitation distribution under initial distribution $\rho$.

The policy gradient functional form (see e.g. Williams (1992); Sutton et al. (1999)) is then:

Furthermore, if we are working with a differentiable parameterization of $\pi_{\theta}(\cdot|s)$ that explicitly constrains $\pi_{\theta}(\cdot|s)$ to be in the simplex, i.e. $\pi_{\theta}\in\Delta(\mathcal{A})^{|\mathcal{S}|}$ for all $\theta$, then we also have:

Note the above gradient expression (Equation 6) does not hold for the direct parameterization, while Equation 5 is valid. This is due to $\sum_{a}\nabla_{\theta}\pi_{\theta}(a|s)=0$ not explicitly being maintained by the direct parameterization.

The following lemma is helpful throughout:

(The performance difference lemma (Kakade and Langford, 2002)) For all policies $\pi,\pi^{\prime}$ and states $s_{0}$,

For completeness, we provide a proof in Appendix A.

We often characterize the difficulty of the exploration problem faced by our policy optimization algorithms when maximizing the objective $V^{\pi}(\mu)$ through the following notion of distribution mismatch coefficient.

Given a policy $\pi$ and measures $\rho,\mu\in\Delta(\mathcal{S})$, we refer to \Bigl{\lVert}\frac{d_{\rho}^{\pi}}{\mu}\Bigr{\rVert}_{\infty} as the distribution mismatch coefficient of $\pi$ relative to $\mu$. Here, $\frac{d_{\rho}^{\pi}}{\mu}$ denotes componentwise division.

We often instantiate this coefficient with $\mu$ as the initial state distribution used in a policy optimization algorithm, $\rho$ as the distribution to measure the sub-optimality of our policy (this is the start state distribution of interest), and where $\pi$ above is often chosen to be $\pi^{\star}\in\operatorname{argmax}_{\pi\in\Pi}V^{\pi}(\rho)$, given a policy class $\Pi$.

Following convention, we use $V^{\star}$ and $Q^{\star}$ to denote $V^{\pi^{\star}}$ and $Q^{\pi^{\star}}$ respectively. For iterative algorithms which obtain policy parameters $\theta^{(t)}$ at iteration $t$, we let $\pi^{(t)}$, $V^{(t)}$ and $A^{(t)}$ denote the corresponding quantities parameterized by $\theta^{(t)}$, i.e. $\pi_{\theta^{(t)}}$, $V^{\theta^{(t)}}$ and $A^{\theta^{(t)}}$, respectively. For vectors $u$ and $v$, we use $\tfrac{u}{v}$ to denote the componentwise ratio; $u\geq v$ denotes a componentwise inequality; we use the standard convention where $\|v\|_{2}=\sqrt{\sum_{i}v_{i}^{2}}$, $\|v\|_{1}=\sum_{i}|v_{i}|$, and $\|v\|_{\infty}=\max_{i}|v_{i}|$.

Warmup: Constrained Tabular Parameterization

Our starting point is, arguably, the simplest first-order method: we directly take gradient ascent updates on the policy simplex itself and then project back onto the simplex if the constraints are violated after a gradient update. This algorithm is projected gradient ascent on the direct policy parametrization of the MDP, where the parameters are the state-action probabilities, i.e. $\theta_{s,a}=\pi_{\theta}(a|s)$ (see (2)). As noted in Lemma 3.1, $V^{\pi_{\theta}}(s)$ is non-concave in the parameters $\pi_{\theta}$. Here, we first prove that $V^{\pi_{\theta}}(\mu)$ satisfies a Polyak-like gradient domination condition (Polyak, 1963), and this tool helps in providing convergence rates. The basic approach was also used in the analysis of CPI (Kakade and Langford, 2002); related gradient domination-like lemmas also appeared in Scherrer and Geist (2014).

It is instructive to consider this special case due to the connections it makes to the non-convex optimization literature. We also provide a lower bound that rules out algorithms whose runtime appeals to the curvature of saddle points (e.g. (Nesterov and Polyak, 2006; Ge et al., 2015; Jin et al., 2017)).

For the direct policy parametrization where $\theta_{s,a}=\pi_{\theta}(a|s)$, the gradient is:

using (5). In particular, for this parameterization, we may write $\nabla_{\pi}V^{\pi}(\mu)$ instead of $\nabla_{\theta}V^{\pi_{\theta}}(\mu)$.

Informally, we say a function $f(\theta)$ satisfies a gradient domination property if for all $\theta\in\Theta$,

where $\theta^{\star}\in\operatorname{argmax}_{\theta^{\prime}\in\Theta}f(\theta^{\prime})$ and where $G({\theta})$ is some suitable scalar notion of first-order stationarity, which can be considered a measure of how large the gradient is (see (Karimi et al., 2016; Bolte et al., 2007; Attouch et al., 2010)). Thus if one can find a $\theta$ that is (approximately) a first-order stationary point, then the parameter $\theta$ will be near optimal (in terms of function value). Such conditions are a standard device to establishing global convergence in non-convex optimization, as they effectively rule out the presence of bad critical points. In other words, given such a condition, quantifying the convergence rate for a specific algorithm, like say projected gradient ascent, will require quantifying the rate of its convergence to a first-order stationary point, for which one can invoke standard results from the optimization literature.

The following lemma shows that the direct policy parameterization satisfies a notion of gradient domination. This is the basic approach used in the analysis of CPI (Kakade and Langford, 2002); a variant of this lemma also appears in Scherrer and Geist (2014). We give a proof for completeness.

Even though we are interested in the value $V^{\pi}(\rho)$, it is helpful to consider the gradient with respect to another state distribution $\mu\in\Delta(\mathcal{S})$.

For the direct policy parameterization (as in (2)), for all state distributions $\mu,\rho\in\Delta(\mathcal{S})$, we have

where the max is over the set of all policies, i.e. $\bar{\pi}\in\Delta(\mathcal{A})^{|\mathcal{S}|}$.

Before we provide the proof, a few comments are in order with regards to the performance measure $\rho$ and the optimization measure $\mu$. Subtly, note that although the gradient is with respect to $V^{\pi}(\mu)$, the final guarantee applies to all distributions $\rho$. The significance is that even though we may be interested in our performance under $\rho$, it may be helpful to optimize under the distribution $\mu$. To see this, note the lemma shows that a sufficiently small gradient magnitude in the feasible directions implies the policy is nearly optimal in terms of its value, but only if the state distribution of $\pi$, i.e. $d_{\mu}^{\pi}$, adequately covers the state distribution of some optimal policy $\pi^{\star}$. Here, it is also worth recalling the theorem of Bellman and Dreyfus (1959) which shows there exists a single policy $\pi^{\star}$ that is simultaneously optimal for all starting states $s_{0}$. Note that the hardness of the exploration problem is captured through the distribution mismatch coefficient (Definition 3.1).

Proof:[of Lemma 4.1] By the performance difference lemma (Lemma 3.2),

where the last inequality follows since $\max_{\bar{a}}A^{\pi}(s,\bar{a})\geq 0$ for all states $s$ and policies $\pi$. We wish to upper bound (8). We then have:

where the first step follows since $\max_{\bar{\pi}}$ is attained at an action which maximizes $A^{\pi}(s,\cdot)$ (per state); the second step follows as $\sum_{a}{\pi}(a|s)A^{\pi}(s,a)=0$; the third step uses $\sum_{a}(\bar{\pi}(a|s)-\pi(a|s))V^{\pi}(s)=0$ for all $s$; and the final step follows from the gradient expression (see (7)). Using this in (8),

In a sense, the use of an appropriate $\mu$ circumvents the issues of strategic exploration. It is natural to ask whether this additional term is necessary, a question which we return to. First, we provide a convergence rate for the projected gradient ascent algorithm.

2 Convergence Rates for Projected Gradient Ascent

Using this notion of gradient domination, we now give an iteration complexity bound for projected gradient ascent over the space of stochastic policies, i.e. over $\Delta(\mathcal{A})^{|\mathcal{S}|}$. The projected gradient ascent algorithm updates

where $P_{\Delta(\mathcal{A})^{|\mathcal{S}|}}$ is the projection onto $\Delta(\mathcal{A})^{|\mathcal{S}|}$ in the Euclidean norm.

The projected gradient ascent algorithm (9) on $V^{\pi}(\mu)$ with stepsize $\eta=\frac{(1-\gamma)^{3}}{2\gamma|\mathcal{A}|}$ satisfies for all distributions $\rho\in\Delta(\mathcal{S})$,

A proof is provided in Appendix B.1. The proof first invokes a standard iteration complexity result of projected gradient ascent to show that the gradient magnitude with respect to all feasible directions is small. More concretely, we show the policy is $\epsilon$-stationarySee Appendix B.1 for discussion on this definition., that is, for all ${\pi_{\theta}}+\delta\in\Delta(\mathcal{A})^{|\mathcal{S}|}$ and $\|\delta\|_{2}\leq 1$, $\delta^{\top}\nabla_{\pi}V^{{\pi_{\theta}}}(\mu)\leq\epsilon$. We then use Lemma 4.1 to complete the proof.

Note that the guarantee we provide is for the best policy found over the $T$ rounds, which we obtain from a bound on the average norm of the gradients. This type of a guarantee is standard in the non-convex optimization literature, where an average regret bound cannot be used to extract a single good solution, e.g. by averaging. In the context of policy optimization, this is not a serious limitation as we collect on-policy trajectories for each policy in doing sample-based gradient estimation, and these samples can be also used to estimate the policy’s value. Note that the evaluation step is not required for every policy, and can also happen on a schedule, though we still need to evaluate $O(T)$ policies to obtain the convergence rates described here.

3 A Lower Bound: Vanishing Gradients and Saddle Points

To understand the necessity of the distribution mismatch coefficient in Lemma 4.1 and Theorem 4.1, let us first give an informal argument that some condition on the state distribution of $\pi$, or equivalently $\mu$, is necessary for stationarity to imply optimality. For example, in a sparse-reward MDP (where the agent is only rewarded upon visiting some small set of states), a policy that does not visit any rewarding states will have zero gradient, even though it is arbitrarily suboptimal in terms of values. Below, we give a more quantitative version of this intuition, which demonstrates that even if $\pi$ chooses all actions with reasonable probabilities (and hence the agent will visit all states if the MDP is connected), then there is an MDP where a large fraction of the policies $\pi$ have vanishingly small gradients, and yet these policies are highly suboptimal in terms of their value.

Concretely, consider the chain MDP of length $H+2$ shown in Figure 2. The starting state of interest is state $s_{0}$ and the discount factor $\gamma=H/(H+1)$. Suppose we work with the direct parameterization, where $\pi_{\theta}(a|s)=\theta_{s,a}$ for $a=a_{1},a_{2},a_{3}$ and $\pi_{\theta}(a_{4}|s)=1-\theta_{s,a_{1}}-\theta_{s,a_{2}}-\theta_{s,a_{3}}$. Note we do not over-parameterize the policy. For this MDP and policy structure, if we were to initialize the probabilities over actions, say deterministically, then there is an MDP (obtained by permuting the actions) where all the probabilities for $a_{1}$ will be less than $1/4$.

The following result not only shows that the gradient is exponentially small in $H$, it also shows that many higher order derivatives, up to $O(H/\log H)$, are also exponentially small in $H$.

This lemma also suggests that results in the non-convex optimization literature, on escaping from saddle points, e.g. (Nesterov and Polyak, 2006; Ge et al., 2015; Jin et al., 2017), do not directly imply global convergence due to that the higher order derivatives are small.

(Exact vs. Approximate Gradients) The chain MDP of Figure 2, is a common example where sample based estimates of gradients will be under random exploration strategies; there is an exponentially small in $H$ chance of hitting the goal state under a random exploration strategy. Note that this lemma is with regards to exact gradients. This suggests that even with exact computations (along with using exact higher order derivatives) we might expect numerical instabilities.

(Comparison with the upper bound) The lower bound does not contradict the upper bound of Theorem 4.1 (where a small gradient is turned into a small policy suboptimality bound), as the distribution mismatch coefficient, as defined in Definition 3.1, could be infinite in the chain MDP of Figure 2, since the start-state distribution is concentrated on one state only. More generally, for any policy with $\theta_{s,a_{1}}<1/4$ in all states $s$, $\left\|\frac{d^{\pi^{\star}}_{\rho}}{d^{\pi_{\theta}}_{\rho}}\right\|_{\infty}=\Omega(4^{H})$.

(Comparison with information-theoretic lower bounds) The lower bound here is not information theoretic, in that it does not present a hard problem instance for all algorithms. Indeed, exploration algorithms for tabular MDPs starting from $E^{3}$ (Kearns and Singh, 2002), RMAX (Brafman and Tennenholtz, 2003) and several subsequent works yield polynomial sample complexities for the chain MDP. Proposition 4.1 should be interpreted as a hardness result for the specific class of policy gradient like approaches that search for a policy with a small policy gradient, as these methods will find the initial parameters to be valid in terms of the size of (several orders of) gradients. In particular, it precludes any meaningful claims on global optimality, based just on the size of the policy gradients, without additional assumptions as discussed in the previous remark.

The proof is provided in Appendix B.2. The lemma illustrates that lack of good exploration can indeed be detrimental in policy gradient algorithms, since the gradient can be small either due to $\pi$ being near-optimal, or, simply because $\pi$ does not visit advantageous states often enough. In this sense, it also demonstrates the necessity of the distribution mismatch coefficient in Lemma 4.1.

The Softmax Tabular Parameterization

We now consider the softmax policy parameterization (3). Here, we still have a non-concave optimization problem in general, as shown in Lemma 3.1, though we do show that global optimality can be reached under certain regularity conditions. From a practical perspective, the softmax parameterization of policies is preferable to the direct parameterization, since the parameters $\theta$ are unconstrained and standard unconstrained optimization algorithms can be employed. However, optimization over this policy class creates other challenges as we study in this section, as the optimal policy (which is deterministic) is attained by sending the parameters to infinity.

We study three algorithms for this problem. The first performs direct policy gradient ascent on the objective without modification, while the second adds a log barrier regularizer to keep the parameters from becoming too large, as a means to ensure adequate exploration. Finally, we study the natural policy gradient algorithm and establish a global optimality result with no dependence on the distribution mismatch coefficient or dimension-dependent factors.

For the softmax parameterization, the gradient takes the form:

Due to the exponential scaling with the parameters $\theta$ in the softmax parameterization, any policy that is nearly deterministic will have gradients close to . In spite of this difficulty, we provide a positive result that gradient ascent asymptotically converges to the global optimum for the softmax parameterization.

Assume we follow the gradient ascent update rule as specified in Equation (11) and that the distribution $\mu$ is strictly positive i.e. $\mu(s)>0$ for all states $s$. Suppose $\eta\leq\frac{(1-\gamma)^{3}}{8}$, then we have that for all states $s$, $V^{(t)}(s)\rightarrow V^{\star}(s)$ as $t\rightarrow\infty$.

(Strict positivity of $\mu$ and exploration) Theorem 5.1 assumed that optimization distribution $\mu$ was strictly positive, i.e. $\mu(s)>0$ for all states $s$. We leave it is an open question of whether or not gradient ascent will globally converge if this condition is not met. The concern is that if this condition is not met, then gradient ascent may not globally converge due to that $d^{\pi_{\theta}}_{\mu}(s)$ effectively scales down the learning rate for the parameters associated with state $s$ (see (10)).

The complete proof is provided in the Appendix C.1. We now discuss the subtleties in the proof and show why the softmax parameterization precludes a direct application of the gradient domination lemma. In order to utilize the gradient domination property (in Lemma 4.1), we would desire to show that: $\nabla_{\pi}V^{\pi}(\mu)\rightarrow 0$. However, using the functional form of the softmax parameterization (see Lemma C.1) and (7), we have that:

Hence, we see that even if $\nabla_{\theta}V^{\pi_{\theta}}(\mu)\rightarrow 0$, we are not guaranteed that $\nabla_{\pi}V^{\pi_{\theta}}(\mu)\rightarrow 0$.

We now briefly discuss the main technical challenges in the proof. The proof first shows that the sequence $V^{(t)}(s)$ is monotone increasing pointwise, i.e. for every state $s$, $V^{(t+1)}(s)\geq V^{(t)}(s)$ (Lemma C.2). This implies the existence of a limit $V^{(\infty)}(s)$ by the monotone convergence theorem (Lemma C.3). Based on the limiting quantities $V^{(\infty)}(s)$ and $Q^{(\infty)}(s,a)$, which we show exist, define the following limiting sets for each state $s$:

The challenge is to then show that, for all states $s$, the set $I^{s}_{+}$ is the empty set, which would immediately imply $V^{(\infty)}(s)=V^{\star}(s)$. The proof proceeds by contradiction, assuming that $I^{s}_{+}$ is non-empty. Using that $I^{s}_{+}$ is non-empty and that the gradient tends to zero in the limit, i.e. $\nabla_{\theta}V^{\pi_{\theta}}(\mu)\rightarrow 0$, we have that for all $a\in I^{s}_{+}$, $\pi^{(t)}(a|s)\rightarrow 0$ (see (10)). This, along with the functional form of the softmax parameterization, implies that there must be divergence (in magnitude) among the set of parameters associated with some action $a$ at state $s$, i.e. that $\max_{a\in\mathcal{A}}|\theta^{(t)}_{s,a}|\to\infty$. The primary technical challenge in the proof is to then use this divergence, along with the dynamics of gradient ascent, to show that $I^{s}_{+}$ is empty via a contradiction.

We leave it as a question for future work as to characterizing the convergence rate, which we conjecture is exponentially slow in some of the relevant quantities, such as in terms of the size of state space. Here, we turn to a regularization based approach to ensure convergence at a polynomial rate in all relevant quantities.

2 Polynomial Convergence with Log Barrier Regularization

where $\lambda$ is a regularization parameter. The constant (i.e. the last term) is not relevant with regards to optimization. This regularizer is different from the more commonly utilized entropy regularizer as in Mnih et al. (2016), a point which we return to in Remark 5.2.

The policy gradient ascent updates for $L_{\lambda}(\theta)$ are given by:

Our next theorem shows that approximate first-order stationary points of the entropy-regularized objective are approximately globally optimal, provided the regularization is sufficiently small.

(Log barrier regularization) Suppose $\theta$ is such that:

and $\epsilon_{\text{opt}}\leq\lambda/(2|\mathcal{S}|\,|\mathcal{A}|)$. Then we have that for all starting state distributions $\rho$:

Proof: The proof consists of showing that $\max_{a}A^{\pi_{\theta}}(s,a)\leq 2\lambda/(\mu(s)|\mathcal{S}|)$ for all states. To see that this is sufficient, observe that by the performance difference lemma (Lemma 3.2),

We now proceed to show that $\max_{a}A^{\pi_{\theta}}(s,a)\leq 2\lambda/(\mu(s)|\mathcal{S}|)$. For this, it suffices to bound $A^{\pi_{\theta}}(s,a)$ for any state-action pair $s,a$ where $A^{\pi_{\theta}}(s,a)\geq 0$ else the claim is trivially true. Consider an $(s,a)$ pair such that $A^{\pi_{\theta}}(s,a)>0$. Using the policy gradient expression for the softmax parameterization (see Lemma C.1),

The gradient norm assumption $\|\nabla_{\theta}L_{\lambda}(\theta)\|_{2}\leq\epsilon_{\text{opt}}$ implies that:

where we have used $A^{\pi_{\theta}}(s,a)\geq 0$. Rearranging and using our assumption $\epsilon_{\text{opt}}\leq\lambda/(2|\mathcal{S}|\,|\mathcal{A}|)$,

Solving for $A^{\pi_{\theta}}(s,a)$ in (14), we have:

where the penultimate step uses $\epsilon_{\text{opt}}\leq\lambda/(2|\mathcal{S}|\,|\mathcal{A}|)$ and the final step uses $d^{\pi_{\theta}}_{\mu}(s)\geq(1-\gamma)\mu(s)$. This completes the proof.

By combining the above theorem with standard results on the convergence of gradient ascent (to first order stationary points), we obtain the following corollary.

(Iteration complexity with log barrier regularization) Let $\beta_{\lambda}:=\frac{8\gamma}{(1-\gamma)^{3}}+\frac{2\lambda}{|\mathcal{S}|}$. Starting from any initial $\theta^{(0)}$, consider the updates (13) with $\lambda=\frac{\epsilon(1-\gamma)}{2\left\lVert\frac{d^{\pi^{\star}}_{\rho}}{\mu}\right\rVert_{\infty}}$ and $\eta=1/\beta_{\lambda}$. Then for all starting state distributions $\rho$, we have

See Appendix C.2 for the proof. The corollary shows the importance of balancing how the regularization parameter $\lambda$ is set relative to the desired accuracy $\epsilon$, as well as the importance of the initial distribution $\mu$ to obtain global optimality.

(Entropy vs. log barrier regularization) The more commonly considered regularizer is the entropy (Mnih et al., 2016) (also see Ahmed et al. (2019) for a more detailed empirical investigation), where the regularizer would be:

Note the entropy is far less aggressive in penalizing small probabilities, in comparison to the log barrier, which is equivalent to the relative entropy. In particular, the entropy regularizer is always bounded between and $\log|\mathcal{A}|$, while the relative entropy (against the uniform distribution over actions), is bounded between and infinity, where it tends to infinity as probabilities tend to . We leave it is an open question if a polynomial convergence rate Here, ideally we would like to be poly in $|\mathcal{S}|$, $|\mathcal{A}|$, $1/(1-\gamma)$, $1/\epsilon$, and the distribution mismatch coefficient, which we conjecture may not be possible. is achievable with the more common entropy regularizer; our polynomial convergence rate using the KL regularizer crucially relies on the aggressive nature in which the relative entropy prevents small probabilities (the proof shows that any action, with a positive advantage, has a significant probability for any near-stationary policy of the regularized objective).

3 Dimension-free Convergence of Natural Policy Gradient Ascent

We now show the Natural Policy Gradient algorithm, with the softmax parameterization (3), obtains an improved iteration complexity. The NPG algorithm defines a Fisher information matrix (induced by $\pi$), and performs gradient updates in the geometry induced by this matrix as follows:

We leverage a particularly convenient form the update takes for the softmax parameterization (see Kakade (2001)). For completeness, we provide a proof in Appendix C.3.

(NPG as soft policy iteration) For the softmax parameterization (3), the NPG updates (5.3) take the form:

where $Z_{t}(s)=\sum_{a\in\mathcal{A}}\pi^{(t)}(a|s)\exp(\eta A^{(t)}(s,a)/(1-\gamma))$.

The updates take a strikingly simple form in this special case; they are identical to the classical multiplicative weights updates (Freund and Schapire, 1997; Cesa-Bianchi and Lugosi, 2006) for online linear optimization over the probability simplex, where the linear functions are specified by the advantage function of the current policy at each iteration. Notably, there is no dependence on the state distribution $d^{(t)}_{\rho}$, since the pseudoinverse of the Fisher information cancels out the effect of the state distribution in NPG. We now provide a dimension free convergence rate of this algorithm.

Suppose we run the NPG updates (5.3) using $\rho\in\Delta(\mathcal{S})$ and with $\theta^{(0)}=0$. Fix $\eta>0$. For all $T>0$, we have:

In particular, setting $\eta\geq(1-\gamma)^{2}\log|\mathcal{A}|$, we see that NPG finds an $\epsilon$-optimal policy in a number of iterations that is at most:

which has no dependence on the number of states or actions, despite the non-concavity of the underlying optimization problem.

The proof strategy we take borrows ideas from the online regret framework in changing MDPs (in (Even-Dar et al., 2009)); here, we provide a faster rate of convergence than the analysis implied by Even-Dar et al. (2009) or by Geist et al. (2019). We also note that while this proof is obtained for the NPG updates, it is known in the literature that in the limit of small stepsizes, NPG and TRPO updates are closely related (e.g. see Schulman et al. (2015); Neu et al. (2017); Rajeswaran et al. (2017)).

First, the following improvement lemma is helpful:

For the iterates $\pi^{(t)}$ generated by the NPG updates (5.3), we have for all starting state distributions $\mu$

Proof: First, let us show that $\log Z_{t}(s)\geq 0$. To see this, observe:

where the inequality follows by Jensen’s inequality on the concave function $\log x$ and the final equality uses $\sum_{a}\pi^{(t)}(a|s)A^{(t)}(s,a)=0$. Using $d^{(t+1)}$ as shorthand for $d^{(t+1)}_{\mu}$, the performance difference lemma implies:

where the last step uses that $d^{(t+1)}=d^{(t+1)}_{\mu}\geq(1-\gamma)\mu$, componentwise (by (4)), and that $\log Z_{t}(s)\geq 0$.

With this lemma, we now prove Theorem 5.3.

Proof:[of Theorem 5.3] Since $\rho$ is fixed, we use $d^{\star}$ as shorthand for $d^{\pi^{\star}}_{\rho}$; we also use $\pi_{s}$ as shorthand for the vector of $\pi(\cdot|s)$. By the performance difference lemma (Lemma 3.2),

where we have used the closed form of our updates from Lemma 5.1 in the second step.

By applying Lemma 5.2 with $d^{\star}$ as the starting state distribution, we have:

Using the above equation and that $V^{(t+1)}(\rho)\geq V^{(t)}(\rho)$ (as $V^{(t+1)}(s)\geq V^{(t)}(s)$ for all states $s$ by Lemma 5.2), we have:

The proof is completed using that $V^{(T)}(\rho)\geq V^{(T-1)}(\rho)$.

Function Approximation and Distribution Shift

We now analyze the case of using parametric policy classes:

where $\Pi$ may not contain all stochastic policies (and it may not even contain an optimal policy). In contrast with the tabular results in the previous sections, the policy classes that we are often interested in are not fully expressive, e.g. $d\ll|\mathcal{S}||\mathcal{A}|$ (indeed $\lvert\mathcal{S}\rvert$ or $\lvert\mathcal{A}\rvert$ need not even be finite for the results in this section); in this sense, we are in the regime of function approximation.

We focus on obtaining agnostic results, where we seek to do as well as the best policy in this class (or as well as some other comparator policy). While we are interested in a solution to the (unconstrained) policy optimization problem

(for a given initial distribution $\rho$), we will see that optimization with respect to a different distribution will be helpful, just as in the tabular case,

We will consider variants of the NPG update rule (5.3):

Our analysis will leverage a close connection between the NPG update rule (5.3) with the notion of compatible function approximation (Sutton et al., 1999), as formalized in Kakade (2001). Specifically, it can be easily seen that:

where $w^{\star}$ is a minimizer of the following regression problem:

The above is a straightforward consequence of the first order optimality conditions (see (50)). The above regression problem can be viewed as “compatible” function approximation: we are approximating $A^{\pi_{\theta}}(s,a)$ using the $\nabla_{\theta}\log\pi_{\theta}(\cdot|s)$ as features. We also consider a variant of the above update rule, $Q$-NPG, where instead of using advantages in the above regression we use the $Q$-values.

This viewpoint provides a methodology for approximate updates, where we can solve the relevant regression problems with samples. Our main results establish the effectiveness of NPG updates where there is error both due to statistical estimation (where we may not use exact gradients) and approximation (due to using a parameterized function class); in particular, we provide a novel estimation/approximation decomposition relevant for the NPG algorithm. For these algorithms, we will first consider log linear policies classes (as a special case) and then move on to more general policy classes (such as neural policy classes). Finally, it is worth remarking that the results herein provide one of the first provable approximation guarantees where the error conditions required do not have explicit worst case dependencies over the state space.

In practice, the most common policy classes are of the form:

where $f_{\theta}$ is a differentiable function. For example, the tabular softmax policy class is one where $f_{\theta}(s,a)=\theta_{s,a}$. Typically, $f_{\theta}$ is either a linear function or a neural network. Let us consider the NPG algorithm, and a variant $Q$-NPG, in each of these two cases.

With regards to compatible function approximation for the log-linear policy class, we have:

that is, $\overline{\phi}^{\ \theta}_{s,a}$ is the centered version of $\phi_{s,a}$. With some abuse of notation, we accordingly also define $\bar{\phi}^{\pi}$ for any policy $\pi$. Here, using (17), the NPG update rule (16) is equivalent to:

(We have rescaled the learning rate $\eta$ in comparison to (16)). Note that we recompute $w_{\star}$ for every update of $\theta$. Here, the compatible function approximation error measures the expressivity of our parameterization in how well linear functions of the parameterization can capture the policy’s advantage function.

We also consider a variant of the NPG update rule (16), termed $Q$-NPG, where:

Note we do not center the features for $Q$-NPG; observe that $Q^{\pi}(s,a)$ is also not 0 in expectation under $\pi(\cdot|s)$, unlike the advantage function.

(NPG/$Q$-NPG and Soft-Policy Iteration) We now see how we can view both NPG and $Q$-NPG as an incremental (soft) version of policy iteration, just as in Lemma 5.1 for the tabular case. Rather than writing the update rule in terms of the parameter $\theta$, we can write an equivalent update rule directly in terms of the (log-linear) policy $\pi$:

where $Z_{s}$ is normalization constant. While the policy update uses the original features $\phi$ instead of $\overline{\phi}^{\ \pi}$, whereas the quadratic error minimization is terms of the centered features $\overline{\phi}^{\ \pi}$, this distinction is not relevant due to that we may also instead use $\overline{\phi}^{\ \pi}$ (in the policy update) which would result in an equivalent update; the normalization makes the update invariant to (constant) translations of the features. Similarly, an equivalent update for $Q$-NPG, where we update $\pi$ directly rather than $\theta$, is:

(On the equivalence of NPG and $Q$-NPG) If it is the case that the compatible function approximation error is , then it straightforward to verify that the NPG and $Q$-NPG are equivalent algorithms, in that their corresponding policy updates will be equivalent to each other.

1.2 Neural Policy Classes

and, using (17), the NPG update rule (16) is equivalent to:

(Again, we have rescaled the learning rate $\eta$ in comparison to (16)).

The $Q$-NPG variant of this update rule is:

2 Q𝑄Q-NPG: Performance Bounds for Log-Linear Policies

For a state-action distribution $\upsilon$, define:

The iterates of the $Q$-NPG algorithm can be viewed as minimizing this loss under some (changing) distribution $\upsilon$.

We now specify an approximate version of $Q$-NPG. It is helpful to consider a slightly more general version of the algorithm in the previous section, where instead of optimizing under a starting state distribution $\rho$, we have a different starting state-action distribution $\nu$. Analogous to the definition of the state visitation measure, $d_{\mu}^{\pi}$, we can define a visitation measure over states and actions induced by following $\pi$ after $s_{0},a_{0}\sim\nu$. We overload notation using $d_{\nu}^{\pi}$ to also refer to the state-action visitation measure; precisely,

$Q$-NPG will be defined with respect to the on-policy state action measure starting with $s_{0},a_{0}\sim\nu$. As per our convention, we define

The approximate version of this algorithm is:

Note that $w_{\star}^{(t)}$ depends on the current parameter $\theta^{(t)}$.

Our analysis will take into account both the excess risk (often also referred to as estimation error) and the transfer error. Here, the excess risk will be due to that $w^{(t)}$ may not be equal $w_{\star}^{(t)}$, and the approximation error will be due to that even the best linear fit using $w_{\star}^{(t)}$ may not perfectly match the $Q$-values, i.e. $L(w_{\star}^{(t)};\theta^{(t)};d^{(t)})$ is unlikely to be in practical applications.

We now formalize these concepts in the following assumption:

Fix a state distribution $\rho$; a state-action distribution $\nu$; an arbitrary comparator policy $\pi^{\star}$ (not necessarily an optimal policy). With respect to $\pi^{\star}$, define the state-action measure $d^{\star}$ as

i.e. $d^{\star}$ samples states from the comparators state visitation measure, $d_{\rho}^{\pi^{\star}}$ and actions from the uniform distribution. Let us permit the sequence of iterates $w^{(0)},w^{(1)},\ldots w^{(T-1)}$ used by the $Q$-NPG algorithm to be random, where the randomness could be due to sample-based, estimation error. Suppose the following holds for all $t<T$:

(Excess risk) Assume that the estimation error is bounded as follows:

In both conditions, the expectations are with respect to the randomness in the sequence of iterates $w^{(0)},w^{(1)},\ldots w^{(T-1)}$, e.g. the approximate algorithm may be sample based.

Shortly, we discuss how the transfer error relates to the more standard approximation-estimation decomposition. Importantly, with the transfer error, it is always defined with respect to a single, fixed measure, $d^{\star}$.

Consider the same $\rho$, $\nu$, and $\pi^{\star}$ as in Assumption 6.1. With respect to any state-action distribution $\upsilon$, define:

(Agnostic learning with $Q$-NPG) Fix a state distribution $\rho$; a state-action distribution $\nu$; an arbitrary comparator policy $\pi^{\star}$ (not necessarily an optimal policy). Suppose Assumption 6.2 holds and $\|\phi_{s,a}\|_{2}\leq B$ for all $s,a$. Suppose the $Q$-NPG update rule (in (20)) starts with $\theta^{(0)}=0$, $\eta=\sqrt{2\log|\mathcal{A}|/(B^{2}W^{2}T)}$, and the (random) sequence of iterates satisfies Assumption 6.1. We have that

The usual approximation-estimation error decomposition is that we can write our error as:

As we obtain more samples, we can drive the excess risk (the estimation error) to (see Corollary 6.2). The approximation error above is due to modeling error. Importantly, for our $Q$-NPG performance bound, it is not this standard approximation error notion which is relevant, but it is this error under a different measure $d^{\star}$, i.e. $L(w_{\star}^{(t)};\theta^{(t)},d^{\star})$. One appealing aspect about the transfer error is that this error is with respect to a fixed measure, namely $d^{\star}$. Furthermore, in practice, modern machine learning methods often performs favorably with regards to transfer learning, substantially better than worst case theory might suggest.

The following corollary provides a performance bound in terms of the usual notion of approximation error, at the cost of also depending on the worst case distribution mismatch ratio. The corollary disentangles the estimation error from the approximation error.

(Estimation error/Approximation error bound for $Q$-NPG) Consider the same setting as in Theorem 6.1. Rather than assuming the transfer error is bounded (part 2 in Assumption 6.1), suppose that, for all $t\leq T$,

Proof: We have the following crude upper bound on the transfer error:

The above also shows the striking difference between the effects of estimation error and approximation error. The proof shows how the transfer error notion is weaker than previous conditions based on distribution mistmatch coefficients or concentrability coefficients. Also, as discussed in Scherrer (2014), the (distribution mismatch) coefficient \Bigl{\lVert}\frac{d^{\star}}{\nu}\Bigr{\rVert}_{\infty} is already weaker than the more standard concentrability coefficients.

A few additional remarks are now in order. We now make a few observations with regards to $\kappa$.

Proof: The distribution can be found through constructing the minimal volume ellipsoid containing $\Phi$, i.e. the Loẅner-John ellipsoid (John, 1948). In particular, this $\nu$ is supported on the contact points between this ellipsoid and $\Phi$; the lemma immediately follows from properties of this ellipsoid (e.g. see Ball (1997); Bubeck et al. (2012)).

(Comparison with Politex and EE-Politex) Compared with Politex (Abbasi-Yadkori et al., 2019a), Assumption 6.2 is substantially milder, in that it just assumes a good relative condition number for one policy rather than all possible policies (which cannot hold in general even for tabular MDPs). Changing this assumption to an analog of Assumption 6.2 is the main improvement in the analysis of the EE-Politex (Abbasi-Yadkori et al., 2019b) algorithm. They provide a regret bound for the average reward setting, which is qualitatively different from the suboptimality bound in the discounted setting that we study. They provide a specialized result for linear function approximation, similar to Theorem 6.1.

For a fixed state-action distribution $\nu$, we assume the ability to: start at $s_{0},a_{0}\sim\nu$; continue to act thereafter in the MDP according to any policy $\pi$; and terminate this “rollout” when desired. With this oracle, it is straightforward to obtain unbiased samples of $Q^{\pi}(s,a)$ (or $A^{\pi}(s,a)$) under $s,a\sim d_{\nu}^{\pi}$ for any $\pi$; see Algorithms 1 and 3.

Algorithm 2 provides a sample based version of the $Q$-NPG algorithm; it simply uses stochastic projected gradient ascent within each iteration. The following corollary shows this algorithm suffices to obtain an accurate sample based version of $Q$-NPG.

(Sample complexity of $Q$-NPG) Assume we are in the setting of Theorem 6.1 and that we have access to an episodic sampling oracle (i.e. Assumption 6.3). Suppose that the Sample Based $Q$-NPG Algorithm (Algorithm 2) is run for $T$ iterations, with $N$ gradient steps per iteration, with an appropriate setting of the learning rates $\eta$ and $\alpha$. We have that:

Furthermore, since each episode has expected length $2/(1-\gamma)$, the expected number of total samples used by $Q$-NPG is $2NT/(1-\gamma)$.

Proof: Note that our sampled gradients are bounded by $G:=2B(BW+\frac{1}{1-\gamma})$. Using $\alpha=\frac{W}{G\sqrt{N}}$, a standard analysis for stochastic projected gradient ascent (Theorem E.3) shows that:

3 NPG: Performance Bounds for Smooth Policy Classes

We now return to the analyzing the standard NPG update rule, which uses advantages rather than $Q$-values (see Section 6.1). It is helpful to define

We now consider an approximate version of the NPG update rule:

where again we use the on-policy, fitting distribution $d^{(t)}$. As with $Q$-NPG, we also permit the use of a starting state-action distribution $\nu$ as opposed to just a starting state distribution (see Remark 6.3). Again, we let $w_{\star}^{(t)}$ denote the minimizer, i.e. $w_{\star}^{(t)}\in\operatorname{{argmin}}_{\|w\|_{2}\leq W}L_{A}(w;\theta^{(t)},d^{(t)})$.

For this section, our analysis will focus on more general policy classes, beyond log-linear policy classes. In particular, we make the following smoothness assumption on the policy class:

(Policy Smoothness) Assume for all $s\in\mathcal{S}$ and $a\in\mathcal{A}$ that $\log\pi_{\theta}(a|s)$ is a $\beta$-smooth function of $\theta$ (to recall the definition of smoothness, see (24)).

It is not to difficult to verify that the tabular softmax policy parameterization is a $1$-smooth policy class in the above sense. The more general class of log-linear policies is also smooth as we remark below.

(Smoothness of the log-linear policy class) For the log-linear policy class (see Section 6.1.1), smoothness is implied if the features $\phi$ have bounded Euclidean norm. Precisely, if the feature mapping $\phi$ satisfies $\|\phi_{s,a}\|_{2}\leq B$, then it is not difficult to verify that $\log\pi_{\theta}(a|s)$ is a $B^{2}$-smooth function.

For any state-action distribution $\upsilon$, define:

and, again, we use $\Sigma^{(t)}_{\upsilon}$ as shorthand for $\Sigma^{\theta^{(t)}}_{\upsilon}$.

(Estimation/Transfer/Conditioning) Fix a state distribution $\rho$; a state-action distribution $\nu$; an arbitrary comparator policy $\pi^{\star}$ (not necessarily an optimal policy). With respect to $\pi^{\star}$, define the state-action measure $d^{\star}$ as

Note that, in comparison to Assumption 6.1, $d^{\star}$ is the state-action visitation measure of the comparator policy. Let us permit the sequence of iterates $w^{(0)},w^{(1)},\ldots w^{(T-1)}$ used by the NPG algorithm to be random, where the randomness could be due to sample-based, estimation error. Suppose the following holds for all $t<T$:

(Excess risk) Assume the estimation error is bounded as:

(Relative condition number) For all iterations $t$, assume the average relative condition number is bounded as follows:

Note that term inside the expectation is a random quantity as $\theta^{(t)}$ is random.

In the above conditions, the expectation is with respect to the randomness in the sequence of iterates $w^{(0)},w^{(1)},\ldots w^{(T-1)}$.

(Agnostic learning with NPG) Fix a state distribution $\rho$; a state-action distribution $\nu$; an arbitrary comparator policy $\pi^{\star}$ (not necessarily an optimal policy). Suppose Assumption 6.4 holds. Suppose the NPG update rule (in (21)) starts with $\pi^{(0)}$ being the uniform distribution (at each state), $\eta=\sqrt{2\log|\mathcal{A}|/(\beta W^{2}T)}$, and the (random) sequence of iterates satisfies Assumption 6.5. We have that

(The $|\mathcal{A}|$ dependence: NPG vs. $Q$-NPG) Observe there is no polynomial dependence on $|\mathcal{A}|$ in the rate for NPG (in constrast to Theorem 6.1); also observe that here we define $d^{\star}$ as the state-action distribution of $\pi^{\star}$ in Assumption 6.5, as opposed to a uniform distribution over the actions, as in Assumption 6.1. The main difference arises in the analysis in that, even for $Q$-NPG, we need to bound the error in fitting the advantage estimates; this leads to the dependence on $|\mathcal{A}|$ (which can be removed with a path dependent bound, i.e. a bound which depends on the sequence of iterates produced by the algorithm) For $Q$-NPG, we have to bound two distribution shift terms to both $\pi^{\star}$ and $\pi^{(t)}$ at step $t$ of the algorithm.. For NPG, the direct fitting of the advantage function sidesteps this conversion step. Note that the relative condition number assumption in $Q$-NPG (Assumption 6.2) is a weaker assumption, due to that it can be bounded independently of the path of the algorithm (see Remark 6.2), while NPG’s centering of the features makes the assumption on the relative condition number depend on the path of the algorithm.

(Generalizing $Q$-NPG for smooth policies) A similar reasoning as the analysis here can be also used to establish a convergence result for the $Q$-NPG algorithm in this more general setting of smooth policy classes. Concretely, we can analyze the $Q$-NPG update described for neural policy classes in Section 6.1.2, assuming that the function $f_{\theta}$ is Lipschitz-continuous in $\theta$. Like for Theorem 6.2, the main modification is that Assumption 6.2 on relative condition numbers is now defined using the covariance matrix for the features $f_{\theta}(s,a)$, which depend on $\theta$, as opposed to some a feature map $\phi(s,a)$ in the log-linear case. The rest of the analysis follows with an appropriate adaptation of the results above.

Algorithm 4 provides a sample based version of the NPG algorithm, again using stochastic projected gradient ascent; it uses a slight modification of the $Q$-NPG algorithm to obtain unbiased gradient estimates. The following corollary shows that this algorithm provides an accurate sample based version of NPG.

(Sample complexity of NPG) Assume we are in the setting of Theorem 6.2 and that we have access to an episodic sampling oracle (i.e. Assumption 6.3). Suppose that the Sample Based NPG Algorithm (Algorithm 4) is run for $T$ iterations, with $N$ gradient steps per iteration. Also, suppose that $\|\nabla_{\theta}\log\pi^{(t)}(a|s)\|_{2}\leq B$ holds with probability one. There exists a setting of $\eta$ and $\alpha$ such that:

Furthermore, since each episode has expected length $2/(1-\gamma)$, the expected number of total samples used by NPG is $2NT/(1-\gamma)$.

Proof: Let us see that the update direction in Step 7 of Algorithm 4 uses an unbiased estimate of the true gradient of the loss function $L_{A}$:

where the last step follows due to that sampling procedure in Algorithm 3 produces a conditionally unbiased estimate.

Since $\|\nabla_{\theta}\log\pi^{(t)}(a|s)\|_{2}\leq B$ and since $\widehat{A}(s,a)\leq 2/(1-\gamma)$, our sampled gradients are bounded by $G:=8B(BW+\frac{1}{1-\gamma})$. The remainder of the proof follows that of Corollary 6.2

4 Analysis

We first proceed by providing a general analysis of NPG, for arbitrary sequences. We then specialize it to complete the proof of our two main theorems in this section.

It is helpful for us to consider NPG more abstractly, as an update rule of the form

We will now provide a lemma where $w^{(t)}$ is an arbitrary (bounded) sequence, which will be helpful when specialized.

and, due to Taylor’s theorem, recall that this implies:

The following analysis of NPG is based on the mirror-descent approach developed in (Even-Dar et al., 2009), which motivates us to refer to it as a “regret lemma”.

(NPG Regret Lemma) Fix a comparison policy $\widetilde{\pi}$ and a state distribution $\rho$. Assume for all $s\in\mathcal{S}$ and $a\in\mathcal{A}$ that $\log\pi_{\theta}(a|s)$ is a $\beta$-smooth function of $\theta$. Consider the update rule (23), where $\pi^{(0)}$ is the uniform distribution (for all states) and where the sequence of weights $w^{(0)},\ldots,w^{(T)}$, satisfies $\|w^{(t)}\|_{2}\leq W$ (but is otherwise arbitrary). Define:

We use $\widetilde{d}$ as shorthand for $d^{\widetilde{\pi}}_{\rho}$ (note $\rho$ and $\widetilde{\pi}$ are fixed); for any policy $\pi$, we also use $\pi_{s}$ as shorthand for the vector $\pi(\cdot|s)$. Using the performance difference lemma (Lemma 3.2),

4.2 Proofs of Theorem 6.1 and 6.2

Proof: (of Theorem 6.1) Using the NPG regret lemma (Lemma 6.2) and the smoothness of the log-linear policy class (see Example 6.7),

where we have used our setting of $\eta$.

where we have used the definition of $d^{\star}$ and $L(w_{\star}^{(t)};\theta^{(t)},d^{\star})$ in the last step.

For the second term, let us now show that:

To see this, first observe that a similar argument to the above leads to:

where we use the notation $\|x\|_{M}^{2}:=x^{\top}Mx$ for a matrix $M$ and a vector $x$. From the definition of $\kappa$,

using that $(1-\gamma)\nu\leq d_{\nu}^{\pi^{(t)}}$ (see (19)). Due to that $w_{\star}^{(t)}$ minimizes $L(w;\theta^{(t)},d^{(t)})$ over the set $\mathcal{W}:=\{w:\|w\|_{2}\leq W\}$, for any $w\in\mathcal{W}$ the first-order optimality conditions for $w_{\star}^{(t)}$ imply that:

Noting that $w^{(t)}\in\mathcal{W}$ by construction in Algorithm 4 yields the claimed bound on the second term in (26).

Using the bounds on the first and second terms in (25) and (26), along with concavity of the square root function, we have that:

The following proof for the NPG algorithm follows along similar lines.

Proof: (of Theorem 6.2) Using the NPG regret lemma and our setting of $\eta$,

where the expectation is with respect to the sequence of iterates $w^{(0)},w^{(1)},\ldots w^{(T-1)}$.

where we have used the definition of $L_{A}(w_{\star}^{(t)};\theta^{(t)},d^{\star})$ in the last step.

For the second term, a similar argument leads to:

Define $\kappa^{(t)}:=\|(\Sigma^{(t)}_{\nu})^{-1/2}\Sigma_{d^{\star}}(\Sigma^{(t)}_{\nu})^{-1/2}\|_{2}$, which is the relative condition number at iteration $t$. We have

where the last step uses that $w_{\star}^{(t)}$ is a minimizer of $L_{A}$ over $\mathcal{W}$ and that $w^{(t)}$ is feasible as before (see the proof of Theorem 6.1). Now taking an expectation we have:

The proof is completed by substitution and using the concavity of the square root function.

Discussion

This work provides a systematic study of the convergence properties of policy optimization techniques, both in the tabular and the function approximation settings. At the core, our results imply that the non-convexity of the policy optimization problem is not the fundamental challenge for typical variants of the policy gradient approach. This is evidenced by the global convergence results which we establish and that demonstrate the relative niceness of the underlying optimization problem. At the same time, our results highlight that insufficient exploration can lead to the convergence to sub-optimal policies, as is also observed in practice; technically, we show how this is an issue of conditioning. Conversely, we can expect typical policy gradient algorithms to find the best policy from amongst those whose state-visitation distribution is adequately aligned with the policies we discover, provided a distribution-shifted notion of approximation error is small.

In the tabular case, our results show that the nature and severity of the exploration/distribution mismatch term differs in different policy optimization approaches. For instance, we find that doing policy gradient in its standard form for both the direct and softmax parameterizations can be slow to converge, particularly in the face of distribution mismatch, even when policy gradients are computed exactly. Natural policy gradient, on the other hand, enjoys a fast dimension-free convergence when we are in tabular settings with exact gradients. On the other hand, for the function approximation setting, or when using finite samples, all algorithms suffer to some degree from the exploration issue captured through a conditioning effect.

With regards to function approximation, the guarantees herein are the first provable results that permit average case approximation errors, where the guarantees do not have explicit worst case dependencies over the state space. These worst case dependencies are avoided by precisely characterizing an approximation/estimation error decomposition, where the relevant approximation error is under distribution shift to an optimal policies measure. Here, we see that successful function approximation relies on two key aspects: good conditioning (related to exploration) and low distribution-shifted, approximation error. In particular, these results identify the relevant measure of the expressivity of a policy class, for the natural policy gradient.

With regards to sample size issues, we showed that simply using stochastic (projected) gradient ascent suffices for accurate policy optimization. However, in terms of improving sample efficiency and polynomial dependencies, there are number of important questions for future research, including variance reduction techniques along with data re-use.

There are number of compelling directions for further study. The first is in understanding how to remove the density ratio guarantees among prior algorithms; our results are suggestive that the incremental policy optimization approaches, including CPI (Kakade and Langford, 2002), PSDP (Bagnell et al., 2004), and MD-MPI Geist et al. (2019), may permit such an improved analysis. The question of understanding what representations are robust to distribution shift is well-motivated by the nature of our distribution-shifted, approximation error (the transfer error). Finally, we hope that policy optimization approaches can be combined with exploration approaches, so that, provably, these approaches can retain their robustness properties (in terms of their agnostic learning guarantees) while mitigating the need for a well conditioned initial starting distribution.

We thank the anonymous reviewers who provided detailed and constructive feedback that helped us significantly improve the presentation and exposition. Sham Kakade and Alekh Agarwal gratefully acknowledge numerous helpful discussions with Wen Sun with regards to the $Q$-NPG algorithm and our notion of transfer error. We also acknowledge numerous helpful comments from Ching-An Cheng and Andrea Zanette on an earlier draft of this work. We thank Nan Jiang, Bruno Scherrer, and Matthieu Geist for their comments with regards to the relationship between concentrability coefficients, the condition number, and the transfer error; this discussion ultimately lead to Corollary 6.1. Sham Kakade acknowledges funding from the Washington Research Foundation for Innovation in Data-intensive Discovery, the ONR award N00014-18-1-2247, and the DARPA award FA8650-18-2-7836. Jason D. Lee acknowledges support of the ARO under MURI Award W911NF-11-1-0303. This is part of the collaboration between US DOD, UK MOD and UK Engineering and Physical Research Council (EPSRC) under the Multidisciplinary University Research Initiative.

References

Appendix A Proofs for Section 3

Proof:[of Lemma 3.1] Recall the MDP in Figure 2. Note that since actions in terminal states $s_{3}$, $s_{4}$ and $s_{5}$ do not change the expected reward, we only consider actions in states $s_{1}$ and $s_{2}$. Let the ”up/above” action as $a_{1}$ and ”right” action as $a_{2}$. Note that

where $\theta$ is written as a tuple $(\theta_{a_{1},s_{1}},\theta_{a_{2},s_{1}},\theta_{a_{1},s_{2}},\theta_{a_{2},s_{2}})$. Then, for the softmax parameterization, we have

Also, for $\theta^{(\text{mid})}=\frac{\theta^{(1)}+\theta^{(2)}}{2}$,

which shows that $V^{\pi}$ is non-concave.

Proof:[of Lemma 3.2] Let $\Pr^{\pi}(\tau|s_{0}=s)$ denote the probability of observing a trajectory $\tau$ when starting in state $s$ and following the policy $\pi$. Using a telescoping argument, we have:

where $(a)$ rearranges terms in the summation and cancels the $V^{\pi^{\prime}}(s_{0})$ term with the $-V^{\pi^{\prime}}(s)$ outside the summation, and $(b)$ uses the tower property of conditional expectations and the final equality follows from the definition of $d^{\pi}_{s}$.

Appendix B Proofs for Section 4

We first define first-order optimality for constrained optimization.

A policy ${\pi_{\theta}}\in\Delta(\mathcal{A})^{|S|}$ is $\epsilon$-stationary with respect to the initial state distribution $\mu$ if

where $\Delta(\mathcal{A})^{|S|}$ is the set of all policies.

If $\epsilon=0$, then the definition simplifies to $\delta^{\top}\nabla_{\pi}V^{{{\pi}}}(\mu)\leq 0$. Geometrically, $\delta$ is a feasible direction of movement since the probability simplex $\Delta(\mathcal{A})^{|S|}$ is convex. Thus the gradient is negatively correlated with any feasible direction of movement, and so ${\pi}$ is first-order stationary.

Let $V^{\pi}(\mu)$ be $\beta$-smooth in ${\pi}$. Define the gradient mapping

and the update rule for the projected gradient is $\pi^{+}={\pi}+\eta G^{\eta}(\pi)$. If $\|G^{\eta}(\pi)\|_{2}\leq\epsilon$, then

Proof:[of Theorem 4.1] Recall the definition of gradient mapping

From Lemma D.3, we have $V^{\pi}(s)$ is $\beta$-smooth for all states $s$ (and also hence $V^{\pi}(\mu)$ is also $\beta$-smooth) with $\beta=\frac{2\gamma|\mathcal{A}|}{(1-\gamma)^{3}}$. Then, from standard result (Theorem E.1), we have that for $G^{\eta}({\pi})$ with step-size $\eta=\frac{1}{\beta}$,

where the last step follows as $\left\lVert\bar{\pi}-{\pi}\right\rVert_{2}\leq 2\sqrt{|\mathcal{S}|}$. And then using Lemma 4.1 and $\eta\beta=1$, we have

We can get our required bound of $\epsilon$, if we set $T$ such that

Using $V^{\star}(\mu)-V^{(0)}(\mu)\leq\frac{1}{1-\gamma}$ and $\beta=\frac{2\gamma|\mathcal{A}|}{(1-\gamma)^{3}}$ from Lemma D.3 leads to the desired result.

B.2 Proofs for Section 4.3

Recall the MDP in Figure 2. Each trajectory starts from the initial state $s_{0}$, and we use the discount factor $\gamma=H/(H+1)$. Recall that we work with the direct parameterization, where $\pi_{\theta}(a|s)=\theta_{s,a}$ for $a=a_{1},a_{2},a_{3}$ and $\pi_{\theta}(a_{4}|s)=1-\theta_{s,a_{1}}-\theta_{s,a_{2}}-\theta_{s,a_{3}}$. Note that since states $s_{0}$ and $s_{H+1}$ only have once action, therefore, we only consider the parameters for states $s_{1}$ to $s_{H}$. For this policy class and MDP, let $P^{\theta}$ be the state transition matrix under $\pi_{\theta}$, i.e. $[P^{\theta}]_{s,s^{\prime}}$ is the probability of going from state $s$ to $s^{\prime}$ under policy $\pi_{\theta}$:

For the MDP illustrated in Figure 2, the entries of this matrix are given as:

With this definition, we recall that the value function in the initial state $s_{0}$ is given by

For convenience, we also define $\bar{p}$ (resp. $\underline{p}$) to be the largest (resp. smallest) of the probabilities $\theta_{s}$ across the states $s\in[1,H]$ in the MDP.

In this section, we prove Proposition 4.1, that is: for $0<\theta<1$ (componentwise across states and actions), $\bar{p}\leq 1/4$, and for all $k\leq\frac{H}{40\log(2H)}-1$, we have $\lVert\nabla_{\theta}^{k}V^{\pi_{\theta}}(s_{0})\rVert\leq(1/3)^{H/4}$, where $\nabla_{\theta}^{k}V^{\pi_{\theta}}(s_{0})$ is a tensor of the $k_{th}$ order. Furthermore, we seek to show $V^{\star}(s_{0})-V^{\pi_{\theta}}(s_{0})\geq(H+1)/8-(H+1)^{2}/3^{H}$ (where $\theta^{\star}$ are the optimal policy’s parameters).

It is easily checked that $V^{\pi_{\theta}}(s_{0})=M^{\theta}_{0,H+1}$, where

since the only rewards are obtained in the state $s_{H+1}$. In order to bound the derivatives of the expected reward, we first establish some properties of the matrix $M^{\theta}$.

Suppose $\bar{p}\leq 1/4$. Fix any $\alpha\in\left[\tfrac{1-\sqrt{1-4\gamma^{2}\bar{p}(1-\underline{p})}}{2\gamma(1-\underline{p})},\max\left\{\tfrac{1+\sqrt{1-4\gamma^{2}\bar{p}(1-\underline{p})}}{2\gamma(1-\underline{p})},1\right\}\right]$. Then

$M^{\theta}_{a,b}\leq\frac{\alpha^{b-a-1}}{1-\gamma}$ for $0\leq a\leq b\leq H$.

$M^{\theta}_{a,H+1}\leq\frac{\gamma\bar{p}}{1-\gamma}M^{\theta}_{a,H}\leq\frac{\gamma\bar{p}}{(1-\gamma)^{2}}\alpha^{H-a}$ for $0\leq a\leq H$.

Proof: Let $\rho^{k}_{a,b}$ be the normalized discounted probability of reaching $b$, when the initial state is $a$, in $k$ steps, that is

where we recall the convention that $U^{0}$ is the identity matrix for any square matrix $U$. Observe that $0\leq\rho^{k}_{a,b}\leq 1$, and, based on the form (27) of $P^{\theta}$, we have the recursive relation for all $k>0$:

Note that $\rho^{0}_{a,b}=0$ for $a\neq b$ and $\rho^{0}_{a,b}=1-\gamma$ for $a=b$. Now let us inductively prove that for all $k\geq 0$

Clearly this holds for $k=0$ since $\rho^{0}_{a,b}=0$ for $a\neq b$ and $\rho^{0}_{a,b}=1-\gamma$ for $a=b$. Now, assuming the bound for all steps till $k-1$, we now prove it for $k$ case by case.

For $1<b<H$ and $a<b$, observe that the recursion (29) and the inductive hypothesis imply that

where the last inequality follows since $\alpha^{2}\gamma(1-\underline{p})-\alpha+\gamma\bar{p}\leq 0$ due to that $\alpha$ is within the roots of this quadratic equation. Note the discriminant term in the square root is non-negative provided $\bar{p}<1/4$, since the condition along with the knowledge that $\underline{p}\leq\bar{p}$ ensures that $4\gamma^{2}\bar{p}(1-\underline{p})\leq 1$.

This proves the inductive claim (note that the cases of $b=a=1$ and $b=a=H$ are already handled in the first part above). Next, we prove that for all $k\geq 0$

Clearly this holds for $k=0$ and $b\neq 0$ since $\rho^{0}_{0,b}=0$. Furthermore, for all $k\geq 0$ and $b=0$,