O N C OMPUTATION AND G ENERALIZATION OF G ENERATIVE A DVERSARIAL I MITATION L EARNING
Anonymous authors
Paper under double-blind review
A BSTRACT
Generative Adversarial Imitation Learning (GAIL) is a powerful and practical approach for learning sequential decision-making policies.
Different from Reinforcement Learning (RL), GAIL takes advantage of demonstration data by experts (e g., human) and learns both the policy and reward function of the unknown environment.
Despite the signiﬁcant empirical progresses, the theory behind GAIL is still largely unknown.
The major difﬁculty comes from the underlying temporal dependency of the demonstration data and the minimax computational formulation of GAIL without convex-concave structure.
To bridge such a gap between theory and practice, this paper investigates the theoretical properties of GAIL.
Speciﬁcally, we show: (1) For GAIL with general reward parameterization, the generalization can be guaranteed as long as the class of the reward functions is properly controlled; (2) For GAIL, where the reward is parameterized as a reproducing kernel function, GAIL can be efﬁciently solved by stochastic ﬁrst order optimization algorithms, which attain sublinear convergence to a stationary solution.
To the best of our knowledge, these are the ﬁrst results on statistical and computational guarantees of imitation learning with reward/policy function approximation.
Numerical experiments are provided to support our analysis
I NTRODUCTION
As various robots (Tail et al., 2018), self-driving cars (Kueﬂer et al., 2017), unmanned aerial vehicles (Pfeiffer et al., 2018) and other intelligent agents are applied to complex and unstructured environments, programming their behaviors/policy has become increasingly challenging.
These intelligent agents need to accommodate a huge number of tasks with unique environmental demands.
To address these challenges, many reinforcement learning (RL) methods have been proposed for learning sequential decision-making policies (Sutton et al., 1998; Kaelbling et al., 1996; Mnih et al., 2015).
These RL methods, however, heavily rely on human expert domain knowledge to design proper reward functions.
For complex tasks, which are often difﬁcult to describe formally, these RL methods become impractical.
The Imitation Learning (IL, Argall et al.
; Abbeel & Ng (2004)) approach is a powerful and practical alternative to RL.
Rather than having a human expert handcrafting a reward function for learning the desired policy, the imitation learning approach only requires the human expert to demonstrate the desired policy, and then the intelligent agent (a.k.a. learner) learns to match the demonstration.
Most of existing imitation learning methods fall in the following two categories:
• Behavioral Cloning (BC, Pomerleau (1991)).
BC treats the IL problem as supervised learning.
Speciﬁcally, it learns a policy by ﬁtting a regression model over expert demonstrations, which directly maps states to actions.
Unfortunately, BC has a fundamental drawback.
Recall that in supervised learning, the distribution of the training data is decoupled from the learned model, whereas in imitation learning, the agent’s policy affects what state is queried next.
The mismatch between training and testing distributions, also known as covariate shift (Ross & Bagnell, 2010; Ross et al., 2011), yields
signiﬁcant
compounding
errors.
Therefore
, BC often
suffers
from
poor
generalization
• Inverse Reinforcement Learning (IRL, Russell (1998); Ng et al.
Finn et al. (2016); Levine & Koltun (
IRL treats the IL problem as bi-level optimization.
Speciﬁcally, it ﬁnds a reward function, under which the expert policy is uniquely optimal.
Though IRL does not have the error compounding issue, its computation is very inefﬁcient.
Many existing IRL methods need to solve a sequence of computationally expensive reinforcement learning problems, due to their bi-level optimization nature.
Therefore, they often fail to scale to large and high dimensional environments.
More recently, Ho & Ermon (2016) propose a Generative Adversarial Imitation Learning (GAIL) method, which obtains signiﬁcant performance gains over existing IL methods in imitating complex expert policies in large and high-dimensional environments.
GAIL generalizes IRL by formulating the IL problem as minimax optimization, which can be solved by alternating gradient-type algorithms in a more scalable and efﬁcient manner.
Speciﬁcally, we consider an inﬁnite horizon Markov Decision Process (MDP), where S denotes the state space, A denotes the action space, P denotes the Markov transition kernel, r ∗ denotes the reward function, and p 0 denotes the distribution of the initial state.
We assume that the Markov transition kernel P is ﬁxed and there is an unknown expert policy π ∗ : S → P ( A ) , where P ( A ) denotes the set of distributions over the action space.
As can be seen, { s t } T − 1 t =0 essentially forms a
Markov chain with the transition kernel induced by π ∗ as P π ∗ ( s, s (cid:48) )
= (cid:80) a ∈A π ∗ ( a | s ) · P ( s (cid:48) | s, a ) .
Given n demonstration trajectories from π ∗ denoted by { s ( i ) t , a ( i ) t } T − 1 t =0 , where i = 1 , ..., n , s 0 ∼ p 0 , a t ∼ π ∗ ( · | s t ) , and s t +1 ∼ P ( · | s t , a t ) , GAIL aims to learn π ∗ by solving the following minimax optimization problem,
[Equation]
(1) where E π [ r ( s, a )] =
lim T →∞ E [ 1 T (cid:80) T − 1 t =
r
( s t , a t ) | π ] denotes the average reward under the policy π when the reward function is r , and E π ∗ n [ r ( s, a )]
= 1 nT (cid:80) ni
=1 (cid:80) T − 1 t =
[ r ( s ( i ) t , a ( i ) t )] denotes the empirical average reward over the demonstration trajectories.
As shown in (1), GAIL aims to ﬁnd a policy, which attains an average reward similar to that of the expert policy with respect to any reward belonging to the function class R .
For large and high-dimensional imitation learning problems, we often encounter inﬁnitely many states.
To ease computation, we need to consider function approximations.
Speciﬁcally, suppose that for every s ∈ S and a ∈ A , there are feature vectors ψ s ∈ R d S and ψ a ∈ R d A associated with a and s , respectively.
Then we can approximate the policy and reward as
[Equation]
where (cid:101) π and (cid:101) r belong to certain function classes (e.g. reproducing kernel Hilbert space or deep neural networks, Ormoneit & Sen (2002); LeCun et al.
associated with parameters ω and θ , respectively.
Accordingly, we can optimize (1) with respect to the parameters ω and θ by scalable alternating gradient-type algorithms.
Although GAIL has achieved signiﬁcant empirical progresses, its theoretical properties are still largely unknown.
There are three major difﬁculties when analyzing GAIL: 1).
There exists temporal dependency in the demonstration trajectories/data due to their sequential nature (Howard, 1960;
Puterman, 2014; Abounadi et al., 2001); 2).
GAIL is formulated as a minimax optimization problem.
Most of existing learning theories, however, focus on empirical risk minimization problems, and therefore are not readily applicable (Vapnik, 2013; Mohri et al., 2018; Anthony & Bartlett, 2009); 3).
The minimax optimization problem in (1) does not have a convex-concave structure, and therefore existing theories in convex optimization literature cannot be applied for analyzing the alternating stochastic gradient-type algorithms (Willem, 1997; Ben-Tal & Nemirovski, 1998; Murray & Overton, 1980; Chambolle & Pock, 2011; Chen et al., 2014).
Some recent results suggest to use stage-wise stochastic gradient-type algorithms (Raﬁque et al., 2018;
Dai et al., 2017).
More specifically, at every iteration, they need to solve the inner maximization problem up to a high precision, and then apply stochastic gradient update to the outer minimization problem.
Such algorithms, however, are rarely used by practitioners, as they are inefﬁcient in practice (due to the computationally intensive inner maximization).
To bridge such a gap between practice and theory, we establish the generalization properties of
GAIL and the convergence properties of the alternating mini-batch stochastic gradient algorithm for solving (1).
Speciﬁcally, our contributions can be summarized as follows:
• We formally deﬁne the generalization of GAIL under the “so-called” R -reward distance, and then show that the generalization of GAIL can be guaranteed under reward distance as long as the class of the reward functions is properly controlled;
• We provide sufﬁcient conditions, under which an alternating mini-batch stochastic gradient algorithm can efﬁciently solve the minimax optimization in (1), and attains sublinear convergence to a stationary solution.
To the best of our knowledge, these are the ﬁrst results on statistical and computational theories of imitation learning with reward/policy function approximations.
Our work is related to Syed et al.
; Cai et al.
Syed et al.
(2008) study the generalization and computational properties of apprenticeship learning.
Since they assume that the state space of the underlying Markov decision process is ﬁnite, they do not consider any reward/policy function approximations;
Cai et al.
(2019) study the computational properties of imitation learning under a simple control setting.
Their assumption on linear policy and quadratic reward is very restrictive, and does not hold for many real applications.
Notation.
Given a vector x = ( x 1 , ..., x d ) (cid:62) ∈ R d , we deﬁne (cid:107) x (cid:107) 22 = (cid:80) dj
=1 x 2 j .
Given a function f : R d (cid:55)→ R , we denote its (cid:96) ∞ norm as (cid:107) f (cid:107) ∞ =
max x | f ( x ) | .
G ENERALIZATION OF GAIL
To analyze the generalization properties of GAIL, we ﬁrst assume that we can access an inﬁnite number of the expert’s demonstration trajectories (underlying population), and that the reward function is chosen optimally within some large class of functions.
This allows us to remove the maximum operation from (1), which leads to an interpretation of how and in what sense the resulting policy is close to the true expert policy.
Before we proceed, we ﬁrst introduce some preliminaries.
Deﬁnition 1 (Stationary Distribution) .
Note that any policy π induces a Markov chain on S × A .
The transition kernel is given by
[Equation]
When such a Markov chain is aperiodic and recurrent, we denote its stationary distribution as ρ π .
Note that a policy π is uniquely determined by its stationary distribution ρ π in the sense that
[Equation]
Then we can write the expected average reward of r ( s, a ) under the policy π as
[Equation]
We further deﬁne the R -distance between two policies π and π (cid:48) as follows.
Deﬁnition 2.
Let R denote a class of symmetric reward functions from S × A to R , i.e., if r ∈ R , then − r ∈ R .
Given two policy π (cid:48) and π , the R -distance for GAIL is deﬁned as
[Equation]
The R -distance over policies for Markov decision processes is essentially an Integral Probability
Metric (IPM) over stationary distributions (M ¨ uller, 1997).
For different choices of R , we have various R -distances.
For example, we can choose R as the class of all 1 -Lipschitz continuous functions, which yields that d R ( π, π (cid:48) ) is the Wasserstein distance between ρ π and ρ π (cid:48)
(Vallender, 1974).
For computational convenience, GAIL and its variants usually choose R as a class of functions from some reproducing kernel Hilbert space, or a class of neural network functions.
Deﬁnition 3.
Given n demonstration trajectories from time 0 to T − 1 obtained by an expert policy π ∗ denoted by ( s ( i ) t , a ( i ) t ) T − 1 t =0 , where i = 1 , ..., n , a policy (cid:98) π learned by GAIL generalizes under the
R -distance d R ( · , · ) with generalization error (cid:15) , if with high probability, we have
[Equation]
where d R ( π ∗ n , (cid:98) π ) is the empirical R -distance between π ∗ and (cid:98) π deﬁned as d R ( π ∗ n , (cid:98) π ) =
sup r ∈R [ E π ∗ n r ( s, a ) − E (cid:98) π r ( s, a )] with E π ∗ n [ r ( s, a )]
= 1 nT (cid:80) ni
{+=1 (cid:80) T − 1 t =
+}
[ r ( s ( i ) t , a ( i ) t )] .
The generalization of GAIL implies that the R -distance between the expert policy π ∗ and the learned policy (cid:98) π is close to the empirical R -distance between them.
Our analysis aims to prove the former distance to be small, whereas the latter one is what we attempts to minimize in practice.
We then introduce the assumptions on the underlying Markov decision process and expert policy.
Assumption 1.
Under the expert policy π ∗ , ( s t , a t ) T − 1 t =0 forms a stationary and exponentially βmixing Markov chain, i.e.,
[Equation]
where β 0 , β 1 , α are positive constants, and σ ji is the σ -algebra generated by ( s t , a t )
jt
= i for i ≤ j .
Moreover, for every s ∈ S and a ∈ A , there are feature vectors ψ s ∈ R d S and ψ a ∈ R d A associated with a and s , respectively, and ψ s and ψ a are uniformly bounded, where
[Equation]
Assumption 1 requires the underlying MDP to be ergodic (Levin & Peres, 2017), which is a commonly studied assumption in exiting reinforcement learning literature on maximizing the expected average reward (Strehl & Littman, 2005; Li et al., 2011; Brafman & Tennenholtz, 2002;
Kearns &
Singh, 2002).
The feature vectors associated with a and s allow us to apply function approximations to parameterize the reward and policy functions.
Accordingly, we write the reward function as r ( s, a ) =
(cid:101) r ( ψ s , ψ a ) , which is assumed to be bounded.
Assumption 2.
The reward function class is uniformly bounded, i.e., (cid:107) r (cid:107) ∞ ≤ B r for any r ∈ R .
Now we proceed with our main result on generalization properties of GAIL.
We use N ( R , (cid:15), (cid:107) · (cid:107) ∞ ) to denote the covering number of the function class
R under the (cid:96) ∞ distance (cid:107) · (cid:107) ∞ .
Theorem 1 (Main Result) .
Suppose Assumptions 1-2 hold, and the policy learned by GAIL satisﬁes
[Equation]
where the inﬁmum is taken over all possible learned policies.
Then with probability at least 1 − δ over the joint distribution of { ( a ( i ) t , s ( i ) t ) T − 1 t =0 } ni =1 , we have
[Equation]
where ζ = ( β − 1 1 log β 0 Tδ ) 1 α .
Theorem 1 implies that the policy (cid:98) π learned by GAIL generalizes as long as the complexity of the function class R is well controlled.
To the best of our knowledge, this is the ﬁrst result on the generalization of imitation learning with function approximations.
As the proof of Theorem 1 is involved, we only present a sketch due to space limit.
More details are provided in Appendix A.1.
Proof Sketch.
Our analysis relies on characterizing the concentration property of the empirical average reward under the expert policy.
For notational simplicity, we deﬁne
[Equation]
The key challenge comes from the fact that ( s ( i ) t , a ( i ) t ) ’s are dependent.
To handle such a dependency, we adopt the independent block technique from Yu (1994).
Speciﬁcally, we partition every trajectory into disjoint blocks (where the block size is of the order O
((log( T ) + log(1 /δ )) 1 /α ) , and construct two separable trajectories:
One contains all blocks with odd indices (denoted by B odd ), and the other contains all those with even indices (denoted by B even ).
We deﬁne
[Equation]
and analogously for φ 2 with ( s ( i ) t , a ( i ) t ) ∈ B even .
Then we have
P
[Equation]
We consider a block-wise independent counterpart of φ 1 denoted by (cid:101) φ 1 , where each block is sampled independently from the same Markov chain as φ 1 , i.e., (cid:101) φ 1 has independent blocks of samples from the same exponentially β -mixing Markov chain .
Accordingly, we denote (cid:101) φ 1 as
[Equation]
where (cid:101) B odd denotes i.i.d.
blocks of samples.
Now we bound the difference between φ 1 and (cid:101) φ 1 by
[Equation] where C is a constant, and β is the mixing coefﬁcient, and P (sup r ∈R (cid:101) φ 1 − E (cid:101) φ 1 ≥ ε − E (cid:101) φ 1 ) can be bounded using the empirical process technique for independent random variables.
The details of the above inequality can be found in Corollary 3 in Appendix A.1, where the proof technique is adapted from Lemma 1 in Mohri & Rostamizadeh (2009).
Let (cid:101) φ 2 be deﬁned analogously as (cid:101) φ 1 .
With a similar argument further applied to φ 2 and (cid:101) φ 2 , we obtain P
[Equation]
The rest of our analysis follows the PAC-learning framework using Rademacher complexity and is omitted (Mohri et al., 2018).
We complete the proof sketch.
Example 1: Reproducing Kernel Reward Function .
One popular option to parameterize the reward by functions is the reproducing kernel Hilbert space (RKHS, Kim & Park (2018);
Li et al.
There have been several implementations of RKHS, and we consider the feature mapping approach.
Speciﬁcally, we consider g : R d S × R d
A → R q , and the reward can be written as
[Equation]
where θ ∈ R q .
We require g to be Lipschitz continuous with respect to ( ψ a , ψ s ) .
Assumption 3.
The feature mapping g satisﬁes g (0 , 0) = 0 , and there exists a constant ρ g such that for any ψ a , ψ (cid:48) a , ψ s and ψ (cid:48) s , we have
[Equation]
Assumption 3 is mild and satisﬁed by popular feature mappings, e.g., random Fourier feature mapping 1 (Rahimi & Recht, 2008; Bach, 2017).
The next corollary presents the generalization bound of
GAIL using feature mapping.
Corollary 1.
Suppose (cid:107) θ (cid:107) 2 ≤ B θ .
For large enough n and T , with probability at least 1 − δ over the joint distribution of { ( a it , s it )
T − 1 t =0 } ni =1 , we have
[Equation]
Corollary 1 indicates that with respect to a class of properly normalized reproducing kernel reward functions, GAIL generalizes in terms of the R -distance.
Example 2: Neural Network Reward Function .
Another popular option to parameterize the reward function is to use neural networks.
Speciﬁcally, let σ ( v ) =
[max { v 1 , 0 } , ..., max { v d , 0 } ] (cid:62) denote the ReLU activation for v ∈ R d .
We consider a D -layer feedforward neural network with
ReLU activation as follows,
[Equation]
where W = { W i | W i ∈
R d i − 1 ×
d
i , i = 1 , ..., D − 1 , W D ∈ R
d D − 1 } and d 0 = d A + d S .
The next corollary presents the generalization bound of GAIL using neural networks.
Corollary 2.
Suppose (cid:107) W
i
(cid:107) 2 ≤ 1 , where i = 1 , ..., D .
For large enough n and T , with probability at least 1 − δ over the joint distribution of { ( a t , s t ) T − 1 t =0 } ni =1 , we have
[Equation]
Corollary 2 indicates that with respect to a class of properly normalized neural network reward functions, GAIL generalizes in terms of the R -distance.
Remark 1 ( The Tradeoff between Generalization and Representation of GAIL).
As can be seen from Deﬁnition 2, the R -distances are essentially differentiating two policies.
For the Wassersteintype distance, i.e., R contains all 1 -Lipschitz continuous functions, if d R ( π, π (cid:48) ) is small, it is safe to conclude that two policies π and π (cid:48) are nearly the same almost everywhere.
However, when we choose R to be the reproducing kernel Hilbert space or the class of neural networks with relatively small complexity, d R ( π, π (cid:48) ) can be small even if π and π (cid:48) are not very close.
Therefore, we need to choose a sufﬁciently diverse class of reward functions to ensure that we recover the expert policy.
As Theorem 1 suggests, however, that we need to control the complexity of the function class R to guarantee the generalization.
This implies that when parameterizing the reward function, we need to carefully choose the function class to attain the optimal tradeoff between generalization and representation of GAIL.
C OMPUTATION OF GAIL
To investigate the computational properties of GAIL, we parameterize the reward by functions belonging to some reproducing kernel Hilbert space.
The implementation is based on feature mapping, as mentioned in the previous section.
The policy can be parameterized by functions belonging to some reproducing kernel Hilbert space or some class of deep neural networks with parameter ω .
Speciﬁcally, we denote π ( a | s ) = (cid:101) π ω ( ψ s ) , where (cid:101) π ω ( ψ s ) is the parametrized policy mapping from R d S to a simplex in R d
A with |A| =
d .
For computational convenience, we consider solving a slightly modiﬁed minimax optimization problem:
[Equation]
where (cid:101) r θ ( s, a )
= θ (cid:62) g
( ψ s , ψ a ) , H ( (cid:101) π ω ) is some regularizer for the policy (e.g., causal entropy regularizer, Ho & Ermon (2016)), and λ > 0 and µ > 0 are tuning parameters.
Compared with (1), the additional regularizers in (2) can improve the optimization landscape, and help mitigate computational instability in practice.
We apply the alternating mini-batch stochastic gradient algorithm to (2).
Speciﬁcally, we denote the objective function in (2) as F ( ω, θ ) for notational simplicity.
At the ( t + 1) -th iteration, we take
[Equation]
where η θ and η ω are learning rates, the projection Π κ ( v ) = 1 ( (cid:107) v (cid:107) 2 ≤ κ ) · v + 1 ( (cid:107) v (cid:107) 2 > κ ) · κ · v/ (cid:107) v (cid:107) 2 , ∇ f j ’s and ∇ (cid:101)
f j ’s are independent stochastic approximations of ∇ F (Sutton et al., 2000), and M ( t ) θ , M ( t ) ω are mini-batches with sizes q θ and q ω , respectively.
Before we proceed with the convergence analysis, we impose the follow assumptions on the problem.
Assumption 4.
There are two positive constants M ω and M θ such that for any ω and (cid:107) θ (cid:107) 2 ≤ κ ,
Unbiased : E ∇ f j ( ω, θ ) =
E ∇ (cid:101) f
j ( ω, θ ) = ∇ F ( ω, θ ) ,
Bounded : E (cid:107)∇ ω (cid:101)
f j ( ω, θ ) − ∇ ω F ( ω, θ ) (cid:107) 2 2 ≤ M ω and E (cid:107)∇ θ f j ( ω, θ ) − ∇ θ F ( ω, θ ) (cid:107) 2 2 ≤ M θ .
Assumption 4 requires the stochastic gradient to be unbiased with a bounded variance, which is a common assumption in existing optimization literature (Nemirovski et al., 2009; Ghadimi & Lan, 2013; Duchi et al., 2011; Bottou, 2010).
Assumption 5.
(i)
For any ω , there exists some constant χ > 0 and υ ∈
(0 , 1) such that
[Equation]
where P (cid:101) π ω ( s (cid:48) , a (cid:48) | s, a ) = (cid:101) π ω ( a (cid:48) | s (cid:48) ) P ( s (cid:48) | s, a ) is the transition kernel induced by (cid:101) π ω , ρ 0 is the initial distribution of ( s 0 , a 0 ) , and ρ (cid:101) π ω is the stationary distribution induced by (cid:101) π ω . (ii) There exist constants S (cid:101) π , B ω , L ρ , L Q > 0 such that for any ω, ω (cid:48) , we have
[Equation]
where Q
[Equation]
a
[Equation]
= [Equation] 0 E [ [Equation]
E
[Equation]
ω
[Equation]
] |
[Equation]
0 =
[Equation]
a
[Equation]
a [Equation] ] is the action-value function.
(iii) There exist constants B H and S H > 0 such that for any ω, ω (cid:48) , we have
[Equation]
and
[Equation]
Note that (i) of Assumption 5 requires the Markov Chain to be geometrically mixing.
(ii) and (iii) state some commonly used regularity conditions for policies (Sutton et al., 2000; Pirotta et al., 2015).
We then deﬁne L -stationary points of F .
Speciﬁcally, we say that ( ω ∗ , θ ∗ ) is a stationary point of
F , if and only if, for any ﬁxed α > 0 ,
[Equation]
The L -stationarity is a generalization of the stationary point for unconstrained optimization, and is a necessary condition for optimality.
Accordingly, we take α = 1 and measure the sub-stationarity of the algorithm at the iteration N by
[Equation]
We then state the global convergence of the alternating mini-batch stochastic gradient algorithm.
Theorem 2.
Suppose Assumptions 1-5 hold.
We choose step sizes η θ , η ω satisfying η ω
[Equation]
, and meanwhile η ω /η θ ≤ µ/ (30 L ω + 5) , where L ω = 2 √ 2( S (cid:101) π
+ 2 B ω L ρ )
κρ g χ/ (1 − υ ) + B ω L Q , and S ω
= 2 √ 2 qκρ g χB ω / (1 − υ ) .
Given any (cid:15) > 0 , we choose batch sizes q θ = (cid:101) O
(1 /(cid:15) ) and q ω = (cid:101) O
(1 /(cid:15) ) .
Then we need at most
[Equation]
iterations such that J N ≤ (cid:15) , where C 0 depends on the initialization, and η depends on η ω and η θ .
Here (cid:101)
O hides linear or quardic dependence on some constants in Assumptions 1-5.
Theorem 2 shows that though the minimax optimization problem in (2) does not have a convex-concave structure, the alternating mini-batch stochastic gradient algorithm still guarantees to converge to a stationary point.
We are not aware of any similar results for GAIL in existing literature.
Proof Sketch.
We prove the convergence by showing
[Equation]
[Equation]
where C is a constant and N(cid:15)/ 2 is the accumulation of noise in stochastic approximations of ∇ F .
Then we straightforwardly have NJ N ≤ C + N(cid:15)/ 2 .
Dividing both sizes by N , we can derive the desired result.
The main difﬁculty of showing (5) comes from the fact that the outer minimization problem is nonconvex and we cannot solve the inner maximization problem exactly.
To overcome this difﬁculty, we construct a monotonically decreasing potential function:
[Equation]
for a constant s to be chosen later.
Denote ξ ( t ) θ and ξ ( t ) ω as the i.i.d. noise of the stochastic gradients.
The following lemma characterizes the decrement of the potential function at each iteration.
Lemma 1.
With the step sizes η θ and η ω chosen as in Theorem 2, we have
[Equation]
where ν is a constant depending on F , η θ , and η ω .
Moreover, we have constants k 1 , k 2 , k 3 , k 4 , k 5 > 0 for s = 8 / ( η 2 ω (58 L ω + 9)) .
Let k = 1 / min { k 1 , k 4 } and φ = max { 1 , 1 /η 2 θ , 1 /η 2 ω } .
We obtain
[Equation]
t ) + ∇ θ F ( ω ( t ) , θ ( t ) )) (cid:13)(cid:13) 2 2 + E (cid:13)(cid:13) ∇ ω F ( ω ( t ) , θ ( t +1) ) (cid:13)(cid:13) 2(i) ≤ φ (cid:80)
Nt =1 E (cid:2) (cid:107) θ ( t +1) − θ ( t ) (cid:107) {+
 ++} (cid:107) ω ( t +1) − ω ( t ) (cid:107) 22 (cid:3) (ii) ≤
kφ ( E (1) − E ( N ) ) + kφNν E (cid:2) (cid:107) ξ ( t ) ω {+(cid:107) 
 + (cid:107) ξ (+} t ) θ (cid:107) 22 (cid:3) , where (i) follows from plugging in the update (3) as well as the contraction property of projection, and (ii) follows from Lemma 1.
Choosing q θ = 4 kφνM θ /(cid:15) and q ω = 4 kφνM ω /(cid:15) , we obtain (cid:80)
Nt =1 E (cid:13)(cid:13)
θ ( t ) − Π κ ( θ ( t )
+ ∇ θ F ( ω ( t ) , θ ( t ) ))
(cid:13)(cid:13) 2 2 + E (cid:13)(cid:13) ∇ ω F ( ω ( t ) , θ ( t +1) )
(cid:13)(cid:13) 2 2 ≤ kφ ( E (1) − E ( N ) ) + N(cid:15) 2 .
We have E ( N ) ≥ E F ( ω ( N ) , θ ( N ) ) by the construction of E ( N ) .
It is easy to verify that F is lower bounded (Lemma 10 in Appendix B).
Eventually, we complete the proof by substituting the lower bound and choosing N = kφ (2 E (1) + 4 √
ρ g κ + µκ 2 + 2 λB H ) (cid:15) − 1 .
E XPERIMENT
To verify our theory in Section 3, we conduct experiments in three reinforcement learning tasks:
Acrobot, MountainCar, and Hopper.
For each task, we ﬁrst train an expert policy using the proximal policy optimization (PPO) algorithm in (Schulman et al., 2017) for 500 iterations, and then use the expert policy to generate the demonstration data.
The demonstration data for every task containstrajectories, each of which is a series of state action pairs throughout one episode in the environment.
When training GAIL, we randomly select a mini-batch of trajectories, which contain at leaststate action pairs.
We use PPO to update the policy parameters.
This avoids the instability of the policy gradient algorithm, and improves the reproducibility of our experiments.
[Figure]
We use the same neural network architecture for all the environments.
For policy, we use a fully connected neural network with two hidden layers of 128 neurons in each layer and tanh activation.
For reward, we use a fully connected ReLU neural network with two hidden layers of 1024 and 512 neurons, respectively.
To implement the kernel reward, we ﬁx the ﬁrst two layers of the neural network after random initialization and only update the third layer, i.e., the ﬁrst two layers mimic the random feature mapping.
We choose κ = 1 and µ = 0 .
When updating the neural network reward, we use weight normalization in each layer (Salimans & Kingma, 2016).
When updating the kernel reward at each iteration, we choose to take the stochastic gradient ascent step for either once (i.e., alternating update in Section 3) or 10 times.
When updating the neural network reward at each iteration, we choose to take the stochastic gradient ascent step for only once.
We tune step size parameters for updating the policy and reward, and summarize the numerical results of the step sizes attaining the maximal average episode reward in Figure 1.
As can be seen, using multiple stochastic gradient ascent steps for updating the reward at each iteration yields similar performance as that of one step.
We present the convergence analysis of using multiple stochastic gradient ascent steps for updating the reward in
Appendix C. Moreover, we observe that parameterizing the reward by neural networks slightly outperform that of the kernel reward.
However, its training process tends to be unstable and takes longer time to converge.
D ISCUSSIONS
Our proposed theories of GAIL are closely related to Generative Adversarial Networks (Goodfellow et al., 2014; Arjovsky et al., 2017): (1) The generalization of GANs is deﬁned based on the integral probabilistic metric (IPM) between the synthetic distribution obtained by the generator network and the distribution of the real data (Arora et al., 2017).
As the real data in GANs are considered as independent realizations of the underlying distribution, the generalization of GANs can be analyzed using commonly used empirical process techniques for i.i.d.
random variables.
GAIL, however, involves dependent demonstration data from experts, and therefore the analysis is more involved.
Our computational theory of GAIL can be applied to MMD-GAN and its variants, where the IPM is induced by some reproducing kernel Hilbert space (Li et al., 2017; Bi´nkowski et al., 2018; Arbel et al., 2018).
The alternating mini-batch stochastic gradient algorithm attains a similar sublinear rate of convergence to a stationary solution.
Moreover, our computational theory of GAIL only considers the policy gradient update when learning the policy (Sutton et al., 2000).
Extending to other types of updates such as natural policy gradient (Kakade, 2002), proximal policy gradient (Schulman et al., 2017) and trust region policy optimization (Schulman et al., 2015) is a challenging, but important future direction.
A P ROOFS IN S ECTIONA. P ROOF OF T HEOREM
We ﬁrst consider n = 1 .
For notational simplicity, we denote x t =
( s t , a t ) and τ = { x t } T − 1 t =0 .
Then the generalization gap is bounded by
[Equation]
Denote Φ( τ ) = sup
[Equation]
x
[Equation]
We utilize the independent block technique ﬁrst proposed in Yu (1994) to show the concentration of Φ( τ )
[Equation]
Speciﬁcally
[Equation]
we partition τ into 2 m blocks of equal size b .
We denote two alternating sequences as
[Equation]
We now deﬁne a new sequence
[Equation]
where (cid:101) X ’s are i.i.d blocks of size b , and each block (cid:101)
X
i follows the same distribution as X i .
We deﬁne r b : X → R as r b ( X ) = 1 b (cid:80) bk =1 r ( x k ) .
Note that r b is essentially the average reward on a block.
Accordingly, we denote R b as the set of all r b ’s induced by r ∈ R .
Before we proceed, we need to introduce a lemma which characterizes the relationship between the expectations of a bounded measurable function with respect to τ 0 and (cid:101) τ 0 .
Lemma 2.
(Yu, 1994)
Suppose h is a measurable function bounded by M > 0 deﬁned over the blocks
X i , then the following inequality holds:
[Equation]
where E τ 0 denotes the expectation with repect to τ 0 , and E (cid:101) τ 0 denotes the expectation with respect to (cid:101) τ 0 .
Corollary 3.
Applying Lemma 2, we have P
[Equation]
Proof.
Consider P (Φ( τ ) > (cid:15) ) ,we have
[Equation]
where the ﬁrst inequality (8) follows from the convexity of supremum.
Applying Lemma 2 and setting h = 1 { (Φ( (cid:101) τ 0 ) − E (cid:101) τ
[Φ( (cid:101) τ 0 )] > (cid:15) − E (cid:101) τ
[Φ( (cid:101) τ 0 )] } , we obtain P
[Equation]
Now since (cid:101) τ 0 consists of independent blocks, we can apply McDiarmid’s inequality to r b by viewing (cid:101)
X i ’s as i.i.d samples.
We rewrite Φ( (cid:101) τ 0 ) as
[Equation]
Given samples (cid:101) X 1 , · · · , (cid:101)
X i , · · · , (cid:101)
X m and (cid:101) X 1 , · · · , (cid:101) X (cid:48)
i , · · · , (cid:101)
X m , we have
[Equation]
Then by McDiarmid’s inequality, we have
[Equation]
Now combining (7) and (9), we obtain
[Equation]
By the argument of symmetrization, we have
[Equation]
where σ i ’s are i.i.d.
Rademacher random variables.
Now we relate the Rademacher complexity (11) to its counterpart taking i.i.d samples.
Speciﬁcally, we denote x ( t ) j as the j -th point of the t -th block.
Denote (cid:101) τ
j 0 as the collection of the j -th sample from each independent block (cid:101) X i for i = 1 , . . .
, m .
Plugging in the deﬁnition of r b , we have
[Equation]
Setting the right-hand side of (10) to be δ 2 and substituting (12), we obtain, with probability at least 1 − δ 2 , for all r ∈ R ,
[Equation] [Equation]
where δ (cid:48)
=
δ − 4( m − 1) β
( b ) .
Then we denote
Rademacher complexity for (cid:101)
X i ’s: R
[Equation]
E
[Equation]
( x (
[Equation] [Equation]
(cid:35) ,
Empirical Rademacher complexity for (cid:101)
X i ’s:
(cid:98) R (cid:101) τ 10 =
E σ
[Equation]
m sup r ∈R m (cid:88) t =
[Equation]
x ( t ) 1 ) (cid:35) ,
Empirical Rademacher complexity for X
i ’s:
(cid:98) R m =
E σ (cid:34)m sup
[Equation]
∈R m (cid:88) t =
[Equation] [Equation]
( t )
[Equation]
(cid:35) .
Applying Lemma 2 to the indicator function 1 { R
[Equation]
Dm − [Equation] R m >
[Equation]
} , we obtain
[Equation]
It is straightforward to verify that given x (1)1 , . . .
, x ( i ) 1 , . . .
, x ( m )and x (1)1 , . . .
, x (cid:48) ( i ) 1 , . . .
, x ( m ) 1 , the
Rademacher complexity satisﬁes
[Equation]
Then by applying McDiarmid’s Inequality again, we obtain
[Equation]
Thus with probability at least 1 − δ 2 , we have
[Equation]
[Equation]
Combining (13) and (14), we have with probability 1 − δ ,
[Equation]
We apply the Dudley’s entropy integral to bound (cid:98) R m .
Speciﬁcally, we have
[Equation]
It sufﬁces to pick α = 1 √ m .
By combining (6)
[Equation] [Equation]
and (16), we have with probability at least 1 − δ ,
[Equation]
m + (cid:15)
[Equation]
where δ (cid:48)
=
δ − 4( m − 1) β
( b ) and 2 bm = T .
Substituting m =
T/ 2 b
[Equation]
we have
[Equation] [Equation]
Now we instantiate a choice of b and m for expotentially β -mixing sequences, where the mixing coefﬁcient β ( b ) ≤ β 0 exp( − β 1 b α ) for constants β 0 , β 1 , α > 0 .
We set δ (cid:48)
[Equation]
By a simple calculation, it is enough to choose b = ( log(4 β 0 T/δ ) β) 1 /α .
Substituting such a b into (17).
We have with probability at least 1 − δ :
[Equation]
where ζ = (
[Equation]
When n > 1 , we concatenate n trajectories to form a sequence of length nT , and such a sequence is still exponentially β -mixing.
Applying the same technique, we partition the whole sequence into 2 nm blocks of equal size b .
Then with probability at least 1 − δ , we have
[Equation]
A. P ROOF OF C OROLLARY
The reward function can be bounded by
[Equation]
where the ﬁrst inequality comes from Cauchy-Schwartz inequality
[Equation]
To compute the covering number
[Equation]
we exploit the Lipschitz continuity of r ( s, a ) with respect to parameter θ .
Speciﬁcally, for two different parameters θ and θ (cid:48) , we have
[Equation]
where (i) comes from Cauchy-Schwartz inequality, (ii) comes from the Lipschitz continuity of g , and (iii) comes from the boundedness of ψ s and ψ a .
Denote Θ = { θ ∈ R q : (cid:107) θ (cid:107) 2 ≤ B θ } .
By the standard argument of the volume ratio, we have
[Equation]
Accordingly, we have
N
[Equation]
Plugging (19) into (18), we have
[Equation]
hold
[Equation]
with probability at least 1 − δ .
A. P ROOF OF C OROLLARY
We investigate the Lipschitz continuity of r with respect to the weight matrices W 1 , · · · , W D .
Speciﬁcally, given two different sets of matrices W 1 , · · · , W D and W (cid:48) 1 , · · · , W (cid:48) D , we have
[Equation]
r
[Equation]
s
[Equation]
a )
[Equation]
r
[Equation]
s, a )
[Equation]
∞ ≤
[Equation]
[Table]
≤ (cid:13)(cid:13) W
[Table]
σ ( W
[Table]
σ (
[Table]
σ ( W
[Table]
[ ψ
[Table]
, ψ
[Table] [Table]
[Table]
)) − ( W
[Table]
σ ( W
[Table]
σ (
[Table]
σ
[Table]
W
[Table]
ψ
[Table]
, ψ
[Table]
+ (cid:13)(cid:13) ( W (cid:48) D ) (cid:62) σ ( W D − 1 σ ( ...σ ( W 1 [ ψ (cid:62) a , ψ (cid:62) s ] (cid:62) ) ... ))
− ( W (cid:48) D ) (cid:62) σ ( W (cid:48)
D − 1 σ ( ...σ ( W (cid:48)
[ ψ (cid:62) a , ψ (cid:62) s ] (cid:62) ) ... ))
(cid:13)(cid:13)≤ (cid:107) W D − W (cid:48)
D (cid:107) 2 (cid:13)(cid:13) σ ( W D − 1 σ ( ...σ ( W 1 [ ψ (cid:62) a , ψ (cid:62) s ] (cid:62) ) ... ))
(cid:13)(cid:13)+ (cid:107) W (cid:48) D (cid:107) 2 (cid:13) (cid:13) σ ( W D − 1 σ ( ...σ ( W
[Table]
Note that we have
[Equation]
where (i) comes from the deﬁnition of the ReLU activation, (ii) comes from (cid:107) W i (cid:107) 2 ≤ 1 and recur- sion, and (iii) comes from the boundedness of ψ s and ψ a .
Accordingly, we have
[Equation]
where (i) comes from the Lipschitz continuity of the ReLU activation
[Equation]
and (ii) comes from the recursion.
We then derive the covering number of R by the Cartesian product of the matrix covering of W 1 , ..., W D :
[Equation]
where the second inequality comes from the standard argument of the volume ratio.
Plugging (20) into (18), we have
[Equation]
hold, with probability at least 1 − δ .
B P ROOF OF T HEOREM
The proof is arranged as follows.
We ﬁrst prove the bounded ness of Q fucntion and characterize the Lipschitz properties of the gradients of F with respect to ω and θ , respectively, in Section B.2.
Then we provide some important lemmas in Section B.3.
Using these lemmas, we prove Lemmain Section B.4.
We prove Theorem 2 in Section B.5.
For notational simplicity, we denote (cid:104)· , ·(cid:105) as the vector inner product throughout the rest of our analysis.
B. B OUNDEDNESS OF Q FUNCTION
Lemma 3.
For any ω , we have
[Equation]
where B Q = 2 √ 2 κρ g χ 1 − υ .
Proof.
[Equation]
where the ﬁrst inequality comes from the deﬁnition of Total Variance distance of probability measures and the second inequality results from (i) of Assumption 5.
B. L IPSCHITZ PROPERTIES OF THE GRADIENTS
Lemma 4.
Suppose that Assumption 1, 3 and 5 hold.
For any ω , ω (cid:48) , θ and θ
[Equation]
, we have
[Equation]
where L
[Equation]
=
[Equation]
Proof.
By the Policy Gradient Theorem (Sutton et al., 2000)
[Equation]
we have
[Equation]
Therefore,
[Equation]
(21) where the second and the third inequality results from (ii) of Assumption 5.
Plugging B Q = 2 √ 2 κρ g χ 1 − υ into (21) yields the desired result.
Similarly, we have
[Equation]
We then characterize the Lipschitz continuity of ∇ θ F with respect to ω .
Lemma 5.
Suppose Assumptions 1, 3 and 5 hold.
For any ω, ω (cid:48) and θ , we have
[Equation]
where S ω = 2 √ 2 qκρ g χB ω 1 − υ .
Proof.
We have
[Equation]
Therefore
[Equation]
we have
[Equation] [Equation]
Suppose j
[Equation]
= argmax [Equation] ( ψ s t , ψ a t )
[Equation]
−
[Equation]
g
[Equation]
ψ s t
[Equation]
ψ
[Equation]
t ) j (cid:12)(cid:12)(cid:12) , by Mean Value Theorem, there exists vector (cid:101) ω , which is some interpolation between vectors ω and ω (cid:48) , such that E
[Equation]
By Policy Gradient Theorem, we have
[Equation]
(24) where Q (cid:101) π (cid:101) ω
g ( s, a ) =
(cid:80) ∞ t =
E [ g ( s t , a t ) j ∗ − E (cid:101) π (cid:101) ω
g j ∗ | s
= s, a 0 = a, (cid:101) π (cid:101) ω ] .
Combining (22), (23), (24) and using Cauchy-Schwartz Inequality, we prove the lemma.
B. S OME I MPORTANT L EMMAS FOR P ROVING L EMMA
We denote
[Equation]
and ξ ( t
[Equation]
= ∇
[Equation]
as the i
[Equation]
i
[Equation]
d. noise of the stochastic gradient
[Equation]
respectively.
Throughout the rest of the analysis, the expectation E is taken with respect to all the noise in each iteration of the alternating mini-batch stochastic gradient descent algorithm.
The next lemma characterizes the progress at the ( t + 1) -th iteration.
For notational simplicity, we deﬁne vector function
[Equation]
Lemma 6.
At the ( t + 1) -th iteration, we have
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
[Equation] [Equation]
(cid:13)(cid:13)(cid:13)2 .
Proof.
We have
[Equation]
By the mean value theorem, we have
[Equation]
ω
[Equation]
where (cid:101) ω ( t ) is some interpolation between ω ( t +1) and ω ( t ) .
Then we have
[Equation]
By Cauchy- Swartz inequality
[Equation]
we have
[Equation]
where the last inequality comes from Lemma 5.
Morever
[Equation]
(4) implies ω ( t +1)
[Equation]
Therefore
[Equation]
we have
[Equation]
Thus, we have
[Equation] [Equation]
By (3)
[Equation]
the increment of F ( ω, θ ) takes the form
[Equation]
For notational simplicity
[Equation]
we deﬁne
[Equation]
Note that we have
[Equation]
Plugging (29) and (30) into (28), we obtain
[Equation]
Since θ belongs to the convex set { θ | (cid:107) θ (cid:107) 2 ≤ κ } , we have
[Equation]
Then we obtain
F
[Equation]
By the deﬁnition of (cid:15) ( t ) in (29)
[Equation]
we have
[Equation]
32) and
[Equation] [Equation]
Subtracting (33) from (32),we obtain (cid:15) (
[Equation]
Plugging (34) into the ﬁrst term on the right hand side of (31), we obtain
[Equation]
For term ( A )
[Equation]
we apply the Cauchy-Schwarz inequality to obtain an upper bound as follows.
[Equation]
[Table]
To derive the upper bound of ( B ) , we apply Lemma 5 to obtain
[Equation]
[Table]
[Equation]
[Equation]
Plugging (36) and (37) into (35), we obtain
[Equation]
Further plugging (38) into (31)
[Equation]
we obtain
[Equation]
[Equation]
Finally, taking expectation of (39) with respect to the noise and together with (27) , we prove the ﬁnal result.
[Equation]
2 η θ +
S ω 2 + µ (cid:19) E (cid:13) (cid:13)(cid:13)
θ ( t +1) − θ ( t ) (cid:13) (cid:13)(cid:13)
+ (cid:18)2 η θ + µ (cid:19) E (cid:13)(cid:13)(cid:13) θ ( t ) − θ ( t − 1) (cid:13)(cid:13)(cid:13) 2 2 + η ω E (cid:13)(cid:13)(cid:13) ξ ( t ) ω
(cid:13)(cid:13)(cid:13) 2 2 + 1 2 µ E (cid:13)(cid:13)(cid:13) ξ ( t − 1) θ (cid:13)(cid:13)(cid:13)2 .
We then characterize the update of ω.
Lemma 7.
The update of ω satisﬁes
[Equation]
Proof.
By the update policy
[Equation]
we have
[Equation]
Then for ( C )
[Equation]
by the deﬁnition of objective function
[Equation]
we have
[Equation] [Equation]
where for every j , (cid:101) π (cid:101) ω ( t ) j is some interpolation between vectors (cid:101) π ω ( t ) and (cid:101) π ω ( t +1) such that
[Equation]
[Equation]
For the ﬁrst term in the right hand side of (41), by Lemma 4, we have
[Equation]
For the second term in the right hand side of (41), by (42) we have
[Equation]
Then by the deﬁntion of objective function, we have
[Equation]
For ( D ) , applying Cauchy-Schwartz inequality we have
[Equation]
Finally, combining (40)-(45), we prove Lemma
[Equation]
For notational simplicity
[Equation]
we deﬁne
[Equation] [Equation]
Lemma 8.
The ﬁrst term on the right hand side of Lemma (7) satisﬁes
[Equation] [Equation]
Proof.
Plugging (46) into (47)
[Equation]
we obtain
[Equation]
By applying the equality
[Equation]
[Equation]
Recall that
[Equation]
and
[Equation]
Following from the convexity of { θ | (cid:107) θ (cid:107) 2 ≤ κ } , we have (cid:104) (cid:15) ( t +2) , θ ( t +2) − θ ( t +1) (cid:105) ≤ 0 and (cid:104) (cid:15) ( t +1) , θ ( t +2) − θ ( t +1) (cid:105) ≥ 0 .
Thus, the last term on the right side of (50) is negative.
By rearranging the terms in (50), we obtain −
[Equation]
[Equation]
By deﬁnition of δ ( t +2) in (46), we have
[Equation]
Using the Cauchy-Schwarz inequality
[Equation]
we obtain
[Equation]
( t
[Equation]
[Equation]
Plugging (52) into (51) yields (48), which concludes the proof of Lemma
[Equation]
Lemma 9.
For the update of ω
[Equation]
we have
[Equation]
[Table]
Proof.
Combing Lemma 7 and Lemma 8, we obtain
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
Note that we have
[Equation]
Combining (53) and (54), we conclude the proof of Lemma
[Equation]
Lemma 10.
Suppose Assumption 1, 3 and 5 hold.
F ( ω ( t ) , θ ( t ) ) is lower bounded throughout all iterations.
Proof.
By deﬁnition of F in (2)
[Equation]
we have
[Equation]
B. P ROOF OF L EMMA
Recall that we construct a potential function that dacays monotonically along the solution path
[Equation]
which takes the form
[Equation]
(56) for some constant s > 0 .
We deﬁne ﬁve constants k 1 , k 2 , k 3 , k 4 , k 5 as
[Equation]
Here, we restate Lemma 1 and then prove it
[Equation]
Lemma 1.
We choose step sizes η θ , η ω satisfying
[Equation]
[Table] [Table]
[Equation] [Equation]
Moreover, we have k 1 , k 2 , k 3 , k 4 , k 5 > 0 for s =
η 2 ω (58 L ω + 9) .
Proof.
For notational simplicity, we deﬁne
[Equation]
By rearranging the inequality in Lemma (9)
[Equation]
we obtain
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
By deﬁnition of P ( t ) in (56), we have
E
[Equation]
for some constant s > 0 .
Combining (59) and Lemma 6, since η ω < 12 L ω , we obtain
E ( t +1) − E (
[Equation]
−
[Equation] [Equation]
(cid:13)(cid:13)(cid:13) ω ( t +1)
− ω ( t ) (cid:13)(cid:13)(cid:13) 2− (cid:16) s · η ω L ω −
S ω (cid:17) (cid:13)(cid:13)(cid:13) ω ( t ) − ω ( t − 1) (cid:13)(cid:13)(cid:13) 2 2 − s · (cid:16) µη ω 4 − 3 µ 2 η ω η θ(cid:17) (cid:13)(cid:13)(cid:13) θ ( t +2) − θ ( t +1)
(cid:13)(cid:13)(cid:13) 2− (cid:16) s · µη ω 8 − (cid:0)2 η θ + 2 µ + S ω (cid:1)(cid:17) (cid:13) (cid:13)(cid:13)
θ ( t +1) − θ ( t ) (cid:13)
(cid:13)(cid:13) 2− (cid:16) s · µη ω 8 − (cid:0)2 η θ + µ (cid:1)(cid:17) (cid:13)(cid:13)(cid:13) θ ( t ) − θ ( t − 1) (cid:13)(cid:13)(cid:13) 2+ η ω E
(cid:13)(cid:13)(cid:13) ξ ( t ) ω (cid:13)(cid:13)(cid:13) 2 2 + 1 2 µ E (cid:13)(cid:13)(cid:13) ξ ( t − 1) θ (cid:13)(cid:13)(cid:13) + s · (cid:16) ( η 2 ω + η ω 2 µ ) E (cid:13)(cid:13)(cid:13) ξ ( t ) ω (cid:13)(cid:13)(cid:13) 2 2 + η ω 2 E (cid:13)(cid:13)(cid:13)
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
( t )
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
Since η θ < 1 150 µ , we have k 3 > 0 .
Now we choose a proper constant s such that k 1 , k 2 , k 4 , k 5 are positive.
Note that they are positive if and only if
[Equation]
[Equation]
Rearranging the terms in (61), (62), (63), (64) , we obtain
S ω η [Equation] ω < [Equation]
<
[Equation] [Equation]
) η
[Equation]
S
[Equation]
ω ) /
[Equation] [Equation] [Equation] + (
[Equation] [Equation]
µη
[Equation] [Equation]
< 1 / (
[Equation]
) η
[Equation] [Equation]
S
[Equation] [Equation]
Since η θ ≤ 7 L
[Equation]
+150 S 2 ω and η θ < 100(2 µ + S ω ) , by rearranging the terms in (65) and (66) and taking the leading terms, we obtain η ω <
L ω
S ω (8 L ω + 2) and η ω η θ < µ 30 L ω + 5 .
Therefore, we have η ω η θ < µ 30 L ω + 5 , η ω < η ω and η θ < η θ , where η ω = min (cid:26)
L ω
S ω (8 L ω + 2) , 2 L ω (cid:27) , η θ = min (cid:26)150 µ, 7 L ω +150 S 2 ω , 100(2 µ + S ω ) (cid:27) .
B. P ROOF OF T HEOREM
Let k = 1 / min { k 1 , k 4 } , and φ = max { 1 , 1 /η 2 ω , 1 /η 2 θ } .
Then we have
NJ N ≤
N (cid:88) t = φ · E ( (cid:13)(cid:13)(cid:13) ω ( t +1) − ω ( t ) (cid:13)(cid:13)(cid:13) 2 2 + (cid:13)(cid:13)(cid:13) θ ( t +1) − θ ( t ) (cid:13)(cid:13)(cid:13) 2 2 ) ≤ φk (cid:16) N (cid:88) t
=
( E ( t ) − E ( t +1) )
+
N (cid:88) t = (cid:0) ( η ω + sη 2 ω + sη ω 2 µ ) E (cid:13)(cid:13)(cid:13) ξ ( t ) ω (cid:13)(cid:13)(cid:13) 2 2 + 1 2 µ · E (cid:13)(cid:13)(cid:13) ξ ( t − 1) θ (cid:13)(cid:13)(cid:13) + s · ( η ω / 2 · E (cid:13)(cid:13)(cid:13) ξ ( t − 1) ω (cid:13)(cid:13)(cid:13)2 + 3 η ω η θ / 2 · ( E (cid:13)(cid:13)(cid:13) ξ ( t +1) θ (cid:13)(cid:13)(cid:13)2 + E (cid:13)(cid:13)(cid:13) ξ ( t ) θ (cid:13)(cid:13)(cid:13) 2 2 ) (cid:1)(cid:17) ≤ φk (cid:16) (
E (1) − E ( N ) ) + N (cid:88) t = (cid:0) 2 max (cid:26) η ω + sη 2 ω + sη ω 2 µ , sη ω 2 (cid:27) E (cid:13)(cid:13)(cid:13) ξ ( t ) ω (cid:13)(cid:13)(cid:13) 2+ 3 max (cid:26)2 µ, 3 η ω η θ (cid:27) E (cid:13)(cid:13)(cid:13) ξ ( t ) θ (cid:13)(cid:13)(cid:13) (cid:1)(cid:17) .
Now [Table] ν = max (cid:26)
[Equation]
and divide both sides of (67) by N , we have
[Equation]
By deﬁnition of E in (56) and Lemma 10, we have
[Equation]
Now for any given (cid:15) > 0 , we take
[Equation]
and N = kφ
[Equation]
and obtain
J N ≤ (cid:15).
C E MPIRICAL M AXIMIZER C ASE
Notice that the object function (2) is quadratic in θ , and thus we are able to get the empirical maximizer by simple calculation instead of performing stochastic gradient ascent at each iteration.
This allows us to avoid using large batches while achieving the same sample complexity (cid:101) O ( 1 (cid:15) 2 ) .
Given a ﬁxed ω ( t ) , by deﬁnition of G in (25), we can get the optimal θ ∗ ( ω ( t ) ) in population form:
[Equation]
At the ( t + 1) -th iteration, we can achieve the closed form of the empirical maximizer by sampling one trajectory randomly and then apply stochastic gradient descent to the outer minimization.
More speciﬁcally, the updating rule is deﬁned as
[Equation]
71) where (cid:101) G is the empirical version of G obtained by sampling, and ∇ ω (cid:101) f is the stochastic approximation of ∇ ω F at the ( t + 1) -the iteration.
We deﬁne the stationary point as follow.
Deﬁnition 4.
We call ω ∗ an stationary point if ∇ ω F ( ω ∗ , θ ∗ ( ω ∗ ))
= 0 .
Before we proceed with the convergence analysis, we impose the following assumptions on the problem.
Assumption 6.
There is some constant M G > 0 s.t.
for any ω, (cid:101) θ and π , the following two conditions hold.
[Equation]
Assumption 6 requires the stochastic gradient to be unbiased with bounded second moment.
Corollary 4.
Under Assumption 1, 3 and 5, there exists B F = 12 ρ 2 g µ + λB H such that for any ω, we have | F ( ω, θ ∗ ( ω )) | < B F .
Proof.
By Equation (69),we have (cid:107) θ ∗ ( ω ) (cid:107) 2 ≤√ 2 ρ g µ .
Plugging this into F ( ω, θ ∗ ( ω )) , we have for any ω ,
[Equation]
For notational simplicity, We deﬁne ξ ( t ) ω
= ∇ θ F ( ω ( t ) , (cid:101) θ ( t +1) )
− ∇ θ (cid:101) f t ( ω ( t ) , (cid:101) θ ( t +1) )
as the i.i.d stochastic noise and σ ( t ) θ =
θ ∗ ( ω ( t ) ) − (cid:101) θ ( t ) as the i.i.d error of empirical maximizer.
We denote E as the expectation over ξ ( t ) ω ( t ≥ 1) and σ ( t ) θ ( t ≥ 1) .
We measure the sub-stationarity of the algotithm at the iteration N by
[Equation]
Then we state the global convergence of the above mentioned optimization method.
Theorem 3.
Suppose Assumptions 1, 3, 5, 6 hold.
Given any (cid:15) > 0 , we take η ω = (cid:15) √ 2(
L ω + S 2 ω /µ )
M G , then we need at most
[Equation]
iterations to have I N < (cid:15).
Proof.
By employing the ﬁrst inequality in Lemma 4, we have
[Equation]
Note that
[Equation]
where ( i ) comes from the unbiased property of (cid:101) θ ( t ) and the fact that ∇ ω F ( ω, θ ) is linear in θ , and ( ii ) comes from the unbiased property of (cid:101)
f j ( ω, (cid:101) θ ) .
Now taking the expectation on both sides of (72) and plugging (73) in
[Equation]
we obtain
[Equation] [Equation]
Dividing both sides by η ω and rearranging the terms in (74)
[Equation]
we get
[Equation]
, θ ∗ ( ω ( t +1) ))
− F ( ω ( t +1) , θ ∗ ( ω ( t ) ))
η ω .
Now consider F ( ω ( t +1) , θ ∗ ( ω ( t +1) ))
− F ( ω ( t +1) , θ ∗ ( ω ( t ) )) , we have
F ( ω ( t +1) , θ ∗ ( ω ( t +1) ))
− F ( ω ( t +1) , θ ∗ ( ω ( t ) ))
= (cid:104) G ( π ω ( t +1) )
− G ( π ∗ ) , θ ∗ ( ω ( t +1) )
(cid:105) − (cid:104) G
( π ω ( t +1) )
[Equation]
G ( π
[Equation]
) , θ ∗ (
[Equation]
( t ) ) (cid:105) − µ
[Equation]
( (cid:13)(cid:13)(cid:13) θ
[Equation] [Equation]
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
2 − (cid:13)(cid:13)(cid:13)
[Equation] [Equation]
( t
[Equation]
(cid:13)(cid:13)(cid:13) 2 2 ) = (cid:104) µθ ∗ (
[Equation] [Equation]
[Equation] [Equation]
[Equation] [Equation]
(cid:105) − µ
[Equation]
[Equation] [Equation]
1) ) −
[Equation] [Equation]
) ) (cid:105) = µ (cid:13) (cid:13)(cid:13)
[Equation]
∗ ( ω ( t
[Equation]
1) ) − θ ∗ (
[Equation]
( t ) (cid:13) (cid:13)(cid:13)
2= µ 2 (cid:13)(cid:13)(cid:13)(cid:13) 1 µ ( G ( π ω ( t +1) )
− G ( π ω ( t ) ))
(cid:13)(cid:13)(cid:13)(cid:13)
[Equation]
Taking expectation on both sides of (76) with respect to the noise introduced by SGD, we have
[Equation]
Summing the equation(75) up, we have
[Equation]
Dividing both sides of the above equation by N , we get
[Equation]
2 2 ≤| F ( ω (1) , θ ∗ ( ω (1) ))
− E F ( ω
[Equation] [Equation]
Nη
[Equation]
Since | F ( ω (1) , θ ∗ ( ω (1) ))
− E F ( ω ( N +1) , θ ∗ ( ω ( N +1) ))
| ≤ 2 B F .
Take η ω = (cid:113) 2 B F ( L ω / 2+ S 2 ω / 2) M G N , then we have min
[Equation]
N
[Equation]
where B F = 12 ρ 2 g µ + λM H .
This implies that when η ω = (cid:15) √ 2(2 L ω + S 2 ω /µ )
M G , we need at most
N =
[Equation]
O
[Equation]
such that I N < (cid:15) .