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Momentum in Reinforcement Learning Momentum in Reinforcement Learning Nino Vieillard1,2 Bruno Scherrer2 Olivier Pietquin1 Matthieu Geist1 1Google Research, Brain Team 2Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France Abstract state-action dependant noise. The resulting sequence of policies will be unstable and suboptimal, even with We adapt the optimization’s concept of mo- centered and bounded noise. Dealing with errors is how- mentum to reinforcement learning. Seeing ever crucial to RL, as we hope to tackle problems with the state-action value functions as an analog large states spaces that require function approximation. to the gradients in optimization, we inter- Indeed, many recent RL successes are algorithms that pret momentum as an average of consecutive instantiate ADP schemes with neural networks for func- q-functions. We derive Momentum Value It- tion approximation. Deep Q-Networks (DQN, Mnih eration (MoVI), a variation of Value iteration et al. [2015]) for example, can be seen as an extension that incorporates this momentum idea. Our of AVI with neural networks. analysis shows that this allows MoVI to av- In optimization, a common strategy to stabilize the erage errors over successive iterations. We descent direction, known as momentum, is to average show that the proposed approach can be read- the successive gradients instead of considering the last ily extended to deep learning. Specifically,we one. In reinforcement learning, the state-action value propose a simple improvement on DQN based function can be seen informally as a kind of gradient, as on MoVI, and experiment it on Atari games. it gives an improvement direction for the policy. Hence, we propose to bring the concept of momentum to rein- forcement learning by basically averaging q-values in a 1 Introduction DP scheme. Reinforcement Learning (RL) is largely based on Ap- We introduce Momentum Value Iteration (MoVI) in proximate Dynamic Programming (ADP), that pro- Section 4. It is Value Iteration, up to the fact that vides algorithms to solve Markov Decision Processes the policy, instead of being greedy with respect to the (MDP, Puterman [1994]) under approximation. In the last state-action value function, is greedy with respect exact case, where there is no approximation, classic to an average of the past value functions. We analyze algorithms such as Value Iteration (VI) or Policy Iter- the propagation of errors of this scheme. In AVI, the ation (PI) are guaranteed to converge to the optimal performance bound will depend on a weighted sum of solution, that is find an optimal policy that dominates the norms of the errors at each iteration. For MoVI, we every policy in terms of value. These algorithms rely on show that this depends on the norms of the cumulative solving fixed-point problems: in VI, one tries to reach errors of previous iteration. This means that it allows the fixed point of the Bellman optimality operator by for a compensation of errors along different iterations, an iterative method. We focus on VI for the rest of the and even convergence in the case of zero-mean and paper, but the principle we propose can be extended bounded noises, under some assumption. This compen- beyond this. Approximate Value Iteration (AVI) is a sation property is shared by a few algorithms that will VI scheme with approximation errors. It is well known be discussed in Section 6. We also show that MoVI [Bertsekas and Tsitsiklis, 1996] that if the errors do can be successfully combined with powerful function not vanish, AVI does not converge. To get some in- approximation by proposing Momentum-DQN in Sec- tuition, consider a sequence of policies being greedy tion 5, an extension of MoVI with neural networks according to the optimal q-function, with an additional based on DQN. It provides a strong performance im- provement over DQN on the standard Arcade Learning Environment (ALE) benchmark [Bellemare et al., 2013]. rd Proceedings of the 23 International Conference on Artificial All stated results are proven in the appendix. Intelligence and Statistics (AISTATS) 2020, Palermo, Italy. PMLR: Volume 108. Copyright 2020 by the author(s). Momentum in Reinforcement Learning 2 Background the partial evaluation step. AVI satisfies the following bound for the quality of the policy πk Markov Decision Processes. We consider the RL 2γ max k k setting where an agent interacts with an environment kq − q k ≤ 2γkq + j<k j ∞ . (1) modeled as an infinite discounted horizon MDP. An ∗ πk ∞ max (1 − γ)2 MDP is a quintuple {S, A, P, r, γ}, where S is a finite1 S×A This explains why AVI is not resistant to errors: state space, A a finite action space, P ∈ ∆S is a Markovian transition kernel (writing ∆X the simplex maxj<k kjk∞ can be high even if each k is zero-mean. S×A over the set X), r ∈ [−rmax, rmax] a reward func- tion and γ ∈ (0, 1) the discount factor. A policy π maps 3 Momentum Value Iteration the state space to distributions over actions π(·|s). We define the q-value qπ of a policy π as, for each s ∈ S In the context of optimization, momentum aims at and a ∈ A, stabilizing gradient ascent (or descent) methods. Con- " ∞ # sider we want to maximize a concave function f whose X t qπ(s, a) = Eπ γ r(st, at) s0 = s, a0 = a , gradient is not known analytically, and we use a classic t=0 (stochastic) gradient ascent algorithm. This algorithm iterates from a value x by computing an approxima- where E denotes the expected value over all trajec- 0 π tion g of ∇f(x ), and updating x = x +ηg . One tories (s , a , s , a ,...) produced by π. The value k k k+1 k k 1 1 2 2 can then use momentum [Qian, 1999] to stabilize the is bounded by q = r /(1 − γ). Let us de- max max process through a smoothing function h = ρh + g , fine the transition kernel operator associated to π k k k with ρ ∈ R, and an update x = x + ηh . This can as, for each q ∈ RS×A and for each (s, a) ∈ S × A, k+1 k k 0 0 stabilize the ascent as the gradient may vary greatly as [P q](s, a) = E 0 0 0 [q(s , a )]. The q- π s ∼P (·|s,a),a ∼π(·|s ) from step to step. function of a policy is the fixed point of its Bellman evaluation operator, defined for each q ∈ RS×A as In the context of ADP, the q-function intuitively gives Tπq = r + γPπq. An optimal policy π∗ is such that for the direction that guides the policy, in the same way any other policy π, we have that, for each (s, a) ∈ S×A, that the gradient is the improvement direction of a qπ∗ (s, a) ≥ qπ(s, a). The Bellman optimality operator variable. In particular, we can rewrite the greedy step is defined as T q = max T q, and we have that q is ∗ π π ∗ (in AVI) as πk(·|s) ∈ argmaxπ(·|s)∈∆A hqk(s, ·), π(·|s)i, the unique fixed point of T∗. A policy is said to be thus seeing this step as finding the policy being state- S×A greedy with respect to q ∈ R if T∗q = Tπq. We wise the most colinear with qk. This is also reminiscent denote the set of these policies G(q). Note that such a of the direction finding subproblem of Frank and Wolfe policy can be computed without accessing to the model [1956]. Consequently, the greedy step can be seen as (the transition kernel). an analog of the update in gradient ascent (the policy π is analog to the variable x), the differences being Finally, for µ ∈ ∆ we write d = (1 − γ)µ(I − S×A π,µ (i) that q in AVI is not a gradient, but the result of γP )−1 the discounted cumulative occupancy measure k π an iterative process, q = T q , and (ii) that the induced by π when starting from the distribution µ k πk k−1 policy is not updated, but replaced. (distributions being written as row vectors). We define S×A the µ-weighted `p-norm as, for each q ∈ R , kqkp,µ = This analogy is thus quite limited (qk is not really a p 1 gradient, there is no optimized function, the policy is E [|q(s, a)| ] p . (s,a)∼µ replaced rather than updated). However, it is sufficient to adapt the momentum idea to AVI, by replacing the Approximate Value Iteration. Approximate Dy- q-function in the improvement step by a smoothing namic Programming provides algorithms to solve an of the q-functions, h = ρh + q . We can then MDP under some errors. One classic algorithm is Ap- k k−1 k notice that G(h ) = G( hk ), allowing us to compute a proximate Value Iteration. It looks directly for the k 1+ρ moving average instead of a smoothing, hk = βkhk−1 + fixed point of T∗ with an iterative process (1 − βk)qk, which leads to the following ADP scheme, ( π ∈ G(q ) initialized with h0 = q0, k+1 k (AVI) qk+1 = Tπ qk + k+1. k+1 π = G(h ) k+1 k (MoVI) (2) Notice that here, Tπk+1 qk = T∗qk. In this scheme, we qk+1 = Tπk+1 qk + k+1 call the first line the greedy step, and the second line hk+1 = βk+1hk + (1 − βk+1)qk+1. 1This is for ease and clarity of exposition, the proposed algorithm and analysis can be extended to continuous state We call this scheme Momentum Value Iteration (MoVI), spaces.
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