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In this paper we proved tight last-iterate convergence rates for smooth monotone games when all players act according to the optimistic gradient algorithm, which is no-regret

Tight last-iterate convergence rates for no-regret learning in multi-player games

NIPS 2020, (2020)

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Abstract

We study the question of obtaining last-iterate convergence rates for no-regret learning algorithms in multi-player games. We show that the optimistic gradient (OG) algorithm with a constant step-size, which is no-regret, achieves a last-iterate rate of $O(1/\sqrt{T})$ with respect to the gap function in smooth monotone games. This resu...More

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Introduction
  • In the setting of multi-agent online learning ([SS11, CBL06]), K players interact with each other over time.

    At each time step t, each player k ∈ {1, . . . , K} chooses an action z(kt); z(kt) may represent, for instance, the bidding strategy of an advertiser at time t.
  • Fails to capture the game dynamics over time ([MPP17]), and both types of guarantees use newly acquired information with decreasing weight, which, as remarked by [LZMJ20], is very unnatural from an economic perspective.1 the following question is of particular interest ([MZ18, LZMJ20, MPP17, DISZ17]): Can the authors establish last-iterate rates if all players act according to a no-regret learning algorithm with constant step size?
Highlights
  • In the setting of multi-agent online learning ([SS11, CBL06]), K players interact with each other over time.

    At each time step t, each player k ∈ {1, . . . , K} chooses an action z(kt); z(kt) may represent, for instance, the bidding strategy of an advertiser at time t
  • A fundamental quantity used to measure the performance of an online learning algorithm is the regret of player k, which is the difference between the total loss of player k over T time steps and the loss of the best possible action in hindsight: formally, the regret at time T is
  • We show in Theorem 5 and Corollary 6 that the actions taken by learners following the optimistic gradient (OG) algorithm, which is no-regret√, exhibit last-iterate convergence to a Nash equilibrium in smooth, monotone games at a rate of O(1/ T ) in terms of the global gap function
  • As in prior work proving lower bounds for p-stationary canonical linear iterative methods (p-SCLIs) ([ASSS15, IAGM19]), we reduce the problem of proving a lower bound on TGapGDD (z(t)) to the problem of proving a lower bound on the supremum of the spectral norms of a family of polynomials
  • In this paper we proved tight last-iterate convergence rates for smooth monotone games when all players act according to the optimistic gradient algorithm, which is no-regret
  • As for l√ower bounds, it would be interesting to determine whether an algorithm-independent lower bound of Ω(1/ T ) in the context of Theorem 7 could be proven for stationary p-SCLIs
Results
  • The authors show in Theorem 5 and Corollary 6 that the actions taken by learners following the optimistic gradient (OG) algorithm, which is no-regret√, exhibit last-iterate convergence to a Nash equilibrium in smooth, monotone games at a rate of O(1/ T ) in terms of the global gap function.
  • This algorithm exhibits last-iterate convergence at a rate of O(1/ T ) in smooth monotone games when all players play according to it [GPDO20], it is straightforward to see that it is not a no-regret learning algorithm, i.e., for an adversarial loss function the regret can be linear in T.
  • It has been shown ([DP18, LNPW20]) that a modification of OG known as optimistic multiplicativeweights update exhibits last-iterate convergence to Nash equilibria in two-player zero-sum monotone games, but as with the unconstrained case ([MOP19a]) non-asymptotic rates are unknown.
  • To the best of the knowledge, the only work proving last-iterate convergence rates for general smooth monotone VIs was [GPDO20], which only treated the EG algorithm, which is not no-regret.
  • The following essentially optimal regret bound is well-known for the OG algorithm, when the actions of the other players z(−t)k are adversarial: Proposition 3.
  • The following Theorem 7 uses functions in Fnb,ill,D as “hard instances” to show that the O(1/ T ) rate of Corollary 5 cannot be improved by more than an algorithm-dependent constant factor.
  • Using Proposition 8, the authors show that any p-SCLI algorithm must have a rate of at least ΩA(1/T ) for smooth convex function minimization.9 This is slower than the O(1/T 2) error achievable with Nesterov’s AGD with a time-varying learning rate.
Conclusion
  • In this paper the authors proved tight last-iterate convergence rates for smooth monotone games when all players act according to the optimistic gradient algorithm, which is no-regret.
  • [DP18, LNPW20] showed that OMWU exhibits last-iterate convergence, but non-asymptotic rates remain unknown even for the case that FG(·) is linear, which includes finite-action polymatrix games.
Tables
  • Table1: Known last-iterate convergence rates for learning in smooth monotone games with perfect gradient feedback (i.e., deterministic algorithms). We specialize to the 2-player 0-sum case in presenting prior work, since some papers in the literature only consider this setting. Recall that a game G has a γ-singular value lower bound if for all z, all singular values of ∂FG(z) are ≥ γ. l, Λ are the Lipschitz constants of FG, ∂FG, respectively, and c, C > 0 are absolute constants where c is sufficiently small and C is sufficiently large. Upper bounds in the left-hand column are for the EG algorithm, and lower bounds are for a general form of 1-SCLI methods which include EG. Upper bounds in the right-hand column are for algorithms which are implementable as online no-regret learning algorithms
  • Table2: Known upper bounds on last-iterate convergence rates for learning in smooth monotone games with noisy gradient feedback (i.e., stochastic algorithms). Rows of the table are as in Table 1; l, Λ are the Lipschitz constants of FG, ∂FG, respectively, and c > 0 is a sufficiently small absolute constant. The right-hand column contains algorithms implementable as online no-regret learning algorithms: stochastic optimistic gradient (Stoch. OG) or stochastic gradient descent (SGD). The left-hand column contains algorithms not implementable as no-regret algorithms, which includes stochastic extragradient (Stoch. EG), stochastic forward-backward (FB) splitting, double stepsize extragradient (DSEG), and stochastic variance reduced extragradient (SVRE). SVRE only applies in the finite-sum setting, which is a special case of (Abs) in which fk is a sum of m individual loss functions fk,i, and a noisy gradient is obtained as ∇fk,i for a random i ∈ [m]. Due to the stochasticity, many prior works make use of a step size ηt that decreases with t; we make note of whether this is the case (“ηt decr.”) or whether the step size ηt can be constant (“ηt const.”). For simplicity of presentation we assume Ω(1/t) ≤ {τt, σt} ≤ O(1) for all t ≥ 0 in all cases for which σt, τt vary with t. Reported bounds are stated for the total gap function (Definition 3); leading constants and factors depending on distance between initialization and optimum are omitted
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Related work
  • Multi-agent learning in games. In the constrained setting, many papers have studied conditions under which the action profile of no-regret learning algorithms, often variants of Follow-The-Regularized-Leader (FTRL), converges to equilibrium. However, these works all assume either a learning rate that decreases over time ([MZ18, ZMB+17, ZMA+18, ZMM+17]), or else only apply to specific types of potential games ([KKDB15, KBTB18, PPP17, KPT09, CL16, BEDL06, PP14]), which significantly facilitates the analysis of last-iterate convergence.3

    Such potential games are in general incomparable with monotone games, and do not even include finitestate two-player zero sum games (i.e., matrix games). In fact, [BP18] showed that the actions of players following FTRL in two-player zero-sum matrix games diverge from interior Nash equilibria. Many other works ([HMC03, MPP17, KLP11, DFP+10, BCM12, PP16]) establish similar non-convergence results in both discrete and continuous time for various types of monotone games, including zero-sum polymatrix games. Such non-convergence includes chaotic behavior such as Poincaré recurrence, which showcases the insufficiency of on-average convergence (which holds in such settings) and so is additional motivation for the question (⋆).
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Noah Golowich
Noah Golowich
Sarath Pattathil
Sarath Pattathil
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