# Multi-agent online learning in time-varying games

Benoît Duvocelle, Panayotis Mertikopoulos, Mathias Staudigl, Dries Vermeulen

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## To cite this version:

Benoît Duvocelle, Panayotis Mertikopoulos, Mathias Staudigl, Dries Vermeulen. Multi-agent online learning in time-varying games. Mathematics of Operations Research, 2023, 48 (2), pp.914-941. 10.1287/moor.2022.1283. hal-01891545

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Submitted on 21 Dec 2023

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MULTI-AGENT ONLINE LEARNING IN TIME-VARYING GAMES

BENOIT DUVOCELLE$^{‡}$, PANAYOTIS MERTIKOPOULOS$^{∗}$,
MATHIAS STAUDIGL$^{◦}$, AND DRIES VERMEULEN$^{‡}$

ABSTRACT. We examine the long-run behavior of multi-agent online learning in games that evolve over time. Specifically, we focus on a wide class of policies based on mirror descent, and we show that the induced sequence of play (a) converges to Nash equilibrium in time-varying games that stabilize in the long run to a strictly monotone limit; and (b) it stays asymptotically close to the evolving equilibrium of the sequence of stage games (assuming they are strongly monotone). Our results apply to both gradient-based and payoff-based feedback – i.e., when players only get to observe the payoffs of their chosen actions.

# 1. INTRODUCTION

Consider a repeated multi-agent decision process that unfolds as follows:

(1) At each stage $t = 1, 2, \dots$, every agent selects an action from some continuous set.

(2) Each agent receives a reward based on their chosen action and the actions of all other players. These rewards are determined by a normal form game $G_t$ that evolves over time and is a priori unknown to the players.

(3) Based on the reward that they received (and/or any other payoff-relevant information), the players update their actions and the process repeats.

The main questions that we seek to address in this general context are the following: First, are there online learning policies that allow players to track a Nash equilibrium over time (or to converge to one if the stage games stabilize)? And, if so, what is the impact of the information available to the players and the variability of the sequence of stage games?

**Background.** One of the most widely used policies for learning in games is the *mirror descent* (MD) class of algorithms and its variants, cf. Bubeck and Cesa-Bianchi [13], Shalev-Shwartz [54], and references therein. This family of first-order methods dates back to Nemirovski and Yudin [43], and contains as special cases standard (sub)gradient descent methods, entropic gradient descent [4], the “Hedge” (or exponential/multiplicative weights) algorithm in finite games [3, 37, 67], and, in games with a linear payoff structure, the “follow the regularized leader” (FTRL) class of policies [55]. These methods have been applied to a wide range of games – from min-max to potential games – leading to a vast corpus of literature that

<footnote>$^{‡}$ MAASTRICHT UNIVERSITY, DEPARTMENT OF QUANTITATIVE ECONOMICS, P.O. Box 616, NL-6200 MD MAASTRICHT, THE NETHERLANDS.</footnote>

<footnote>$^{∗}$ UNIV. GRENOBLE ALPES, CNRS, INRIA, LIG, 38000, GRENOBLE, FRANCE.</footnote>

<footnote>$^{◦}$ MAASTRICHT UNIVERSITY, DEPARTMENT OF DATA SCIENCE AND KNOWLEDGE ENGINEERING, P.O. Box 616, NL-6200 MD MAASTRICHT, THE NETHERLANDS</footnote>

<footnote>E-mail addresses: b.duvocelle@maastrichtuniversity.nl, panayotis.mertikopoulos@imag.fr, m.staudigl@maastrichtuniversity.nl, d.vermeulen@maastrichtuniversity.nl.</footnote>

<footnote n="2010">2010 Mathematics Subject Classification. Primary 91A10, 91A26; secondary 68Q32, 68T02.</footnote>

<footnote>Key words and phrases. Nash equilibrium; mirror descent; time-varying games.</footnote>
is impossible to survey here; for an appetizer, see [9, 29, 33, 36, 41, 44, 48] and references therein.

In the single-player case, the standard figure of merit is the minimization of the learner's regret, i.e., the cumulative payoff difference between the player's chosen policy and the "best policy in hindsight" (static or dynamic, depending on the precise notion of regret under consideration). In this context, when the payoff functions encountered by the learner are concave, MD methods guarantee an $O(\sqrt{T})$ static regret bound which is well known to be order-optimal [1]; moreover, if the problem has a favorable geometry (e.g., when the learner's action set is a simplex or a spectrahedron), these bounds are "almost" dimension-free, a fact which is of crucial importance in practical applications.

In view of these appealing guarantees, one might expect this picture to carry over effortlessly to multi-agent decision problems as well. However, game-theoretic learning can be considerably more involved because, in addition to the *exogenous* variability of the stage game $G_t$ as a function of $t$, the players' individual reward functions also vary *endogenously* as a function of the actions chosen by the other players at any given time $t$. Moreover, the standard solution concept in game theory is that of a *Nash equilibrium* – not the players' regret (external, internal, or dynamic). As a result, even though the algorithms under study are essentially the same in both single- and multi-agent environments, the analysis and the results obtained in these two settings are often markedly different.

Our paper focuses on multi-agent problems and aims to analyze the equilibrium tracking and convergence properties of MD-based policies in time-varying games. In so doing, we seek to partially fill a gap in the existing literature on game-theoretic learning, which has focused almost exclusively on the case where there are no exogenous variations in the players' payoff functions – i.e., when the stage game $G_t$ remains *fixed* for all $t$. To provide the necessary context, we begin by discussing below some relevant works, and we outline our main contributions right after.

**Related work.** Starting with mixed-strategy learning in finite games, a “folk” result in the field states that the empirical frequency of no-regret play converges to the game’s *Hannan set* (also known as the set of coarse correlated equilibria). However, as was shown by Viossat and Zaechelnyuk [66], the Hannan set of a game may contain strategies that assign positive weight *only* to dominated strategies, a point which is clearly incompatible with Nash play. More to the point, the impossibility result of Hart and Mas-Colell [26] shows that there are no uncoupled dynamics leading to Nash equilibrium in all games: since no-regret dynamics are unilateral by construction – and hence uncoupled a fortiori – it is not possible to establish a blanket causal link between no-regret play and convergence to Nash equilibrium.

For this reason, deriving the equilibrium convergence properties of multi-agent learning processes require a more specialized look, typically zooming in on specific classes of games. In the case of mixed extensions of finite games, Leslie and Collins [35, 36] and Coucheney et al. [16] showed that a variant of the exponential weights algorithm converges to an $\varepsilon$-perturbed equilibrium with probability 1 in potential and $2 \times 2 \times \cdots \times 2$ games. More recently, in the case of *continuous* potential games, Perkins and Leslie [47] showed that a lifted variant of MD-based methods converges weakly to an $\varepsilon$-neighborhood of the game’s set of Nash equilibria. Importantly, in all these works, convergence is established by first showing that a naturally associated continuous-time dynamical system converges, and then using the so-called ordinary differential equation (ODE) method of stochastic approximation [6, 7] to translate this result to discrete time.

More relevant for our purposes is the recent work of Mertikopoulos and Zhou [39] who focused on the class of *monotone games*, i.e., continuous games that satisfy the so-called *diagonal strict concavity* (DSC) condition of Rosen [50]. Specifically, using the same ODE
stochastic approximation tools discussed above, [39] showed that the sequence of play generated by a specific version of the dual averaging algorithm of Nesterov [44] converges to Nash equilibrium with probability 1, even with in the presence of noise and uncertainty. The analysis of [39] was subsequently extended by Bravo et al. [12] to learning with payoff-based, “bandit feedbak” – i.e., when players observe only the payoff of the action that they played. At around the same time – and always in the context of monotone games – Tatarenko and Kamgarpour [61, 62] used a Tikhonov regularization approach to obtain a series of comparable results for “merely monotone” games (i.e., monotone games that are not necessarily *strictly* monotone). Finally, in a very recent paper, Bervoets et al. [9] used stochastic approximation methodologies to prove the convergence of a payoff-based, *dampened gradient approximation* (DGA) scheme in two other classes of one-dimensional concave games – games with strategic complements, and ordinal potential games with isolated equilibria.

**Our contributions.** In all the works described above, the game faced by the players remains *fixed* throughout the learning process, and the variation in the players’ individual payoff functions is strictly *endogenous* – i.e., it is only due to the other players’ evolving action choice. By contrast, our paper seeks to tackle problems where the sequence of games encountered by the players also evolves *exogenously* – i.e., players encounter a *time-varying game*.

In this general context, we consider two distinct regimes: (a) when the sequence of stage games converges to some well-defined limit (in our case, a strictly monotone game); and (b) when $G_t$ evolves over time without converging. In terms of feedback, we consider a flexible oracle model which provides noisy payoff gradient estimates to the players based on the actions that they chose at each stage of the process. We then show that, if the sequence of stage games stabilizes to some well-defined limit, the induced sequence of play converges to a Nash equilibrium of the limit game with probability 1, irrespective of the magnitude of the noise entering the players’ gradient signals. On the other hand, if the stage games do not stabilize, there is no equilibrium state to converge to (either static or in the mean); in this case, we focus on the players’ ability to track the equilibrium of $G_t$ as it evolves over time. More precisely, we show that the average distance from equilibrium vanishes over time, and we provide an explicit estimate for this “tracking error” in terms of the variation of the sequence of stage games (assuming they are strongly monotone).

Finally, to account for environments where gradient information is not available to the players, we also consider the case of learning with *payoff-based* feedback. By considering a one-shot gradient estimation process based on single-point stochastic approximation techniques [12, 23, 59], we map the problem of payoff-based learning to our generic oracle model, and we show that our convergence and equilibrium tracking results still apply in this case (though the corresponding rates are reduced as a consequence of the players’ having even less information at their disposal).

In terms of proof techniques, the exogenous dependence of $G_t$ on $t$ means that the continuous-time limit of the players’ learning process is likewise non-autonomous (i.e., it also depends on $t$). As a result, there is no longer a well-defined “mean field equation” to approximate, so it is not possible to employ the ODE method of Benaïm [6] that underlies the series of papers discussed above. Instead, to establish convergence to an equilibrium in the “stable limit” regime, we work directly in discrete time and we employ a mix of submartingale limit theory and quasi-Fejér arguments. Finally, our equilibrium tracking result relies on decomposing the horizon of play into batches of appropriately chosen length and subsequently utilizes a batch comparison technique that was introduced by Besbes et al. [10] to analyze the dynamic regret of *single-agent* online learning algorithms.
## 2. PRELIMINARIES

2.1. **Notation.** Let $\mathcal{X}$ be a $d$-dimensional real space with norm $\|\cdot\|$, and let $\mathcal{C}$ be a compact convex subset of $\mathcal{X}$. In what follows, we will write $\mathcal{Y} := \mathcal{X}^*$ for the dual of $\mathcal{X}$, $\langle y, x \rangle$ for the duality pairing between $y \in \mathcal{Y}$ and $x \in \mathcal{X}$, and $\|y\|_* = \sup\{\langle y, x \rangle : \|x\| \le 1\}$ for the dual norm of $y \in \mathcal{Y}$. We will also write $ri(\mathcal{C})$ for the relative interior of $\mathcal{C}$, $bd(\mathcal{C})$ for its boundary, and $\text{diam}(\mathcal{C}) = \sup\{\|x' - x\| : x, x' \in \mathcal{C}\}$ for its diameter. Finally, for concision, we will write $[a..b] = \{a, a+1, \dots, b\}$ for the set of positive integers spanned by $a, b \in \mathbb{N}$.

2.2. **Continuous games.** Throughout our paper, we focus on games with a finite number of players and continuous action sets. Specifically, every player $i \in \mathcal{N} = \{1, \dots, N\}$ is assumed to select an action $x_i$ from a compact convex subset $\mathcal{K}_i$ of a finite-dimensional normed space $\mathcal{X}_i$; subsequently, every player receives a reward based on each player's individual objective and the action profile $x = (x_i; x_{-i}) \equiv (x_1, \dots, x_i, \dots, x_N)$ of all players' actions. In more detail, writing $\mathcal{K} := \prod_{i \in \mathcal{N}} \mathcal{K}_i$ for the game's action space, we assume that each player's reward is determined by an associated payoff (or utility) function $u_i: \mathcal{K} \to \mathbb{R}$. The tuple $\mathcal{G} = \mathcal{G}(\mathcal{N}, \mathcal{K}, u)$ will then be referred to as a continuous game.

In terms of regularity, we will assume throughout that the players' payoff functions are continuously differentiable, and we will write $v_i(x)$ for the individual payoff gradient of the $i$-th player, i.e.,

$$v_i(x) = \nabla_{x_i} u_i(x_i; x_{-i}) \quad (1)$$

or, putting all players together,

$$v(x) = (v_1(x), \dots, v_N(x)). \quad (2)$$

In the above, we are tacitly assuming that $u_i$ is defined on an open neighborhood of $\mathcal{K}$ in the ambient space $\mathcal{X} := \prod_{i \in \mathcal{N}} \mathcal{X}_i$ of the game; none of our results depend on this device, so we do not make this assumption explicit. We will also adopt the established convention of treating $v_i(x)$ as an element of the dual space $\mathcal{Y}_i := \mathcal{X}_i^*$ of $\mathcal{X}_i$. Finally, we will assume that $\mathcal{X}$ is endowed with the norm $\|x\|^2 = \sum_i \|x_i\|^2$ where, for ease of notation, we write $\|\cdot\|$ for the norm of each factor space $\mathcal{X}_i$ and rely on the context to resolve any ambiguities.

2.3. **Nash equilibria and monotonicity.** The most prevalent solution concept in game theory is that of a Nash equilibrium (NE). This is an action profile $x^* \in \mathcal{K}$ that is resilient to unilateral deviations, i.e.,

$$u_i(x_i^*; x_{-i}^*) \ge u_i(x_i; x_{-i}^*) \quad \text{for all } x_i \in \mathcal{K}_i \text{ and all } i \in \mathcal{N}. \quad (NE)$$

The set of Nash equilibria of $\mathcal{G}$ will be denoted in the sequel as $\mathcal{K}^* := \text{NE}(\mathcal{G})$.

By virtue of their definition, it is straightforward to check that Nash equilibria satisfy the Stampacchia variational inequality

$$\langle v(x^*), x - x^* \rangle \le 0 \quad \text{for all } x \in \mathcal{K}. \quad (\text{SVI})$$

As a result, finding a Nash equilibrium of a continuous game typically involves solving the Stampacchia problem (SVI). This observation forms the basis of an important link between game theory and optimization, cf. Facchinei and Pang [21], Facchinei and Kanzow [20], Laraki et al. [34], and references therein.

Now, starting with the seminal work of Rosen [50], much of the literature has focused on games that satisfy the diagonal concavity (DC) condition

$$\langle v(x') - v(x), x' - x \rangle \le 0 \quad \text{for all } x, x' \in \mathcal{K}. \quad (\text{DC})$$
Owing to the link between (DC) and the theory of monotone operators in optimization, games that satisfy (DC) are commonly referred to as monotone games.<fnref n="1" /> In particular, mirroring the corresponding terminology from convex analysis, we will say that $\mathcal{G}$ is:

(1) Strictly monotone if (DC) holds as a strict inequality when $x' \neq x$.

(2) Strongly monotone if there exists a positive constant $\mu > 0$ such that

$$ \langle v(x') - v(x), x' - x \rangle \leq -\mu \|x' - x\|^2 \quad \text{for all } x, x' \in \mathcal{K}. \quad (3) $$

Obviously, we have the inclusions “strongly monotone” $\subsetneq$ “strictly monotone” $\subsetneq$ “monotone”, mirroring the corresponding chain of inclusions “strongly concave” $\subsetneq$ “strictly concave” $\subsetneq$ “concave” for concave functions.

Examples of monotone games include Cournot oligopolies [40], Kelly auctions and Tullock competitions [42], signal covariance and power control problems in wireless communications [18, 38], atomic splittable congestion games in networks with parallel links [46, 57], and many other problems where online decision-making is the norm. For an extensive list of applications in different contexts, see Facchinei and Kanzow [20] and Scutari et al. [52].

### 3. THE LEARNING MODEL

To account for the possibility of exogenous variations in the game-theoretic setup of the previous section, we will assume that the players face a different stage game $\mathcal{G}_t$ at each decision opportunity. More explicitly, the envisioned sequence of play unfolds as follows:

(1) At each stage $t = 1, 2, \dots$, every agent $i \in \mathcal{N}$ selects an action $X_{i,t} \in \mathcal{K}_i$.

(2) Each player receives their associated reward based on $\mathcal{G}_t$, and they observe – or otherwise construct – an estimate $V_{i,t} \in \mathcal{Y}_i$ of their individual payoff gradients.

(3) Subsequently, players update their actions and the process repeats.

The core ingredients of the above framework are (a) the sequence of stage games $\mathcal{G}_t$ encountered by the players; (b) the sequence of gradient signals $V_{i,t} \in \mathcal{Y}_i$ observed (or inferred) at each stage; and (c) the way that players update their actions as a function of the observed information. We discuss each of these elements in detail below.

#### 3.1. The stage game sequence.
The only blanket assumption that we will make for the sequence of stage games $\mathcal{G}_t$ is that the players’ payoff functions are Lipschitz continuous and smooth. More precisely, we will posit the following requirement for the players’ t-th stage payoff field $v_t(x) = (v_{i,t}(x))_{i \in \mathcal{N}}$:

**Assumption 1.** The game’s payoff functions are $C^2$-smooth; in particular, there exist constants $G_i, L_i > 0$ such that

$$ \|v_{i,t}(x)\|_{*} \leq G_i \quad (4a) $$

$$ \|v_{i,t}(x') - v_{i,t}(x)\|_{*} \leq L_i \|x' - x\| \quad (4b) $$

for all $t = 1, 2, \dots$, and all $i \in \mathcal{N}$, $x, x' \in \mathcal{K}$.

<footnote n="1">¹More precisely, Rosen [50] uses the name diagonal strict concavity (DSC) for a weighted variant of (DC) which holds as a strict inequality when $x' \neq x$. Hofbauer and Sandholm [27] use the term “stable” to refer to a class of population games that satisfy a condition similar to (DC), while Sandholm [51] and Sorin and Wan [57] respectively call such games “contractive” and “dissipative”. We use the term “monotone” throughout to underline the connection of (DC) with operator theory and variational inequalities.</footnote>
For posterity, we will also write $G := \max_i G_i$ and $L_i := \max_i L_i$. Beyond this mild regularity assumption, the sequence of stage games is assumed arbitrary. For instance, the evolution of $G_t$ could be random (i.e., $G_t$ could be determined by some randomly drawn parameter $\theta_t$ at each stage), it could be governed by an underlying (hidden) Markov chain model, etc. In particular, we do not assume that the stage game $G_t$ is revealed to the players before choosing an action: from their individual viewpoint, the players are involved in a repeated decision process where the choice of an action returns a reward, but they have no knowledge of the game generating this reward. This “agnostic” approach is motivated by the fact that the standard rationality postulates of game theory (full rationality, common knowledge of rationality, etc.) are not satisfied in many cases of practical interest. We briefly discuss two concrete examples of this framework below:

**Example 3.1 (Repeated Kelly auctions).** Consider a Kelly auction where a splittable resource (advertising time on a website, a catch of fish in a fish market, etc.) is auctioned off, day after day, to a set of $N$ buyers [31, 65]. In more detail, each player can place a monetary bid $x_i \in [0, b_i]$ to acquire a unit of said resource, up to the player’s total budget $b_i$. Then, once all bids are in, the resource is allocated proportionally to each player’s bid, i.e., the $i$-th player gets a fraction $\rho_i = x_i/[c + \sum_{j \in N} x_j]$ of the auctioned resource (with $c > 0$ denoting an “entry barrier” for participating in the auction). Thus, if $g_{i,t}$ denotes the marginal gain that the $i$-th player acquires per resource unit, the player’s prorated utility at the $t$-th epoch will be

$$u_{i,t}(x_i; x_{-i}) = \frac{g_{i,t}x_i}{c + \sum_{j \in N} x_j} - x_i. \qquad (5)$$

Clearly, the players’ utility functions evolve as a function of the intrinsic value $g_{i,t}$ associated to a unit of the auctioned resource. Since this value may be subject to arbitrary exogenous fluctuations (for instance, depending on the traffic coming to the website at any given time in the advertising example), we obtain a time-varying game as above. §

**Example 3.2 (Power control).** As another example, consider $N$ wireless users transmitting a stream of packets to a common receiver over a shared wireless channel [64]. If the channel gain for the $i$-th user at the $t$-th frame is $g_{i,t}$ and the user transmits with power $p_i \in [0, P_{\max}]$, the user’s information transmission rate is given by the celebrated Shannon formula

$$R_{i,t}(p_i; p_{-i}) = \log \left( 1 + \frac{g_{i,t}p_i}{\sigma + \sum_{j \neq i} g_{j,t}p_j} \right), \qquad (6)$$

where $\sigma > 0$ denotes the ambient noise in the channel [56]. Since the users’ channel gains evolve over time (e.g., due to fading, user mobility, or other fluctuations in the wireless medium), we obtain a time-varying game where each user seeks to maximize their individual communication rate. §

**3.2. The feedback signal.** The second basic ingredient of our model is the feedback available to the players after choosing an action. In tune with the limited information setting outlined above, we only posit that, at each stage $t = 1, 2, \dots$, every player $i \in \mathcal{N}$ receives – or otherwise constructs – a “gradient signal” $V_{i,t} \in \mathcal{Y}_i$. Analytically, this signal will be treated as if generated from a stochastic first-order oracle (SFO), i.e., an abstract mechanism that provides an estimate of each player’s individual payoff gradient at the chosen action profile. Specifically, if called at $X_t = (X_{1,t}, \dots, X_{N,t}) \in \mathcal{K}$, we assume that $V_{i,t}$ is of the form

$$V_{i,t} = v_{i,t}(X_t) + Z_{i,t} \qquad (SFO)$$

where the “observational error” $Z_{i,t}$ captures all sources of uncertainty in the received input.
To differentiate further between “random” (zero-mean) and “systematic” (non-zero-mean) errors in $V_{i,t}$, it will be convenient to decompose the error process $Z_{i,t}$ as

$$Z_{i,t} = U_{i,t} + b_{i,t} \qquad (7)$$

where $U_{i,t}$ is zero-mean and $b_{i,t}$ denotes the mean of $Z_{i,t}$. Formally, writing $\mathcal{F}_t = \sigma(X_1, \dots, X_t)$ for the natural filtration of $X_t$, we set

$$b_{i,t} = \mathbb{E}[Z_{i,t} | \mathcal{F}_t] \quad \text{and} \quad U_{i,t} = Z_{i,t} - b_{i,t} \qquad (8)$$

so, by definition, $\mathbb{E}[U_{i,t} | \mathcal{F}_t] = 0$. In this way, the oracle feedback received by each player $i \in \mathcal{N}$ can be classified according to the following statistics:

$$(1) \text{ Bias:} \qquad \|b_{i,t}\|_* \le B_{i,t}. \qquad (9a)$$

$$(2) \text{ Variance:} \qquad \mathbb{E}[\|U_{i,t}\|_*^2 | \mathcal{F}_t] \le \sigma_{i,t}^2. \qquad (9b)$$

$$(3) \text{ Second moment:} \qquad \mathbb{E}[\|V_{i,t}\|_*^2 | \mathcal{F}_t] \le M_{i,t}^2. \qquad (9c)$$

Finally, to simplify notation later on, we will also consider the “signal plus noise” error bound

$$S_{i,t}^2 = M_{i,t}^2 + \sigma_{i,t}^2. \qquad (9d)$$

In the above, $B_{i,t}$, $\sigma_{i,t}$ and $M_{i,t}$ are to be construed as deterministic upper bounds on the bias, variance, and magnitude of the oracle signal $V_{i,t}$ that player $i \in \mathcal{N}$ received at time $t$. We will also assume throughout that $B_{i,t}$ is non-increasing while $\sigma_{i,t}$ and $M_{i,t}$ are non-decreasing. Finally, in obvious notation, we will write $V_t$, $b_t$, $U_t$ and so forth for the corresponding profiles $V_t = (V_{i,t})_{i \in \mathcal{N}}$ and the like.

*Remark.* To streamline our presentation, we will first present our results in a model-agnostic manner, i.e., without specifying the origins of the oracle model (SFO); subsequently, in Section 5, we provide an explicit construction of such an oracle from payoff-based observations, and we discuss in detail what this entails for our analysis and results. §

### 3.3. Learning via mirror descent.
The last element of the players’ learning process concerns the way that players update their actions based on the received feedback. For concreteness, we will focus throughout on the widely used family of algorithms known as mirror descent (MD), which posits that players updates their actions by taking a “proximal” gradient step from their current action.<fnref n="2" /> Formally, this can be modeled via the basic recursion

$$X_{i,t+1} = P_i(X_{i,t}; \gamma_{i,t} V_{i,t}) \qquad (MD)$$

where:

1. $t = 1, 2, \dots$ denotes the stage of the process.

2. $X_{i,t}$ denotes the action chosen by player $i$ at stage $t$.

3. $V_{i,t}$ is the oracle signal of player $i$ at stage $t$.

4. $\gamma_{i,t} > 0$ is a player-specific step-size sequence (assumed non-increasing).

5. $P_i$ denotes the “prox-mapping” of player $i \in \mathcal{N}$ (see below for a detailed definition).

<footnote n="2">²The terminology “descent” alludes to the fact that (MD) was originally studied in the context of convex minimization (as opposed to reward maximization). We should also mention here that “mirror descent” is sometimes used synonymously with the popular “follow the regularized leader” (FTRL) protocol of Shalev-Shwartz and Singer [55]. The two methods coincide in linear problems, but not otherwise; in general, FTRL requires access to a best-response oracle, so it is beyond the scope of this paper.</footnote>
<table><thead><tr><th>Algorithm 1: Learning via mirror descent</th><th>[player indices suppressed]</th></tr></thead><tbody><tr><td>Require: prox-mapping <i>P</i>, step-size <i>γ</i><sub>t</sub> > 0</td><td></td></tr><tr><td>1: initialize <i>X</i><sub>1</sub> ← arg min <i>h</i></td><td># initialization</td></tr><tr><td>2: for <i>t</i> = 1, 2, ..., do</td><td></td></tr><tr><td>3: play <i>X</i><sub><i>t</i></sub> ∈ <i>K</i></td><td># play action</td></tr><tr><td>4: get gradient signal <i>V</i><sub><i>t</i></sub></td><td># get feedback</td></tr><tr><td>5: set <i>X</i><sub><i>t</i>+1</sub> ← <i>P</i>(<i>X</i><sub><i>t</i></sub>; <i>γ</i><sub><i>t</i></sub><i>V</i><sub><i>t</i></sub>)</td><td># update action</td></tr><tr><td>6: end for</td><td></td></tr></tbody></table>

For a pseudocode implementation from the viewpoint of a generic player, see Algorithm 1.

Methods based on mirror descent have received intense scrutiny ever since the pioneering work of Nemirovski and Yudin [43]; for an appetizer, see [4, 13, 39, 41, 44, 54, 63] and references therein. For intuition, the archetypal example of the method is based on the Euclidean prox-mapping

$$ \mathcal{P}(x; y) = \Pi_C(x + y) = \arg \min_{x' \in C} \{\|x + y - x'\|_2^2\} = \arg \min_{x' \in C} \{\langle y, x - x' \rangle + \frac{1}{2} \|x' - x\|_2^2\} \quad (10) $$

where $\Pi_C$ denotes the closest-point projection onto a given convex set $C$. Going beyond this familiar example, the key novelty of mirror descent is to replace the quadratic term in (10) by the so-called *Bregman divergence*

$$ D(x', x) = h(x') - h(x) - \langle \nabla h(x), x' - x \rangle, \quad (11) $$

induced by a “distance-generating function” $h$ on $C$. This function plays the role of the squared Euclidean norm in (14) and, following Juditsky et al. [29], we define it as follows:

**Definition 1.** Let $C$ be a compact convex subset of $\mathcal{X} \cong \mathbb{R}^d$. A convex function $h: \mathcal{X} \to \mathbb{R} \cup \{\infty\}$ is said to be a *distance-generating function* (DGF) on $C$ if

(1) $h$ is continuous and supported on $C$, i.e., $\text{dom } h := \{x \in \mathcal{X} : h(x) < \infty\} = C$.

(2) $h$ is $K$-strongly convex relative to $\|\cdot\|$ on $C$, i.e.,

$$ h(\lambda x + (1 - \lambda)x') \leq \lambda h(x) + (1 - \lambda)h(x') - \frac{1}{2} K \lambda (1 - \lambda) \|x' - x\|^2 \quad (12) $$

for all $x, x' \in C$ and all $\lambda \in [0, 1]$.

(3) The subdifferential $\partial h$ of $h$ admits a *continuous selection*, i.e., there exists a continuous mapping $\nabla h: \text{dom } \partial h \to \mathcal{Y}$ such that $\nabla h(x) \in \partial h(x)$ for all $x \in \text{dom } \partial h$.<fnref n="3" />

For conclusion, given a DGF $h$ on $C$, we will refer to $C_h := \text{dom } \partial h$ as the *prox-domain* of $h$. The *Bregman divergence* $D: C_h \times C \to \mathbb{R}$ induced by $h$ is then given by (11), and the associated *prox-mapping* $\mathcal{P}: C_h \times \mathcal{Y} \to C$ is defined as

$$ \mathcal{P}(x; y) = \arg \min_{x' \in C} \{\langle y, x - x' \rangle + D(x', x)\} \quad \text{for all } x \in C_h, y \in \mathcal{Y}. \quad (13) $$

Finally, we say that $h$ is Lipschitz if $\sup_{x \in C_h} \|\nabla h(x)\|_* < \infty$. §

Throughout the sequel, we will assume that each player $i \in \mathcal{N}$ is endowed with their individual distance-generating function $h_i: K_i \to \mathbb{R}$. In obvious notation, we will also write $K_i$ for the strong convexity modulus of $h_i$, $K_{h_i}$ for its prox-domain, $D_i: K_i \times K_{h_i} \to \mathbb{R}$ for the associated Bregman divergence, and $\mathcal{P}_i: K_{h_i} \times \mathcal{Y}_i \to K_i$ for the induced prox-mapping. For concreteness, we provide two standard examples below:

<footnote n="3">³We recall here that the subdifferential $\partial h$ of $h$ at $x$ is defined as $\partial h(x) = \{y \in \mathcal{Y} : h(x') \geq h(x) + \langle y, x' - x \rangle$ for all $x' \in \mathcal{X}\}$. The notation $\text{dom } \partial h := \{x \in \text{dom } h : \partial h(x) \neq \emptyset\}$ stands for the domain of subdifferentiability of $h$ and, by standard results in convex analysis, we have $\text{dom } h \subseteq \text{dom } \partial h \subseteq \text{dom } h$.</footnote>
**Example 3.3** (Euclidean projections). We begin by revisiting Euclidean projections on a compact convex subset $\mathcal{C}$ of $\mathbb{R}^d$. The corresponding DGF is $h(x) = \frac{1}{2}\|x\|^2$ for $x \in \mathcal{X}$, so $\mathcal{C}_h = \mathcal{C}$ and $\nabla h(x) = x$ for all $x \in \mathcal{C}$. Hence, the associated Bregman divergence is

$$D(x', x) = \frac{1}{2}\|x'\|_2^2 - \frac{1}{2}\|x\|_2^2 - \langle x, x' - x \rangle = \frac{1}{2}\|x' - x\|_2^2 \quad (14)$$

and the resulting recursion $x^+ = \Pi(x + \gamma v)$ is just a standard projected forward step. §

**Example 3.4** (Entropic regularization). Let $\mathcal{C} = \Delta_d := \{x \in \mathbb{R}_+^d : \sum_{j=1}^d x_j = 1\}$ denote the unit simplex of $\mathcal{X} = \mathbb{R}^d$. A very widely used distance-generating function for this geometry is the (negative) Gibbs-Shannon entropy $h(x) = \sum_{j=1}^d x_j \log x_j$ (with the standard notational convention $0 \cdot \log 0 = 0$). By inspection, the prox-domain of $h$ is $\mathcal{C}_h := \text{ri} \mathcal{C}$, and the resulting Bregman divergence is just the Kullback-Leibler (KL) divergence

$$D(x', x) = D_{\text{KL}}(x', x) := \sum_{j=1}^{d} x'_{j} \log \left( \frac{x'_{j}}{x_{j}} \right) \quad \text{for all } x \in \mathcal{C}_h, x' \in \mathcal{C}. \quad (15)$$

In turn, a standard calculation leads to the prox-mapping

$$\mathcal{P}(x; y) = \frac{(x_1 e^{y_1}, \dots, x_n e^{y_n})}{x_1 e^{y_1} + \dots + x_n e^{y_n}} \quad (16)$$

for all $x \in \mathcal{C}_h$, $y \in \mathcal{Y}$. The corresponding update rule $x^+ = \mathcal{P}(x; \gamma v)$ is widely known in optimization as entropic gradient descent [4, 32], and as “Hedge” (or exponential/multiplicative weights update) in game theory and online learning [2, 3, 24, 37, 67]. §

## 4. EQUILIBRIUM TRACKING AND CONVERGENCE ANALYSIS

We are now in a position to state our main results for the equilibrium tracking and convergence properties of (MD) in time-varying games. For concreteness, we will focus below on two distinct – and, to a large extent, complementary – regimes: a) when the sequence of stage games $\mathcal{G}_t$ converges to some limit game $\mathcal{G} \equiv \mathcal{G}_\infty$; and b) when $\mathcal{G}_t$ evolves over time without converging. In both cases, we will treat the process defining the time-varying game as a “black box” and we will not scruitinize its origins in detail; we do so in order to focus on the interplay between the variability of the sequence $\mathcal{G}_t$ and the induced sequence of play.

**4.1. Stabilization and convergence to equilibrium.** We begin with the case where the sequence of stage games stabilizes to some monotone limit game $\mathcal{G} \equiv \mathcal{G}(\mathcal{N}, \mathcal{K}, u)$. Formally, it will be convenient to characterize this stabilization in terms of the quantity

$$R_{i,t} = \max_{x \in \mathcal{K}} \|v_{i,t}(x) - v_i(x)\|_{*}, \quad (17)$$

and we will say that the sequence of games $\mathcal{G}_t$, $t = 1, 2, \dots$, converges to $\mathcal{G}$ if

$$\lim_{t \to \infty} R_{i,t} = 0 \quad \text{for all } i \in \mathcal{N}. \quad (18)$$

To state our equilibrium convergence result, we will require two further assumptions. The first is a technical “reciprocity condition” for the players’ DGF, namely

$$D(p, x_t) \to 0 \quad \text{whenever} \quad x_t \to p \quad (\text{RC})$$

for every sequence of actions $x_t \in \mathcal{K}_h$. This requirement is fairly standard in the trajectory analysis of mirror descent algorithms [4, 14] and, taken together with the strong convexity of $h$, it implies that $x_t \to p$ if and only if $D(p, x_t) \to 0$ (hence the name).<fnref n="4" /> In particular, if $h$ is Lipschitz, we have

$$D(p, x_t) \leq h(p) - h(x_t) + \|\nabla h(x_t)\|_{*} \|x_t - p\| = \mathcal{O}(\|x_t - p\|) \quad (19)$$

<footnote n="4">⁴Indeed, $D(p, x_t) = h(p) - h(x_t) - \langle \nabla h(x_t), p - x_t \rangle \geq (K/2)\|x_t - p\|^2$, so $x_t \to p$ whenever $D(p, x_t) \to 0$.</footnote>
so (RC) always holds in that case. A further easy check shows that Example 3.4 also satisfies this condition, so (RC) is not restrictive in this regard.

The second set of conditions concerns the players’ step-size sequence. First, we will assume throughout that

$$ \sum_{t=1}^{\infty} \gamma_{i,t} = \infty \quad \text{for all } i \in \mathcal{N}, \qquad (S1) $$

i.e., each player’s learning process cannot stop prematurely. Second, we will assume that the step-size policies of any two players $i, j \in \mathcal{N}$ are *mutually compatible* in the sense that

$$ \sum_{t=1}^{\infty} |\gamma_{i,t} - \lambda_{ij}\gamma_{j,t}| < \infty \quad \text{for some } \lambda_{ij} > 0. \qquad (S2) $$

Informally, the compatibility assumption (S2) means that the players’ step-size policies exhibit a comparable asymptotic behavior as $t \to \infty$, i.e., $\gamma_{i,t}/\gamma_{j,t} = \Theta(1)$ for all $i, j \in \mathcal{N}$. The rationale for this is fairly straightforward: if a player employs a step-size policy that vanishes much faster than that of all other players, this player would effectively become a “constant externality” in the time-scale of the other players. On that account, it would make more sense to consider convergence in a “reduced” game where this player has been effectively removed from the game – and so on, until only the “slower” time-scale players remain. Assumption (S2) rules out such cases and ensures that all players remain active throughout the horizon of play.

With all this in hand, we have the following equilibrium convergence result:

**Theorem 1.** Let $\mathcal{G}_t$ be a time-varying game converging to a strictly monotone game $\mathcal{G}$. Suppose further that each player $i \in \mathcal{N}$ runs Algorithm 1 with a DGF satisfying (RC) and a step-size policy satisfying (S1), (S2), and

$$ \sum_{t=1}^{\infty} \gamma_{i,t}(R_{i,t} + B_{i,t}) < \infty \quad \text{and} \quad \sum_{t=1}^{\infty} \gamma_{i,t}^2 S_{i,t}^2 < \infty. \qquad (S3) $$

Then, with probability 1, the sequence of realized actions $X_t$ converges to the (necessarily unique) Nash equilibrium $x^*$ of $\mathcal{G}$.

In particular, if the feedback and stabilization metrics $B_{i,t}$, $S_{i,t}$ and $R_{i,t}$ behave asymptotically as $B_{i,t} = \mathcal{O}(1/t^{b_i})$, $S_{i,t} = \mathcal{O}(t^{s_i})$ and $R_{i,t} = \mathcal{O}(1/t^{r_i})$ for some $b_i, s_i, r_i \ge 0$, we have the following immediate corollaries:

**Corollary 1.** With assumptions as above, if each player follows Algorithm 1 with $\gamma_{i,t} \propto 1/t^p$ for some $p > \max\{1 - r_i, 1 - b_i, 1/2 + s_i\}$, $p \le 1$, the induced sequence of play $X_t$ converges to Nash equilibrium with probability 1.

**Corollary 2.** If Algorithm 1 is run with perfect oracle feedback and assumptions as above, taking $p > \max_i r_i$ guarantees that $X_t$ converges to Nash equilibrium with probability 1.

To streamline our discussion, we postpone the proof of Theorem 1 until later in this section and we proceed below with some remarks.

**Learning in static games and stochastic approximation.** The special case $\mathcal{G}_t \equiv \mathcal{G}$ for all $t = 1, 2, \dots$ can be seen as learning in a repeated, *static* game. As we discussed in the introduction, this case has been extensively studied in the literature, usually via the so-called ODE method of stochastic approximation [6–8]. In this literature, convergence of a learning process is typically established by showing that an underlying “mean field” dynamical system converges, and then using a series of asymptotic pseudotrajectory (APT) approximation results to infer that the same applies to the discrete-time algorithm under study as well.
In this direction, the closest result to our own is the recent paper of Mertikopoulos and Zhou [39] where the authors showed that a specific, multi-agent version of Nesterov's [44] dual averaging algorithm converges to Nash equilibrium in static, strictly monotone games. However, there is a number of key obstacles that arise when trying to adapt the proof techniques of Mertikopoulos and Zhou [39] to our setting. First and foremost, the prox-mappings $P_i$ are, in general, discontinuous across different faces of $K_i$, so (MD) cannot be seen as the discretization of an ODE (consider for example the Euclidean case where $P_i$ is the closest-point projection to $K_i$). An approach based on the theory of differential inclusions (DIS) [7] could help overcome this obstacle but, even then, the exogenous dependence of $G_t$ on $t$ means that the DI approximation of the players' learning process would be likewise non-autonomous. Thus, given that there is no longer a well-defined continuous-time system to approximate, it is not possible to employ a dynamical systems approach as in [39].

Finally, we should also note that the use of player-specific step-size sequences compli-
cates the discretization landscape even further. In the stochastic approximation literature,
coordinate-specific step-sizes are usually treated within a multiple time-scales framework,
e.g., as in [11, 35, 36, 47]. However, in this case, the underlying ODE must also separate the
faster from the slower time-scales, which means that the players with the smaller step-sizes
end up being effectively removed from the game. This is an important part of the reason that
the literature on learning in static games has traditionally focused on learning algorithms
with the same step-size across players – and also an important reason that the stochastic
approximation approach of [39] does not apply in our setting.

**Step-size requirements and tuning.** In the literature on learning in games, a common choice for the step-size of iterative methods is the policy $\gamma_{i,t} \propto 1/t$, cf. Beggs [5], Bervoets et al. [9], Cominetti et al. [15], Coucheney et al. [16], Erev and Roth [19], Hofbauer and Sandholm [27], and references therein. In view of Corollary 1, if the players’ oracle feedback is unbiased and bounded in mean square (i.e., $b_i = \infty$, $s_i = 0$ for all $i \in \mathcal{N}$), this step-size policy guarantees convergence to a Nash equilibrium as long as the game stabilizes at a power law rate – i.e., provided that $R_t := \max_i R_{i,t} = \mathcal{O}(1/t^r)$ for some $r > 0$.<fnref n="5" /> In fact, if (MD) is run with $\gamma_{i,t} \propto 1/(t \log t)$, convergence is guaranteed even if the game stabilizes at a slower, sub-logarithmic rate $R_t = \mathcal{O}(1/(\log t)^\epsilon)$ for some $\epsilon > 0$.

The policies $\gamma_{i,t} \propto 1/t$ and $\gamma_{i,t} \propto 1/(t \log t)$ should be seen as conservative “fail-safes”: it stands to reason that, if more information about the asymptotic behavior of $R_{i,t}$ is available, a more aggressive step-size policy (as per Corollary 1) might be more efficient. Specifically, if we focus as above on the case where the players’ oracle feedback is unbiased and bounded in mean square ($b = \infty$, $s = 0$), the second-moment term $\sum_t \gamma_{i,t}^2 S_{i,t}^2$ will be subleading in (S3) relative to the stabilization error term $\sum_t \gamma_{i,t} R_{i,t}$ whenever $p \ge r_i$ for some $i \in \mathcal{N}$. Since the summability condition (S3) further requires $p > 1 - r_i$ for all $i \in \mathcal{N}$, this would suggest taking $p = \min_i r_i$ if $\min_i r_i > 1/2$, and $p$ larger than $1/2$ by an arbitrarily small amount otherwise.

By contrast, if no prior information on $R_{i,t}$ is available, it is not clear how to choose the
exponent $p$ in an optimal manner relative to the variability of $G_t$. In particular, since $r_i$
depends on the entire (infinite) tail of $R_{i,t}$, adaptive policies that rely on the (finite) history
of play up to time $t$ – e.g., in the spirit of Rakhlin and Sridharan [48] and Syrgkanis et al.
[60] – do not seem well-suited for this purpose. We are not aware of any way to circumvent
this difficulty in terms of almost sure convergence of the sequence of play.

<footnote n="5">⁵More generally, the policy $\gamma_{i,t} \propto 1/t$ guarantees convergence as long as the bias decays as $B_{i,t} = \mathcal{O}(1/t^{b_i})$ for some $b_i > 0$ and the variance grows at most sublinearly ($\sigma_{i,t}^2 = \mathcal{O}(t^{2s_i})$ for some $s_i < 1/2$).</footnote>
4.2. **Tracking Nash equilibria.** We now turn to the study of time-varying games that evolve without converging. In this case, any notion of convergence for $X_t$ is meaningless because there is no equilibrium state to converge to, either static or in the mean. As a result, we will focus instead on whether $X_t$ is capable of “tracking” the game’s set of Nash equilibria over a given horizon of play.

To that end, let $\mathcal{G}_t$ be a sequence of strongly monotone games, and consider the equilibrium tracking error

$$ \mathrm{err}(\mathcal{T}) := \sum_{t \in \mathcal{T}} \|X_t - x_t^*\|^2 = \sum_{t \in \mathcal{T}} \sum_{i \in \mathcal{N}} \|X_{i,t} - x_{i,t}^*\|^2 \quad (20) $$

where $x_t^*$ is the (unique) Nash equilibrium of $\mathcal{G}_t$ and $\mathcal{T} = [\tau_{\mathrm{start}}..\tau_{\mathrm{end}}]$, $\tau_{\mathrm{start}}, \tau_{\mathrm{end}} \in \mathbb{N}$, denotes the playing window of interest.<fnref n="6" /> By construction, if $\mathrm{err}(\mathcal{T})$ is small relative to $|\mathcal{T}| = \tau_{\mathrm{end}} - \tau_{\mathrm{start}}$, the sequence of chosen action $X_t$ will be close to equilibrium for most of the window of interest. However, if the variability of $\mathcal{G}_t$ (and, in particular, of the equilibrium $x_t^*$) is too high, it is not reasonable to expect a tracking error that grows sublinearly in $|\mathcal{T}|$, even in the single-player case.

To quantify this, we will also consider the game’s equilibrium variation (or drift) as

$$ V(\mathcal{T}) := \sum_{t \in \mathcal{T}} \|x_{t+1}^* - x_t^*\|, \quad (21) $$

where as before, $\mathcal{T} = [\tau_{\mathrm{start}}..\tau_{\mathrm{end}}]$ denotes the window of interest. For concision, if $\mathcal{T}$ is of the form $\mathcal{T} = [1..T]$, we will simply write $\mathrm{err}(T)$ and $V(T)$ instead of $\mathrm{err}(\mathcal{T})$ and $V(\mathcal{T})$ respectively. In this case, we will say that the equilibrium variation of $\mathcal{G}_t$ is *tame* if

$$ V(T) = o(T) \quad \text{as } T \to \infty \quad (22) $$

and we will seek to establish conditions under which Algorithm 1 guarantees $\mathrm{err}(T) = o(T)$ when (22) holds. Our main result in this direction is as follows:

**Theorem 2.** Let $\mathcal{G}_t$ be a sequence of strongly monotone games satisfying Assumption 1. Suppose further that each player $i \in \mathcal{N}$ runs Algorithm 1 with step-size $\gamma_{i,t} \propto t^{-p_i}$, $p_i \in (0, 1)$, a Lipschitz distance-generating function, and feedback of the form (SFO) with $B_{i,t} = O(1/t^{b_i})$ and $S_{i,t}^2 = O(t^{2s_i})$ for some $b_i, s_i \ge 0$, $i \in \mathcal{N}$. Then the players’ tracking error is bounded as

$$ \mathbb{E}[\mathrm{err}(T)] = O(T^{1-\min_i(p_i-2s_i)} + T^{1-\min_i b_i} + T^{\max_i p_i+\min_i(p_i-2s_i)} V(T)). \quad (23) $$

**Corollary 3.** Suppose that the players’ oracle feedback is unbiased and bounded in mean square ($b_i = \infty, s_i = 0$ for all $i \in \mathcal{N}$). If the equilibrium variation of the game is $V(T) = O(T^r)$ for some $r > 0$, Algorithm 1 enjoys the bound

$$ \mathbb{E}[\mathrm{err}(T)] = O(T^{1-p_{\min}} + T^{2p_{\max}+r}). \quad (24) $$

where $p_{\min} = \min_i p_i$ and $p_{\max} = \max_i p_i$. In particular, if each player runs Algorithm 1 with $\gamma_{i,t} \propto 1/t^{(1-r)/3}$, then

$$ \mathbb{E}[\mathrm{err}(T)] = O(T^{\frac{2+r}{3}}). \quad (25) $$

Theorem 2 is our basic equilibrium tracking result, so we proceed with some remarks:

<footnote n="6">⁶In games with multiple equilibria, the norm should be replaced by the Hausdorff distance of the corresponding equilibrium sets; we focus on strongly monotone games to avoid such complications.</footnote>
**Step-size requirements and tuning.** If the players' gradient oracle is unbiased and bounded in mean square ($b_i = \infty$ and $s_i = 0$ for all $i \in \mathcal{N}$), Corollary 3 shows that equilibrium tracking is possible as long as

$$p_i < \frac{1-r}{2} \quad \text{for all } i \in \mathcal{N}. \qquad (26)$$

Comparing this condition with the step-size requirements for equilibrium convergence (cf. Theorem 1 and Corollary 1), we may infer that equilibrium tracking is more lightweight in terms of prerequisites: specifically, since Theorem 2 does not require the step-size compatibility condition (S2), each player can pick $p_i$ independently of one another. The reason for this difference has to do with the fact that equilibrium tracking focuses on the players' average behavior over the horizon of play; by contrast, the convergence of the sequence of play depends on the entire tail of $\gamma_{i,t}$, so the asymptotic behavior of the players' step-size policies cannot be too different.

**Equilibrium tracking and dynamic regret minimization: similarities.** In our setup, the dynamic regret incurred by the $i$-th player up to time $T$ under the sequence of play $X_t \in \mathcal{K}$, $t = 1, 2, \dots$, can be defined as

$$\text{DynReg}_i(T) = \sum_{t=1}^{T} [u_{i,t}(\hat{x}_{i,t}; X_{-i,t}) - u_{i,t}(X_t)] = \sum_{t=1}^{T} [\tilde{u}_{i,t}(\hat{x}_{i,t}) - \tilde{u}_{i,t}(X_t)] \quad (27)$$

where $\tilde{u}_{i,t} := u_i(\cdot; X_{-i,t})$ denotes the effective payoff function encountered by player $i \in \mathcal{N}$ at stage $t$ given the chosen action profile $X_{-i,t}$ of all other players, and

$$\hat{x}_{i,t} \in \arg\max_{x_i \in \mathcal{K}_i} u_{i,t}(x_i; X_{-i,t}) = \arg\max_{x_i \in \mathcal{K}_i} \tilde{u}_{i,t}(x_i) \quad (28)$$

denotes the $i$-th player's best response to $X_{-i,t}$ in the game $\mathcal{G}_t$ (the latter assumed fixed as a sequence but otherwise arbitrary and unknown to the players). Obviously, if there are no other players in the game, $\hat{x}_t$ coincides with the Nash equilibrium of the $t$-th stage game against nature, so a natural question that arises is whether the equilibrium tracking guarantees of Theorem 2 can be related to a dynamic regret bound.

In this regard, a slight modification of the proof of Theorem 2 yields the following: if an agent with a convex compact action set $\mathcal{K}$ runs Algorithm 1 with step-size $\gamma_t \propto 1/t^p$ against a stream of concave – though not necessarily strongly concave – payoff functions $u_t: \mathcal{K} \to \mathbb{R}$ with drift $V(T)$, then

$$\mathbb{E}[\text{DynReg}(T)] = \mathcal{O}(T^{1+2s-p} + T^{1-b} + T^{2p-2s} V(T)). \quad (29)$$

In particular, if $V(T) = \mathcal{O}(T^r)$ and the player's oracle feedback is unbiased and bounded in mean square ($b = \infty$, $s = 0$), the choice $p = (1-r)/3$ guarantees

$$\mathbb{E}[\text{DynReg}(T)] = \mathcal{O}(T^{\frac{2+r}{3}}). \quad (30)$$

For a precise statement and proof, we refer the reader to Section 6.2.<fnref n="7" />

**Equilibrium tracking and dynamic regret minimization: differences.** Going back to the multi-agent case, the sequence $\hat{x}_t = (\hat{x}_{i,t})_{i \in \mathcal{N}}$ with $\hat{x}_{i,t}$ given by (28) may be very different from the Nash equilibrium sequence $x_t^*$: the former best responds to the actual sequence of play $X_t$, while the latter best responds to itself (so it depends only on $\mathcal{G}_t$ and is otherwise independent of $X_t$). As we saw above, this distinction is redundant in the single-player case, but it is crucial in the multi-agent one: the sequence $\hat{x}_t$ may vary rapidly even if $x_t^*$ is constant. For example, even if the sequence of base payoff functions $u_{i,t}$ does not depend on $t$

<footnote n="7">⁷The guarantee (29) is not a consequence of Theorem 2 because it concerns function values and it makes no strong concavity assumptions for the payoff functions faced by the agent; the proof, however, is similar.</footnote>
explicitly (i.e., $u_{i,t} \equiv u_i$ for all $t$), the effective payoff functions $\tilde{u}_{i,t} := u_i(\cdot; X_{-i,t})$ encountered individually by each agent depend on $t$ implicitly via $X_{-i,t}$.

This subtlety is also reflected on the strong monotonicity assumption in Theorem 2 which invites the question whether the bound (23) is tight. To wit, when faced with a sequence of strongly concave payoff functions, Besbes et al. [10] showed that an adversary can always impose DynReg($T$) = $\Omega(V(T)^{1/2}T^{1/2})$. This bound is strictly better than the $O(T^{1-p} + V(T)T^{2p})$ guarantee of Corollary 3, suggesting that there may be room for improvement. Nevertheless, there are two important roadblocks to achieve this:

(1) First, in the single-agent case, the key to attaining faster regret minimization is the basic inequality

$$u_t(x_t^*) - u_t(x) \le \langle v_t(x), x_t^* - x \rangle - \frac{\mu}{2} \|x - x_t^*\|^2 \quad (31)$$

where $x_t^*$ denotes the (necessarily unique) maximizer of $u_t$. As a result, the growth of Gap($T$) – which is driven by gradient terms of the form $\langle v_t(X_t), x_t^* - X_t \rangle$ – is mitigated by the quadratic correction terms: by balancing these two terms, it is possible to obtain sharper bounds for DynReg($T$) when each $u_t$ is strongly concave. On the other hand, in a multi-agent, game-theoretic setting, (31) becomes

$$u_{i,t}(x_{i,t}^*; x_{-i}) - u_{i,t}(x) \le \langle v_{i,t}(x), x_{i,t}^* - x_i \rangle - \frac{\mu}{2} \|x_i - x_{i,t}^*\|^2 \quad (32)$$

where $x_t^*$ now denotes the (necessarily unique) Nash equilibrium of the strongly monotone stage game $G_t \equiv G_t(\mathcal{N}, \mathcal{K}, u_t)$. Arguing as in the single-agent setting would indeed yield a sharper bound on the quantity

$$\sum_{t \in T} \sum_{i \in \mathcal{N}} [u_{i,t}(x_{i,t}^*; X_{-i,t}) - u_{i,t}(X_t)] \quad (33)$$

but, in general, the minimization of this quantity does not provide a certificate that $X_t$ is in any way close to equilibrium. In particular, in contrast to the single-agent case, (32) could be either positive or negative, so it cannot act as a merit function for tracking an evolving equilibrium.

(2) Second, the optimal static regret minimization rate in strongly convex problems is attained when $\gamma_t \propto 1/t$. However, Besbes et al. [10] provide a counterexample where this step-size policy produces linear dynamic regret. In view of this, achieving an $O(V(T)^{1/2}T^{1/2})$ dynamic regret minimization rate would seem to require a different approach and/or assumptions – for example, an adaptive policy in the spirit of Jadbabaie et al. [28] in the case of perfect gradient feedback.

We mention the above to emphasize that bounding the equilibrium tracking error $\text{err}(T)$ is significantly different than bounding the dynamic regret of an individual agent in the unilateral setting (even though the obtained guarantees look similar). It is reasonable to conjecture that the bound (23) may be improved in games that admit a strongly concave potential function, but the general case seems considerably more difficult.

**Legendre DGFs.** We should also note that the reciprocity condition (RC) has been replaced in the statement of Theorem 2 by the stronger requirement $\sup_{x_i} \|\nabla h_i(x_i)\|_* < \infty$ which rules out Legendre-like DGFs (such as the entropic setup of Example 3.4). This condition is needed in Proposition 3, which requires a finite Bregman diameter $D_i := \sup_{x_i, x'_i} D_i(x_i, x'_i)$ to bound the “regret-like” quantity $\sum_{t=1}^T \langle v_{i,t}(X_t), x_i - X_{i,t} \rangle$. Orabona and Pál [45] recently showed that (MD) may incur linear regret when run with a variable step-size in problems with infinite Bregman diameter, so this requirement is not an artifact of the analysis.
That being said, there are several ways to overcome this hurdle: First, the players could run (MD) with a constant step-size over windows of a specified length and use a restart mechanism to achieve a sublinear equilibrium tracking error; this approach was proposed by Besbes et al. [10] for the minimization of dynamic regret and we discuss it in more detail in Section 6.2. Another way is to add an “anchoring term” in the definition of the prox-mapping $P_i$ and play the so-called dual-stabilized mirror descent policy

$$X_{i,t+1} = \arg \min_{x_i \in \mathcal{K}_i} \{ \gamma_{i,t} \langle V_{i,t}, X_{i,t} - x_i \rangle + D_i(x_i, X_{i,t}) + (\gamma_{i,t+1}^{-1} - \gamma_{i,t}^{-1}) D_i(x_i, X_{i,1}) \} \quad (\text{DS-MD})$$

This policy was introduced by Fang et al. [22] who showed that (DS-MD) achieves sublinear regret even in domains with an infinite Bregman diameter. Finally, another – and arguably simpler – approach is to switch to the dual averaging policy of Nesterov [44] which instead prescribes

$$X_{i,t+1} = \arg \max_{x_i \in \mathcal{K}_i} \left\{ \sum_{s=1}^{t} \langle V_{i,s}, x_i \rangle - \gamma_{i,t} h_i(x_i) \right\}. \quad (\text{DA})$$

This algorithm has the advantage of attaining order-optimal regret guarantees with the Bregman diameter $D_i$ replaced by the range $\mathcal{R}_i := \max h_i - \min h_i$ of $h_i$ (which is always finite since $\mathcal{K}_i$ is compact and the domain of $h_i$ contains $\mathcal{K}_i$). Either of these algorithmic tweaks would ultimately yield a sublinear tracking error in domains with an infinite Bregman diameter, but the details lie beyond the scope of our work so we do not discuss them here.

### 4.3. Proof of Theorem 1.
The rest of this section is devoted to proving the results stated above, starting with the proof of Theorem 1. The first key step in this direction is the definition of a suitable “energy-like” function that is – on average and up to small, second-order errors – decreasing along the trajectory of play $X_t$. In the analysis of mirror descent algorithms, this role is usually played by the Bregman divergence relative to the target point under study (in our case, the Nash equilibrium of $\mathcal{G}$). However, because each player $i \in \mathcal{N}$ now learns at a different pace (as determined by their individual step-size policy $\gamma_{i,t}$), the definition of a suitable energy function for Algorithm 1 is not as straightforward.

To that end (and with a fair amount of hindsight), we begin by introducing the player-specific weights

$$\lambda_i = \left( \prod_{j \in \mathcal{N}} \lambda_{ij} \right)^{1/N} \quad \text{for all } i \in \mathcal{N}, \quad (34)$$

with $\lambda_{ij} > 0$, $i, j \in \mathcal{N}$, given by the mutual compatibility condition (S2). As we show below, these weights enjoy a decomposition property that is key for the sequel:

**Lemma 1.** Suppose that $\gamma_{i,t}$ satisfies (S1) and (S2). Then $\lambda_{ij} = \lambda_i/\lambda_j$ for all $i, j \in \mathcal{N}$.

*Proof.* Our proof relies on the two intermediate claims below:

**Claim 1:** The weights $\lambda_{ij}$ are uniquely defined. Indeed, suppose that (S2) holds also with $\lambda'_{ij} \neq \lambda_{ij}$ for some $i, j \in \mathcal{N}$. Then, for all $t = 1$, we have:

$$|\lambda_{ij} - \lambda'_{ij}|\gamma_{j,t} = |\lambda_{ij}\gamma_{j,t} - \lambda'_{ij}\gamma_{j,t}| \leq |\gamma_{i,t} - \lambda_{ij}\gamma_{j,t}| + |\gamma_{i,t} - \lambda'_{ij}\gamma_{j,t}| \quad (35)$$

so $|\lambda_{ij} - \lambda'_{ij}|\gamma_{j,t}$ is summable given that both $|\gamma_{i,t} - \lambda_{ij}\gamma_{j,t}|$ and $|\gamma_{i,t} - \lambda'_{ij}\gamma_{j,t}|$ are summable (by assumption). This contradicts (S1) so our claim follows.
**Claim 2:** The weights $\lambda_{ij}$ satisfy the chain rule $\lambda_{ik} = \lambda_{ij}\lambda_{jk}$ for all $i, j, k \in \mathcal{N}$. Indeed:

$$ 
\begin{aligned}
\sum_{t=1}^{\infty} |\gamma_{i,t} - \lambda_{ij}\lambda_{jk}\gamma_{k,t}| &= \sum_{t=1}^{\infty} |\gamma_{i,t} - \lambda_{ij}\gamma_{j,t} + \lambda_{ij}\gamma_{j,t} - \lambda_{ij}\lambda_{jk}\gamma_{k,t}| \\
&\leq \sum_{t=1}^{\infty} |\gamma_{i,t} - \lambda_{ij}\gamma_{j,t}| + \lambda_{ij}\sum_{t=1}^{\infty} |\gamma_{j,t} - \lambda_{jk}\gamma_{k,t}| \\
&< \infty
\end{aligned} 
\quad (36) $$

with the last inequality following from (S2). Our claim then follows from the definition of $\lambda_{ik}$ and our uniqueness claim above.

Thus, with these two claims in hand, we readily obtain

$$ \frac{\lambda_i}{\lambda_j} = \frac{(\prod_{k \in \mathcal{N}} \lambda_{ik})^{1/N}}{(\prod_{k \in \mathcal{N}} \lambda_{jk})^{1/N}} = \prod_{k \in \mathcal{N}} (\lambda_{ik} / \lambda_{jk})^{1/N} = \prod_{k \in \mathcal{N}} \lambda_{ij}^{1/N} = \lambda_{ij} \quad (37) $$

where, in the third step, we used the chain rule above to write $\lambda_{ij} = \lambda_{ik}/\lambda_{jk}$. This establishes our assertion and completes our proof. ■

This lemma shows that (S2) can be rewritten as $\sum_{t=1}^{\infty} |\gamma_{i,t}/\lambda_i - \gamma_{j,t}/\lambda_j| < \infty$, which in turn implies that $\lambda_i$ can be interpreted as the relative “learning speed” of player $i \in \mathcal{N}$. In view of this, we will consider the effective step-size

$$ \gamma_t = \frac{1}{N} \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}}{\lambda_i} \quad (38) $$

and the energy function

$$ E(x) = \sum_{i \in \mathcal{N}} \frac{D_i(x_i^*, x_i)}{\lambda_i} \quad (39) $$

where $x_i^* \in \mathcal{K}_i$ denotes the $i$-th component of the Nash equilibrium $x^*$ of $\mathcal{G}$. We then have the following quasi-descent inequality for $E$ under (MD):

**Lemma 2.** Suppose that each player $i \in \mathcal{N}$ runs Algorithm 1 with a step-size policy $\gamma_{i,t}$ satisfying (S1) and (S2). Then the iterates $E_t := E(X_t)$ of $E$ under $X_t$ enjoy the bound

$$ 
\begin{aligned}
E_{t+1} \leq E_t + \gamma_t \langle v(X_t), X_t - x^* \rangle &+ \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}}{\lambda_i} \langle r_{i,t} + Z_{i,t}, X_{i,t} - x_i^* \rangle + \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}^2}{2\lambda_i K_i} \|V_{i,t}\|_*^2 \\
&+ \frac{\max_i G_i \operatorname{diam}(\mathcal{K}_i)}{N} \sum_{i,j \in \mathcal{N}} \left| \frac{\gamma_{i,t}}{\lambda_i} - \frac{\gamma_{j,t}}{\lambda_j} \right|
\end{aligned} 
\quad (40) $$

with $r_{i,t} = v_{i,t}(X_t) - v_i(X_t)$.

*Proof.* By Lemma A.4 in Appendix A, the Bregman divergence $D_{i,t} := D_i(x_i^*, X_{i,t})$ satisfies the inequality

$$ D_{i,t+1} \leq D_{i,t} + \gamma_{i,t} \langle V_{i,t}, X_{i,t} - x_i^* \rangle + \frac{\gamma_{i,t}^2}{2K_i} \|V_{i,t}\|_*^2. \quad (41) $$

Therefore, with $V_{i,t} = v_{i,t}(X_t) + Z_{i,t} = v_i(X_t) + r_{i,t} + Z_{i,t}$ and $E_t = \sum_{i \in \mathcal{N}} \lambda_i^{-1} D_{i,t}$, we get

$$ E_{t+1} \leq E_t + \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}}{\lambda_i} \langle Z_{i,t} + r_{i,t}, X_{i,t} - x_i^* \rangle + \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}^2}{2\lambda_i K_i} \|V_{i,t}\|_*^2. \quad (42a) $$

$$ + \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}}{\lambda_i} \langle v_i(X_t), X_{i,t} - x_i^* \rangle \quad (42b) $$
so it suffices to upper bound the term (42b) of the above inequality. To that end, we have

$$
\begin{align}
(42b) &= \gamma_t \langle v(X_t), X_t - x^* \rangle + \sum_{i \in \mathcal{N}} \left( \frac{\gamma_{i,t}}{\lambda_i} - \gamma_t \right) \langle v_i(X_t), X_{i,t} - x_i^* \rangle \nonumber \\
&\le \gamma_t \langle v(X_t), X_t - x^* \rangle + \sum_{i \in \mathcal{N}} \left| \frac{\gamma_{i,t}}{\lambda_i} - \gamma_t \right| \cdot G_i \operatorname{diam}(\mathcal{K}_i) \nonumber \\
&= \gamma_t \langle v(X_t), X_t - x^* \rangle + \sum_{i \in \mathcal{N}} \frac{G_i \operatorname{diam}(\mathcal{K}_i)}{N} \left| \sum_{j \in \mathcal{N}} \left( \frac{\gamma_{i,t}}{\lambda_i} - \frac{\gamma_{j,t}}{\lambda_j} \right) \right| \nonumber \\
&\le \gamma_t \langle v(X_t), X_t - x^* \rangle + \frac{\max_i G_i \operatorname{diam}(\mathcal{K}_i)}{N} \sum_{i,j \in \mathcal{N}} \left| \frac{\gamma_{i,t}}{\lambda_i} - \frac{\gamma_{j,t}}{\lambda_j} \right|. \tag{43}
\end{align}
$$

Our claim then follows by substituting this bound back in (42). ■

The importance of the energy-like bound (40) lies in that the “drift term” $\gamma_t \langle v(X_t), X_t - x^* \rangle$ provides a leading negative contribution to $E_t$ (since $x^*$ is a Nash equilibrium of $\mathcal{G}$), while all other terms become vanishingly small over time. The proposition below formalizes this idea and shows that $E_t$ converges to some (random) finite value:

**Proposition 1.** Suppose that each player $i \in \mathcal{N}$ runs Algorithm 1 with a step-size $\gamma_{i,t}$ satisfying (S1), (S2) and (S3). Then $E_t$ converges (a.s.) to a random variable $E_\infty$ with $\mathbb{E}[E_\infty] < \infty$.

*Proof.* We begin by decomposing each player’s oracle signal as

$$V_{i,t} = v_{i,t}(X_t) + b_{i,t} + U_{i,t} = v_i(X_t) + r_{i,t} + b_{i,t} + U_{i,t} \quad (44)$$

and we set respectively

$$\rho_{i,t} = \langle r_{i,t}, X_{i,t} - x_i^* \rangle \qquad \rho_t = \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}}{\lambda_i \gamma_t} \rho_{i,t} \quad (45a)$$

$$\beta_{i,t} = \langle b_{i,t}, X_{i,t} - x_i^* \rangle \qquad \beta_t = \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}}{\lambda_i \gamma_t} \beta_{i,t} \quad (45b)$$

and

$$\psi_{i,t} = \langle U_{i,t}, X_{i,t} - x_i^* \rangle \qquad \psi_t = \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}}{\lambda_i \gamma_t} \psi_{i,t} \quad (45c)$$

with $\gamma_t$ given by (38) and $r_{i,t} = v_{i,t}(X_{i,t}) - v_i(X_t)$ defined as in Lemma 2. The energy inequality (40) then gives

$$E_{t+1} \le E_t + \gamma_t \langle v(X_t), X_t - x^* \rangle + \gamma_t (\rho_t + \beta_t + \psi_t) + \chi_t + \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}^2 \|V_{i,t}\|_*^2}{2\lambda_i K_i} \quad (46)$$

where we set

$$\chi_t = \frac{\max_i G_i \operatorname{diam}(\mathcal{K}_i)}{N} \sum_{i,j \in \mathcal{N}} \left| \frac{\gamma_{i,t}}{\lambda_i} - \frac{\gamma_{j,t}}{\lambda_j} \right|. \quad (47)$$

Therefore, conditioning on the history $\mathcal{F}_t$ of $X_t$ up to stage $t$ (inclusive) and taking expectations, we get

$$
\begin{align}
\mathbb{E}[E_{t+1} | \mathcal{F}_t] &\le \mathbb{E} \left[ E_t + \gamma_t \langle v(X_t), X_t - x^* \rangle + \gamma_t (\rho_t + \beta_t + \psi_t) + \chi_t + \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}^2 \|V_{i,t}\|_*^2}{2\lambda_i K_i} \middle| \mathcal{F}_t \right] \nonumber \\
&\le E_t + \gamma_t (\rho_t + \beta_t) + \chi_t + \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}^2}{2\lambda_i K_i} M_{i,t}^2 \tag{48}
\end{align}
$$
where we used the definition (9c) of $M_{i,t}$ and the fact that a) $x^*$ is a Nash equilibrium of $\mathcal{G}$ (so $\langle v(X_t), X_t - x^* \rangle \le 0$); b) $\rho_t$ and $\beta_t$ are both $\mathcal{F}_t$-measurable (by definition); and c) $\mathbb{E}[\psi_t | \mathcal{F}_t] = \mathbb{E}[U_t | \mathcal{F}_t], X_t - x^* = 0$.

To proceed, note that

$$ \rho_{i,t} = \langle r_{i,t}, X_{i,t} - x_i^* \rangle \le \|r_{i,t}\|_* \|X_{i,t} - x_i^*\| \le \mathrm{diam}(\mathcal{K}_i) R_{i,t} \quad (49) $$

and, similarly, $\beta_{i,t} \le \mathrm{diam}(\mathcal{K}_i)B_{i,t}$. The bound (48) may then be written as $\mathbb{E}[E_{t+1} | \mathcal{F}_t] \le E_t + \varepsilon_t$ where

$$ \varepsilon_t = \sum_{i \in \mathcal{N}} \left[ \frac{\gamma_{i,t}}{\lambda_i} \mathrm{diam}(\mathcal{K}_i) \cdot (R_{i,t} + B_{i,t}) + \frac{\gamma_{i,t}^2}{2\lambda_i K_i} M_{i,t}^2 \right]. \quad (50) $$

Consider now the auxiliary process $\zeta_t = E_{t+1} + \sum_{s=t+1}^{\infty} \varepsilon_s$. Taking expectations yields

$$ \mathbb{E}[\zeta_t | \mathcal{F}_t] \le E_t + \varepsilon_t + \sum_{s=t+1}^{\infty} \varepsilon_s = E_t + \sum_{s=t}^{\infty} \varepsilon_s = \zeta_{t-1}, \quad (51) $$

i.e., $\zeta_t$ is a supermartingale relative to $\mathcal{F}_t$. Moreover, since $\sum_{t=1}^{\infty} \varepsilon_t < \infty$ by (S3) and Lemma 1, we also get $\mathbb{E}[\zeta_t] \le \mathbb{E}[\zeta_1] < \infty$, i.e., $\zeta_t$ is bounded in $L^1$. Therefore, by Doob's (sub)martingale convergence theorem [25, Theorem 2.5], it follows that $\zeta_t$ converges almost surely to some random variable $\zeta$ that is itself finite (almost surely and in $L^1$). Since $E_t = \zeta_{t-1} - \sum_{s=t}^{\infty} \varepsilon_s$ and $\lim_{t \to \infty} \sum_{s=t}^{\infty} \varepsilon_s = 0$, we conclude that $E_t$ converges (a.s.) to $\zeta$ and our proof is complete. ■

Moving forward, our next result shows that we can extract a subsequence of $X_t$ that converges to a Nash equilibrium of the limit game $\mathcal{G}$:

**Proposition 2.** With assumptions as in Proposition 1, we have $\lim_{t \to \infty} \inf_t \|X_t - x^*\| = 0$ (a.s.).

*Proof.* We begin by showing that, for all $\varepsilon > 0$, the hitting time

$$ \tau_{\varepsilon} = \inf\{t \in \mathbb{N} : \|X_t - x^*\| \ge \varepsilon\} \quad (52) $$

is finite with probability 1; formally, we will show that the event $\mathcal{N}_{\varepsilon} = \{\tau_{\varepsilon} = \infty\}$ has $\mathbb{P}(\mathcal{N}_{\varepsilon}) = 0$ for all $\varepsilon > 0$.

To do so, fix some $\varepsilon > 0$ and let $c_{\varepsilon} = -\inf\{\langle v(x), x - x^* \rangle : \|x - x^*\| \ge \varepsilon\}$, so $c_{\varepsilon} > 0$ by the strict monotonicity of $\mathcal{G}$ and the fact that $v$ is continuous and $\mathcal{K}$ is compact. Then, with notation as in the proof of Proposition 1, telescoping the bound (46) yields

$$ E_{t+1} \le E_1 - c_{\varepsilon} \sum_{s=1}^{t} \gamma_s + \underbrace{\sum_{s=1}^{t} \gamma_s (\rho_s + \beta_s)}_{I_t} + \underbrace{\sum_{s=1}^{t} \chi_s}_{\Pi_t} + \underbrace{\sum_{s=1}^{t} \gamma_s \psi_s}_{\Pi_t} + \underbrace{\sum_{s=1}^{t} \sum_{i \in \mathcal{N}} \frac{\gamma_{i,s}^2}{2\lambda_i K_i} \|V_{i,s}\|_*^2}_{\Pi_t} \quad (53) $$

for all $t \le \tau_{\varepsilon}$. We now proceed to bound each of the underscored terms above:

1. First, for the term $I_t$, we have shown in the proof of Proposition 1 that

$$ \gamma_t(\rho_t + \beta_t) \le \sum_{i \in \mathcal{N}} \frac{\gamma_{i,t}}{\lambda_i} \mathrm{diam}(\mathcal{K}_i) \cdot (R_{i,t} + B_{i,t}) \quad (54) $$

so $\sum_{t=1}^{\infty} \gamma_t(\rho_t + \beta_t) < \infty$ by (S3). Condition (S2) further gives $\sum_{t=1}^{\infty} \chi_t < \infty$, so $I_t$ is uniformly bounded from above by $I_{\infty} := \sum_{t=1}^{\infty}[\gamma_t(\rho_t + \beta_t) + \chi_t] < \infty$.
2. For the noise term $\Pi_t = \sum_{s=1}^t \gamma_s \psi_s$, we have $\mathbb{E}[\psi_t | \mathcal{F}_t] = 0$, so $\Pi_t$ is a martingale. Furthermore, by (9b) and the step-size assumption (S3) of Theorem 1, we have

$$ \sum_{t=1}^{\infty} \gamma_{i,t}^2 \mathbb{E}[\psi_{i,t}^2 | \mathcal{F}_t] \le \sum_{t=1}^{\infty} \gamma_{i,t}^2 \|X_{i,t} - x_i^*\|^2 \mathbb{E}[\|U_{i,t}\|_*^2 | \mathcal{F}_t] \\ \le \mathrm{diam}(\mathcal{K}_i)^2 \sum_{t=1}^{\infty} \gamma_{i,t}^2 \sigma_{i,t}^2 < \infty. \quad (55) $$

In turn, this implies that $\sum_{t=1}^{\infty} \gamma_t^2 \mathbb{E}[\psi_t^2 | \mathcal{F}_t] < \infty$ so, by the law of large numbers for martingale difference sequences [25, Theorem 2.18], we conclude that $\sum_{s=1}^t \gamma_s \psi_s / \sum_{s=1}^t \gamma_s \to 0$ (a.s.).

3. Finally, for the last term, let $\Psi_{i,t} = \sum_{s=1}^t \gamma_{i,s}^2 \|V_{i,s}\|_*^2$ so $\Pi_t = \sum_{i \in \mathcal{N}} (2\lambda_i K_i)^{-1} \Psi_{i,t}$. We then have

$$ \mathbb{E}[\Psi_{i,t} | \mathcal{F}_t] = \mathbb{E}\left[\sum_{s=1}^{t-1} \gamma_{i,s}^2 \|V_{i,s}\|_*^2 + \gamma_{i,t}^2 \|V_{i,t}\|_*^2 \mid \mathcal{F}_t\right] \\ = \Psi_{i,t-1} + \gamma_{i,t}^2 \mathbb{E}[\|V_{i,t}\|_*^2 | \mathcal{F}_t] \ge \Psi_{i,t-1}, \quad (56) $$

i.e., $\Psi_{i,t}$ is a submartingale relative to $\mathcal{F}_t$ (recall that $V_t$ is generated after $X_t$ so it is not $\mathcal{F}_t$-measurable). Furthermore, by the law of total expectation, we also have

$$ \mathbb{E}[\Psi_{i,t}] = \mathbb{E}[\mathbb{E}[\Psi_{i,t} | \mathcal{F}_t]] \le \sum_{t=1}^{\infty} \gamma_{i,t}^2 M_{i,t}^2 < \infty. \quad (57) $$

This shows that $\Psi_{i,t}$ is uniformly bounded in $L^1$ so, by Doob's (sub)martingale convergence theorem [25, Theorem 2.5], it follows that $\Psi_{i,t}$ converges to some (almost surely finite) random variable $\Psi_{i,\infty}$ with $\mathbb{E}[\Psi_{i,\infty}] < \infty$. We thus conclude that $\Pi_t$ is likewise bounded from above by $\Pi_\infty = \sum_{i \in \mathcal{N}} (2\lambda_i K_i)^{-1} \Psi_{i,\infty} < \infty$ (a.s.).

Suppose now that $\mathbb{P}(\mathcal{N}_\varepsilon) = \mathbb{P}(\tau_\varepsilon = \infty) > 0$. Then there exists a realization of $X_t$ such that

$$ E_{t+1} \le E_t - \left[ c_\varepsilon - \frac{I_t + \Pi_t + \Pi_\infty}{\sum_{s=1}^t \gamma_s} \right] \cdot \sum_{s=1}^t \gamma_s \quad \text{for all } t = 1, 2, \dots \quad (58) $$

and, in addition, $(I_t + \Pi_t + \Pi_\infty) / \sum_{s=1}^t \gamma_s \to 0$ (since we have shown that this last event occurs w.p.1). However, by (S1), this gives $\lim_{t \to \infty} E_t = -\infty$, a contradiction which shows that $\tau_\varepsilon < \infty$ w.p.1 for all $\varepsilon > 0$. Hence, given that each $\mathcal{N}_{1/k}$ is a zero-probability event and there is a countable number thereof, we conclude that

$$ \begin{aligned} \mathbb{P}(\liminf_t \|X_t - x^*\| = 0) &= \mathbb{P}(\tau_{1/k} < \infty \text{ for all } k = 1, \dots, \infty) \\ &= \mathbb{P}(\bigcap_{k=1}^\infty \{\tau_{1/k} < \infty\}) = 1 - \mathbb{P}(\bigcup_{k=1}^\infty \mathcal{N}_{1/k}) = 1 \end{aligned} \quad (59) $$

and our proof is complete.

<figure><img src="image_3.png" /></figure>

With these two intermediate results at hand, we are finally in a position to prove Theorem 1:

*Proof of Theorem 1.* By Proposition 2, $X_t$ admits a (possibly random) subsequence $X_{t_k}$ such that $X_{t_k} \to x^*$ (a.s.). By the reciprocity condition (RC), this further implies that $\liminf_{t \to \infty} E_t = 0$ (a.s.). However, since $\lim_{t \to \infty} E_t$ exists (by Proposition 1), we conclude that

$$ \mathbb{P}\left(\lim_{t \to \infty} X_t = x^*\right) = \mathbb{P}\left(\lim_{t \to \infty} E_t = 0\right) = \mathbb{P}\left(\liminf_{t \to \infty} E_t = 0\right) = 1 \quad (60) $$

and our proof is complete.

<figure><img src="image_4.png" /></figure>
4.4. **Proof of Theorem 2.** We now proceed to prove the equilibrium tracking guarantees of Algorithm 1. To that end, given a sequence of action profiles $X_t \in \mathcal{K}$, $t = 1, 2, \dots$, and a window of interest $\mathcal{T} = [\tau_{\text{start}} \dots \tau_{\text{end}}]$, it will be useful to consider the gap functions

$$ \text{Gap}_{x_i}(\mathcal{T}) = \sum_{t \in \mathcal{T}} \langle v_{i,t}(X_t), x_i - X_{i,t} \rangle \qquad \text{Gap}_x(\mathcal{T}) = \sum_{i \in \mathcal{N}} \text{Gap}_{x_i}(\mathcal{T}) \qquad (61a) $$

and

$$ \text{Gap}_i(\mathcal{T}) = \max_{x_i \in \mathcal{K}_i} \text{Gap}_{x_i}(\mathcal{T}) \qquad \text{Gap}(\mathcal{T}) = \sum_{i \in \mathcal{N}} \text{Gap}_i(\mathcal{T}) \qquad (61b) $$

and we will likewise write $\text{Gap}_{x_i}(\mathcal{T})$, $\text{Gap}_x(\mathcal{T})$, etc. when the window of interest is of the form $\mathcal{T} = [1 \dots T]$. By the strong monotonicity of $\mathcal{G}_t$, we have $\mu\|X_t - x_t^*\|^2 \le \langle v_t(X_t), x_t^* - X_t \rangle$, so $\text{Gap}(\mathcal{T})$ will act as a surrogate for bounding the equilibrium tracking error $\text{err}(\mathcal{T})$ of Algorithm 1. In view of this, we begin with a technical bound for the gap under (MD):

**Proposition 3.** Suppose that player $i \in \mathcal{N}$ runs Algorithm 1 with step-size $\gamma_{i,t}$ and oracle feedback of the form (SFO). Then, for any window of the form $\mathcal{T} = [\tau_{\text{start}} \dots \tau_{\text{end}}]$, we have:

$$ \text{Gap}_{x_i}(\mathcal{T}) \le \sum_{t \in \mathcal{T}} \left( \frac{1}{\gamma_{i,t}} - \frac{1}{\gamma_{i,t-1}} \right) D_i(x_i, X_{i,t}) + \sum_{t \in \mathcal{T}} \langle Z_{i,t}, X_{i,t} - x_i \rangle + \frac{1}{2K_i} \sum_{t \in \mathcal{T}} \gamma_{i,t} \|V_{i,t}\|_*^2 \quad (62) $$

with the convention $\gamma_{i,\tau_{\text{start}}-1} = \infty$ in the sum above. In addition, if $\gamma_{i,t}$ is non-increasing, then

$$ \mathbb{E}[\text{Gap}_i(\mathcal{T})] \le \frac{2H_i(x_i)}{\gamma_{i,\tau_{\text{end}}} + 2 \text{diam}(\mathcal{K}_i)} \sum_{t \in \mathcal{T}} B_{i,t} + \frac{1}{2K_i} \sum_{t \in \mathcal{T}} \gamma_{i,t} S_{i,t}^2 \quad (63) $$

where $H_i(x_i) = \sup_{x'_i \in \mathcal{K}_{h_i}} D(x_i, x'_i)$.

*Proof.* We first focus on the pointwise bound (62). To that end, since $X_{i,t+1} = \mathcal{P}(X_{i,t}; \gamma_{i,t}V_{i,t})$ for all $t = 1, 2, \dots$, invoking Lemma A.4 with $Y_t \leftarrow \gamma_{i,t}V_{i,t}$ and $\alpha_t \leftarrow 1/\gamma_{i,t}$ yields

$$ \sum_{t \in \mathcal{T}} \langle V_{i,t}, x_i - X_{i,t} \rangle \le \sum_{t \in \mathcal{T}} \left( \frac{1}{\gamma_{i,t}} - \frac{1}{\gamma_{i,t-1}} \right) D_i(x_i, X_{i,t}) + \frac{1}{2K_i} \sum_{t \in \mathcal{T}} \gamma_{i,t} \|V_{i,t}\|_*^2. \quad (64) $$

By the feedback model (SFO), we have $V_{i,t} = v_{i,t}(X_t) + Z_{i,t}$ so

$$ \text{Gap}_{x_i}(\mathcal{T}) = \sum_{t \in \mathcal{T}} \langle v_{i,t}(X_t), x_i - X_{i,t} \rangle = \sum_{t \in \mathcal{T}} \langle V_{i,t}, x_i - X_{i,t} \rangle + \sum_{t \in \mathcal{T}} \langle Z_{i,t}, X_{i,t} - x_i \rangle. \quad (65) $$

Our claim then follows by adding (64) and (65).

For the bound (63), maximizing over $x_i \in \mathcal{K}_i$ in (62) and taking expectations, we get

$$ \mathbb{E}[\text{Gap}_i(\mathcal{T})] = \mathbb{E} \left[ \max_{x_i \in \mathcal{K}_i} \text{Gap}_{x_i}(\mathcal{T}) \right] \le \mathbb{E} \left[ \sum_{t \in \mathcal{T}} \left( \frac{1}{\gamma_{i,t}} - \frac{1}{\gamma_{i,t-1}} \right) D_i(x_i, X_{i,t}) \right] \quad (66a) $$

$$ + \frac{1}{2K_i} \sum_{t \in \mathcal{T}} \gamma_{i,t} \mathbb{E}[\|V_{i,t}\|_*^2] \quad (66b) $$

$$ + \mathbb{E} \left[ \max_{x_i \in \mathcal{K}_i} \sum_{t \in \mathcal{T}} \langle Z_{i,t}, X_{i,t} - x_i \rangle \right] \quad (66c) $$

With $\gamma_{i,t}$ non-increasing, the first two terms above are readily bounded as

$$ (66a) \le \sum_{t \in \mathcal{T}} \left( \frac{1}{\gamma_{i,t}} - \frac{1}{\gamma_{i,t-1}} \right) H_i(x_i) \le \frac{H_i(x_i)}{\gamma_{i,\tau_{\text{end}}} \quad (67a) $$
and

$$ (66b) \leq \frac{K_i}{2} \sum_{t \in \mathcal{T}} \gamma_{i,t} M_{i,t}^2 \qquad (67b) $$

so we are left to bound (66c). To that end, introduce the auxiliary process

$$ \tilde{X}_{i,t+1} = \mathcal{P}(\tilde{X}_{i,t}; -\gamma_{i,t} U_{i,t}) \qquad (68) $$

with $\tilde{X}_1 = X_1$. We then have

$$ \begin{aligned} \sum_{t \in \mathcal{T}} \langle Z_{i,t}, X_{i,t} - x_i \rangle &= \sum_{t \in \mathcal{T}} \langle Z_{i,t}, (X_{i,t} - \tilde{X}_{i,t}) + (\tilde{X}_{i,t} - x_i) \rangle \\ &= \sum_{t \in \mathcal{T}} \langle Z_{i,t}, X_{i,t} - \tilde{X}_{i,t} \rangle + \sum_{t \in \mathcal{T}} \langle b_{i,t}, \tilde{X}_{i,t} - x_i \rangle + \sum_{t \in \mathcal{T}} \langle U_{i,t}, \tilde{X}_{i,t} - x_i \rangle \quad (69) \end{aligned} $$

so it suffices to derive a bound for each of these terms. This can be done as follows:

(1) The first term of (69) does not depend on $x_i$, so we have

$$ \begin{aligned} \mathbb{E} \left[ \max_{x_i \in \mathcal{K}_i} \sum_{t \in \mathcal{T}} \langle Z_{i,t}, X_{i,t} - \tilde{X}_{i,t} \rangle \right] &= \sum_{t \in \mathcal{T}} \mathbb{E} \left[ \mathbb{E} \left[ \langle Z_{i,t}, X_{i,t} - \tilde{X}_{i,t} \rangle \mid \mathcal{F}_t \right] \right] \\ &= \sum_{t \in \mathcal{T}} \mathbb{E} \left[ \langle b_{i,t}, X_{i,t} - \tilde{X}_{i,t} \rangle \right] \leq \mathrm{diam}(\mathcal{K}_i) B_{i,t} \quad (70) \end{aligned} $$

where, in the last step, we used the definition (9a) of $B_{i,t}$ and the bound

$$ \langle b_{i,t}, X_{i,t} - \tilde{X}_{i,t} \rangle \leq \|X_{i,t} - \tilde{X}_{i,t}\|_{*} \leq \mathrm{diam}(\mathcal{K}_i) \|b_{i,t}\|_{*}. \qquad (71) $$

(2) The second term of (69) can be bounded in a similar way as

$$ \mathbb{E} \left[ \max_{x_i \in \mathcal{K}_i} \sum_{t \in \mathcal{T}} \langle b_{i,t}, \tilde{X}_{i,t} - x_i \rangle \right] \leq \mathbb{E}[\mathrm{diam}(\mathcal{K}_i) \|b_{i,t}\|_*] \leq \mathrm{diam}(\mathcal{K}_i) B_{i,t}. \qquad (72) $$

(3) Finally, for the last term, Lemma A.4 with $Y_t \leftarrow -\gamma_{i,t}U_{i,t}$ and $\alpha_t = 1/\gamma_{i,t}$ gives

$$ \begin{aligned} \sum_{t \in \mathcal{T}} \langle U_{i,t}, \tilde{X}_{i,t} - x_i \rangle &= \sum_{t \in \mathcal{T}} \alpha_{i,t} \langle -\gamma_{i,t} U_{i,t}, x_i - \tilde{X}_{i,t} \rangle \\ &\leq \sum_{t \in \mathcal{T}} \left( \frac{1}{\gamma_{i,t}} - \frac{1}{\gamma_{i,t-1}} \right) D(x_i, \tilde{X}_{i,t}) + \frac{1}{2K_i} \sum_{t \in \mathcal{T}} \gamma_{i,t} \|U_{i,t}\|_*^2. \end{aligned} \qquad (73) $$

Thus, after taking expectations and telescoping, we obtain

$$ \mathbb{E} \left[ \max_{x_i \in \mathcal{K}_i} \langle U_{i,t}, \tilde{X}_{i,t} - x_i \rangle \right] \leq \frac{H_i(x_i)}{\gamma_{i,t_{\mathrm{end}}}} + \frac{1}{2K_i} \sum_{t \in \mathcal{T}} \gamma_{i,t} \sigma_{i,t}^2. \qquad (74) $$

The bound (63) then follows by plugging back all of the above in (66c). ■

We are now in a position to prove our equilibrium tracking result. Our proof strategy will be to leverage the gap minimization guarantees of Algorithm 1 (as encoded in Proposition 3) together with a batch comparison idea due to Besbes et al. [10].

*Proof of Theorem 2.* For the sake of the analysis (and only the analysis), partition the horizon of play $\mathcal{T} = [1..T]$ in $m$ contiguous batches $\mathcal{T}_k$, $k = 1, \dots, m$, each of length $\Delta$ (except possibly the $m$-th one, which might be smaller). We will prove the error bound (23) by linking $\mathrm{err}(\mathcal{T}_k)$ to $\mathrm{Gap}(\mathcal{T}_k) = \sum_{i \in \mathcal{N}} \mathrm{Gap}_i(\mathcal{T}_k)$ for all $k = 1, \dots, m = \lceil T/\Delta \rceil$.

More explicitly, take the batch length to be of the form $\Delta = \lceil T^q \rceil$ for some constant $q \in [0, 1]$ to be determined later. In this way, the number of batches is $m = \lceil T/\Delta \rceil = \Theta(T^{1-q})$ and the $k$-th batch will be of the form $\mathcal{T}_k = [(k-1)\Delta + 1..k\Delta]$ for all $k = 1, \dots, m-1$ (the
value $k=m$ is excluded as the $m$-th batch might be smaller). Then, to bound the players' equilibrium tracking error within $\mathcal{T}_k$, the strong monotonicity property (3) for $\mathcal{G}_t$ gives

$$ \mu \|X_t - x_t^*\|^2 \leq \langle v_t(X_t), x_t^* - X_t \rangle = \langle v_t(X_t), \hat{x} - X_t \rangle + \langle v_t(X_t), x_t^* - \hat{x} \rangle \quad (75) $$

for every reference action profile $\hat{x} \in \mathcal{K}$ and all $t \in \mathcal{T}$. We thus obtain the batch bound

$$ \begin{aligned} \mu \operatorname{err}(\mathcal{T}_k) &= \mu \sum_{t \in \mathcal{T}_k} \|X_t - x_t^*\|^2 \le \sum_{t \in \mathcal{T}_k} \langle v_t(X_t), x_t^* - X_t \rangle \\ &= \sum_{t \in \mathcal{T}_k} \langle v_t(X_t), \hat{x} - X_t \rangle + \sum_{t \in \mathcal{T}_k} \langle v_t(X_t), x_t^* - \hat{x} \rangle \\ &\le \operatorname{Gap}(\mathcal{T}_k) + \sum_{t \in \mathcal{T}_k} \langle v_t(X_t), x_t^* - \hat{x} \rangle. \end{aligned} \quad (76) $$

To proceed, pick a batch-specific reference action $\hat{x}_k \in \mathcal{K}$ for each $k = 1, \dots, m$, and write

$$ C_k = \sum_{t \in \mathcal{T}_k} \langle v_t(X_t), x_t^* - \hat{x}_k \rangle, \quad (77) $$

for the last term of (76). A meaningful bound for $C_k$ can then be obtained by taking $\hat{x}_k$ to be the (unique) Nash equilibrium of the first game encountered in the batch $\mathcal{T}_k$, i.e., setting $\hat{x}_k = x_{\min \mathcal{T}_k}^*$. Doing this, we obtain the series of estimates:

$$ \begin{aligned} C_k &\le \sum_{t \in \mathcal{T}_k} \|v_t(X_t)\|_* \cdot \|x_t^* - \hat{x}_k\| & \{\text{by Cauchy-Schwarz}\} \\ &\le \sum_{t \in \mathcal{T}_k} G \|x_t^* - \hat{x}_k\| & \{\text{by Assumption 1}\} \\ &\le G \Delta \max_{t \in \mathcal{T}_k} \|x_t^* - \hat{x}_k\| & \{\text{term-by-term bound}\} \\ &\le G \Delta \sum_{t \in \mathcal{T}_k} \|x_{t+1}^* - x_t^*\| & \{\text{by definition of } \hat{x}_k\} \\ &= G \Delta V(\mathcal{T}_k). \end{aligned} \quad (78) $$

Thus, plugging everything back in (76) and summing over all batches $k = 1, \dots, m$, we get the total bound

$$ \mathbb{E}[\operatorname{err}(T)] \le \frac{1}{\mu} \mathbb{E}[\operatorname{Gap}(T)] + \frac{G\Delta}{\mu} \operatorname{V}(T). \quad (79) $$

With this estimate in hand, let $\mathcal{D}_i := \sup_{x_i, x'_i} \mathcal{D}_i(x_i, x'_i) = \max_{x_i \in \mathcal{K}_i} H_i(x_i)$, so $\mathcal{D}_i < \infty$ by Lemma A.5. Then, with $\gamma_{i,t}$ decreasing, summing the second part of Proposition 3 over all $i \in \mathcal{N}$ yields

$$ \begin{aligned} \sum_{k=1}^{m} \mathbb{E}[\operatorname{Gap}(\mathcal{T}_k)] &\le \sum_{i \in \mathcal{N}} \left[ \sum_{k=1}^{m} \frac{2\mathcal{D}_i}{\gamma_{i,k}\Delta} + 2 \operatorname{diam}(\mathcal{K}_i) \sum_{t=1}^{T} B_{i,t} + \frac{1}{2K_i} \sum_{t=1}^{T} \gamma_{i,t} S_{i,t}^2 \right] \\ &= \mathcal{O} \left( \Delta^{\max_i p_i} \sum_{k=1}^{m} k^{\max_i p_i} p_i + \sum_{t=1}^{T} t^{-\min_i b_i} + \sum_{t=1}^{T} t^{-\min_i (p_i - 2s_i)} \right) \\ &= \mathcal{O} \left( \Delta^{\max_i p_i} m^{1+\max_i p_i} + T^{1-\min_i b_i} + T^{1-\min_i (p_i - 2s_i)} \right) \end{aligned} \quad (80) $$

where, in the second line, we used the fact that $\gamma_{i,t} = \Theta(1/t^{p_i})$. Since $\Delta = \mathcal{O}(T^q)$ and $m = \mathcal{O}(T/\Delta) = \mathcal{O}(T^{1-q})$, we get

$$ \Delta^{\max_i p_i} m^{1+\max_i p_i} = \mathcal{O}(T^q \max_i p_i T^{(1-q)(1+\max_i p_i)}) = \mathcal{O}(T^{1+\max_i p_i - q}) \quad (81) $$
In turn, this yields the error bound

$$ \mathbb{E}[\text{err}(T)] = \mathcal{O}\left(T^{1+\max_i p_i - q} + T^{1-\min_i b_i} + T^{1-\min_i(p_i - 2s_i)} + T^q V(T)\right) \quad (82) $$

so the guarantee (23) follows by setting $q = \max_i p_i + \min_i(p_i - 2s_i)$.

<figure><img src="image_5.png" /></figure>

## 5. LEARNING WITH PAYOFF-BASED INFORMATION

In this section, we proceed to examine a “payoff-based” learning scheme, i.e., a method that relies only on observations of the players’ realized, in-game payoffs (the so-called “bandit setting”). The first step will be to introduce a payoff-based stochastic first-order oracle in the spirit of Spall [58, 59]; subsequently, by mapping this oracle to the general feedback model of Section 3, we will leverage the analysis of Section 4 to derive the algorithm’s properties in time-varying games.

### 5.1. Payoff-based feedback and estimation of payoff gradients.
Heuristically, the main idea of the player’s gradient estimation process is easiest to describe in one-dimensional environments. In particular, suppose that an agent wishes to estimate the derivative of an unknown function $f: \mathbb{R} \to \mathbb{R}$ at some point $x \in \mathbb{R}$. Then, by definition, given an accuracy target $\delta$, the derivative of $f$ at $x$ can be approximated by two queries of $f$ as

$$ f'(x) \approx \frac{f(x + \delta) - f(x - \delta)}{2\delta}. \quad (83) $$

Building on this idea, $f'(x)$ can be estimated from a single function evaluation as follows: let $w$ be a random variable taking the value $+1$ or $-1$ with probability $1/2$, and consider the estimator

$$ V = \frac{f(x + \delta w)}{\delta} w. \quad (84) $$

In expectation, this gives:

$$ \mathbb{E}[V] = \frac{1}{2\delta}f(x + \delta) - \frac{1}{2\delta}f(x - \delta). \quad (85) $$

Thus, if $f'$ is Lipschitz continuous, we readily get $\mathbb{E}[V - f'(x)] = \mathcal{O}(\delta)$, i.e., the estimator (84) is accurate up to $\mathcal{O}(\delta)$.

This idea is the starting point of the so-called single-point stochastic approximation (SPSA) method that was pioneered by Spall [58, 59]. Its extension to a multi-dimensional setting is straightforward: If an agent seeks to estimate the gradient of a function $f: \mathbb{R}^d \to \mathbb{R}$, it suffices to sample a perturbation direction $w$ uniformly at random from $\mathcal{E} \equiv \{\pm e_1, \dots, \pm e_d\}$ and consider the estimator

$$ V = \frac{d}{\delta} f(x + \delta w) w. \quad (86) $$

The only difference between (84) and (86) is the dimensional scaling factor $d$ which compensates for the fact that each principal direction of $\mathbb{R}^d$ is sampled with probability $1/d$. Then the same reasoning as above shows that $\mathbb{E}[\|V - \nabla f(x)\|] = \mathcal{O}(\delta)$.

In the presence of constraints, a caveat that arises is that the query point $\hat{x} = x + \delta w$ must remain feasible. To guarantee this, let $C$ be a convex body in $\mathbb{R}^d$, and let $f: C \to \mathbb{R}$ be a function whose gradient we want to estimate at some point $x \in C$. To avoid the occurrence $x + \delta w \notin C$, we first transfer $x$ towards the interior of $C$ by a homothetic transformation of the form

$$ x \mapsto x^{\delta} \equiv x - \frac{\delta}{r}(x - p) \quad (87) $$
<caption>**Algorithm 2:** Payoff-based learning via mirror descent</caption>

Require: step-size $\gamma_{i,t} > 0$; sampling radius $\delta_{i,t} > 0$; homothety parameters $p_i \in \mathcal{K}_i$, $r_i > 0$
1: initialize $X_{i,1} \in \mathcal{K}_{h_i}$ \quad # initialize pivot
2: **for** $t = 1, 2, \dots$ **do** simultaneously **for** all $i = 1, \dots, N$
3: &nbsp;&nbsp;&nbsp;&nbsp;draw $W_{i,t}$ uniformly from $\{\pm e_1, \dots, \pm e_{d_i}\}$ \quad # random perturbation
4: &nbsp;&nbsp;&nbsp;&nbsp;play $\hat{X}_{i,t} = X_{i,t} + \delta_{i,t}W_{i,t} + (\delta_{i,t}/r_{i,t})(p_i - X_{i,t})$ \quad # select action
5: &nbsp;&nbsp;&nbsp;&nbsp;receive $\hat{u}_{i,t} \equiv u_{i,t}(\hat{X}_{i,t}; \hat{X}_{-i,t})$ \quad # get payoff
6: &nbsp;&nbsp;&nbsp;&nbsp;set $V_{i,t} = (d_i/\delta_{i,t})\hat{u}_{i,t}W_{i,t}$ \quad # estimate gradient
7: &nbsp;&nbsp;&nbsp;&nbsp;set $X_{i,t+1} \leftarrow \mathcal{P}_i(X_{i,t}; \gamma_{i,t}V_{i,t})$ \quad # update pivot
8: **end for**

where $p \in \text{int}(\mathcal{C})$ is an interior point of $\mathcal{C}$ and $r > 0$ is such that a) the ball $\mathcal{B}_r(p)$ is entirely contained in $\mathcal{C}$; and b) $\delta/r < 1$. Taken together, these conditions ensure that the query point

$$ \hat{x} = x^{\delta} + \delta w = (1 - \delta/r)x + (\delta/r)(p + r w) \quad (88) $$

belongs itself to $\mathcal{C}$ (simply note that $p + rw \in \mathcal{B}_r(p) \subseteq \mathcal{C}$).

With all this in mind, we obtain the following process for estimating individual payoff gradients in the context of a continuous game $\mathcal{G} \equiv \mathcal{G}(\mathcal{N}, \mathcal{K}, u)$:

1) Every player $i \in \mathcal{N}$ selects a pivot point $x_i \in \mathcal{K}_i$ and draws a perturbation vector $w_i$ uniformly at random from $\mathcal{E}_i := \{\pm e_1, \dots, \pm e_{d_i}\}$. Subsequently, each player plays

$$ \hat{x}_i = x_i + \delta_i w_i + (\delta_i / r_i)(p_i - x_i) \quad (89) $$

and they receive the associated payoffs $\hat{u}_i := u_i(\hat{x}_1, \dots, \hat{x}_N)$, $i \in \mathcal{N}$.

2) Each player constructs the single-point stochastic approximation estimate

$$ V_i = \frac{d_i}{\delta_i} \hat{u}_i \cdot w_i \quad (90) $$

and the process repeats.

In the above, the sampling radius $\delta_i$ and the homothety parameters $p_i \in \mathcal{K}_i$, $r_i > 0$, are chosen arbitrarily by each player $i \in \mathcal{N}$, only subject to the requirements $\mathcal{B}_{r_i}(p_i) \subseteq \mathcal{K}_i$ and $\delta_i/r_i < 1$ (to guarantee that $\hat{x}_i$ is a feasible action). Also, when unfolding over the course of a learning process, we will assume that players employ a variable sampling radius $\delta_{i,t}$ (similar to the players' individual step-size policy $\gamma_{i,t}$). In this way, the estimator (90) can be seen as a payoff-based oracle which can be coupled with Algorithm 1 to generate a new candidate action and continue playing. For a pseudocode implementation of the resulting policy, see Algorithm 2.

**Remark.** Throughout this section, we tacitly assume that the players' action spaces are convex bodies, i.e., they have nonempty topological interior. This assumption is only made for convenience: if this is not the case, it suffices to replace the basis vectors $\{\pm e_k\}$ with a basis of the affine hull of each player's action space and proceed in the same way. §

5.2. **Analysis and results.** The first step in the analysis of Algorithm 2 consists of quantifying the statistics of the players' gradient estimation process:

**Lemma 3.** The SPSA estimator (90) satisfies:

$$ \|\mathbb{E}[V_i - v_i(x)]\|_* = \mathcal{O}(\delta_{\max}^2/\delta_i) \quad \text{and} \quad \mathbb{E}[\|V_i\|_*^2] = \mathcal{O}(1/\delta_i^2). \quad (91) $$

where $\delta_{\max} = \max_i \delta_i$.
Proof. The second moment bound $\mathbb{E}[\|V_i\|_*^2] = \mathcal{O}(1/\delta_i^2)$ follows trivially from the definition (90) of $V$ and the boundedness of $u_i$. As for our first claim, let

$$ \xi_i = \hat{x}_i - x_i = \delta_i w_i + (\delta_i/r_i)(p_i - x_i). \quad (92) $$

and set $\xi = (\xi_i)_{i \in \mathcal{N}}$. Then, by the smoothness of $u_i$, a first-order Taylor expansion with integral remainder gives

$$ V_i = \frac{d_i}{\delta_i} u_i(\hat{x}) \cdot w_i = \frac{d_i}{\delta_i} u_i(x) \cdot w_i + \frac{d_i}{\delta_i} \sum_{j \in \mathcal{N}} \langle \nabla_{x_j} u_i(x), \xi_j \rangle w_i \quad (93a) $$

$$ + \sum_{j,k \in \mathcal{N}} \int_0^1 (1-t) \xi_j^\top \nabla^2_{x_j x_k} u_i(x+t\xi) \xi_k \, dt \cdot w_i \quad (93b) $$

Hence, taking expectations, the first term above becomes

$$ \begin{aligned} \mathbb{E}[(93a)] &= \frac{d_i}{\delta_i} \mathbb{E}[\langle v_i(x), \xi_i \rangle w_i] + \frac{d_i}{\delta_i} \sum_{j \neq i} \langle \nabla_{x_j} u_i(x), \mathbb{E}[\xi_j] \rangle \mathbb{E}[w_i] \\ &= d_i \mathbb{E}[\langle v_i(x), w_i \rangle w_i] = d_i \cdot \frac{1}{2d_i} \sum_{\ell=1}^{d_i} [v_{i\ell}(x)e_\ell - v_{i\ell}(x)(-e_\ell)] \\ &= v_i(x) \end{aligned} \quad (94) $$

where we used the fact that $\mathbb{E}[w_i] = 0$ for all $i \in \mathcal{N}$ and that $w_i$ and $w_j$ are independent for all $i, j \in \mathcal{N}, i \neq j$. As for the second term, we have

$$ \mathbb{E}[(93b)] = \frac{d_i}{\delta_i} \sum_{j,k \in \mathcal{N}} \delta_j \delta_k \mathbb{E} \left[ \int_0^1 (1-t) \xi_j^\top \nabla^2_{x_j x_k} u_i(x+t\xi) \xi_k \, dt \cdot w_i \right] = \mathcal{O}(\delta_{\max}^2 / \delta_i), \quad (95) $$

where we used the fact that $\mathcal{K}$ is compact and $u_i$ is $C^2$-smooth over $\mathcal{K}$. Our claim then follows by combining the bounds (94) and (95). ■

We are now in a position to state and prove our main result for the payoff-based learning policy outlined in Algorithm 2:

**Theorem 3.** Let $\mathcal{G}_t$ be a time-varying game satisfying Assumption 1. Suppose further that each player $i \in \mathcal{N}$ runs Algorithm 1 with step-size $\gamma_{i,t} \propto t^{-p_i}$ and sampling radius $\delta_{i,t} \propto t^{-q_i}$ for some $p_i, q_i \in (0, 1]$. Then:

(1) If $\mathcal{G}_t$ stabilizes to a strictly monotone game $\mathcal{G}$ at a rate $R_{i,t} = \mathcal{O}(1/t^{r_i})$, $r_i > 0$, and $p_i = p > \max\{1-r_i, 1+q_i-2q_{\min}, 1/2+q_i\}$ for all $i \in \mathcal{N}$, the sequence of chosen actions $\tilde{X}_t$, $t = 1, 2, \dots$, converges to the Nash equilibrium of $\mathcal{G}$ with probability 1. In particular, convergence to Nash equilibrium is guaranteed under the choice $p_i = 1$, $q_i = 1/3$.

(2) If $\mathcal{G}_t$ is strongly monotone and its drift is bounded as $V(T) = \mathcal{O}(T^r)$ for some $r < 1$, the sequence of chosen actions $\tilde{X}_t$, $t = 1, 2, \dots$, enjoys the equilibrium tracking guarantee:

$$ \mathbb{E} \left[ \sum_{t=1}^{T} \| \hat{X}_t - x_t^* \|^2 \right] = \mathcal{O} \left( T^{1-\min_i(p_i-2q_i)} + T^{1+q_{\max}-2q_{\min}} + T^{r+p_{\max}+\min_i(p_i-2q_i)} \right) \quad (96) $$

where $x_t^*$ denotes the (necessarily unique) Nash equilibrium of $\mathcal{G}_t$, and we set $p_{\min/\max} = \min/\max_i p_i$ and $q_{\min/\max} = \min/\max_i q_i$. In particular, for $p_i = 3(1-r)/5$ and $q_i = (1-r)/5$, we get the optimized tracking guarantee:

$$ \mathbb{E} \left[ \sum_{t=1}^{T} \| \hat{X}_t - x_t^* \|^2 \right] = \mathcal{O} \left( T^{\frac{4+r}{5}} \right). \quad (97) $$
**Theorem 3** combines two regimes: Part (1) treats time-varying games that stabilize to a well-defined limit, while Part (2) concerns the case where the game evolves without converging. This is in direct analogy to Theorems 1 and 2 for the case of generic SFO feedback and, indeed, Theorem 3 draws heavily on these results. However, there is now a discrepancy between the actions $\hat{X}_t$ chosen by the players and the candidate actions $X_t$ on which the SPSA estimator (90) returns feedback. We explain this difference in the proof of Theorem 3 below:

*Proof of Theorem 3.* Let $X_t$, $t = 1, 2, \dots$, be the sequence of pivot points generated by Algorithm 2: specifically, $X_t$ is given by (MD), but the players' realized action profile $\hat{X}_t$ is given by (88). Then, by Lemma 3, it follows that the SPSA estimator $V_t$ of (90) returns feedback of the form (SFO) on $X_t$ with bias and variance bounded as $B_t = O(\delta_{i,t}) = O(1/t^{q_i})$ and $M_t^2 = O(1/\delta_{i,t}^2) = O(t^{2q_i})$ respectively. Since the sequence $X_t$ is generated via the prox-rule $X_{t+1} = \mathcal{P}(X_t; \gamma_t V_t)$ of Algorithm 1, we have:

(1) If $\mathcal{G}_t$ stabilizes to a strictly monotone game $\mathcal{G}$, invoking Corollary 1 with $b_i = s_i = q_i$ shows that the sequence of pivot points $X_t$ converges (a.s.) to the (necessarily unique) equilibrium of $\mathcal{G}$ as long as $p_i = p > \max\{1 - r_i, 1 - q_i, 1/2 + q_i\}$ for all $i \in \mathcal{N}$. Since $\|\hat{X}_t - X_t\| = O(\delta_{i,t})$ and $\delta_{i,t} \to 0$, our claim follows.

(2) If $\mathcal{G}_t$ is strongly monotone with drift $V(T) = O(T^r)$, Theorem 2 gives

$$ \mathbb{E}[\text{err}(T)] = O\left(T^{1-\min_i(p_i-2q_i)} + T^{1+q_{\max}-2q_{\min}} + T^{p_{\max}+\min_i(p_i-2q_i)} V(T)\right) \quad (98) $$

where, by virtue of Lemma 3, we set $s_i = q_i$ and $b_i = 2q_{\min} - q_i$ in (23). However, by (88) and the compactness of $\mathcal{K}$, we also have $\|\hat{X}_t - X_t\| = O(\delta_{i,t}) = O(1/t^{q_{\min}})$, implying in turn that

$$ \begin{aligned} \frac{1}{2} \sum_{t=1}^{T} \|\hat{X}_t - x_t^*\|^2 &\le \sum_{t=1}^{T} \|\hat{X}_t - X_t\|^2 + \sum_{t=1}^{T} \|X_t - x_t^*\|^2 \\ &= O(T^{1-2q_{\min}}) + \sum_{t=1}^{T} \|X_t - x_t^*\|^2. \end{aligned} \quad (99) $$

Putting all this together, we conclude that $\mathbb{E}[\sum_{t=1}^{T} \|\hat{X}_t - x_t^*\|^2]$ is bounded as per (96), and our proof is complete. ■

As a special case, Part (1) of Theorem 3 implies that the sequence of play induced by Algorithm 2 in a fixed strictly monotone game $\mathcal{G}_t \equiv \mathcal{G}$ converges to Nash equilibrium with probability 1 as long as $p > \max\{1 - q, 1/2 + q\}$. In this way, we recover a recent result by Bravo et al. [12] who used a different form of the SPSA estimator (90) to establish the convergence of payoff-based no-regret learning in constant, monotone games. It is also possible to undertake a finer analysis for the method's rate of convergence in the case where the limit game $\mathcal{G}$ is strongly monotone, but this lies beyond the scope of this work.

## 6. FURTHER RESULTS AND DISCUSSION

In this section, we proceed to discuss some extensions and applications of our results that would have otherwise disrupted the flow of our paper.
6.1. **Games with randomly evolving payoffs.** We begin by discussing some applications of our results to games that evolve randomly over time – i.e., when $\mathcal{G}_t$ is determined by some randomly drawn parameter $\omega_t$ describing the “state of the world”. Randomly evolving games of this type are commonly referred to as stochastic Nash games in the mathematical optimization, control and engineering literatures [17, 49], where they are sometimes analyzed within a more general framework featuring joint coupling constraints. For example, in the wireless communications problem we described earlier (Example 3.2 in Section 3), this would correspond to the case where the users’ channel gains $g_{i,t}$ fluctuate randomly between transmission frames – the so-called “fast-fading” channel model [64].

To define this game-theoretic setting in detail, suppose that the players’ utilities are determined by an ensemble of random functions of the form $\tilde{u}_i: \mathcal{K} \times \Omega \to \mathbb{R}$ where $\Omega$ has the structure of a complete probability space and each $\tilde{u}_i(x; \omega)$ is assumed to be a) measurable in $\omega$; b) $C^2$-smooth in $x$ with uniformly bounded derivatives; and c) individually concave in the $i$-th component of $x$. Then, at each stage $t = 1, 2, \dots$, an i.i.d. state variable $\omega_t$ is drawn from $\Omega$ according to $\mathbb{P}$, and the players face the game $\mathcal{G}_t$ with payoff functions

$$u_{i,t}(x) = \tilde{u}_i(x; \omega_t) \quad \text{for all } i \in \mathcal{N}. \qquad (100)$$

Given the randomness involved, it is meaningful to consider the associated mean game $\mathcal{G} = \mathcal{G}(\mathcal{N}, \mathcal{K}, u)$ with payoff functions

$$u_i(x) = \mathbb{E}[\tilde{u}_i(x; \omega)] \quad \text{for all } i \in \mathcal{N} \qquad (101)$$

where the expectation $\mathbb{E}[\cdot]$ is taken relative to the (common) law of the state variables $\omega_t$. It is then natural to ask whether the players’ behavior under Algorithm 1 approaches a Nash equilibrium of the mean $\mathcal{G}$ as the game unfolds. Our next result provides a positive result in this direction under the assumption that the players’ individual payoff gradients have finite variance, i.e.,

$$\mathbb{E}[\|\nabla_{x_i} \tilde{u}_i(x; \omega) - \nabla_{x_i} u_i(x)\|_*^2] \le \Sigma^2 \quad \text{for all } x \in \mathcal{K}. \qquad (102)$$

Under this assumption, we have the following equilibrium convergence guarantee:

**Theorem 4.** Let $\mathcal{G}_t$, $t = 1, 2, \dots$, be a sequence of random games as above, and assume that the mean game $\mathcal{G}$ is strictly monotone. Suppose further that each player $i \in \mathcal{N}$ runs Algorithm 1 with a DGF satisfying (RC) and a step-size policy satisfying (S1), (S2), and

$$\sum_{t=1}^{\infty} \gamma_{i,t} B_{i,t} < \infty \quad \text{and} \quad \sum_{t=1}^{\infty} \gamma_{i,t}^2 S_{i,t}^2 < \infty. \qquad (S3')$$

Then, with probability 1, the sequence of realized actions $X_t$ converges to the (necessarily unique) Nash equilibrium $x^*$ of $\mathcal{G}$.

**Remark.** There are two distinct and conditionally independent sources of stochasticity in Theorem 4: a) the randomness coming from $\omega_t$ (which determines the $t$-th stage game $\mathcal{G}_t$); and b) the randomness in the players’ oracle feedback. In particular, we tacitly assume here that the filtration $\mathcal{F}_t$ underlying the definition (9) of the players’ feedback process refers to the joint history of $X_t$ and $\omega_t$, and the statement “with probability 1” likewise refers to both sources of randomness taken together.

**Proof.** Let $\tilde{v}_i(x; \omega) = \nabla_{x_i} \tilde{u}_i(x; \omega)$ and $v_i(x) = \nabla_{x_i} u_i(x)$. Then, by differentiating under the integral sign, we have $\mathbb{E}[\tilde{v}_i(x; \omega)] = \mathbb{E}[\nabla_{x_i} \tilde{u}_i(x; \omega)] = \nabla_{x_i} \mathbb{E}[\tilde{v}_i(x; \omega)] = v_i(x)$, so the players’ oracle signal may be decomposed as

$$V_{i,t} = v_i(X_t; \omega_t) + U_{i,t} + b_{i,t} = v_i(X_t) + \bar{U}_{i,t} + b_{i,t} \qquad (103)$$
where $\bar{U}_{i,t} = U_{i,t} + v_i(X_t; \omega_t) - v_i(X_t)$. Then, in a slight abuse of notation, we obtain

$$ \mathbb{E}[\bar{U}_{i,t} | X_t, \dots, X_1] = \mathbb{E}[\mathbb{E}[\bar{U}_{i,t} | \mathcal{F}_t]] = 0 + \mathbb{E}[v_i(X_t; \omega_t) - v_i(X_t)] = 0 \quad (104) $$

and, furthermore

$$ \begin{aligned} \mathbb{E}[\|\bar{U}_{i,t}\|_{*}^{2} | X_t, \dots, X_1] &\le 2 \mathbb{E}[\|U_{i,t}\|_{*}^{2} + \|v_i(X_t; \omega_t) - v_i(X_t)\|_{*}^{2} | X_t, \dots, X_1] \\ &\le 2\sigma_{i,t}^{2} + 2\Sigma^{2} = \mathcal{O}(S_{i,t}^{2}). \end{aligned} \quad (105) $$

Finally, letting $\bar{b}_{i,t} = \mathbb{E}[b_{i,t}]$, we also get $\|\bar{b}_{i,t}\|_* \le B_{i,t}$ by definition. Accordingly, given that $\mathbb{E}[V_{i,t} | X_t, \dots, X_1] = v_i(X_t) + \bar{b}_{i,t}$, our claim follows by applying Theorem 1 to the sequence of (strictly monotone) games $\bar{G}_t \equiv \mathcal{G}$ for all $t \ge 1$. ■

Even though Theorem 1 plays a major role in the proof of Theorem 4, the latter is conceptually distinct from the former because it provides an equilibrium convergence result in a setting where the sequence of stage games does not stabilize over time. Analogous results for equilibrium tracking or payoff-based learning (in the direction of Theorem 2 or Theorem 3 respectively) can also be derived, but this would take us too far afield, so we do not carry out the detailed analysis here.

**6.2. Regret bounds.** We close this section with a precise statement and derivation of the dynamic regret bound (29) that was alluded to in Section 4.4.

**Proposition 4.** Suppose that a single player runs Algorithm 1 against a sequence of concave payoff functions $u_t: \mathcal{K} \to \mathbb{R}$ with a Lipschitz DGF and step-size and oracle feedback parameters as in Theorem 2. Then the player's dynamic regret is bounded as

$$ \mathbb{E}[\mathrm{DynReg}(T)] = \mathcal{O}(T^{1+2s-p} + T^{1-b} + T^{2p-2s} \mathrm{V}(T)). \quad (29, \mathrm{redux}) $$

In particular, if $\mathrm{V}(T) = \mathcal{O}(T^r)$ and the algorithm's feedback is unbiased and bounded in mean square ($b = \infty, s = 0$), the player enjoys the bound $\mathbb{E}[\mathrm{DynReg}(T)] = \mathcal{O}(T^{1-p} + T^{2p+r})$. Hence, for $p = (1-r)/3$, the player achieves

$$ \mathbb{E}[\mathrm{DynReg}(T)] = \mathcal{O}(T^{\frac{2+r}{3}}). \quad (30, \mathrm{redux}) $$

**Proof of Proposition 4.** As in the proof of Theorem 2, partition the horizon of play $\mathcal{T} = [1..T]$ in $m$ contiguous batches $\mathcal{T}_k$, $k = 1, \dots, m$, each of length $\Delta$ (except possibly the $m$-th one, which might be smaller). We then have

$$ \begin{aligned} \mathrm{DynReg}(\mathcal{T}_k) &= \sum_{t \in \mathcal{T}_k} [u_t(x_t^*) - u_t(X_t)] \\ &\le \sum_{t \in \mathcal{T}_k} \langle v_t(X_t), x_t^* - X_t \rangle \\ &= \sum_{t \in \mathcal{T}_k} \langle v_t(X_t), \hat{x}_k - X_t \rangle + \sum_{t \in \mathcal{T}_k} \langle v_t(X_t), x_t^* - \hat{x}_k \rangle \\ &\le \mathrm{Gap}(\mathcal{T}_k) + \sum_{t \in \mathcal{T}_k} \langle v_t(X_t), x_t^* - \hat{x}_k \rangle, \end{aligned} \quad (106) $$

where $\hat{x}_k \in \mathcal{K}$ is a test action specific to each batch $k = 1, \dots, m$. Then, repeating the series of arguments leading up to (79), we get the dynamic regret bound

$$ \mathbb{E}[\mathrm{DynReg}(T)] \le \mathbb{E}[\mathrm{Gap}(T)] + G\Delta \mathrm{V}(T) \quad (107) $$

and our claim by invoking the bounds (80) and (81). ■

Dynamic regret guarantees of the form (30) already exist in the literature. Specifically, Besbes et al. [10] obtained a similar bound by exploiting the following meta-principle: (i) first, break the horizon of play into batches of size $\Delta$; (ii) over each batch, run an algorithm that guarantees low static regret relative to $\Delta$; then (iii) finetune these steps in terms of the horizon $T$ and the variation $\mathrm{V}(T)$ of the agent's payoff functions in order to get low dynamic
regret. In our setting, if Algorithm 1 is rebooted every $\Delta \sim [T/V(T)]^{2/3}$ iterations and is run with constant step-size $\gamma \sim 1/\sqrt{\Delta}$ between reboots, the meta-principle of Besbes et al. [10] guarantees the dynamic regret bound

$$ \mathbb{E}[\text{DynReg}(T)] = \mathcal{O}(T^{2/3} V(T)^{1/3}). \qquad (108) $$

Besbes et al. [10] further showed that this bound is unimprovable under the blanket feedback model (SFO), so (30) is tight in this regard.<fnref n="8" />

A disadvantage of this restart approach is that (i) the batch length $\Delta$ must be chosen carefully relative to the total variation of the sequence of payoff functions encountered; and (ii) at every reboot, the algorithm begins tabula rasa, essentially forgetting all knowledge it had accumulated up to the point in question. Besbes et al. [10] already discuss some possible ways to avoid restarts, and we are aware of at least two related approaches in the literature: Jun et al. [30] proposed a meta-aggregator based on coin betting, while Jadbabaie et al. [28] and Shahrampour and Jadbabaie [53] take an approach based on optimistic mirror descent. Importantly, both policies achieve DynReg($T$) = $\mathcal{O}(V(T)^{1/2}T^{1/2})$ without prior knowledge of $V(T)$: since $V(T)^{1/2}T^{1/2} = V(T)^{1/3}V(T)^{1/6}T^{1/2} = o(V(T)^{1/3}T^{2/3})$ whenever $V(T) = o(T)$, these guarantees would seem to contradict the optimality of the bound $\mathcal{O}(T^{2/3}V(T)^{1/3})$. The resolution of this apparent incongruity is that Jun et al. [30] and Jadbabaie et al. [28] assume access to a perfect gradient oracle, while the discussion above only assumes access to a stochastic one.

To the best of our knowledge, the perfect oracle requirement cannot be relaxed: if the players' gradient feedback is noisy, successive oracle calls cannot provide reliable information about the variation of the agent's payoff functions from one stage to the next, so the learning process cannot adapt to $V(T)$. Designing a policy that provably interpolates between the stochastic and deterministic regimes is a very fruitful question for further research, but one which lies beyond the scope of this paper.

## 7. CONCLUDING REMARKS

There are many interesting points for future research. A particularly promising one is to bridge the gap between the step-size policies that guarantee an optimal equilibrium tracking error and the policies that guarantee convergence to a Nash equilibrium in the case where $G_t$ stabilizes to a well-defined limit. As we saw, these considerations are not always in tune: when the rules of the game fluctuate constantly, players can use very different step-sizes, and still track the game's equilibrium on average; by contrast, when the game stabilizes, convergence to Nash equilibrium requires a certain compatibility between the players' step-size policies (and requires finer tuning). Balancing these two objectives in an adaptive, context-agnostic manner is a rich and promising direction for future research.

While on the topic of adaptivity, it should be recalled that players with access to perfect gradient information can achieve better rates of dynamic regret minimization, without any prior knowledge of the game's drift over time [28, 53]. Whether this is still possible in the stochastic (or, worse, bandit) case is another fruitful open question for further research.

## APPENDIX A. BASIC PROPERTIES OF BREGMAN PROXIMAL MAPPINGS

In this appendix we collect some basic technical facts on distance-generating functions and prox-mappings. These results are not new, but given the range of conventions and definitions in the literature, we find it useful to provide here precise statements and proofs.

<footnote n="8">⁸Strictly speaking, Besbes et al. [10] define $V(T)$ as $V(T) = \sum_{t=1}^{T} \|u_{t+1} - u_t\|_{\infty}$, but this distinction is not important for our purposes.</footnote>
For a detailed discussion, we refer the reader to Juditsky et al. [29], Nemirovski et al. [41], and references therein.

In what follows, $h$ will denote a distance-generating function on a compact convex subset $C$ of an $d$-dimensional normed space $\mathcal{X} \cong \mathbb{R}^d$ with dual $\mathcal{Y} = \mathcal{X}^*$, as per Definition 1. We begin with a basic subgradient comparison lemma:

**Lemma A.1.** For all $p \in C$ and all $y \in \partial h(x)$, $x \in C_h$, we have:

$$ \langle \nabla h(x), x - p \rangle \leq \langle y, x - p \rangle. \qquad (A.1) $$

*Proof.* By continuity, it suffices to show that (A.1) holds for all $p \in \text{ri} C$. To show this, fix $p \in \text{ri} C$, and let

$$ \phi(t) = h(x + t(p - x)) - [h(x) + \langle y, x + t(p - x) \rangle] \quad \text{for all } t \in [0, 1]. \qquad (A.2) $$

Given that $h$ is strongly convex and $y \in \partial h(x)$, it follows that $\phi(t) \ge 0$ with equality if and only if $t = 0$. Since $\psi(t) = \langle \nabla h(x + t(p - x)) - y, p - x \rangle$ is a continuous selection of subgradients of $\phi$ and both $\phi$ and $\psi$ are continuous over $[0, 1]$, it follows that $\phi$ is continuously differentiable with $\phi' = \psi$ on $[0, 1]$. Hence, with $\phi$ convex and $\phi(t) \ge 0 = \phi(0)$ for all $t \in [0, 1]$, we conclude that $\phi'(0) = \langle \nabla h(x) - y, p - x \rangle \ge 0$, and our proof is complete. ■

We continue with a basic property of Bregman divergences known as the “three-point identity” [14]:

**Lemma A.2 (3-point identity).** For all $p \in C$ and all $x, x' \in C_h$, we have:

$$ D(p, x) = D(p, x') + D(x', x) + \langle \nabla h(x) - \nabla h(x'), x' - p \rangle. \qquad (A.3) $$

The proof of this lemma is a straightforward expansion, so we omit it. Below, we employ this identity to estimate the Bregman divergence relative to a base point $p \in C$ before and after a prox-step.

**Lemma A.3.** Fix some $p \in C$ and consider the recursive update rule

$$ x^+ = \mathcal{P}(x; y) \qquad (A.4) $$

for $x \in C_h$, $y \in \mathcal{Y}$. Then:

$$ D(p, x^+) \leq D(p, x) - D(x^+, x) + \langle y, x^+ - p \rangle \qquad (A.5a) $$

$$ \leq D(p, x) + \langle y, x - p \rangle + \frac{1}{2K} \|y\|_*^2. \qquad (A.5b) $$

*Proof.* By the definition (13) of $\mathcal{P}$, we have $y + \nabla h(x) \in \partial h(x^+)$. This means that $x^+ \in \text{dom} \partial h \equiv C_h$, so the three-point identity (Lemma A.2) applies. We thus get

$$ D(p, x) = D(p, x^+) + D(x^+, x) + \langle \nabla h(x) - \nabla h(x^+), x^+ - p \rangle \qquad (A.6) $$

or, after rearranging:

$$ D(p, x^+) = D(p, x) - D(x^+, x) + \langle \nabla h(x^+) - \nabla h(x), x^+ - p \rangle. \qquad (A.7) $$

Since $\nabla h(x) + y \in \partial h(x^+)$, Lemma A.1 yields $\langle \nabla h(x^+), x^+ - p \rangle \le \langle y + \nabla h(x), x^+ - p \rangle$, so (A.5a) follows by plugging this bound back to (A.7).

For the second part of the lemma, first rewrite (A.5a) as

$$ D(p, x^+) \leq D(p, x) + \langle y, x - p \rangle + \langle y, x^+ - x \rangle - D(x^+, x). \qquad (A.8) $$

By Young's inequality, we also have

$$ \langle y, x^+ - x \rangle \leq \frac{1}{2K} \|y\|_*^2 + \frac{K}{2} \|x^+ - x\|^2 \qquad (A.9) $$
so (A.8) becomes

$$D(p, x^+) \le D(p, x) + \langle y, x-p \rangle + \frac{1}{2K} \|y\|_*^2 + \frac{K}{2} \|x^+ - x\|^2 - D(x^+, x). \quad (A.10)$$

Then, by the strong convexity of $h$, we obtain $D(x^+, x) = h(x^+) - h(x) - \langle \nabla h(x), x^+ - x \rangle \ge (K/2) \|x^+ - x\|^2$, and our claim follows. ■

This basic lemma allows us to derive the following “template inequality” for processes of the general form (A.5):

**Lemma A.4.** Consider a sequence of dual vectors $Y_t \in \mathcal{Y}$, $t = 1, 2, \dots$, and let

$$X_{t+1} = \mathcal{P}(X_t; Y_t) \quad (A.11)$$

with $X_1 \in C_h$ initialized arbitrarily. Then, for all $x \in C$ and every nonnegative sequence $\alpha_t \ge 0$ defined over the window $T = [\tau_{\text{start}} \dots \tau_{\text{end}}]$, we have

$$\sum_{t \in \mathcal{T}} \alpha_t \langle Y_t, x - X_t \rangle \le \sum_{t \in \mathcal{T}} (\alpha_t - \alpha_{t-1}) D(x, X_t) + \frac{1}{2K} \sum_{t \in \mathcal{T}} \alpha_t \|Y_t\|_*^2, \quad (A.12)$$

with the convention that $\alpha_{\tau_{\text{start}}-1} = 0$ in the above sum.

*Proof.* Let $D_t = D(x, X_t)$. Then (A.5b) readily yields

$$D_{t+1} \le D_t + \langle Y_t, X_t - x \rangle + \frac{1}{2K} \|Y_t\|_*^2 \quad (A.13)$$

so, after multiplying by $\alpha_t \ge 0$ and rearranging, we get

$$\alpha_t \langle Y_t, x - X_t \rangle \le \alpha_t (D_t - D_{t+1}) + \frac{\alpha_t}{2K} \|Y\|_*^2. \quad (A.14)$$

Therefore, by bringing $\langle Y_t, X_t - x \rangle$ to the left-hand side and summing over $t \in \mathcal{T}$, we get

$$\begin{aligned} \sum_{t \in \mathcal{T}} \alpha_t \langle Y_t, x - X_t \rangle &\le \sum_{t \in \mathcal{T}} \alpha_t (D_t - D_{t+1}) + \frac{1}{2K} \sum_{t \in \mathcal{T}} \alpha_t \|Y\|_*^2 \\ &= \sum_{t \in \mathcal{T}} (\alpha_t - \alpha_{t-1}) D_t - \alpha_{\tau_{\text{end}}} D_{\tau_{\text{end}}+1} + \frac{1}{2K} \sum_{t \in \mathcal{T}} \alpha_t \|Y\|_*^2. \end{aligned} \quad (A.15)$$

Since $D_{\tau_{\text{end}}+1} \ge 0$, our claim follows. ■

Finally, we will make frequent use of the following straightforward result:

**Lemma A.5.** Suppose that $h$ is Lipschitz. Then $\sup_{x \in C, x' \in C_h} D(x, x') < \infty$.

*Proof.* For all $x \in C$ and all $x' \in C_h$, we have:

$$D(x, x') = h(x) - h(x') - \langle \nabla h(x'), x - x' \rangle \le h(x) - h(x') + \|\nabla h(x')\|_* \|x - x'\|. \quad (A.16)$$

By assumption, $L \equiv \sup_{x'} \|\nabla h(x')\|_* < \infty$. Hence, with $C$ compact, we readily get

$$D(x, x') \le h(x) - h(x') + L \operatorname{diam}(C). \quad (A.17)$$

Since $h(x) - h(x') \le \max h - \min h < \infty$, our assertion follows. ■

## ACKNOWLEDGMENTS

This research was partially supported by the COST Action CA16228 “European Network for Game Theory” (GAMENET). P. Mertikopoulos is grateful for financial support by the French National Research Agency (ANR) in the framework of the “Investissements d’avenir” program (ANR-15-IDEX-02), the LabEx PERSYVAL (ANR-11-LABX-0025-01), MIAI@Grenoble Alpes (ANR-19-P3IA-0003), and the grants ORACLESS (ANR-16-CE33-0004) and ALIAS (ANR-19-CE48-0018-01).
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