Stable stochastic FTRL limit sets: boundary-containment agenda

GT-020 | Stochastic regularized learning
R | Source conjectural agenda and an editorial deterministic-set benchmark.

Definitions and mathematical statement

Let $G$ be a finite game, $X=\prod_i\Delta(S_i)$, and $v_i(x)=(u_i(a,x_{-i}))_{a\in S_i}$. Set $h_i(x_i)=\sum_a\theta_i(x_{i,a})$, where $\theta_i$ is $C^3$ on $(0,1)$, $\inf\theta_i''>0$, $\theta_i'(z)\to-\infty$ as $z\downarrow0$, $\theta_i'''<0$, and $\sup|\theta_i'''|/(\theta_i'')^2<\infty$. At endpoints take the lower-semicontinuous extended-real limits, so that $h_i$ is a proper convex function on the closed simplex, possibly $+\infty$ on its boundary. Define \[ Q_i(y_i)=\operatorname{argmax}_{p\in\Delta(S_i)} \{\langle y_i,p\rangle-h_i(p)\}. \] Consider the score process and induced strategy process \[ dY(t)=v(X(t))\,dt+\sigma(X(t))\,dW(t),\qquad X_i(t)=Q_i(Y_i(t)), \] where $W$ is a finite-dimensional standard Brownian motion, $\sigma$ is Lipschitz on $X$, and $\sigma(x)\sigma(x)^{\mathsf T}\succeq cI$ for some $c>0$ and every $x\in X$. Boundary points use the continuous face-wise extension of the induced strategy dynamics, when well-defined. A compact set $K\subset X$ is stochastically asymptotically stable if for every neighborhood $U$ of $K$ and $\delta\in(0,1)$ there is a neighborhood $V$ of $K$ such that, for all $x\in V$, \[ \Pr_x\{X(t)\in U\ \forall t\geq0,\ \operatorname{dist}(X(t),K)\to0\}\geq1-\delta. \] For a trajectory, its omega-limit set is $\Omega=\bigcap_{T\geq0}\overline{\{X(t):t\geq T\}}$. An explicit deterministic-set benchmark is the implication \[ \left. \begin{gathered} \varnothing\ne K\subsetneq X\text{ is compact and stochastically asymptotically stable},\\ \Pr_x(\Omega=K)>0\text{ for some initialization }x \end{gathered}\right\} \quad\Longrightarrow\quad K\subseteq\partial X, \] where $\partial X=X\setminus\operatorname{relint}X$ is the relative boundary. This benchmark is not asserted to be an unresolved problem: for a process irreducible in the relative interior, stability of a deterministic closed set from every nearby initialization entails invariance, and invariance may already exclude proper such sets meeting the interior. The source's broader agenda is to make its stable-limit-set notion precise, including the treatment of trajectory-dependent random omega-limit sets, and to determine the resulting boundary-containment statement. The benchmark supplies one explicitly quantified interpretation without claiming equivalence to that agenda.

Short English statement

Which precise notions of stable limiting sets of noisy regularized learning imply boundary containment?

Variants and scope boundaries

Nondegenerate noise, regularizer assumptions, face extension, and the definition of a limit set are part of the statement. Stability alone allows $K=X$ trivially. A deterministic compact set stable from every nearby initialization and a random limiting set of a single trajectory are distinct objects. Vanishing learning rates, degenerate noise, and stochastic population replicator models change the problem.

Source relationship and status

Section 6 of [S1] explicitly proposes boundary containment of stable limit sets. Definition 1 supplies the stability criterion, and the uniformly positive score covariance above is the source's Assumption 1. The deterministic-set benchmark, proper-set exclusion, and explicit omega-limit convention are editorial, not a source-exact open conjecture. In particular, the benchmark's possible reduction to interior irreducibility must not be presented as a resolution of, or an unresolved equivalent to, the source's informal limiting-set agenda. A source-exact fully quantified random-set conjecture was not recovered.

References

[S1] Pierre-Louis Cauvin, Davide Legacci, and Panayotis Mertikopoulos (2025). The impact of uncertainty on regularized learning in games. arXiv:2506.13286v1. Locator: Section 2.3 and Equation (7), Section 3.1, Assumptions 1--2, Definition 1, and Section 6, first conjectural paragraph. https://arxiv.org/html/2506.13286v1

Common conventions: Source mathematical conventions

Finite games. For a finite set $A$, $\Delta(A)$ is its probability simplex. Players choose independent mixed strategies in Nash equilibrium; correlation is allowed only when explicitly part of the solution concept. In a game with payoff functions $u_i$ and strategy profile $x$, an additive $\varepsilon$ Nash equilibrium satisfies \[ u_i(x)\geq u_i(x_i',x_{-i})-\varepsilon \quad\text{for every player $i$ and every admissible deviation $x_i'$}. \] For numerical approximation, payoffs are normalized as locally specified. An $\varepsilon$ payoff residual is distinct from distance at most $\varepsilon$ to the set of exact equilibria. Point convergence, convergence of empirical play, and convergence in distance to an equilibrium set are also distinct.

Computational representations. Unless stated otherwise, a complexity question takes rational data with binary encoding; polynomial time is measured in total bit length. A fixed-accuracy polynomial algorithm is not necessarily polynomial in $1/\varepsilon$, and neither is necessarily polynomial in $\log(1/\varepsilon)$. Succinct games and value, demand, or sampling oracles must declare their interfaces. Exact real solutions require an explicit representation; an unspecified list of arbitrary real numbers is not a finite computational output. A claimed algorithmic barrier is meaningful only for its specified model and reductions.

Dynamic games. Histories, information, observation, and allowable randomization are part of the model. A stationary strategy depends only on the current state; a positional strategy in a deterministic graph game chooses one action at each controlled vertex. Nash equilibrium at an initial state and equilibrium after every history are different requirements. An expectation of a pathwise $\liminf$ payoff, a $\liminf$ of expected averages, a limiting expected average, and a uniform finite-horizon equilibrium are different criteria. None is silently substituted for another.

Allocation and mechanisms. A complete allocation assigns every item exactly once unless otherwise stated. Zero-valued goods, negative values, capacity constraints, feasibility, lotteries, and copies of goods affect fairness definitions and are specified locally. Dominant-strategy incentive compatibility, Bayesian incentive compatibility, universal truthfulness, and truthfulness in expectation impose different inequalities. Whenever an objective ratio has zero denominator, its convention is stated or those instances are excluded.

Infinite-dimensional models. Expectations, integrals, stochastic equations, and optimization objectives require their local regularity, integrability, filtrations, and admissible domains. A backward--forward mean-field system is a boundary-value problem; it is not an initial-value semiflow without an additional construction. Long-time, large-population, and vanishing-noise limits require an order of limits and a convergence topology. If a minimum need not be attained, the objective is its infimum or supremum and the approximation requirement is stated separately.