SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Guillaume Wang
Station Z, CH-1015 Lausanne
guillaume.wang@epfl.ch
Lenaic Chizat
lenaic.chizat@epfl.ch
We study the convergence to local Nash equilibria of gradient methods for two-player zero-sum differentiable games. It is well-known that such dynamics converge locally when  $S \succ 0$  and may diverge when  $S = 0$ , where  $S \succeq 0$  is the symmetric part of the Jacobian at equilibrium that accounts for the "potential" component of the game. We show that these dynamics also converge as soon as  $S$  is nonzero (partial curvature) and the eigenvectors of the antisymmetric part  $A$  are in general position with respect to the kernel of  $S$ . We then study the convergence rates when  $S \ll A$  and prove that they typically depend on the average of the eigenvalues of  $S$ , instead of the minimum as an analogy with minimization problems would suggest. To illustrate our results, we consider the problem of computing mixed Nash equilibria of continuous games. We show that, thanks to partial curvature, conic particle methods - which optimize over both weights and supports of the mixed strategies - generically converge faster than fixed-support methods. For min-max games, it is thus beneficial to add degrees of freedom "with curvature": this can be interpreted as yet another benefit of over-parameterization.
Min-max optimization is notoriously subtler than minimization, even in convex-concave settings. While many of the proof techniques for minimization have a natural equivalent in the min-max world, some common intuitions fail to transfer. The picture is clear for strongly convex-strongly concave (SC-SC) min-max games: all the classical gradient methods converge exponentially (for small enough step-sizes) with worst-case convergence rates dependent on the strong convexity and strong concavity parameters  $\mu_x, \mu_y$ . But, most famously perhaps, for bilinear min-max games the last iterate of simultaneous Gradient Descent-Ascent (GDA) diverges, while the Proximal Point (PP) method converges, and alternating GDA and the continuous-time limit of all of these algorithms - Gradient Flow (GF) - exhibit a cycling behavior.
Based on those two extreme cases, it still seems that part of our intuition from minimization, where the last-iterate convergence rate is indeed determined by the strong convexity parameter, is preserved. In this paper, we argue that this intuition is in fact overly pessimistic, and that gradient methods behave in general more favorably in the min-max setting than in the minimization setting.
Let us first look at a basic example. Consider a bilinear min-max game regularized by a quadratic term only in one scalar variable:
$$
\min_{x\in \mathbb{R}^{d}}\max_{y\in \mathbb{R}^{d}}x^{\top}Py + \frac{\alpha}{2} x_{1}^{2}.
$$
(a) Local convergence of the GF iterates for a fixed draw of  $P \in \mathbb{R}^{2 \times 2}$  and various values of  $\alpha$ . Only the final phase of the dynamics is shown, so here we see the iterates evolve along the "dominant" eigenspace of  $M$  (the subspace along which convergence is the slowest).
(b) Observed and predicted  $(\tilde{\mu}_M)$  normalized convergence rate  $r / \eta$  of GDA with a small step-size  $\eta$  ,i.e.,  $\left\| x^{k}\right\| +\left\| y^{k}\right\| = \Theta ((1 - r)^{k})$  , vs. regularization strength  $\alpha$  (the higher the faster). Each color represents one draw of  $P\in \mathbb{R}^{2\times 2}$
(c) Spectrum of the Jacobian  $M$  for a fixed draw of  $P\in \mathbb{R}^{3\times 3}$  (green) and its approximation by Equation (3.1) (gray)
Figure 1: Convergence of gradient methods on a random bilinear game regularized by  $\frac{\alpha}{2} x_1^2$  for small step-sizes. The fact that  $\tilde{\mu}_M$ , and so  $r / \eta$ , scale linearly with  $\alpha$  (for small  $\alpha$ ) is explained by Prop. 3.1.
Here there is no strong convexity in either player (so  $\mu_x = \mu_y = 0$ ), yet as observed on Fig. 1a, Fig. 1b, when  $P \in \mathbb{R}^{d \times d}$  is random with independent standard normal entries, GDA (with a small step-size) typically converges at an exponential rate that scales linearly with  $\alpha$ .<sup>1</sup>
As we will see, this phenomenon is a consequence of the existing theory for general smooth min-max games
$$
\min  _ {x \in \mathbb {R} ^ {n}} \max  _ {y \in \mathbb {R} ^ {m}} f (x, y) \tag {1.1}
$$
with a local Nash equilibrium (NE), or local saddle point,  $z^{*} = (x^{*}, y^{*})$ . It is well-known from the dynamical systems literature that GF converges locally to  $z^{*}$  if
$$
\tilde {\mu} _ {M} := \min _ {\lambda \in \mathrm {S p} (M)} \Re (\lambda) > 0, \qquad \mathrm {w i t h} M = \left[ \begin{array}{c c} \nabla_ {x x} ^ {2} f & \nabla_ {x y} ^ {2} f \\ - \nabla_ {x y} ^ {2} f ^ {\top} & - \nabla_ {y y} ^ {2} f \end{array} \right] (z ^ {*}) \in \mathbb {R} ^ {(n + m) \times (n + m)}
$$
the Jacobian of the skewed gradient field at  $z^{*}$ , and where  $\mathrm{Sp}(\cdot)$  denotes the spectrum, i.e., the set of eigenvalues. For our starting basic example, this quantity can be visualized in Fig. 1c. Moreover,  $\tilde{\mu}_M$  also characterizes the convergence behavior of gradient methods in the leading order in the step-size  $\eta$ : they all converge to  $z^{*}$  with the rate  $\eta \tilde{\mu}_M + O(\eta^2)$ , see the Appendix for a review. $^3$
While the condition  $\tilde{\mu}_M > 0$  is very general and tight, it is not obvious how to control or interpret it. Our purpose with this paper is to explore what this condition entails and to emphasize some of its surprising consequences.
<sup>1</sup>Julia code for the numerical experiments is available at https://github.com/guillaumew16/local_cvgce_minmax.
${}^{2}$  In numerical analysis and stability theory,the quantity  $- {\widetilde{\mu }}_{M}$  is known as the spectral abscissa of  $- M$  .
3More precisely, gradient methods converge locally (for small enough step-sizes) if  $\min_{\lambda \in \mathrm{Sp}(M)}\Re (\lambda) > 0$  and only if  $\min_{\lambda \in \mathrm{Sp}(M)\backslash \{0\}}\Re (\lambda)\geq 0$  . Throughout this paper we will assume for simplicity  $M$  invertible so that those two quantities are equal.
The rest of the paper is organized as follows. In Sec. 2 we show that, generically,  $\tilde{\mu}_M > 0$  as long as the problem has partial curvature, i.e. the diagonal blocks of  $M$  are non-zero. In Sec. 3, we study more precisely the case of games where the "interaction" component dominates the "potential" component, i.e., when  $\nabla_{xx}^2 f(z^*),\nabla_{yy}^2 f(z^*)\ll \nabla_{xy}^2 f(z^*)$ . For such games, in a certain random setting, we show that  $\tilde{\mu}_M$  scales as the average of the eigenvalues of the potential part, instead of the minimum as an analogy with minimization problems would suggest. In Sec. 4 we consider the computation of mixed Nash equilibria of continuous games using particle methods, a setting where the convergence under partial curvature has a striking consequence. By optimizing over both the weights and supports of the mixed strategies, one obtains dynamics that generically converge faster than fixed-support methods, even when the latter are using the optimal supports.
For ease of exposition, in most of the paper we focus on the local convergence of GF. We discuss the convergence of discrete time algorithms in Sec. 3.2.
Throughout this paper, by "convergence" we mean convergence of the last iterate to a (local) NE, while a different line of work also considers convergence of the averaged iterate [Nem04], and another considers convergence to any critical point [ALW21]. Also to prevent possible confusion, let us mention a related but distinct line of work [DH19; Doa22] that considers using different step-sizes to update the  $x$  and the  $y$  variables. The resulting two-timescale dynamics can be analyzed globally assuming a min-max analog of the Polyak-Łojasiewicz inequality, using quite different considerations than the ones developed in this paper.
Analysis of GDA and PP for SC-SC, bilinear or convex-SC games. The fact that PP converges for SC-SC and bilinear min-max games is well-established since at least [Roc76]; for a modern reformulation, see e.g. [MOP20]. The convergence of simultaneous GDA for SC-SC games and its divergence for bilinear games is also a classical fact, see e.g. [LS19; Lu22]. The cycling behavior of alternating GDA for bilinear games is proved in [BGP20, Theorem 4].
As shown more recently in [NK17, Appendix G] and [ZWLG22, Theorem 6], convexity in  $x$ , strong concavity in  $y$  and non-degeneracy of the interaction component are in fact sufficient to ensure  $\tilde{\mu}_M > 0$ , and so local convergence of GDA. The setting of the former work corresponds to  $\nabla_{xx}^2 f(z^*) = 0$ ,  $-\nabla_{yy}^2 f(z^*) \succ 0$  and  $\nabla_{xy}^2 f(z^*)$  full-row-rank, extended in the latter work to  $\nabla_{xx}^2 f(z^*) \succeq 0$ . Those two works also provide bounds on  $\tilde{\mu}_M$  for those specific cases in terms of the least eigenvalues of  $-\nabla_{yy}^2 f(z^*)$  and of  $\nabla_{xy}^2 f(z^*)\nabla_{xy}^2 f(z^*)^\top$ .
Spectral analysis-based convergence analyses. Our approach to analyzing min-max algorithms is to directly study the properties of the update operator (e.g., of  $T(z) = z - \left( \begin{array}{c} \nabla_x f \\ -\nabla_y f \end{array} \right)(z)$  for simultaneous GDA). Specifically, by a classical result on discrete-time dynamical systems, the local exponential convergence is characterized by the spectral radius of that operator's Jacobian. This approach is used for example in [Gid+19] to analyze the local convergence behavior of alternating GDA with negative momentum, and in [AMLG20] to derive tight convergence bounds – dependent explicitly on  $\eta$  and  $\operatorname{Sp}(M)$  – for simultaneous GDA and multi-step Extra-Gradient.
Average-case analysis of the (local) convergence rate. In [PS20; DPS21], the authors analyze the convergence of gradient methods for affine operator problems - i.e., for finding  $z$  such that  $M(z - z^{*}) = 0$  - when  $M$  is a random normal matrix with a known spectral distribution. They derive average-case optimal methods for this setting. Our analysis in Sec. 3 also has an average-case flavor, but our random model is different: we assume that the symmetric  $S$  and antisymmetric parts  $A$  of  $M$  are independent and that  $S \ll A$ . In particular we do not require  $M$  to be normal (i.e.,  $S$  and  $A$  do not commute a priori, which is the typical case in min-max optimization).
Hypocoercivity. In the context of partial differential equations (PDEs), the phenomenon that a linear PDE  $\partial_t u_t = -Lu_t$  may exhibit linear convergence to 0 even when the generator  $L$  is not coercive, is called hypocoercivity and is studied in detail in [Vil09]. This is precisely the infinite-dimensional analog of the phenomenon studied in the present paper: still denoting by  $S$ ,  $A$  the symmetric resp. antisymmetric parts of  $M$ , coercivity of  $L$  corresponds to  $S\succ 0$ , while hypocoercivity corresponds to
$\tilde{\mu}_M > 0$ . Specifically, [Vil09] shows how to construct Lyapunov functions to establish hypocoercivity and convergence rates of certain PDEs, by exploiting properties of the iterated commutators of  $S$  and  $A$ . By contrast our focus is on the finite-dimensional case, where it is easier and more natural to directly study the spectrum of  $M$ .
1.2 Notation
For any matrix  $T \in \mathbb{R}^{d \times d}$  or  $\mathbb{C}^{d \times d}$ , denote by  $\operatorname{Sp}(T) \subset \mathbb{C}$  its spectrum, i.e., the set of its eigenvalues, and by  $\rho(T) = \max_{\lambda \in \operatorname{Sp}(T)} |\lambda|$  its spectral radius. Recall that the spectral radius is distinct from the operator norm, which is the largest singular value, although they coincide for Hermitian matrices. Denote eigenspaces as  $E_{\lambda}(T) = \{z \in \mathbb{C}^d; Tz = \lambda z\}$  and let Eigvecs  $(T) = \bigcup_{\lambda \in \operatorname{Sp}(T)} E_{\lambda}(T)$  the set of all (complex) eigenvectors.  $\| \cdot \|$  denotes Euclidean or Hermitian norm. For a collection of square matrices (resp. scalars)  $(C_k)_k$ ,  $\operatorname{Diag}((C_k))$  denotes the (block-)diagonal matrix with blocks (resp. coefficients)  $(C_k)_k$ .
2 Characterization of local convergence
Consider the general smooth min-max game of Equation (1.1) and assume  $M$  is invertible. We decompose  $M$  into its symmetric and antisymmetric parts as
$$
M = \left[ \begin{array}{c c} \nabla_ {x x} ^ {2} f & \nabla_ {x y} ^ {2} f \\ - \nabla_ {x y} ^ {2} f ^ {\top} & - \nabla_ {y y} ^ {2} f \end{array} \right] (z ^ {*}) =: \left[ \begin{array}{c c} Q & P \\ - P ^ {\top} & R \end{array} \right], \qquad S := \left[ \begin{array}{c c} Q & 0 \\ 0 ^ {\top} & R \end{array} \right], \qquad A := \left[ \begin{array}{c c} 0 & P \\ - P ^ {\top} & 0 \end{array} \right].
$$
By the second-order optimality condition in the definition of NE,  $Q\succeq 0$  and  $R\succeq 0$
Following [Let+19],  $S$  and  $A$  can be thought of intuitively as the "potential" resp. "interaction" (or Hamiltonian) components of the two-player zero-sum game (1.1). Indeed, consider the quadratic
game  $\min_{x\in \mathbb{R}^n}\max_{y\in \mathbb{R}^m}\left\{\frac{1}{2}\binom{x}{y}^\top\nabla^2f(z^*)(\binom{x}{y}) = \frac{1}{2}x^\top Qx - \frac{1}{2}y^\top Ry + x^\top Py\right\}$ , which is essentially sufficient to understand the local behavior of GF for (1.1) around  $z^{*}$ . Then we can interpret the terms in  $Q$  and  $R$  as quadratic potentials to be optimized independently by each player, with the bilinear term in  $P$  capturing all of the interaction between the players.
Partial curvature generically suffices. Let us recall that  $\tilde{\mu}_M = \min_{\lambda \in \mathrm{Sp}(M)}\Re (\lambda)$  governs the local convergence of GF around the local NE  $z^{*}$ . It is a general fact that this quantity is nonnegative (the proof is included below). In the following theorem, we give necessary and sufficient conditions for it to be positive, in terms of geometric conditions on  $Q,R$  and  $P$ .
Theorem 2.1. The following conditions are equivalent:
(i)  $\tilde{\mu}_M > 0$
(ii) Eigvecs(A)  $\cap$  Ker  $S = \{0\}$
(iii) For any eigenvector  $x$  of  $PP^{\top}$  (i.e., left-singular vector of  $P$ ),  $x \notin \operatorname{Ker} Q$  or  $P^{\top}x \notin \operatorname{Ker} R$ .
(iv) For any eigenvector  $y$  of  $P^{\top}P$  (i.e., right-singular vector of  $P$ ),  $Py \notin \operatorname{Ker} Q$  or  $y \notin \operatorname{Ker} R$ .
As a consequence of Thm. 2.1, the condition  $\tilde{\mu}_M > 0$  holds generically in the following sense: for any fixed  $S \neq 0$ , the set of matrices  $P$  such that  $\tilde{\mu}_M > 0$  is dense and open in  $\mathbb{R}^{n \times m}$ . In particular, this property holds with probability 1 if  $P$  is drawn from an absolutely continuous distribution and is independent from  $Q$  and  $R$ , as in the experiment of Fig. 1.
Proof. Let  $\lambda \in \mathrm{Sp}(M)$  and  $z\in \mathbb{C}^{n + m}$  non-zero such that  $Mz = (S + A)z = \lambda z$ . Since  $S$  and  $A$  are real and symmetric resp. antisymmetric,
$$
\overline {{z}} ^ {\top} S z = z ^ {\top} S \bar {z} = (z ^ {\top} S \bar {z}) ^ {\top} = \bar {z} ^ {\top} S z, \text {s o} \bar {z} ^ {\top} S z \in \mathbb {R}
$$
and  $\overline{z}^{\top}Az = z^{\top}A\overline{z} = (z^{\top}A\overline{z})^{\top} = -\overline{z}^{\top}Az$  so  $\overline{z}^{\top}Az\in i\mathbb{R}$
So by taking the real part in  $\overline{z}^{\top}(S + A)z = \lambda \| z\|^{2}$ ,
$$
\Re (\lambda) \| z \| ^ {2} = \bar {z} ^ {\top} S z = \Re (z) ^ {\top} S \Re (z) + \Im (z) ^ {\top} S \Im (z) \geq 0
$$
since  $S = \operatorname{Diag}(Q, R) \succeq 0$ . This shows that  $\tilde{\mu}_M \geq 0$ .
Now let us show the equivalence of the conditions.
$(ii) \Rightarrow (i)$ : By contraposition, suppose there exists  $\lambda \in \mathrm{Sp}(M)$  with  $\Re(\lambda) = 0$ , and let  $z$  non-zero such that  $Mz = \lambda z$ . Then  $\overline{z}^{\top}Sz = \Re(\lambda)\|z\|^{2} = 0$ , so  $z \in \operatorname{Ker} S$ . So  $Mz = Az = \lambda z$ , and  $z \in (E_{\lambda}(A) \cap \operatorname{Ker} S) \setminus \{0\}$ .
$(i)\Rightarrow (ii)$  : By contraposition, suppose there exists  $\lambda \in \mathrm{Sp}(A)$  and a non-zero  $z\in E_{\lambda}(A)\cap \mathrm{Ker}S$  Since  $A$  is antisymmetric, then  $\Re (\lambda) = 0$  . On the other hand,  $Mz = (S + A)z = Az = \lambda z$  i.e.,  $\lambda \in \operatorname {Sp}(M)$  . So  $\tilde{\mu}_M\leq \Re (\lambda) = 0$
(i)  $\Rightarrow$  (iii), (iv): By contraposition, suppose there exists an eigenvector  $x$  of  $PP^{\top}$  such that  $x \in \operatorname{Ker} Q$  and  $P^{\top} x \in \operatorname{Ker} R$ , and denote  $\sigma \in \mathbb{R}$  such that  $PP^{\top} x = \sigma^{2} x$  (since  $PP^{\top} \succeq 0$ ). Then
$$
M \left( \begin{array}{c} i \sigma   x \\ P ^ {\top}   x \end{array} \right) = \left[ \begin{array}{c c} Q & P \\ - P ^ {\top} & R \end{array} \right] \left( \begin{array}{c} i \sigma   x \\ P ^ {\top}   x \end{array} \right) = \left( \begin{array}{c} \sigma^ {2} x \\ - i \sigma P ^ {\top}   x \end{array} \right) = - i \sigma \left( \begin{array}{c} i \sigma   x \\ P ^ {\top}   x \end{array} \right)
$$
and so  $-i\sigma \in \mathrm{Sp}(M)$ . This shows  $(i) \Longrightarrow (iii)$ , and  $(i) \Longrightarrow (iv)$  follows analogously.
$(iii),(iv)\Rightarrow (ii)$  : By contraposition, suppose there exists  $\lambda = i\sigma \in \operatorname {Sp}(A)$  and a non-zero  $z = (x,y)\in$ $E_{\lambda}(A)\cap \mathrm{Ker}S$  . Expanding the blocks in  $Az = \lambda z$
$$
\left\{ \begin{array}{l} P y = i \sigma x \\ - P ^ {\top} x = i \sigma y \end{array} \right. \quad \text {a n d s o} \quad \left\{ \begin{array}{l} P ^ {\top} P y = \sigma^ {2} y \\ P P ^ {\top} x = \sigma^ {2} x. \end{array} \right.
$$
Moreover, this implies that  $x = 0 \iff y = 0$ , and since  $z \neq 0$ , then both  $x \neq 0$  and  $y \neq 0$ . So  $x$  is an eigenvector of  $PP^{\top}$  and, since  $Sz = 0$ ,  $x \in \mathrm{Ker}Q$  and  $P^{\top}x = -i\sigma y \in \mathrm{Ker}R$ , which contradicts (iii). Likewise,  $y$  is an eigenvector of  $P^{\top}P$  and  $y \in \mathrm{Ker}R$  and  $Py = i\sigma x \in \mathrm{Ker}Q$ , which contradicts (iv).
Geometric interpretation using real vectors. We draw the attention of the reader to the fact that  $(ii)$  involves complex eigenvectors. For a rephrasing in terms of real objects, note that if  $z\in E_{i\sigma}(A)$  with  $\sigma \in \mathbb{R}$ , then  $A[\Re (z)\Im (z)] = [\Re (z)\Im (z)]\left[ \begin{array}{cc}0 & \sigma \\ -\sigma & 0 \end{array} \right]$ ; geometrically,  $F_{i\sigma}(A) = \mathrm{span}(\Re (z),\Im (z))$  is a "rotation plane" of the GF for the bilinear game  $\min_x\max_yx^\top Py$ , in the sense that the projection of GF on  $F_{i\sigma}(A)$  is a circular motion with constant speed  $\sigma$ . Condition  $(ii)$  expresses that for each such  $F_{i\sigma}(A)$ , there exists an eigenspace  $E_{\mu}(S)$  of  $S$  (for a  $\mu >0$ ) that is not orthogonal to it. This causes the GF for  $\min_x\max_y\frac{1}{2} x^\top Qx - \frac{1}{2} y^\top Ry + x^\top Py$  projected on  $F_{i\sigma}(A)$  to spiral down to 0 instead of cycling around it.
One may naturally wonder whether a notion of non-orthogonality between the potential  $(S)$  and interaction components  $(A)$  can be used to bound the convergence quantitatively; this is developed in the next section in the particular case  $S \ll A$ .
3 Convergence rate when interaction dominates  $(S\ll A)$
Let us now discuss the case of games with a small symmetric part, that is, whose Jacobian at optimum is  $M_{\alpha} = A + \alpha S$  for some symmetric  $S = \begin{bmatrix} Q & 0 \\ 0^{\top} & R \end{bmatrix}$ , antisymmetric  $A = \begin{bmatrix} 0 & P \\ -P^{\top} & 0 \end{bmatrix}$  and some small  $\alpha > 0$ . In this section we assume  $n = m$  (the general case is technically more challenging as Prop. 3.1 requires  $A$  to have distinct eigenvalues, which requires  $|n - m| \leq 1$ ).
3.1 Convergence rate of Gradient Flow
As discussed previously, the normalized local exponential convergence rate  $r / \eta$  of gradient methods in the asymptotic regime  $\eta \rightarrow 0$  – or equivalently, the convergence rate of GF – for a game with Jacobian at optimum  $M_{\alpha}$  is equal to  $\tilde{\mu}_{M_{\alpha}}$ . We can estimate this quantity using the standard formula for the asymptotic expansion of the eigenvalues of a perturbed matrix, which takes an interesting form in our context.
Proposition 3.1. Suppose that  $P$  is full-rank and has distinct singular values, and let  $P = U\Sigma V^{\top} = \sum_{j=1}^{n} \sigma_{j} u_{j} v_{j}^{\top}$  be its singular value decomposition. Then it holds
$$
\tilde {\mu} _ {M _ {\alpha}} = \frac {1}{2} \alpha \Big (\min  _ {1 \leq j \leq n} u _ {j} ^ {\top} Q u _ {j} + v _ {j} ^ {\top} R v _ {j} \Big) + O \left(\alpha^ {3}\right).
$$
This expansion explains in particular why the normalized convergence rate  $r / \eta \sim \tilde{\mu}_{M_{\alpha}}$  is approximately proportional to  $\alpha$  in Fig. 1. Interestingly, the error term is  $O(\alpha^3)$ , which suggests that the approximation can be reasonably accurate even for quite large values of  $\alpha$ , as illustrated in Fig. 1c.
Proof. Here  $M_0 = A$  has distinct eigenvalues  $\{is\sigma_j, s \in \{-1, 1\}, 1 \leq j \leq n\}$ , with unit-norm eigenvectors  $A\left(\frac{-isu_j / \sqrt{2}}{v_j / \sqrt{2}}\right) = is\sigma_j\left(\frac{-isu_j / \sqrt{2}}{v_j / \sqrt{2}}\right)$ . By the calculation of the eigenvalue derivatives from [Tao08], we obtain the following expansion for  $\mathrm{Sp}(M_\alpha)$ :
$$
\begin{array}{l} \operatorname {S p} (M _ {\alpha}) = \left\{i s \sigma_ {j} + \frac {1}{2} \alpha \left(u _ {j} ^ {\top} Q u _ {j} + v _ {j} ^ {\top} R v _ {j}\right) + \frac {1}{4} \alpha^ {2} \sum_ {(s ^ {\prime}, j ^ {\prime}) \neq (s, j)} \frac {1}{i s \sigma_ {j} - i s ^ {\prime} \sigma_ {j ^ {\prime}}} \left(s s ^ {\prime} u _ {j ^ {\prime}} ^ {\top} Q u _ {j} + v _ {j} ^ {\top} R v _ {j}\right) ^ {2} \right. \\ + O \left(\alpha^ {3}\right), s \in \{- 1, 1 \}, 1 \leq j \leq n \}. \tag {3.1} \\ \end{array}
$$
Now the zeroth- and second-order terms are all in  $i\mathbb{R}$ , hence the announced expansion for  $\tilde{\mu}_{M_{\alpha}}$
Estimate of the leading term under a probabilistic model. Assuming the singular vectors  $(u_{1},\dots,u_{n}),(v_{1},\dots,v_{n})$  of  $P$  are distributed uniformly at random - which is the case for example if  $P$  has i.i.d. Gaussian entries by rotational invariance, the leading term in the expansion of  $\tilde{\mu}_{M_{\alpha}}$  can be estimated in expectation as follows. The proof is placed in the Appendix, where we also include a high-probability version of the estimate.
Proposition 3.2. Suppose  $Q, R$  are fixed and  $U, V$  are independently distributed uniformly on the set of  $n \times n$  orthonormal matrices. Then
$$
\frac {\operatorname {T r} (S)}{n} \left(1 - 2 \frac {\| S \| _ {F}}{\operatorname {T r} (S)} \sqrt {\log n}\right) \leq \mathbb {E} \left[ \min  _ {1 \leq j \leq n} u _ {j} ^ {\top} Q u _ {j} + v _ {j} ^ {\top} R v _ {j} \right] \leq \frac {\operatorname {T r} (S)}{n}
$$
where  $\| \cdot \| _F$  denotes the Frobenius norm. In particular, provided that  $\frac{\mathrm{Tr}(S)}{\|S\|_F} \geq 2\sqrt{\log n} (1 + c)$  for some fixed  $c > 0$ , we have  $\mathbb{E}\left[\min_{1\leq j\leq n}u_j^\top Q u_j + v_j^\top R v_j\right] \asymp \frac{\mathrm{Tr}(S)}{n}$  as  $n\to \infty$ .
Note that  $\frac{\mathrm{Tr}(S)}{\|S\|_F}$ , which always lies in the interval  $[1, \sqrt{2n}]$ , is a measure of the effective sparsity of the spectrum of  $S$  (larger meaning less sparse), so the condition  $\frac{\mathrm{Tr}(S)}{\|S\|_F} \geq 2\sqrt{\log n}(1 + c)$  means the spectrum of  $S$  is well spread-out. So the proposition shows that the exponential convergence rate depends on the average of the eigenvalues of  $S$ , when  $\alpha S \ll A$  and the spectrum of  $S$  is well spread-out. This fact should be contrasted with the case of minimization, where the convergence rate scales as the minimum eigenvalue of the Hessian.