SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In this time-scale separated setting, model (5) represents a dynamic maximization of  $G_{c}$  with respect to  $\rho$  in Wasserstein-2 space, and an instantaneous minimization of  $G_{c}$  with respect to the algorithm parameters  $x$ . Model (6) represents an instantaneous maximization of  $G_{c}$  with respect to  $\rho$  and a dynamic minimization of  $G_{c}$  with respect to the algorithm parameters  $x$ . The key results on existence and uniqueness of a ground state as well as the convergence behavior of solutions depend on convexity (concavity) of  $G_{a}$  and  $G_{c}$ . The notion of convexity that we will employ for energy functionals in the Wasserstein-2 geometry is (uniform) displacement convexity, which is analogous to (strong) convexity in Euclidean spaces. One can think of displacement convexity for an energy functional defined on  $\mathcal{P}_2$  as convexity along the shortest path in the Wasserstein-2 metric (linear interpolation in the Wasserstein-2 space) between any two given probability distributions. For a detailed definition of (uniform) displacement convexity and concavity, see Section A.2. In fact, suitable convexity properties of the input functions  $f_{1}, f_{2}, W$  and  $\tilde{\rho}$  will ensure (uniform) displacement convexity of the resulting energy functionals appearing in the gradient flow structure, see for instance [Vil03, Chapter 5.2].
We make the following assumptions in both the competitive case and aligned interest cases. Here,  $\mathrm{I}_d$  denotes the  $d\times d$  identity matrix,  $\operatorname {Hess}(f)$  denotes the Hessian of  $f$  in all variables, while  $\nabla_x^2 f$  denotes the Hessian of  $f$  in the variable  $x$  only.
Assumption 1 (Convexity of  $f_{1}$  and  $f_{2}$ ). The functions  $f_{1}, f_{2} \in C^{2}(\mathbb{R}^{d} \times \mathbb{R}^{d}; [0, \infty))$  satisfy for all  $(z, x) \in \mathbb{R}^{d} \times \mathbb{R}^{d}$  the following:
- There exists constants  $\lambda_1, \lambda_2 \geq 0$  such that  $\text{Hess}(f_1) \succeq \lambda_1 \mathrm{I}_{2d}$  and  $\nabla_x^2 f_2 \succeq \lambda_2 \mathrm{I}_d$ ;
- There exist constants  $a_i > 0$  such that  $x \cdot \nabla_x f_i(z, x) \geq -a_i$  for  $i = 1, 2$ ;
Assumption 2 (Reference Distribution Shape). The reference distribution  $\tilde{\rho} \in \mathcal{P}(\mathbb{R}^d) \cap L^1(\mathbb{R}^d)$  satisfies  $\log \tilde{\rho} \in C^2(\mathbb{R}^d)$  and  $\nabla_z^2 \log \tilde{\rho}(z) \preceq -\tilde{\lambda} I_d$  for some  $\tilde{\lambda} > 0$ .
Assumption 3 (Convex Interaction Kernel). The interaction kernel  $W \in C^{2}(\mathbb{R}^{d};[0,\infty))$  is convex, symmetric  $W(-z) = W(z)$ , and for some  $D > 0$  satisfies
$$
z \cdot \nabla_ {z} W (z) \geq - D, \quad | \nabla_ {z} W (z) | \leq D (1 + | z |) \quad \forall z \in \mathbb {R} ^ {d}.
$$
We make the following observations regarding the assumptions above:
- The convexity in Assumption 3 can be relaxed and without affecting the results outlined below by following a more detailed analysis analogous to the approach in [CMV03].
- If  $f_{1}$  and  $f_{2}$  are strongly convex, the proveable convergence rate increases, but without strict or strong convexity of  $f_{1}$  and  $f_{2}$ , the regularizers  $KL(\rho |\tilde{\rho})$  and  $\int \| x - x_0\| _2^2\mathrm{d}x$  provide the convexity guarantees necessary for convergence.
For concreteness, one can consider the following classical choices of input functions to the evolution:
- Using the log-loss function for  $f_{1}$  and  $f_{2}$  satisfies Assumption 1.
- Taking the reference measure  $\tilde{\rho}$  to be the normal distribution satisfies Assumption 2, which ensures the distribution is not too flat.
- Taking quadratic interactions  $W(z) = \frac{1}{2} |z|^2$  satisfies Assumption 3.
Remark 1 (Cauchy-Problem). To complete the arguments on convergence to equilibrium, we require sufficient regularity of solutions to the PDEs under consideration. In fact, it is sufficient if we can show that equations (4), (5), and (6) can be approximated by equations with smooth solutions. Albeit tedious, these are standard techniques in the regularity theory for partial differential equations, see for example [CMV03, Proposition 2.1 and Appendix A], [OV00], [Vil03, Chapter 9], and the references therein. Similar arguments as in [DV00] are expected to apply to the coupled gradient flows considered here, guaranteeing existence of smooth solutions with fast enough decay at infinity, and we leave a detailed proof for future work.
3.2 Analysis of Case (i): Aligned Objectives
The primary technical contribution of this setting consists of lifting the algorithm dynamics from an ODE to a PDE, which allows us to model the system as a joint gradient flow on the product space of probability measures. The coupling occurs in the potential function, rather than as cross-diffusion or non-local interaction as more commonly seen in the literature for multi-species systems.
Theorem 2. Suppose that Assumptions 1-3 are satisfied and let  $\lambda_{a} \coloneqq \lambda_{1} + \min (\lambda_{2} + \beta, \alpha \tilde{\lambda}) > 0$ . Consider solutions  $\gamma_{t} \coloneqq (\rho_{t}, \mu_{t})$  to the dynamics (4) with initial conditions satisfying  $\gamma_0 \in \mathcal{P}_2(\mathbb{R}^d) \times \mathcal{P}_2(\mathbb{R}^d)$  and  $G_{a}(\gamma_{0}) < \infty$ . Then the following hold:
(a) There exists a unique minimizer  $\gamma_{\infty} = (\rho_{\infty},\mu_{\infty})$  of  $G_{a}$ , which is also a steady state for equation (4). Moreover,  $\rho_{\infty}\in L^{1}(\mathbb{R}^{d})$ , has the same support as  $\tilde{\rho}$ , and its density is continuous.
(b) The solution  $\gamma_t$  converges exponentially fast in  $G_a(\cdot|\gamma_\infty)$  and  $\overline{W}$ ,
$$
\begin{array}{l} G _ {a} \left(\gamma_ {t} \mid \gamma_ {\infty}\right) \leq e ^ {- 2 \lambda_ {a} t} G _ {a} \left(\gamma_ {0} \mid \gamma_ {\infty}\right) \quad a n d \quad \overline {{W}} \left(\gamma_ {t}, \gamma_ {\infty}\right) \leq c e ^ {- \lambda_ {a} t} \quad f o r a l l t \geq 0, \\ w h e r e c > 0 i s a c o n s t a n t o n l y d e p e n d i n g o n \gamma_ {0}, \gamma_ {\infty} a n d t h e p a r a m e t e r \lambda_ {a}. \\ \end{array}
$$
Proof. (Sketch) For existence and uniqueness, we leverage classical techniques in the calculus of variations. To obtain convergence to equilibrium in energy, our key result is a new HWI-type inequality, providing as a consequence generalizations of the log-Sobolev inequality and the Talagrand inequality. Together, these inequalities relate the energy (classically denoted by  $H$  in the case of the Boltzmann entropy), the metric (classically denoted by  $W$  in the case of the Wasserstein-2 metric) and the energy dissipation (classically denoted by  $I$  in the case of the Fisher information). Combining these inequalities with Gronwall's inequality allows us to deduce convergence both in energy and in the metric  $\overline{W}$ .
3.3 Analysis of Case (ii): Competing Objectives
In this setting, we consider the case where the algorithm and the strategic population have goals in opposition to each other; specifically, the population benefits from being classified incorrectly. First, we will show that when the algorithm instantly best-responds to the population, then the distribution of the population converges exponentially in energy and in  $W_{2}$ . Then we will show a similar result for the case where the population instantly best-responds to the algorithm.
In both cases, we begin by proving two Danskin-type results (see [Dan67; Ber71]) which will be used for the main convergence theorem, including convexity (concavity) results. To this end, we make the following assumption ensuring that the regularizing component in the evolution of  $\rho$  is able to control the concavity introduced by  $f_{1}$  and  $f_{2}$ .
Assumption 4 (Upper bounds for  $f_{1}$  and  $f_{2}$ ). There exists a constant  $\Lambda_{1} > 0$  such that
$$
\nabla_ {z} ^ {2} f _ {1} (z, x) \preceq \Lambda_ {1} I _ {d} \quad \text {f o r a l l} (z, x) \in \mathbb {R} ^ {d} \times \mathbb {R} ^ {d},
$$
and for any  $R > 0$  there exists a constant  $c_{2} = c_{2}(R)\in \mathbb{R}$  such that
$$
\sup  _ {x \in B _ {R} (0)} \int f _ {2} (z, x) \mathrm {d} \bar {\rho} (z) <   c _ {2}.
$$
Hence the name HWI inequalities.
Equipped with Assumption 4, we state the result for a best-responding algorithm.
Theorem 3. Suppose Assumptions 1-4 are satisfied with  $\alpha \tilde{\lambda} > \Lambda_1$ . Let  $\lambda_b \coloneqq \alpha \tilde{\lambda} - \Lambda_1$ . Define  $G_b(\rho) \coloneqq G_c(\rho, b(\rho))$ . Consider a solution  $\rho_t$  to the dynamics (5) with initial condition  $\rho_0 \in \mathcal{P}_2(\mathbb{R}^d)$  such that  $G_b(\rho_0) < \infty$ . Then the following hold:
(a) There exists a unique maximizer  $\rho_{\infty}$  of  $G_{b}(\rho)$ , which is also a steady state for equation (5). Moreover,  $\rho_{\infty} \in L^{1}(\mathbb{R}^{d})$ , has the same support as  $\tilde{\rho}$ , and its density is continuous.
(b) The solution  $\rho_t$  converges exponentially fast to  $\rho_{\infty}$  with rate  $\lambda_{b}$  in  $G_{b}(\cdot |\rho_{\infty})$  and  $W_{2}$ ,
$$
G _ {b} \left(\rho_ {t} \mid \rho_ {\infty}\right) \leq e ^ {- 2 \lambda_ {b} t} G _ {a} \left(\rho_ {0} \mid \rho_ {\infty}\right) \quad a n d \quad W _ {2} \left(\rho_ {t}, \rho_ {\infty}\right) \leq c e ^ {- \lambda_ {b} t} \quad f o r a l l t \geq 0,
$$
where  $c > 0$  is a constant only depending on  $\rho_0, \rho_\infty$  and the parameter  $\lambda_b$ .
Proof. (Sketch) The key addition in this setting as compared with Theorem 2 is proving that  $G_{b}(\rho)$  is bounded below, uniformly displacement concave and guaranteeing its smoothness via Berge's Maximum Theorem. This is non-trivial as it uses the properties of the best response  $b(\rho)$ . A central observation for our arguments to work is that  $\delta_{\rho}G_{b}[\rho] = (\delta_{\rho}G_{c}[\rho,x])|_{x=b(\rho)}$ . We can then conclude using the direct method in the calculus of variations and the HWI method.
Here, the condition that  $\alpha \tilde{\lambda}$  must be large enough corresponds to the statement that the system must be subjected to a strong enough regularizing effect.
In the opposite case, where  $\rho$  instantly best-responds to the algorithm, we show Danskin-like results for derivatives through the best response function and convexity of the resulting energy in  $x$  which allows to deduce convergence.
Theorem 4. Suppose Assumptions 1-4 are satisfied with  $\alpha \tilde{\lambda} > \Lambda_1$ , and that  $r(x)$  is differentiable (as shown by example conditions in Lemmas 27 and 28). Define  $G_{d}(x) \coloneqq G_{c}(r(x), x)$ . Then it holds:
(a) There exists a unique minimizer  $x_{\infty}$  of  $G_{d}(x)$  which is also a steady state for (6).
(b) The vector  $x(t)$  solving the dynamics (6) with initial condition  $x(0) \in \mathbb{R}^d$  converges exponentially fast to  $x_{\infty}$  with rate  $\lambda_d \coloneqq \lambda_1 + \lambda_2 + \beta > 0$  in  $G_d$  and in the Euclidean norm:
$$
\begin{array}{l} \left\| x (t) - x _ {\infty} \right\| \leq e ^ {- \lambda_ {d} t} \left\| x (0) - x _ {\infty} \right\|, \\ G _ {d} (x (t)) - G _ {d} (x _ {\infty}) \leq e ^ {- 2 \lambda_ {d} t} \left(G _ {d} (x (0)) - G _ {d} (x _ {\infty})\right) \\ \end{array}
$$
for all  $t \geq 0$ .
These two theorems illustrate that, under sufficient convexity conditions on the cost functions, we expect the distribution  $\rho$  and the algorithm  $x$  to converge to a steady state. In practice, when the distributions are close enough to the steady state there is no need to retrain the algorithm.
While we have proven results for the extreme timescale cases, we anticipate convergence to the same equilibrium in the intermediate cases. Indeed, it is well known [Bor09] (especially for systems in Euclidean space) that for two-timescale stochastic approximations of dynamical systems, with appropriate stepsize choices, converge asymptotically, and finite-time high probability concentration bounds can also be obtained. These results have been leveraged in strategic classification [Zrn+21] and Stackelberg games [FCR20; FR21; Fie+21]. We leave this intricate analysis to future work.
In the following section we show numerical results in the case of a best-responding  $x$ , best-responding  $\rho$ , and in between where  $x$  and  $\rho$  evolve on a similar timescale. Note that in these settings, the dynamics do not have a gradient flow structure due to a sign difference in the energies, requiring conditions to ensure that one species does not dominate the other.
4 Numerical Examples
We illustrate numerical results for the case of a classifier, which are used in scenarios such as loan or government aid applications [CC11], school admissions [PS13], residency match [Ree18], and recommendation algorithms [LSW10], all of which have some population which is incentivized to submit data that will result in a desirable classification. For all examples, we select classifiers
of the form  $x \in \mathbb{R}$ , so that a data point  $z \in \mathbb{R}$  is assigned a label of 1 with probability  $q(z, x) = (1 + \exp(-b^\top z + x))^{-1}$  where  $b > 0$ . Let  $f_1$  and  $f_2$  be given by
$$
f _ {1} (z, x) = - \log (1 - q (z, x)), \qquad f _ {2} (z, x) = - \log q (z, x).
$$
Note that  $\mathrm{Hess}(f_1) \succeq 0$  and  $\nabla_x^2 f_2 \succeq 0$ , so  $\lambda_1 = \lambda_2 = 0$ . Here, the strictness of the convexity of the functional is coming from the regularizers, not the cost functions, with  $\tilde{\rho}$  a scaled normal distribution. We show numerical results for two scenarios with additional settings in the appendix. First we illustrate competitive interests under three different timescale settings. Then we simulate the classifier taking an even more naive strategy than gradient descent and discuss the results. The PDEs were implemented based on the finite volume method from [CCH15].
4.1 Competitive Objectives
In the setting with competitive objectives, we utilize  $G_{c}(\rho ,x)$  with  $W = 0$ ,  $f_{1}$  and  $f_{2}$  as defined above with  $b = 3$  fixed as it only changes the steepness of the classifier for  $d = 1$ , and  $\alpha = 0.1$  and  $\beta = 0.05$ . In Figure 1, we simulate two extremes of the timescale setting; first when  $\rho$  is nearly best-responding and then when  $x$  is best-responding. The simulations have the same initial conditions and end with the same distribution shape; however, the behavior of the strategic population differs in the intermediate stages. When  $\rho$  is nearly best-responding, we see that the distribution quickly shifts
Figure 1: When  $x$  versus  $\rho$  best-responds, we observe the same final state but different intermediate states. Modes appear in the strategic population which simpler models cannot capture.
mass over the classifier threshold. Then the classifier shifts right, correcting for the shift in  $\rho$ , which then incentivizes  $\rho$  to shift more mass back to the original mode. In contrast, when  $x$  best-responds, the right-hand mode slowly increases in size until the system converges.
Figure 2 shows simulation results from the setting where  $\rho$  and  $x$  evolve on the same timescale. We observe that the distribution shift in  $\rho$  appears to fall between the two extreme timescale cases, which we expect. We highlight two important observations for the competitive case. One, a single-mode
Figure 2: In this experiment the population and classifier have similar rates of change, and the distribution change for  $\rho$  exhibits behaviors from both the fast  $\rho$  and fast  $x$  simulations; the right-hand mode does not peak as high as the fast  $\rho$  case but does exceed its final height and return to the equilibrium.
distribution becomes bimodel, which would not be captured using simplistic metrics such as the mean and variance. This split can be seen as polarization in the population, a phenomenon that a mean-based strategic classification model would not capture. Two, the timescale on which the
classifier updates significantly impacts the intermediate behavior of the distribution. In our example, when  $x$  updated slowly relative to the strategic population, the shifts in the population were greater than in the other two cases. This suggests that understanding the effects of timescale separation are important for minimizing volatility of the coupled dynamics.
4.2 Naive Behavior
In this example, we explore the results of the classifier adopting a non-gradient-flow strategy, where the classifier chooses an initially-suboptimal value for  $x$  and does not move, allowing the strategic population to respond. All functions and parameters are the same as in the previous example. When
(a) Both species minimize their respective losses; when the classifier uses a naive strategy, the final performance is better for the classifier and uniformly worse for the population.
(b) The classifier selects a suboptimal initial condition  $x = 2.2$ , instead of  $x = 1.5$  which minimizes the initial loss, and then does not move in response to the population.
Figure 3: Although the classifier starts with a larger cost by taking the naive strategy, the final loss is better. This illustrates how our model can be used to compare robustness of different strategies against a strategic population.
comparing with the gradient descent strategy, we observe that while the initial loss for the classifier is worse for the naive strategy, the final cost is better. While this results is not surprising, because one can view this as a general-sum game where the best response to a fixed decision may be better than the equilibrium, it illustrates how our method provides a framework for evaluating how different training strategies perform in the long run against a strategic population.
5 Future Directions, Limitations, and Broader Impact
Our work presents a method for evaluating the robustness of an algorithm to a strategic population, and investigating a variety of robustness using our techniques opens a range of future research directions. Our application suggests many questions relevant to the PDE literature, such as: (1) Does convergence still hold with the gradient replaced by an estimated gradient? (2) Can we prove convergence in between the two timescale extremes? (3) How do multiple dynamic populations respond to an algorithm, or multiple algorithms? In the realm of learning algorithms, our framework can be extended to other learning update strategies and presents a way to model how we can design these update strategies to induce desired behaviors in the population.
A challenge in our method is that numerically solving high-dimensional PDEs is computationally expensive and possibly unfeasible. Here we note that in many applications, agents in the population do not alter more than a few features due to the cost of manipulation. We are encouraged by the recent progress using deep learning to solve PDEs, which could be used in our application.
Broader Impacts Modeling the full population distribution rather than simple metrics of the distribution is important because not all individuals are affected by the algorithm in the same way. For example, if there are tails of the distribution that have poor performance even if on average the model is good, we need to know how that group is advantaged or disadvantaged relative to the rest of the population. Additionally, understanding how people respond to algorithms offers an opportunity to incentivise people to move in a direction that increases social welfare.
LC is supported by an NDSEG fellowship from the Air Force Office of Scientific Research. FH is supported by start-up funds at the California Institute of Technology. LR is supported by ONR YIP N00014-20-1-2571 P00003 and NSF Awards CAREER 1844729, and CPS 1931718. EM acknowledges support from NSF Award 2240110. We are grateful for helpful discussions with José A. Carrillo.
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A General structure and preliminaries
In this section, we give more details on the models discussed in the main article, and introduce definitions and notation that are needed for the subsequent proofs.
A.1 Structure of the dynamics
For the case of aligned objectives, the full coupled system of PDEs (4) can be written as
$$
\partial_ {t} \rho = \alpha \Delta \rho + \operatorname {d i v} \left(\rho \nabla_ {z} \left(\int f _ {1} \mathrm {d} \mu - \alpha \log \tilde {\rho} + W * \rho\right)\right), \tag {7a}
$$
$$
\partial_ {t} \mu = \operatorname {d i v} \left(\mu \nabla_ {x} \left(\int f _ {1} \mathrm {d} \rho + \int f _ {2} \mathrm {d} \bar {\rho} + \frac {\beta}{2} \| x - x _ {0} \| ^ {2}\right)\right). \tag {7b}
$$