SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Unlike the first setting, we observe in Figure 7 a noticeable difference between the evolution of  $\rho_{t}$  with  $n = 4$  versus  $n = 40$  samples. This is not surprising because optimizing with a very poor estimate of the cost function at each time step would cause  $x_{t}$  to vary wildly, and this method fails to take advantage of correct "average" behavior that gradient descent provides.
Figure 6: When the classifier is updating at the same rate as the population, we do not see a significant change in the evolution of both species, suggesting that as long as the gradient estimate for the classifier is correct on average, the estimate itself does not need to be particularly accurate.
Figure 7: When the classifier is best-responding to the population, we observe that using  $n = 4$  samples leads to different behavior for both the classifier and the population, compared with a more accurate estimate using  $n = 40$  samples.
E.4 Two-dimensional Distributions
In practice, individuals may alter more that one of their attributes in response to an algorithm, for example, both cancelling a credit card and also reporting a different income in an effort to change a credit score. We model this case with  $z \in \mathbb{R}^2$  and  $x \in \mathbb{R}^2$ , and simulate the results for the setting where the classifier and the population are evolving at the same rate. While this setting is not covered in our theory, it interpolates between the two timescale extremes.
We consider the following classifier:
$$
f _ {1} (z, x) = \frac {1}{2} \left(1 - \frac {1}{1 + \exp x ^ {\top} z}\right) \tag {41}
$$
$$
f _ {2} (z, x) = \frac {1}{2} \left(\frac {1}{1 + \exp x ^ {\top} z}\right)
$$
with  $W = 0$ . Again, the reference distribution  $\tilde{\rho}$  corresponds to the initial shape of the distribution, instituting a penalty for deviating from the initial distribution. We use  $\alpha = 0.5$  and  $\beta = 1$  for the penalty weights, run for  $t = 4$  with  $dt = 0.005$  and  $dx = dy = 0.2$  for the discretization. In this case, the strategic population is competing with the classifier, with dynamics given by
$$
\partial_ {t} \rho = - \operatorname {d i v} \left(\rho \nabla_ {z} \left(f _ {1} (z, x) - \alpha \log (\rho / \tilde {\rho})\right)\right)
$$
In Figure 8, we observe the strategic population increasing mass toward the region of higher probability of being labeled "1" while the true underlying label is zero, with the probability plotted at time  $t = 4$ . This illustrates
Figure 8: We use (41) for the classifier functions, using a Gaussian initial condition and regularizer for  $\rho$ . We see the distribution moving toward the region with higher probability of misclassification.
similar behavior to the one-dimensional case, including the distribution splitting into two modes, which is another example of polarization induced by the classifier. Note that while in this example,  $x \in \mathbb{R}^2$  and we use a linear classifier; we could have  $x \in \mathbb{R}^d$  with  $d > 2$  and different functions for  $f_1$  and  $f_2$  which yield a nonlinear classifier; our theory in the timescale-separated case holds as long as the convexity and smoothness assumptions on  $f_1$  and  $f_2$  are satisfied.