SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We conclude this section by comparing our notion of signal to noise ratio to that of Agarwal et al. [7], who define  $\mathrm{snr}_n(a)$  instead as  $\frac{\sigma_r(\mathbf{X}_n(a))}{\sqrt{n} + \sqrt{d}}$ , i.e. the ratio of the "signal" in the covariates to the expected operator norm of covariate noise  $\mathbb{E}\| \mathcal{E}_n\|_{op}$ . Since the goal of our work is high-probability (not in expectation) estimation guarantees for PCR, we believe using high probability bounds on  $\| \mathcal{E}_n\|_{op}$  is more natural when defining  $\mathrm{snr}_n(a)$ .
4 Adaptive bounds for online (regularized) PCR
We now present the main results of this work—high-probability, time- and action-uniform bounds measuring the convergence of the PCR estimates  $\widehat{\theta}_n(a)$  to the true slope vectors  $\theta (a)$ . Unlike existing results [6-8], our bounds are valid when the covariates  $(X_{n})_{n\geq 1}$  and actions  $(a_{n})_{n\geq 1}$  are determined in an online (potentially adversarial) manner.
We first point out why the analysis of Agarwal et al. [7, 8] breaks down in the setting of adaptive (or online) PCR. First, many of the concentration inequalities leveraged in Agarwal et al. [7] do not hold in the adaptive design setting. As a particular example, the authors leverage the Hanson-Wright inequality [82, 73] for quadratic forms to study how the noisy covariate matrix  $\mathbf{Z}_n$  concentrates around the true matrix  $\mathbf{X}_n$ . This inequality fails to hold when the design points  $(X_{n})_{n\geq 1}$  depend on the previous observations. Second, the techniques leveraged by Agarwal et al. [8] to extend the convergence guarantees of PCR to the multiple action setting fail to hold when the  $n$ -th action  $a_{n}$  is selected based on previous observations. Lastly, the bounds presented in [7] are are inherently fixed-time in nature—a simple way to convert existing fixed-time bounds to time-uniform ones would be to perform a union bound over time steps, but that introduces looseness in the bounds.
We are able to construct our bounds by exploiting connections between online PCR and self-normalized martingale concentration [50, 51, 32, 33]. In particular, we combine martingale-based results for constructing confidence ellipsoids for online regression [3, 32, 33] with methods for high-dimensional covariance estimation [83, 80] to prove our results. Exploiting this connection is what allows us to extend the results of Agarwal et al. [7] to the adaptive design, time-uniform setting. We begin with a bound which, up to constants and polylogarithmic factors, captures the rate of convergence of online PCR in terms of (a) the underlying signal to noise ratio and (b) the conditioning of the observed data.
Theorem 4.1 (Rate of Convergence for Online PCR). Let  $\delta \in (0,1)$  be an arbitrary confidence parameter. Let  $\rho > 0$  be chosen to be sufficiently small, as detailed in Appendix F. Further, assume that there is some  $n_0 \geq 1$  such that  $\mathrm{rank}(\mathbf{X}_{n_0}(a)) = r$  and  $\mathrm{snr}_n(a) \geq 2$  for all  $n \geq n_0$ . Then, with probability at least  $1 - O(A\delta)$ , simultaneously for all actions  $a \in [A]$  and time steps  $n \geq n_0$ , we have
$$
\| \widehat {\theta} _ {n} (a) - \theta (a) \| _ {2} ^ {2} = \widetilde {O} \left(\frac {1}{\mathrm {s n r} _ {n} (a) ^ {2}} \kappa (\mathbf {X} _ {n} (a)) ^ {2}\right),
$$
where  $\kappa(\mathbf{X}_n(a)) := \frac{\sigma_1(\mathbf{X}_n(a))}{\sigma_r(\mathbf{X}_n(a))}$  is the condition number (ignoring zero singular values) of  $\mathbf{X}_n(a)$ .
Theorem 4.1 is proved in Appendix E. We begin by comparing our bounds to those of Agarwal et al. [7, 8]. At any fixed time, our bounds take on roughly the same form as those of the aforementioned authors, having an inverse quadratic dependence on the signal to noise ratio. To make their bounds non-vacuous, the authors need to make the "soft sparsity" assumption of  $\| \theta (a)\| _1 = O(\sqrt{d})$ . Our bound, on the other hand, suffers no dependence on the  $\ell_1$ -norm of the  $\theta (a)$ 's. This makes intuitive sense, as the specific choice of a basis should not impact the rate of convergence of PCR. However, our bounds pay a price for adaptivity—in particular, the signal to noise ratio associated with an action is defined with respect to a bound on the total operator norm of the matrix  $\mathcal{E}_n$ . If an action is selected very infrequently, the above bound may become loose.
While the above bound is stated in terms of signal to noise ratio, if we make additional assumptions, we can obtain bounds directly in terms of  $d$ ,  $n$ , and  $r$ . In particular, the following "well-balancing" assumptions suffice.
Assumption 4.2 (Well-balancing assumptions). For all  $n \geq n_0$ , the following hold: (a)  $\sigma_i(\mathbf{X}_n(a)) = \Theta\left(\sqrt{\frac{c_n(a)d}{r}}\right)$  for all  $i \in [r]$ , (b)  $c_n(a) = \Theta(c_n(a'))$  for all  $a, a' \in [A]$ , and (c)  $A = O(r)$ .
Corollary 4.3. Assume the same setup as Theorem 4.1, and further assume Assumption 4.2 holds. Then with probability at least  $1 - O(A\delta)$ , simultaneously for all actions  $a \in [A]$  and time steps  $n \geq n_0$ , we have
$$
\left\| \widehat {\theta} _ {n} (a) - \theta (a) \right\| _ {2} ^ {2} = \widetilde {O} \left(\frac {r ^ {2}}{d \wedge n}\right).
$$
Corollary 4.3 shows that Theorem 4.1 obtains the same estimation rate as Theorem 4.1 of Agarwal et al. [7] if assumption Assumption 4.2 holds. This "well-balancing" assumption says roughly that all non-zero singular values of  $\mathbf{X}_n$  are of the same order, each action is selected with the same frequency, and that the number of actions is, at most, proportional to dimension of the true, unknown subspace. As noted by Agarwal et al. [7], the assumption of a "well-balanced spectrum" (for  $\mathbf{X}_n$ ) is common in many works in econometrics and robust statistics, and additionally holds with high probability if the entries of  $\mathbf{X}_n$  are i.i.d.[62, 21, 37]. Further, it is often the case that there only few available actions (for instance, in the synthetic control literature there are only two actions [2, 1, 38]), justifying the assumption that  $A = O(r)$ . Lastly, ensuring that each action is played (very roughly) the same number of times can be viewed as a price for adaptivity.
The proof of Theorem 4.1 is immediate as a corollary from the following, more complicated bound. Theorem 4.4 below measures the convergence of  $\widehat{\theta}_n(a)$  to  $\theta (a)$  in terms of empirical (i.e. observed) quantities. We imagine this bound to be the most practically relevant of our results, as, unlike the results of Agarwal et al. [7], it is directly computable by the learner, involves known constants, and places minimal conditions on the signal to noise ratio.
Theorem 4.4 (Empirical Guarantees for Online PCR). Let  $\delta \in (0,1)$  be an arbitrary confidence parameter. Let  $\rho > 0$  be chosen to be sufficiently small, as detailed in Appendix F. Further, assume that there is some  $n_0 \geq 1$  such that  $\mathrm{rank}(\mathbf{X}_{n_0}(a)) = r$  and  $\mathrm{snr}_n(a) \geq 2$  for all  $n \geq n_0$ . Then, with probability at least  $1 - O(A\delta)$ , simultaneously for all actions  $a \in [A]$  and time steps  $n \geq n_0$ , we have
$$
\left\| \widehat {\theta} _ {n} (a) - \theta (a) \right\| _ {2} ^ {2} \leq \frac {L ^ {2}}{\widehat {\operatorname {s n r}} _ {n} (a) ^ {2}} \left[ 7 4 + 2 1 6 \kappa (\mathbf {Z} _ {n} (a)) ^ {2} \right] + \frac {2 \operatorname {e r r} _ {n} (a)}{\sigma_ {r} (\mathbf {Z} _ {n} (a)) ^ {2}},
$$
where  $\kappa (\mathbf{Z}_n(a))\coloneqq \frac{\sigma_1(\mathbf{Z}_n(a))}{\sigma_r(\mathbf{Z}_n(a))},\| \theta (a)\| _2\leq L,$  and in the above we define the "error" term  $\mathrm{err}_n(a)$  to be
$$
\begin{array}{l} \operatorname {e r r} _ {n} (a) := 3 2 \rho L ^ {2} + 6 4 \eta^ {2} \left(\log \left(\frac {A}{\delta}\right) + r \log \left(1 + \frac {\sigma_ {1} \left(\mathbf {Z} _ {n} (a)\right) ^ {2}}{\rho}\right)\right) \\ + 6 \eta^ {2} \sqrt {2 c _ {n} (a) \ell_ {\delta} \left(c _ {n} (a)\right)} + 1 0 \eta^ {2} \ell_ {\delta} \left(c _ {n} (a)\right) + 6 c _ {n} (a) \alpha . \\ \end{array}
$$
We see that the above bound, with the exception of the third term, more or less resembles the bound presented in Theorem 4.1, just written in terms of the observed covariates  $\mathbf{Z}_n(a)$  instead of the true covariates  $\mathbf{X}_n(a)$ . We view the third term as a slowly growing "error" term. In particular, all terms in the quantity  $\mathrm{err}_n(a)$  are either constant, logarithmic in the singular values of  $\mathbf{Z}_n(a)$ , or linear in  $c_{n}(a)$ , the number of times by round  $n$  action  $a$  has been selected. This implies that  $\mathrm{err}_n(a) = \widetilde{O} (n + d)$ , ensuring  $\mathrm{err}_n(a)$  is dominated by other terms in the asymptotic analysis. We now provide the proof of Theorem 4.4. The key application of self-normalized, martingale concentration comes into play in bounding the quantities that appear in the upper bounds of terms  $T_{1}$  and  $T_{2}$  (to be defined below).
Proof. Observe the decomposition, for any  $n \geq 1$  and  $a \in [A]$
$$
\widehat {\theta} _ {n} (a) - \theta (a) = \widehat {\mathbf {P}} _ {n} (a) (\widehat {\theta} _ {n} (a) - \theta (a)) + (\mathbf {P} ^ {\perp} - \widehat {\mathbf {P}} _ {n} ^ {\perp} (a)) \theta (a),
$$
where  $\mathbf{P}^{\perp}$  is the projection onto the subspace orthogonal to  $W^{*}$  and  $\widehat{\mathbf{P}}_n^\perp (a)$  is the projection onto the subspace orthogonal to the learned subspace (i.e. that spanned by  $\mathbf{Z}_{n,r}(a)$ ). Since  $\widehat{\mathbf{P}}_n(a)(\widehat{\theta}_n(a) - \theta (a))$  and  $(\mathbf{P}^{\perp} - \widehat{\mathbf{P}}_{n}^{\perp}(a))\theta (a)$  are orthogonal vectors, we have
$$
\left\| \widehat {\theta} _ {n} (a) - \theta (a) \right\| _ {2} ^ {2} = \left\| \widehat {\mathbf {P}} _ {n} (a) (\widehat {\theta} _ {n} (a) - \theta (a)) \right\| _ {2} ^ {2} + \left\| (\widehat {\mathbf {P}} _ {n} ^ {\perp} (a) - \mathbf {P} ^ {\perp}) \theta (a) \right\| _ {2} ^ {2}.
$$
We bound these two terms separately, beginning with the second term. Going forward, fix an action  $a \in [A]$ . Observe that with probability at least  $1 - \delta$ , simultaneously for all  $n \geq n_0(a)$ ,
$$
\begin{array}{l} \left\| (\widehat {\mathbf {P}} _ {n} ^ {\perp} (a) - \mathbf {P} ^ {\perp}) \theta (a) \right\| _ {2} ^ {2} \leq \left\| \widehat {\mathbf {P}} _ {n} ^ {\perp} (a) - \mathbf {P} ^ {\perp} \right\| _ {o p} ^ {2} \| \theta (a) \| _ {2} ^ {2} \\ \leq L ^ {2} \left\| \widehat {\mathbf {P}} _ {n} ^ {\perp} (a) - \mathbf {P} ^ {\perp} \right\| _ {o p} ^ {2} = L ^ {2} \left\| \widehat {\mathbf {P}} _ {n} (a) - \mathbf {P} \right\| _ {o p} ^ {2} \\ \leq \frac {4 L ^ {2} U _ {n} ^ {2}}{\sigma_ {r} (\mathbf {X} _ {n} (a)) ^ {2}} \leq \frac {6 L ^ {2} U _ {n} ^ {2}}{\sigma_ {r} (\mathbf {Z} _ {n} (a)) ^ {2}}, \\ \end{array}
$$
where the equality in the above comes from observing  $\| \widehat{\mathbf{P}}_n^\perp (a) - \mathbf{P}^\perp \|_{op} = \| \widehat{\mathbf{P}}_n(a) - \mathbf{P}\|_{op}$ , the second-to-last last inequality comes from applying Lemma B.4, and the last inequality follows from the second part of Lemma B.6.
We now bound the first term. Observe that we can write
$$
\begin{array}{l} \left\| \widehat {\mathbf {P}} _ {n} (a) \left(\widehat {\theta} _ {n} (a) - \theta (a)\right) \right\| _ {2} ^ {2} \leq \frac {1}{\sigma_ {r} (\mathbf {Z} _ {n} (a)) ^ {2}} \left\| \widehat {\mathbf {Z}} _ {n} (a) \left(\widehat {\theta} _ {n} (a) - \theta (a)\right) \right\| _ {2} ^ {2} \\ \leq \frac {2}{\sigma_ {r} \left(\mathbf {Z} _ {n} (a)\right) ^ {2}} \left[ \underbrace {\left\| \widehat {\mathbf {Z}} _ {n} (a) \widehat {\theta} _ {n} (a) - \mathbf {X} _ {n} (a) \theta (a) \right\| _ {2} ^ {2}} _ {T _ {1}} + \underbrace {\left\| \mathbf {X} _ {n} (a) \theta (a) - \widehat {\mathbf {Z}} _ {n} (a) \theta (a) \right\| _ {2} ^ {2}} _ {T _ {2}} \right], \tag {1} \\ \end{array}
$$
where the first inequality follows from the fact that  $\widehat{\mathbf{P}}_n(a) \preceq \frac{1}{\sigma_r(\widehat{\mathbf{Z}}_n(a))^2} \widehat{\mathbf{Z}}_n(a)^\top \widehat{\mathbf{Z}}_n(a)$  and  $\sigma_r(\mathbf{Z}_n(a)) = \sigma_r(\widehat{\mathbf{Z}}_n(a))$ , and the second inequality comes from applying the parallelogram inequality. First we bound  $T_{1}$ . We have, with probability at least  $1 - O(\delta)$ , simultaneously for all  $n \geq n_0(a)$
$$
\begin{array}{l} T _ {1} \leq 8 \left\| \check {\mathcal {V}} _ {n} (a) ^ {1 / 2} \left(\check {\theta} _ {n} (a) - \theta (a)\right) \right\| _ {2} ^ {2} + 6 \| \Xi_ {n} (a) \| _ {2} ^ {2} + 8 \left\| \widehat {\mathbf {Z}} _ {n} (a) \theta (a) - \mathbf {X} _ {n} (a) \theta (a) \right\| _ {2} ^ {2} \\ \leq 3 2 \rho L ^ {2} + 6 4 \eta^ {2} \left(\log \left(\frac {A}{\delta}\right) + r \log \left(1 + \frac {\sigma_ {1} \left(\mathbf {Z} _ {n} (a)\right) ^ {2}}{\rho}\right)\right) + 1 6 L ^ {2} U _ {n} ^ {2} \tag {2} \\ + 6 \eta^ {2} \sqrt {2 c _ {n} (a) \ell_ {\delta} \left(c _ {n} (a)\right)} + 1 0 \eta^ {2} \ell_ {\delta} \left(c _ {n} (a)\right) + 6 c _ {n} (a) \alpha + 8 T _ {2}, \\ \end{array}
$$
where the first inequality follows from Lemma D.1, and the second inequality follows from applying Lemmas C.1 and D.3.  $\ell_{\delta}(n) = 2\log \log (2n) + \log \left(\frac{d\pi^2}{12\delta}\right)$ , as defined in Lemma A.2. We now bound  $T_{2}$ . With probability at least  $1 - O(\delta)$  simultaneously for all  $n\geq n_0$ , we have
$$
\begin{array}{l} T _ {2} \leq 2 L ^ {2} \sigma_ {1} (\mathbf {Z} _ {n} (a)) ^ {2} \left\| \mathbf {P} - \hat {\mathbf {P}} _ {n} (a) \right\| _ {o p} ^ {2} + 2 L ^ {2} \| \mathcal {E} _ {n} \| _ {o p} ^ {2} \\ \leq \frac {8 L ^ {2} \sigma_ {1} \left(\mathbf {Z} _ {n} (a)\right) ^ {2} U _ {n} ^ {2}}{\sigma_ {r} \left(\mathbf {X} _ {n} (a)\right) ^ {2}} + 2 L ^ {2} U _ {n} ^ {2} \tag {3} \\ \leq \frac {1 2 L ^ {2} \sigma_ {1} (\mathbf {Z} _ {n} (a)) ^ {2} U _ {n} ^ {2}}{\sigma_ {r} (\mathbf {Z} _ {n} (a)) ^ {2}} + 2 L ^ {2} U _ {n} ^ {2}. \\ \end{array}
$$
The first inequality follows from Lemma D.2, the second inequality follows from applying Lemmas B.4 and B.3, and the final inequality follows from Lemma B.6.
Piecing the above inequalities together yields the desired result, which can be checked via the argument at the end of Appendix D. A union bound over actions then yields that the desired inequality holds over all actions  $a \in [A]$  with probability at least  $1 - O(A\delta)$ .
5 Application to causal inference with panel data
We now apply our bounds for adaptive PCR to online experiment design in the context of panel data. In this setting, the learner is interested in estimating unit-specific counterfactuals under different interventions, given a sequence of unit outcomes (or measurements) over time. Units can range from medical patients, to subpopulations or geographic locations. Examples of interventions include medical treatments, discounts, and socioeconomic policies. Synthetic control (SC) is a popular framework used to estimate counterfactual unit outcomes in panel data settings, had they not been treated (i.e. remained under control) [1, 2]. In SC, there is a notion of a pre-intervention time period in which all units are under control, followed by a post-intervention time period, in which every unit undergoes one of several interventions (including control). At a high level, SC fits a model of a unit's pre-treatment outcomes using pre-treatment data from units who remained under control in the post-intervention time period. It then constructs a "synthetic control" by using the learned model to predict the unit's post-intervention outcomes, had they remained under control. Synthetic interventions (SI) is a recent generalization of the SC framework, which allows for counterfactual estimation of unit outcomes under different interventions, in addition to control [8]. Using our
bounds from Section 4, we show how to generalize the SI framework of Agarwal et al. [8] to settings where interventions are assigned via an adaptive intervention assignment policy.
As a motivating example, consider an online e-commerce platform (learner) which assigns discounts (interventions) to users (units) with the goal of maximizing total user engagement on the platform. For concreteness, suppose that the e-commerce platform assigns discounts greedily with respect to the discount level which appears to be best at the current round (i.e. maximizes total engagement for the current user), given the sequence of previously observed (user, discount level, engagement level) tuples. Under such a setting, the intervention assigned at the current round  $n$  will be correlated with the observed engagement levels at previous rounds, thus breaking the requirement of the SI framework [8] that the intervention assignment is not adaptive to previously observed outcomes.
Formally, we consider a panel data setting in which the principal observes units over a sequence of rounds. In each round  $n$ , the learner observes a unit  $n$  under control for  $T_0 \in \mathbb{N}$  time steps, followed by one of  $A$  interventions (including control, which we denote by 0) for the remaining  $T - T_0$  time steps, where  $T \in \mathbb{N}$ . Overloading notation to be consistent with the literature on panel data, we denote the potential outcome of unit  $n$  at time  $t$  under intervention  $a$  by  $Y_{n,t}^{(a)} \in \mathbb{R}$ , the set of unit  $n$ 's pre-treatment outcomes (under control) by  $Y_{n,pre} := [Y_{n,1}^{(0)}, \ldots, Y_{n,T_0}^{(0)}]^T \in \mathbb{R}^{T_0}$ , and their post-intervention potential outcomes under intervention  $a$  by  $Y_{n,post}^{(a)} := [Y_{n,T_0 + 1}^{(a)}, \ldots, Y_{n,T}^{(a)}]^T \in \mathbb{R}^{T - T_0}$ . We use  $a$  to refer to an arbitrary intervention in  $\{0, \ldots, A - 1\}$  and  $a_n$  to denote the realized intervention unit  $n$  actually receives in the post-intervention time period. We posit that potential outcomes are generated by the following latent factor model over units, time steps, and interventions.
Assumption 5.1 (Latent Factor Model). Suppose the outcome for unit  $n$  at time step  $t$  under treatment  $a \in \{0, \dots, A - 1\}$  takes the form
$$
Y _ {n, t} ^ {(a)} = \langle U _ {t} ^ {(a)}, V _ {n} \rangle + \epsilon_ {n, t} ^ {(a)},
$$
where  $U_{t}^{(a)} \in \mathbb{R}^{r}$  is a latent factor which depends only on the time step  $t$  and intervention  $a$ ,  $V_{n} \in \mathbb{R}^{r}$  is a latent factor which only depends on unit  $n$ , and  $\epsilon_{n,t}^{(a)}$  is zero-mean SubGaussian random noise with variance at most  $\sigma^2$ . We assume, without loss of generality, that  $|\langle U_t^{(a)},V_n\rangle |\leq 1$  for all  $n\geq 1$ ,  $t\in [T]$ ,  $a\in \{0,\dots ,A - 1\}$ .
Note that the learner observes  $Y_{n,t}^{(a)}$  for only the intervention  $a_n$  that unit  $n$  is under at time step  $t$ , and never observes  $U_t^{(a)}$ ,  $V_n$ , or  $\epsilon_{n,t}^{(a)}$ . Such "low rank" assumptions are ubiquitous within the panel data literature (see references in Section 2). We assume that  $r$  is known to the learner, although principled heuristics exist for estimating  $r$  in practice from data (see, e.g. Section 3.2 of Agarwal et al. [8]). Additionally, we make the following "causal transportability" assumption on the latent factors.
Assumption 5.2 (Linear span inclusion). For any post-intervention time step  $t \in [T_0 + 1, T]$  and intervention  $a \in \{0, \dots, A - 1\}$ , we assume that  $U_t^{(a)} \in \mathrm{span}(\{U_t^{(0)} : t \in [T_0]\})$ .
Intuitively, Assumption 5.2 allows for information to be inferred about the potential outcomes in the post-intervention time period using pre-treatment observations. The goal of the learner is to estimate unit-specific counterfactual outcomes under different interventions when the sequence of units and interventions is chosen adaptively. In line with previous work in SI and SC, our target causal parameter is the (counterfactual) average expected post-intervention outcome.
Definition 5.3. The average expected post-intervention outcome of unit  $n$  under intervention  $a$  is
$$
\mathbb {E} \bar {Y} _ {n, p o s t} ^ {(a)} := \frac {1}{T - T _ {0}} \sum_ {t = T _ {0} + 1} ^ {T} \mathbb {E} Y _ {n, t} ^ {(a)},
$$
where the expectation is taken with respect to  $(\epsilon_{n,t}^{(a)})_{T_0 < t\leq T}$
While we consider the average post-intervention outcome, our results may be readily extended to settings in which the target causal parameter is any linear combination of post-intervention outcomes. Next we show that under Assumption 5.1 and Assumption 5.2,  $\mathbb{E}\bar{Y}_{n,\text{post}}^{(a)}$  may be written as a linear combination of unit  $n$ 's pre-intervention outcomes. We note that similar observations have previously been made in the panel data literature (e.g. [46]), but we include the following lemma for completeness' sake.
Lemma 5.4 (Reformulation of average expected post-intervention outcome). Under Assumption 5.1 and Assumption 5.2, there exists slope vector  $\theta(a) \in \mathbb{R}^{T_0}$ , such that the average expected post-intervention outcome of unit  $n$  under intervention  $a$  is expressible as
$$
\mathbb {E} \bar {Y} _ {n, p o s t} ^ {(a)} = \frac {1}{T - T _ {0}} \langle \theta (a), \mathbb {E} Y _ {n, p r e} \rangle .
$$
$\theta(a)$  may be interpreted as a unit-independent measure of the causal relationship between pre- and post-intervention outcomes. Using this reformulation, adaptive guarantees for the estimation of causal effects over time may now be obtained by applying our online PCR results of Section 4. Overloading the notation of Section 3, we let  $\mathbf{X}_n = (\mathbb{E}Y_{1,pre},\dots,\mathbb{E}Y_{n,pre})^T$ ,  $\mathbf{Z}_n = (Y_{1,pre},\dots,Y_{n,pre})^T$ ,  $\epsilon_{n,pre} = (\epsilon_{n,1}^{(0)},\dots,\epsilon_{n,T_0}^{(0)})^T$ ,  $\mathcal{E}_n = (\epsilon_{1,pre},\dots,\epsilon_{n,pre})^T$ ,  $\xi_n = \sum_{t=T_0+1}^T \epsilon_{n,t}^{(a_n)}$ ,  $\Xi_n = (\xi_1,\dots,\xi_n)^T$ , and
$$
\mathbf {Y} _ {n} = \left(\frac {1}{T - T _ {0}} \sum_ {t = 1} ^ {T _ {0}} Y _ {1, t} ^ {(a _ {1})}, \dots , \frac {1}{T - T _ {0}} \sum_ {t = 1} ^ {T _ {0}} Y _ {n, t} ^ {(a _ {n})}\right) ^ {T}.
$$
Finally, we define quantities such as  $\mathbf{Z}_n(a),\mathbf{X}_n(a),\mathbf{Y}_n(a)$  analogously to Section 3. We now turn to bounding our primary quantity of interest in the panel data setting: prediction error for the average expected post-intervention outcome.
Theorem 5.5 (Prediction error of average expected post-intervention outcome). Let  $\delta \in (0,1)$  be an arbitrary confidence parameter and  $\rho > 0$  be chosen to be sufficiently small, as detailed in Appendix F. Further, assume that Assumptions 5.1 and 5.2 are satisfied, there is some  $n_0 \geq 1$  such that  $\mathrm{rank}(\mathbf{X}_{n_0}(a)) = r$ , and  $\mathrm{snr}_n(a) \geq 2$  for all  $n \geq n_0$ . If  $T_0 \leq \frac{1}{2} T$  and  $r \leq \sqrt{T_0 \wedge n}$ , then under Assumption 4.2 with probability at least  $1 - O(A\delta)$ , simultaneously for all interventions  $a \in \{0, \dots, A-1\}$
$$
| \widehat {\mathbb {E}} \bar {Y} _ {n, p o s t} ^ {(a)} - \mathbb {E} \bar {Y} _ {n, p o s t} ^ {(a)} | = \widetilde {O} \left(\frac {L}{\sqrt {T - T _ {0}}} + \frac {r}{\sqrt {T _ {0} \wedge n}} + \frac {r (L \vee 1)}{\sqrt {(T - T _ {0}) (T _ {0} \wedge n)}}\right)
$$
where  $\widehat{\mathbb{E}}\bar{Y}_{n,post}^{(a)}\coloneqq \frac{1}{T - T_0}\cdot \langle \widehat{\theta}_n(a),Y_{n,pre}\rangle$  is the estimated average post-intervention outcome for unit  $n$  under intervention  $a$ .
A more complicated expression which does not require Assumption 4.2 or  $T_0 \leq \frac{1}{2} T$  may be found in Appendix G. Observe that  $|\widehat{\mathbb{E}}\bar{Y}_{n,post}^{(a)} - \mathbb{E}\bar{Y}_{n,post}^{(a)}| \to 0$  with high probability as  $T, T_0, n \to \infty$ .