SlowGuess/NeurIPS_2023_OCR huggingface dataset:
- Moreover, the constant  $c_{1}$  appearing in (3.2) is quite benign and can be made arbitrarily close to 2 by suitably inflating our burn-in constants  $c_{2}, c_{3}, c_{4}, c_{5}, c_{6}$ . We have however not been able to approach the optimal leading constant 1 in front of  $\mathrm{edim}(\Sigma)$ . This can be traced to an application of the triangular inequality in Corollary 4.1.
- The moment bound  $\mathbf{E}\langle v,X_i\rangle^4\leq \mathsf{h}^2\langle v,\mathbf{E}[X_iX_i^\top ]v\rangle (v\in \partial \Sigma_X)$  is easily satisfied for e.g., Gaussian or bounded processes but is of course much milder than either assumption. The assumption that  $s\geq 4$  can be relaxed to  $s > 2$  by replacing our result controlling the lower tail, Theorem 4.3. We have chosen to present our result for  $s\geq 4$  to strike a balance between expositional clarity and generality.
- The first burn-in condition of (3.3) is standard for control of the lower tail—beside the deflation factor  $|a_{\mathrm{max}}|$ , it is necessary even for iid data to guarantee that the "denominator" (2.9) is nonsingular. The second condition of (3.3)—in which a ratio of  $s$ :th and 2nd moment of the noise variable appears—is the price we pay in the moderate deviations bandwidth for only having  $s$  moments: it controls the rate at which the random walk (2.6) approaches asymptotic normality and reduces to the condition of Oliveira [2016] in the iid regime. This latter condition can in principle be removed with slightly modified constants if sufficiently many moments of the data-generating process satisfy a sub-Gaussian type moment equivalence condition (in which case the above-mentioned ratio is constant). Without such an assumption, we note that some polynomial dependence on  $1 / \delta$  is necessary under our tail assumptions and is not an artifact of our analysis; OLS is not deviation-optimal in the entire range of  $\delta \in (0,1)$ —due to the presence of the random walk (2.6) in the numerator—unless the noise variables have Gaussian-like tails [cf. Mendelson, 2018, Section 6.4].
- The conditions in (3.4) and (3.5) relate to dependence. The last condition (3.5) simply asks that our process mixes sufficiently fast. If the mixing coefficients  $\beta_{X,Y}(|a_i|)$  are exponential, this amounts to a logarithmic burn-in in  $1/\delta$ . However, we can still handle slow, polynomial mixing rates, at the cost of a polynomial burn-in in  $1/\delta$ . The conditions in (3.4) asks that the odd and even blocks are balanced in terms of their length and second order statistics. It is trivially satisfied for (weakly) stationary processes analyzed using a uniform blocking length (length of the  $a_i$ ).
In light of the above remarks, we are now in position to simplify Theorem 3.1. If we impose stationarity, quite a few terms in the burn-in conditions (3.3),(3.4) and (3.5) either simplify or vanish. Further restricting to the case where targets are 1-dimensional and letting the sample-size be divisible by the block-length yields the corollary below.
Corollary 3.1. Fix  $\delta \in (0,1)$  and  $n,\tau \in \mathbb{N}$  and let  $2\tau$  divide  $n$ . Fix also a joint distribution of 1-dimensional targets and  $d_{\mathsf{X}}$ -dimensional covariates  $\mathsf{P}^{X,Y}$  with at least  $s\in [4,\infty)$  moments. Let  $(X,Y)_{1:n}$  be a stationary  $\beta$ -mixing sequence with marginals equal to  $\mathsf{P}^{X,Y}$ . Assume further that there exists a positive number  $\mathsf{h}\in \mathbb{R}$  such that for every  $v\in \partial \Sigma_X$  we have that  $\mathbf{E}\langle v,X\rangle^4\leq \mathsf{h}^2$  where  $X\sim \mathsf{P}^X$ . Define also the noise interaction term:  $\sigma^2\triangleq \frac{1}{\tau}\sup_{v\in \mathbb{S}^{d_X - 1}}\mathbf{E}\left[\left(\sum_{i = 1}^{\tau}\langle \bar{V}_i,v\rangle^2\right)\right]$ . There exist universal positive constants  $c_{1},c_{2},c_{3},c_{4}$  such that with probability at least  $1 - \delta$ :
$$
\| (\widehat {M} - M _ {\star}) \sqrt {\Sigma_ {X}} \| _ {2} ^ {2} \leq \frac {c _ {1} \sigma^ {2} (d _ {\mathsf {X}} + \log (1 / \delta))}{n}
$$
$$
\frac {n}{\tau} \geq c _ {2} \left(d _ {\mathbb {X}} + \mathrm {h} ^ {2} \log (1 / \delta)\right); \quad \left(\frac {n}{\tau}\right) ^ {1 - 2 / s} \geq c _ {3} s ^ {2} \frac {\left(\mathbf {E} \left\| \frac {1}{\sqrt {\tau d _ {\mathbb {X}}}} \sum_ {i = 1} ^ {\tau} \bar {V} _ {i} \right\| _ {2} ^ {s}\right) ^ {2 / s}}{\sigma^ {2} \delta^ {2 / s}}; \quad \frac {n}{\tau} \beta_ {X, Y} (\tau) \leq c _ {4} \delta . \tag {3.7}
$$
Corollary 3.1 takes a very similar form to—by now—standard results in the iid regime [Hsu et al., 2012, Oliveira, 2016]. Indeed, if the data is stationary the price we pay for dependence is that:
variance and moment terms need to be computed in blocks;
- the burn-in is deflated by a factor of the block-length (the first two parts of (3.7)); and
- we incur an additional burn-in penalizing slow mixing—the last part of (3.7) asks that the block-length is not "too small".
3.1 Comparison to Related Work
Having established our main result, Theorem 3.1, we now provide a more detailed comparison to the relevant literature. Most closely related to our results is Nagaraj et al. [2020], who study bounded linear regression models in which the data comes from an exponentially ergodic Markov chain. They find that strictly realizable linear regression is no harder than its iid counterpart in this setting, and show that a parallelized gradient algorithm achieves the optimal rate. More interestingly, in the absence of realizability, they also establish a lower bound demonstrating that the worst-case (global minimax) excess risk across all Markov chains with a given mixing time is deflated by said mixing time, thereby establishing a gap between realizable and non-realizable learning from dependent data.
Of course, their lower bound is no longer valid if one drops the requirement that the predictor performs uniformly well across all distributions with a prescribed mixing time. It is exactly herein that our analyses differ. While Nagaraj et al. [2020] characterize the worst-case (or global) complexity of linear regression, we focus on the instance-specific (or local) complexity. In other words, they compete against the worst distribution at a given level of mixing, whereas we compete against a fixed distribution. To appreciate this distinction, let us momentarily assume that  $d_{\mathrm{X}} = d_{\mathrm{Y}} = 1$ . The noise term  $\sigma^2$  in Theorem 3.1 can be upper-bounded as:
$$
\sigma^ {2} = \frac {1}{n} \sum_ {i = 1} ^ {2 m} \mathbf {E} \left[ \left(\sum_ {j \in a _ {i}} \bar {V} _ {j}\right) ^ {2} \right] \leq \max  _ {i \in [ 2 m ]} | a _ {i} | \cdot \max  _ {i \in [ n ]} \mathbf {E} \bar {V} _ {i} ^ {2} \tag {3.8}
$$
by the Cauchy-Schwarz inequality. The right hand side of (3.8) is precisely inflated by the (maximal) block-length  $\max_{i\in [2m]}|a_i|$ .
Seen in this light, our results being sharper in terms of the measure of dependency reduces to stating that our results are sharper by an application of the Cauchy-Schwarz Inequality. Moreover, the statement that the global complexity is worse than its iid counterpart by a factor of the mixing time amounts in our setting to stating that there exists a distribution achieving equality in (3.8). We remark that such a distribution is easily constructed by taking  $X_{1:n}$  and  $Y_{1:n}$  to be constant within each block and stationary across the blocks; this is precisely when the application of the Cauchy-Schwarz inequality in (3.8) turns to equality. To further appreciate the distinction between our results, note that our result measures dependence through correlation. By contrast, a result scaling with the mixing time measures dependence in a stronger variational sense. That is, the former measures dependence at the level of orthogonality of the random variables themselves, whereas the latter measures it at the level of orthogonality of all measurable functions of these random variables.
Further Related Work. Another closely related line of work studies parameter identification in auto-regressive models [for a recent survey, see Tsiamis et al., 2022]. When the noise model is strictly realizable—the variables  $W_{1:n}$  form a martingale difference sequence with respect to the filtration generated by  $X_{1:n}$ —identification is possible at the iid rate even in the absence of mixing [Simchowitz et al., 2018, Faradonbeh et al., 2018, Sarkar and Rakhlin, 2019]. Naturally, our results do not cover the mixing-free regime as we consider: (1) the agnostic setting in which self-normalized martingale arguments [Peña et al., 2009, Abbasi-Yadkori et al., 2011] are not available; and (2) excess risk bounds instead of parameter identification—it seems unlikely that (3.2) holds without some notion of stochastic stability due to the presence of  $\Sigma_X$  on the left hand side.
More generally—moving beyond linear time-series models—several authors have considered learning under various weak dependency notions. Kuznetsov and Mohri [2017] give generalization bounds in a more general setting using the same blocking technique—due to Yu [1994]—used here. Statements similar in spirit can also be found in e.g., Steinwart and Christmann [2009], Duchi et al. [2012] and
most recently Roy et al. [2021]. However, they all suffer the dependency deflation discussed above and in our introduction (Section 1). We also note that Ziemann and Tu [2022] obtain rates for strictly realizable square loss that—similar to ours here—relegate mixing times into additive burn-in factors. While they treat more general hypothesis classes, they do not go beyond strict realizability, and their analysis rests on the assumption that the noise interaction term is a martingale difference sequence.
4 Proof Overview
Theorem 3.1 is the direct consequence of two separate results: Theorem 4.2, which controls the centered noise (2.7) term, and Theorem 4.3, which bounds the lower tail of the normalized empirical covariance matrix (2.9). We prove Theorem 4.2 by blocking the Fuk-Nagaev inequality of Einmahl and Li [2008], and Theorem 4.3 by a truncation argument combined with blocking. These results are found in Section 4.2 and Section 4.3. Our proof idea is heavily inspired by that of Oliveira [2016] and the idea is very much to adjust his approach in such a way that blocking does not affect the leading term in the rate. As either result relies on blocking, it is now pertinent to describe this technique in a little more detail.
4.1 Blocking
Recall that we partition  $[n]$  into  $2m$  consecutive intervals, denoted  $a_{j}$  for  $j\in [2m]$ , so that  $\sum_{j = 1}^{2m}|a_j| = n$ . Denote further by  $O$  (resp. by  $E$ ) the union of the oddly (resp. evenly) indexed subsets of  $[n]$ . We further abuse notation by writing  $\beta_Z(a_i) = \beta_Z(|a_i|)$  in the sequel.
We split the process  $Z_{1:n}$  as:
$$
Z _ {1: | O |} ^ {o} \triangleq \left(Z _ {a _ {1}}, \dots , Z _ {a _ {2 m - 1}}\right), \quad Z _ {1: | E |} ^ {e} \triangleq \left(Z _ {a _ {2}}, \dots , Z _ {a _ {2 m}}\right). \tag {4.1}
$$
Let  $\tilde{Z}_{1:|O|}^{o}$  and  $\tilde{Z}_{1:|E|}^{e}$  be blockwise decoupled versions of (4.1). That is we posit that  $\tilde{Z}_{1:|O|}^{o} \sim \mathsf{P}_{\tilde{Z}_{1:|O|}^{o}}$  and  $\tilde{Z}_{1:|E|}^{e} \sim \mathsf{P}_{\tilde{Z}_{1:|E|}^{e}}$ , where:
$$
\mathrm {P} _ {\tilde {Z} _ {1: | O |} ^ {o}} \triangleq \mathrm {P} _ {Z _ {a _ {1}}} \otimes \mathrm {P} _ {Z _ {a _ {3}}} \otimes \dots \otimes \mathrm {P} _ {Z _ {a _ {2 m - 1}}} \quad \text {a n d} \quad \mathrm {P} _ {\tilde {Z} _ {1: | E |} ^ {e}} \triangleq \mathrm {P} _ {Z _ {a _ {2}}} \otimes \mathrm {P} _ {Z _ {a _ {4}}} \otimes \dots \otimes \mathrm {P} _ {Z _ {a _ {2 m}}}. \tag {4.2}
$$
The process  $\tilde{Z}_{1:n}$  with the same marginals as  $\tilde{Z}_{1:|O|}^{o}$  and  $\tilde{Z}_{1:|E|}^{e}$  is said to be the decoupled version of  $Z_{1:n}$ . To be clear:  $\mathsf{P}_{\tilde{Z}_{1:n}} \triangleq \mathsf{P}_{Z_{a_1}} \otimes \mathsf{P}_{Z_{a_2}} \otimes \dots \otimes \mathsf{P}_{Z_{a_{2m}}}$ , so that  $\tilde{Z}_{1:|O|}^{o}$  and  $\tilde{Z}_{1:|E|}^{e}$  are alternatingly embedded in  $\tilde{Z}_{1:n}$ . The following result is key—by skipping every other block,  $\tilde{Z}_{1:n}$  may be used in place of  $Z_{1:n}$  for evaluating scalar functions at the cost of an additive mixing-related term.
Proposition 4.1 (Lemma 2.6 in Yu [1994]; Proposition 1 in Kuznetsov and Mohri [2017]). Fix a  $\beta$ -mixing process  $Z_{1:n}$  and let  $\tilde{Z}_{1:n}$  be its decoupled version. For any measurable function  $f$  of  $Z_{1:|O|}^{o}$  (resp.  $g$  of  $Z_{1:|E|}^{e}$ ) with joint range  $[0,1]$  we have that:
$$
\left| \mathbf {E} \left(f \left(Z _ {1: | O |} ^ {o}\right)\right) - \mathbf {E} \left(f \left(\tilde {Z} _ {1: | O |} ^ {o}\right)\right) \right| \leq \sum_ {i \in E \backslash \{2 m \}} \beta_ {Z} \left(a _ {i}\right),
$$
$$
\left| \mathbf {E} \left(g \left(Z _ {1: | E |} ^ {e}\right)\right) - \mathbf {E} \left(g \left(\tilde {Z} _ {1: | E |} ^ {e}\right)\right) \right| \leq \sum_ {i \in O \backslash \{1 \}} \beta_ {Z} \left(a _ {i}\right). \tag {4.3}
$$
The following corollary to Proposition 4.1 is convenient for controlling norms of random walks.
Corollary 4.1 (Lemma 3 in Kuznetsov and Mohri [2017]). Let  $Z_{1:n}$  be a  $\beta$ -mixing process taking values in a normed space  $(Z, \| \cdot \|)$ , and let  $\tilde{Z}_{1:n}$  be its decoupled version. For any
2At a high level, for iid data, this proof strategy first appeared in the journal version of Oliveira [2016].
$\varepsilon \geq \mathbf{E}\left\| \frac{1}{|O|}\sum_{i\in O}\tilde{Z}_i\right\| \vee \mathbf{E}\left\| \frac{1}{|E|}\sum_{i\in E}\tilde{Z}_i\right\|$  we have that:
$$
\begin{array}{l} \mathbf {P} \left(\left\| \frac {1}{n} \sum_ {i = 1} ^ {n} Z _ {i} \right\| > \varepsilon\right) \leq \sum_ {i = 1} ^ {2 m} \beta_ {Z} (a _ {i}) \\ + \mathbf {P} \left(\left\| \frac {1}{| O |} \sum_ {i \in O} \tilde {Z} _ {i} \right\| > \mathbf {E} \left\| \frac {1}{| O |} \sum_ {i \in O} \tilde {Z} _ {i} \right\| + \varepsilon_ {o}\right) + \mathbf {P} \left(\left\| \frac {1}{| E |} \sum_ {i \in E} \tilde {Z} _ {i} \right\| > \mathbf {E} \left\| \frac {1}{| E |} \sum_ {i \in E} \tilde {Z} _ {i} \right\| + \varepsilon_ {e}\right), \tag {4.4} \\ \end{array}
$$
where  $\varepsilon_{o} = \varepsilon -\mathbf{E}\left\| \frac{1}{|O|}\sum_{i\in O}\tilde{Z}_{i}\right\|$  and  $\varepsilon_{e} = \varepsilon -\mathbf{E}\left\| \frac{1}{|E|}\sum_{i\in E}\tilde{Z}_{i}\right\|$ .
In short, up to a mild failure additional failure probability term, we only need to control the tensor product processes (4.2).
4.2 Dependent Random Walks
Once equipped with Corollary 4.1, we still require control of the independent blocks. The following Fuk-Nagaev inequality due to Einmahl and Li [2008] provides such control.
Theorem 4.1 (Theorem 4 in Einmahl and Li [2008]). Fix  $s > 2$ , a separable normed space  $(\mathsf{U},\| \cdot \|)$  and a U-valued sequence  $U_{1:n}$  of independent random variables. Assume that  $\mathbf{E}\| U_i\| ^s < \infty$  for  $i\in [n]$ . Then for any  $\varepsilon \in (0,\infty)$ ,  $\eta \in (0,1]$ , and  $t\geq 0$ , we have that:
$$
\begin{array}{l} \mathbf {P} \left(\max  _ {k \in [ n ]} \left\| \sum_ {i = 1} ^ {k} U _ {i} \right\| \leq (1 + \eta) \mathbf {E} \left\| \sum_ {i = 1} ^ {n} U _ {i} \right\| + (1 + 9 \varepsilon) t\right) \\ \leq \exp \left(- \frac {t ^ {2}}{(2 + \eta) \Lambda}\right) + C _ {\varepsilon , \eta , s} \sum_ {i = 1} ^ {n} \frac {\mathbf {E} \| U _ {i} \| ^ {s}}{t ^ {s}}, \tag {4.5} \\ \end{array}
$$
where  $\Lambda \triangleq \sup_{v\in \mathcal{S}^*}\mathbf{E}\sum_{i = 1}^{n}v^2 (U_i)$  and where  $\mathcal{S}^*$  is unit disk in the dual space of  $(\mathsf{U},\| \cdot \|)$ . Moreover, we may take  $C_{\varepsilon ,\eta ,s} = \big(1 + (2s / e)^{2s}(2(1 + 2 / \varepsilon)(3 + 4 / \eta))^2 +\varepsilon^{-s}\big)$ .<sup>3</sup>
If our data were drawn independently, Theorem 4.1 would give us the required control of the random walk (2.7). The right hand side of (4.5) consists of a mixed tail: (1) a sub-Gaussian term with a CLT-like weak variance term  $\Lambda$ ; and (2) a polynomial term accounting for the fact that we only imposed the existence of  $s$  moments. With the preliminary results Corollary 4.1 and Theorem 4.1 in place, we are now in position to control dependent random walks of the form (2.7).
Theorem 4.2. Fix  $s > 2$ , a separable normed space  $(Z, \| \cdot \|)$ , constants  $\varepsilon, \eta > 0$ , and set  $C_{\varepsilon, \eta, s} = (1 + (2s / e)^{2s}(2(1 + 2 / \varepsilon)(3 + 4 / \eta))^2 + \varepsilon^{-s})$ . Fix also a consecutive partition  $a_{1:2m}$  of  $[n]$  and let  $Z_{1:n}$  be a mean zero,  $\beta$ -mixing process taking values in  $Z$  with block decoupled version  $\tilde{Z}_{1:n}$ . Let  $O$  be the union of the odd  $a_i$  and  $E$  be the union of the even  $a_i$ . Assume that  $\mathbf{E}\| Z_i\|^s < \infty$  for  $i \in [n]$ . For every  $\varepsilon, \eta > 0$  and  $\delta \in (0,1)$ , we have that:
$$
\begin{array}{l} \mathbf {P} \left(\left\| \frac {1}{n} \sum_ {i = 1} ^ {n} Z _ {i} \right\| \geq \max  _ {\operatorname {s g n} \in \{O, E \}} \sqrt {\frac {\Lambda_ {\operatorname {s g n}}}{| \operatorname {s g n} |}} \left((1 + 2 \eta) \sqrt {r} + (1 + 9 \varepsilon) \sqrt {(2 + \eta) \log (1 / \delta)}\right)\right) \\ \leq 2 \delta + \sum_ {i = 2} ^ {2 m - 1} \beta_ {Z} \left(a _ {i}\right) + \frac {C _ {\varepsilon , \eta , s} (1 + 9 \varepsilon) ^ {s}}{r ^ {s / 2} \eta^ {s}} \sum_ {\operatorname {s g n} \in \{O, E \}} \sum_ {i \in [ 2 m ]: a _ {i} \subset \operatorname {s g n}} \frac {\mathbf {E} \left\| \sum_ {j \in a _ {i}} Z _ {j} \right\| ^ {s}}{\left| \operatorname {s g n} \right| ^ {s / 2} \Lambda_ {\operatorname {s g n}} ^ {s / 2}}, \tag {4.6} \\ \end{array}
$$
where  $\Lambda_{\mathrm{sgn}} \triangleq \sup_{v \in \mathcal{S}^*} \frac{1}{|\mathrm{sgn}|} \sum_{a_i \subset \mathrm{sgn}} \mathbf{E} v^2 \left( \sum_{j \in a_i} Z_j \right)$  for  $\mathrm{sgn} \in \{O, E\}$ ,  $\mathcal{S}^*$  is unit disk in the dual space of  $(Z, \| \cdot \|)$ , and  $\sqrt{r} \geq \max_{\mathrm{sgn} \in \{O, E\}} \frac{\mathbf{E} \left\| \frac{1}{\sqrt{|\mathrm{sgn}|}} \sum_{i \in \mathrm{sgn}} \tilde{Z}_i \right\|}{\sqrt{\Lambda_{\mathrm{sgn}}}}$ .
3The constant  $C_{\varepsilon ,\eta ,s}$  is not specified exactly in Einmahl and Li [2008]. This constant is easy to obtain by observing that their  $K_{s}$ , which may be taken to be the best constant such that  $(\log x)^{2s} \leq K_{s}x$  for  $x \geq 1$ , is upper-bounded by  $(2s / e)^{2s}$ .
In Theorem 4.2 we have combined the blocking technique with the Fuk-Nagaev inequality (4.5). The right hand side of (4.6) is exactly as in (4.5) but instantiated to our setting and with extra additive mixing-related term.
4.3 The Lower Tail of the Empirical Covariance Matrix
We now proceed to analyze the lower tail of the empirical covariance matrix (2.9).
Theorem 4.3. Fix  $\delta >0$  and a consecutive partition  $a_{1:2m}$  of  $[n]$ . Let  $X_{1:n}$  be a sequence of  $\beta$ -mixing random variables taking values in  $\mathbb{R}^{d\times}$  with finite fourth moment. Assume that there exists a positive number  $\mathsf{h}\in \mathbb{R}$  such that for every  $v\in \partial \Sigma_X$  and  $i\in [n]$ , we have  $\mathbf{E}\langle v,X_i\rangle^4\leq \mathsf{h}^2\langle v,\mathbf{E}[X_iX_i^\top ]v\rangle$ . There exists a positive universal constant  $C\in \mathbb{R}$  such that as long as
$$
n \geq C \max  _ {j \in [ 2 m ]} | a _ {j} | \left(d _ {\mathsf {X}} + \mathsf {h} ^ {2} \log (1 / \delta)\right) \quad \text {a n d} \quad \sum_ {i = 2} ^ {2 m - 1} \beta_ {X} \left(a _ {i}\right) \leq \frac {\delta}{2}, \tag {4.7}
$$
$$
\mathbf {P} \left(\forall v \in \mathbb {R} ^ {d _ {\mathrm {X}}} : \frac {1}{n} \sum_ {i = 1} ^ {n} \langle v, X _ {i} \rangle^ {2} \geq \frac {1}{2 n} \sum_ {i = 1} ^ {n} \mathbf {E} \langle v, X _ {i} \rangle^ {2}\right) \geq 1 - \delta . \tag {4.8}
$$
It is by now a well-established fact that lower uniform laws of the form (4.8) hold under mild assumptions for various function classes (linear functions on  $\mathbb{R}^{d_{\times}}$  in this case). Since these assumptions are quite mild and only affect burn-in conditions, deflating the sample-size via blocking does not deflate the final convergence rate. The particular approach we have chosen here to establish Theorem 4.3 is to combine blocking with the approach found in [Wainwright, 2019, Theorem 14.12]. We remark that similar statements hold if one instead blocks the arguments of say Oliveira [2016] or Koltchinskii and Mendelson [2015].
The leading order term of our main result, Theorem 3.1, does not directly depend on any mixing-time type quantities. It mimics the asymptotic rate and scales solely in terms of the second order statistics of the process at hand. To arrive at this result, we rely on two facts:
- The lower tail of the empirical covariance matrix (2.9) is well-behaved under mild assumptions. In an excess risk bound, the contribution of the lower uniform law to the overall error is not of leading order. Hence, incurring a sample size deflation for this purpose is not critical.
- By combining blocking with a version of Bernstein's inequality, we are able to push the effect of blocking to only affect the large deviations regime. In the moderate and small deviations regimes, control of the leading order of the random walk in (2.7) is not directly impacted by slow mixing.
Ingvar Ziemann is supported by a Swedish Research Council international postdoc grant. Nikolai Matni is supported in part by NSF award CPS-2038873, NSF CAREER award ECCS-2045834, and a Google Research Scholar award. The authors thank Bruce Lee for comments on an earlier draft of the manuscript.
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A.1 Proof of Theorem 4.2
Let  $\tilde{Z}_{i:n}$  be the block-decoupled version of  $Z_{1:n}$ . Let  $\operatorname{sgn} \in \{O, E\}$ . For  $i \in O$  we define  $U_i = \frac{1}{|O|} \sum_{j \in a_i} \tilde{Z}_j$  and similarly for  $i \in E$  we define  $U_i = \frac{1}{|E|} \sum_{j \in a_i} \tilde{Z}_j$ .
In light of Corollary 4.1 to control  $\mathbf{P}\left(\left\| \frac{1}{n}\sum_{i = 1}^{n}Z_{i}\right\| >u\right)$  it suffices to control