SlowGuess/NeurIPS_2023_OCR huggingface dataset:
ilias@cs.wisc.edu
dakane@ucsd.edu
Vasilis Kontonis
mailto:vasilis@cs.utexas.edu
Sihan Liu
il046@ucsd.edu
Nikos Zarifis
zarifis@wisc.edu
We give the first polynomial-time algorithm for the testable learning of halfspaces in the presence of adversarial label noise under the Gaussian distribution. In the recently introduced testable learning model, one is required to produce a tester-learner such that if the data passes the tester, then one can trust the output of the robust learner on the data. Our tester-learner runs in time  $\mathrm{poly}(d / \epsilon)$  and outputs a halfspace with misclassification error  $O(\mathrm{opt}) + \epsilon$ , where  $\mathrm{opt}$  is the 0-1 error of the best fitting halfspace. At a technical level, our algorithm employs an iterative soft localization technique enhanced with appropriate testers to ensure that the data distribution is sufficiently similar to a Gaussian. Finally, our algorithm can be readily adapted to yield an efficient and testable active learner requiring only  $d$  polylog  $(1 / \epsilon)$  labeled examples.
A (homogeneous) halfspace is a Boolean function  $h: \mathbb{R}^d \to \{\pm 1\}$  of the form  $h_{\mathbf{w}}(\mathbf{x}) = \mathrm{sign}(\mathbf{w} \cdot \mathbf{x})$ , where  $\mathbf{w} \in \mathbb{R}^d$  is the corresponding weight vector and the function  $\mathrm{sign}: \mathbb{R} \to \{\pm 1\}$  is defined as  $\mathrm{sign}(t) = 1$  if  $t \geq 0$  and  $\mathrm{sign}(t) = -1$  otherwise. Learning halfspaces from random labeled examples is a classical task in machine learning, with history going back to the Perceptron algorithm [Ros58]. In the realizable PAC model [Val84] (i.e., with consistent labels), the class of halfspaces is known to be efficiently learnable without distributional assumptions. On the other hand, in the agnostic (or adversarial label noise) model [Hau92, KSS94] even weak learning is computationally intractable in the distribution-free setting [Dan16, DKMR22, Tie22].
These intractability results have served as a motivation for the study of agnostic learning in the distribution-specific setting, i.e., when the marginal distribution on examples is assumed to be well-behaved. In this context, a number of algorithmic results are known. The  $L_{1}$ -regression algorithm of [KKMS08] agnostically learns halfspaces within near-optimal 0-1 error of  $\mathrm{opt} + \epsilon$ , where  $\mathrm{opt}$  is the 0-1 error of the best-fitting halfspace. The running time of the  $L_{1}$ -regression algorithm is  $d^{\tilde{O}(1/\epsilon^{2})}$  under the assumption that the marginal distribution on examples is the standard Gaussian (and for a few other structured distributions)  $[\mathrm{DGJ}^{+}10, \mathrm{DKN}10]$ . While the  $L_{1}$  regression method leads to improper learners, a proper agnostic learner with qualitatively similar complexity was recently given
in  $\mathrm{[DKK^{+}21]}$ . The exponential dependence on  $1 / \epsilon$  in the running time of these algorithms is known to be inherent, in both the Statistical Query model [DKZ20, GGK20, DKPZ21] and under standard cryptographic assumptions [DKR23].
Interestingly, it is possible to circumvent the super-polynomial dependence on  $1 / \epsilon$  by relaxing the final error guarantee — namely, by obtaining a hypothesis with 0-1 error  $f(\mathrm{opt}) + \epsilon$ , for some function  $f(t)$  that goes to 0 when  $t \to 0$ . (Vanilla agnostic learning corresponds to the case that  $f(t) = t$ .) A number of algorithmic works, starting with [KLS09], developed efficient algorithms with relaxed error guarantees; see, e.g., [ABL17, Dan15, DKS18, DKTZ20b]. The most relevant results in the context of the current paper are the works [ABL17, DKS18] which gave  $\mathrm{poly}(d / \epsilon)$  time algorithms with error  $C\mathrm{opt} + \epsilon$ , for some universal constant  $C > 1$ , for learning halfspaces with adversarial label noise under the Gaussian distribution. Given the aforementioned computational hardness results, these constant-factor approximations are best possible within the class of polynomial-time algorithms.
A drawback of distribution-specific agnostic learning is that it provides no guarantees if the assumption on the marginal distribution on examples is not satisfied. Ideally, one would additionally like an efficient method to test these distributional assumptions, so that: (1) if our tester accepts, then we can trust the output of the learner, and (2) it is unlikely that the tester rejects if the data satisfies the distributional assumptions. This state-of-affairs motivated the definition of a new model — introduced in [RV23] and termed testable learning — formally defined below:
Definition 1.1 (Testable Learning with Adversarial Label Noise [RV23]). Fix  $\epsilon, \tau \in (0,1]$  and let  $f:[0,1] \mapsto \mathbb{R}_+$ . A tester-learner  $\mathcal{A}$  (approximately) testably learns a concept class  $\mathcal{C}$  with respect to the distribution  $D_{\mathbf{x}}$  on  $\mathbb{R}^d$  with  $N$  samples, and failure probability  $\tau$  if the following holds. For any distribution  $D$  on  $\mathbb{R}^d \times \{\pm 1\}$ , the tester-learner  $\mathcal{A}$  draws a set  $S$  of  $N$  i.i.d. samples from  $D$ . In the end, it either rejects  $S$  or accepts  $S$  and produces a hypothesis  $h:\mathbb{R}^d \mapsto \{\pm 1\}$ . Moreover, the following conditions must be met:
- (Completeness) If  $D$  truly has marginal  $D_{\mathbf{x}}$ ,  $\mathcal{A}$  accepts with probability at least  $1 - \tau$ .
- (Soundness) The probability that  $\mathcal{A}$  accepts and outputs a hypothesis  $h$  for which  $\mathbf{Pr}_{(\mathbf{x},y)\sim D}[h(\mathbf{x})\neq y] > f(\mathrm{opt}) + \epsilon$ , where  $\mathrm{opt} \coloneqq \min_{g\in \mathcal{C}}\mathbf{Pr}_{(\mathbf{x},y)\sim D}[g(\mathbf{x})\neq y]$  is at most  $\tau$ .
The probability in the above statements is over the randomness of the sample  $S$  and the internal randomness of the tester-learner  $\mathcal{A}$ .
The initial work [RV23] and the followup paper [GKK22] focused on the setting where  $f(t) = t$  (i.e., achieving optimal error of  $\mathrm{opt} + \epsilon$ ). These works developed general moment-matching based algorithms that yield testable learners for a range of concept classes, including halfspaces. For the class of halfspaces in particular, they gave a testable agnostic learner under the Gaussian distribution with sample complexity and runtime  $d^{\hat{O}(1/\epsilon^2)}$  — essentially matching the complexity of the problem in the standard agnostic PAC setting (without the testable requirement). Since the testable learning setting is at least as hard as the standard PAC setting, the aforementioned hardness results imply that the exponential complexity dependence in  $1/\epsilon$  cannot be improved.
In this work, we continue this line of investigation. We ask whether we can obtain fully polynomial time testable learning algorithms with relaxed error guarantees — ideally matching the standard (non-testable) learning setting. Concretely, we study the following question:
Is there a poly  $(d / \epsilon)$  time tester-learner for halfspaces with error  $f(\mathrm{opt}) + \epsilon$ ? Specifically, is there a constant-factor approximation?
As our main result, we provide an affirmative answer to this question in the strongest possible sense — by providing an efficient constant-factor approximate tester-learner.
Labeling examples often requires hiring expert annotators, paying for query access to powerful language models, etc. On the other hand unlabeled examples are usually easy to obtain in practice. This has motivated the design of active learning algorithms that, given a large dataset of unlabeled examples, carefully choose a small subset to ask for their labels. A long line of research has studied active learning of halfspaces under structured distributions either with clean labels or in the presence of noise [BBL06, BBZ07, BL13, ABL17, She21].
In principle, testable learners could only rely on unlabeled examples during their "testing phase" in order to verify that the x-marginal satisfies the required properties; and, assuming that it does so,
proceed to run an active learning algorithm. However, as the testing process is often coupled with the downstream learning task — that depends on labeled examples — it is an interesting question to investigate the design of testable learners in the active learning setting.
Definition 1.2 (Testable Active Learning). A tester-learner has sample complexity  $N$  and label complexity  $q$  in the active learning model if it draws  $N$  unlabeled samples from  $D_{\mathbf{x}}$ , i.e., the  $\mathbf{x}$ -marginal of  $D$  over  $\mathbb{R}^d$ , and queries the labels of at most  $q$  samples throughout its computation.
Main Result Our main result is the first polynomial-time tester-learner for homogeneous halfspaces with respect to the Gaussian distribution in the presence of adversarial label noise. Formally, we establish the following theorem:
Theorem 1.3 (Testable Active Learning Halfspaces under Gaussian Marginals). Let  $\epsilon, \tau \in (0,1)$  and  $\mathcal{C}$  be the class of homogeneous halfspaces on  $\mathbb{R}^d$ . There exists a tester-learner with sample complexity  $N = \mathrm{poly}(d,1/\epsilon)\log(1/\tau)$  and runtime  $\mathrm{poly}(dN)$  for  $\mathcal{C}$  with respect to  $\mathcal{N}(\mathbf{0},\mathbf{I})$  up to 0-1 error  $O(\mathrm{opt}) + \epsilon$ , where  $\mathrm{opt}$  is the 0-1 error of the best fitting function in  $\mathcal{C}$ , that fails with probability at most  $\tau$ . In addition, in the active learning model, the algorithm has sample complexity  $N$  and label complexity  $q = \widetilde{O}((d\log(1/\epsilon) + \log^2(1/\epsilon))\log(1/\tau))$ .
Before we provide an overview of our technical approach, some remarks are in order. Theorem 1.3 gives the first algorithm for testable learning of halfspaces that runs in  $\mathrm{poly}(d / \epsilon)$  time and achieves dimension-independent error (i.e., error of the form  $f(\mathrm{opt}) + \epsilon$ , where  $f$  satisfies  $\lim_{t\to 0}f(t) = 0$ . Moreover, the constant-factor approximation achieved is best possible, matching the known guarantees without the testable requirement and complexity lower bounds. Prior to our work, the only known result in the testable setting, due to [RV23, GKK22], achieves error  $\mathrm{opt} + \epsilon$  with complexity  $d^{\mathrm{poly}(1 / \epsilon)}$ . A novel (and seemingly necessary) feature of our approach is that the testing components of our algorithm depend on the labels (as opposed to the label-oblivious testers of [RV23, GKK22]). As will be explained in the preceding discussion, to prove Theorem 1.3 we develop a testable version of the well-known localization technique that may be of broader interest.
Independent Work In concurrent and independent work, [GKSV23] gave an efficient tester-learner for homogeneous halfspaces under the Gaussian distribution (and strongly log-concave distributions) achieving dimension-independent error guarantees. Specifically, for strongly log-concave distributions, their algorithm achieves error  $O(k^{1/2} \mathrm{opt}^{1 - 1/k})$  with sample complexity and running time of  $\text{poly}(d^{\widetilde{O}(k)}, (1/\epsilon)^{\widetilde{O}(k)})$ . That is, they obtain error  $O(\mathrm{opt}^c)$ , where  $c < 1$  is a universal constant, in  $\text{poly}_c(d/\epsilon)$  time; and error  $\widetilde{O}(\mathrm{opt})$  in quasi-polynomial  $(d/\epsilon)^{\text{polylog}(d)}$  time. For the Gaussian distribution, they achieve error  $O(\mathrm{opt})$  with complexity  $\text{poly}(d, 1/\epsilon)$ . We remark that since the learner in [GKSV23] corresponds to the stochastic gradient descent algorithm from [DKTZ20b], which requires labeled sample in every iteration, it is not clear whether their approach can be made label-efficient.
1.1 Overview of Techniques
Our tester-learner is based on the well-known localization technique that has been used in the context of learning halfspaces with noise; see, e.g., [ABL17, DKS18]. At a high-level, the idea of localization hinges on updating a given hypothesis by using "the most informative" examples, specifically examples that have very small margin with respect to the current hypothesis. Naturally, the correctness of this geometric technique leverages structural properties of the underlying distribution over examples, namely concentration, anti-concentration, and anti-anti-concentration properties (see, e.g., [DKTZ20a]). While the Gaussian distribution satisfies these properties, they are unfortunately hard to test. In this work, we show that localization can be effectively combined with appropriate efficient testing routines to provide an efficient tester-learner.
Localization and a (Weak) Proper Testable Learner Assume that we are given a halfspace defined by the unit vector  $\mathbf{w}$  with small 0-1 error, namely  $\mathbf{Pr}_{\mathbf{x} \sim D_{\mathbf{x}}}[\mathrm{sign}(\mathbf{v}^{*} \cdot \mathbf{x}) \neq \mathrm{sign}(\mathbf{w} \cdot \mathbf{x})] \leq \delta$ , for some small  $\delta > 0$ , where  $\mathbf{v}^{*}$  is the unit vector defining an optimal halfspace. The localization approach improves the current hypothesis, defined by  $\mathbf{w}$ , by considering the conditional distribution  $D'$  on the points that fall in a thin slice around  $\mathbf{w}$ , i.e., the set of points  $\mathbf{x}$  satisfying  $|\mathbf{w} \cdot \mathbf{x}| \leq O(\delta)$ . The goal is to compute a new (unit) weight vector  $\mathbf{w}'$  that is close to an optimal halfspace, defined by  $\mathbf{v}^{*}$ , with
Figure 1: The disagreement region between a halfspace with normal vector  $\mathbf{w}$  and the target  $\mathbf{v}^*$  is shown in green. The unit direction  $\mathbf{u}$  corresponds to the projection of  $\mathbf{v}^*$  on the orthogonal complement of  $\mathbf{w}$ . We assume that the  $\ell_2$  distance of the two halfspaces is  $\delta$  (and thus their angle is  $\Theta(\delta)$ ). Since the slabs  $S_i = \{i\delta \leq |\mathbf{x} \cdot \mathbf{w}| \leq (i + 1)\delta\}$  have width  $\delta$ , the  $x$ -coordinate of the start of the  $i$ -th box is  $\Theta(i)$ .
respect to  $D^{\prime}$ , i.e.,  $\mathbf{Pr}_{\mathbf{x}^{\prime} \sim D_{\mathbf{x}}^{\prime}}[\mathrm{sign}(\mathbf{v}^{*} \cdot \mathbf{x}^{\prime}) \neq \mathrm{sign}(\mathbf{w}^{\prime} \cdot \mathbf{x}^{\prime})] \leq \alpha$ , for an appropriate  $\alpha > 0$ . We can then show that the halfspace defined by  $\mathbf{w}^{\prime}$  will be closer to the target halfspace (defined by  $\mathbf{v}^{*}$ ) with respect to the original distribution, i.e., we have that  $\mathbf{Pr}_{\mathbf{x} \sim D_{\mathbf{x}}}[\mathrm{sign}(\mathbf{v}^{*} \cdot \mathbf{x}) \neq \mathrm{sign}(\mathbf{w}^{\prime} \cdot \mathbf{x})] \leq O(\delta \alpha)$ . By repeating the above step, we iteratively reduce the disagreement with  $\mathbf{v}^{*}$  until we reach our target error of  $O(\mathrm{opt})$ . Similarly to [DKS18], instead of "hard" conditioning on a thin slice, we perform a "soft" localization step where (by rejection sampling) we transform the  $\mathbf{x}$ -marginal to a Gaussian whose covariance is  $O(\delta^{2})$  in the direction of  $\mathbf{w}$  and identity in the orthogonal directions, i.e.,  $\boldsymbol{\Sigma} = \mathbf{I} - (1 - \delta^{2})\mathbf{ww}^{\top}$ ; see Fact 3.2.
A crucial ingredient of our approach is a proper testable, weak agnostic learner with respect to the Gaussian distribution. More precisely, our tester-learner runs in polynomial time and either reports that the  $\mathbf{x}$ -marginal is not  $\mathcal{N}(\mathbf{0},\mathbf{I})$  or outputs a unit vector  $\mathbf{w}$  with small constant distance to the target  $\mathbf{v}^*$ , i.e.,  $\| \mathbf{w} - \mathbf{v}^*\|_2 \leq 1/100$ ; see Proposition 2.1. Our weak proper tester-learner first verifies that the given  $\mathbf{x}$ -marginal approximately matches constantly many low-degree moments with the standard Gaussian; and if it does, it returns the vector defined by the degree-1 Chow parameters, i.e.,  $\mathbf{c} = \mathbf{E}_{(\mathbf{x},y) \sim D}[y\mathbf{x}]$ . Our main structural result in this context shows that if  $D_{\mathbf{x}}$  approximately matches its low-degree moments with the standard Gaussian, then the Chow parameters of any homogeneous LTF with respect to  $D_{\mathbf{x}}$  are close to its Chow parameters with respect to  $\mathcal{N}(\mathbf{0},\mathbf{I})$ , i.e., for any homogeneous LTF  $f(\mathbf{x})$ , we have that  $\mathbf{E}_{\mathbf{x} \sim D_{\mathbf{x}}} [f(\mathbf{x})\mathbf{x}] \approx \mathbf{E}_{\mathbf{x}^* \sim \mathcal{N}(\mathbf{0},\mathbf{I})} [f(\mathbf{x}^*)\mathbf{x}^*]$ ; see Lemma 2.3. Since the Chow vector of a homogeneous LTF with respect to the Gaussian distribution is parallel to its normal vector  $\mathbf{v}^*$  (see Fact 2.2), it is not hard to show that the Chow vector of the LTF with respect to  $D_{\mathbf{x}}$  will not be very far from  $\mathbf{v}^*$  and will satisfy the (weak) learning guarantee of  $\| \mathbf{c} - \mathbf{v}^*\|_2 \leq 1/100$ . Finally, to deal with label noise, we show that if  $\mathbf{x}'$  has bounded second moments (a condition that we can efficiently test), we can robustly estimate  $\mathbf{E}_{\mathbf{x} \sim D_{\mathbf{x}}} [f(\mathbf{x})\mathbf{x}]$  with samples from  $D$  up to error  $O(\sqrt{\mathrm{opt}})$  (see Lemma 2.4), which suffices for our purpose of weak learning. The detailed description of our weak, proper tester-learner can be found in Section 2.
From Parameter Distance to Zero-One Error Having a (weak) testable proper learner, we can now use it on the localized (conditional) distribution  $D'$  and obtain a vector  $\mathbf{w}'$  that is closer to  $\mathbf{v}^*$  in  $\ell_2$  distance; see Lemma 3.3. However, our goal is to obtain a vector that has small zero-one disagreement with the target halfspace  $\mathbf{v}^*$ . Assuming that the underlying  $\mathbf{x}$ -marginal is a standard normal distribution, and that  $\| \mathbf{w} - \mathbf{v}^* \|_2 = \delta$ , it holds that  $\operatorname{Pr}_{\mathbf{x} \sim D_{\mathbf{x}}}[\operatorname{sign}(\mathbf{w} \cdot \mathbf{x}) \neq \operatorname{sign}(\mathbf{v}^* \cdot \mathbf{x})] = O(\delta)$ , which implies that achieving  $\ell_2$ -distance  $O(\mathrm{opt}) + \epsilon$  suffices. We give an algorithm that can efficiently either certify that small  $\ell_2$ -distance implies small zero-one disagreement with respect to the given marginal  $D_{\mathbf{x}}$  or declare that  $D_{\mathbf{x}}$  is not the standard normal.
The disagreement region of  $\mathbf{v}^*$  and  $\mathbf{w}$  is a union of two "wedges" (intersection of two halfspaces); see Figure 1. In order for our algorithm to work, we need to verify that these wedges do not contain too much probability mass. Similarly to our approach for the weak tester-leaner, one could try a moment-matching approach and argue that if  $D_{\mathbf{x}}$  matches its "low"-degree moments with  $\mathcal{N}(\mathbf{0},\mathbf{I})$ , then small  $\ell_2$ -distance translates to small zero-one disagreement. However, we will need to use this result for vectors that are very close to the target (but still not close enough), namely  $\| \mathbf{w} - \mathbf{v}^{*}\|_{2} = \delta$ ,
where  $\delta = \Theta (\epsilon)$ ; this would require matching  $\mathrm{poly}(1 / \delta)$  many moments (as we essentially need to approximate the wedge of Figure 1 with a polynomial) and would thus lead to an exponential runtime of  $d^{\mathrm{poly}(1 / \delta)}$ .
Instead of trying to approximate the disagreement region with a polynomial, we will make use of the fact that our algorithm knows  $\mathbf{w}$  (but not  $\mathbf{v}^*$ ) and approximate the disagreement region by a union of cylindrical slabs. We consider slabs of the form  $S_{i} = \{\mathbf{x}:i\delta \leq |\mathbf{w}\cdot \mathbf{x}|\leq (i + 1)\delta \}$ . If the target distribution is Gaussian, we know that the set  $|\mathbf{w}\cdot \mathbf{x}|\gg \sqrt{\log(1 / \epsilon)}$  has mass  $O(\delta)$  and we can essentially ignore it. Therefore, we can cover the whole space by considering roughly  $M = O(\sqrt{\log(1 / \delta)} /\delta)$  slabs of width  $\delta$  and split the disagreement region into the disagreement region inside each slab  $S_{i}$ . We have that
$$
\mathbf {P r} _ {\mathbf {x} \sim D _ {\mathbf {x}}} [ \operatorname {s i g n} (\mathbf {w} \cdot \mathbf {x}) \neq \operatorname {s i g n} (\mathbf {v} ^ {*} \cdot \mathbf {x}) ] \leq \sum_ {i = 1} ^ {M} \mathbf {P r} [ | \mathbf {u} \cdot \mathbf {x} | \geq i \mid \mathbf {x} \in S _ {i} ] \mathbf {P r} [ S _ {i} ],
$$
where  $\mathbf{u}$  is the unit direction parallel to the projection of the target  $\mathbf{v}^*$  onto the orthogonal complement of  $\mathbf{w}$ , see Figure 1. By the anti-concentration of the Gaussian distribution we know that each slab should have mass at most  $O(\delta)$ . Note that this is easy to test by sampling and computing empirical estimates of the mass of each slab. Moreover, assuming that underlying distribution is  $\mathcal{N}(0,\mathbf{I})$ , we have that, conditional on  $S_{i}$  the orthogonal direction  $\mathbf{u}\cdot \mathbf{x}\sim \mathcal{N}(0,1)$  (see Figure 1) and in particular  $\mathbf{u}\cdot \mathbf{x}$  has bounded second moment. We do not know the orthogonal direction  $\mathbf{u}$  as it depends on the unknown  $\mathbf{v}^*$  but we can check that, conditional on the slab  $S_{i}$ , the projection of  $D_{\mathbf{x}}$  onto the orthogonal complement of  $\mathbf{w}$  is (approximately) mean-zero and has bounded covariance (i.e., bounded above by 2I). Note that both these conditions hold when  $\mathbf{x}\sim \mathcal{N}(\mathbf{0},\mathbf{I})$  and can be efficiently tested with samples in time poly  $(d,1 / \delta)$ . Under those conditions we have that  $\mathbf{Pr}[S_i] = O(\delta)$  for all  $i$ . Moreover, when the conditional distribution on  $S_{i}$  (projected on the orthogonal complement of  $\mathbf{w}$ ) has bounded second moment, we have that  $\mathbf{Pr}[|\mathbf{u}\cdot \mathbf{x}|\geq i|\mathbf{x}\in S_i]\leq O(1 / i^2)$ . Combining the above, we obtain that under those assumptions the total probability of disagreement is at most  $O(\delta)$ . The detailed analysis is given in Section 3.1.
Obtaining an Active Learner. Our localization-based algorithm can be readily adapted to yield an efficient active learner-tester. We first observe that the tester that verifies that the parameter distance is proportional to  $0/1$  error does not require any labels, as it relies on verifying moments of the  $\mathbf{x}$ -marginal of the underlying distribution. During each iteration of localization, since the learner is only trying to find a constant approximation of the defining vector of the unknown halfspace, the learner only needs roughly  $O(d)$  samples that fall in the slice we localize to. Since we can selectively query only the labels of samples that fall in the right region and the total number of iterations is at most  $O(\log(1/\epsilon))$ , the labels needed to learn the defining vectors is roughly  $O(d\log(1/\epsilon))$ . In the end, the learner needs to find the best halfspace among the ones from  $O(\log(1/\epsilon))$  localization iterations. We remark this can also be done with polylog(1/ $\epsilon$ ) many label queries (see Lemma 3.6).
We use small boldface characters for vectors and capital bold characters for matrices. We use  $[d]$  to denote the set  $\{1,2,\ldots,d\}$ . For a vector  $\mathbf{x} \in \mathbb{R}^d$  and  $i \in [d]$ ,  $\mathbf{x}_i$  denotes the  $i$ -th coordinate of  $\mathbf{x}$ , and  $\|\mathbf{x}\|_2 \coloneqq \sqrt{\sum_{i=1}^d \mathbf{x}_i^2}$  the  $\ell_2$  norm of  $\mathbf{x}$ . We use  $\mathbf{x} \cdot \mathbf{y} \coloneqq \sum_{i=1}^n \mathbf{x}_i \mathbf{y}_i$  as the inner product between them. We use  $\mathbb{1}\{E\}$  to denote the indicator function of some event  $E$ .