achievingminimaxratesinpoolbasedbatchactivelearning/full.md

Achieving Minimax Rates in Pool-Based Batch Active Learning

Claudio Gentile¹ Zhilei Wang² Tong Zhang¹³

Abstract

We consider a batch active learning scenario where the learner adaptively issues batches of points to a labeling oracle. Sampling labels in batches is highly desirable in practice due to the smaller number of interactive rounds with the labeling oracle (often human beings). However, batch active learning typically pays the price of a reduced adaptivity, leading to suboptimal results. In this paper we propose a solution which requires a careful trade off between the informativeness of the queried points and their diversity. We theoretically investigate batch active learning in the practically relevant scenario where the unlabeled pool of data is available beforehand (pool-based active learning). We analyze a novel stage-wise greedy algorithm and show that, as a function of the label complexity, the excess risk of this algorithm matches the known minimax rates in standard statistical learning settings. Our results also exhibit a mild dependence on the batch size. These are the first theoretical results that employ careful trade offs between informativeness and diversity to rigorously quantify the statistical performance of batch active learning in the pool-based scenario.

1. Introduction

The aim of Active Learning is to reduce the data requirement of training processes through the careful selection of informative subsets of the data across several interactive rounds. This increased interactive power enables the adaptation of the sampling process to the actual state of the learning algorithm at hand, yet this benefit comes at the price of frequent re-training of the model and increased interactions with the labeling oracle (which is often just a

¹Google Research, New York ²Citadel Securities, New York ³The Hong Kong University of Science and Technology, Hong Kong. Correspondence to: Zhilei Wang zhilei-wang92@gmail.com.

Proceedings of the $39^{th}$ International Conference on Machine Learning, Baltimore, Maryland, USA, PMLR 162, 2022. Copyright 2022 by the author(s).

pool of human labelers).

The batch mode of active learning is one where labels are queried in batches of suitable size, and the models are re-trained/updated either after each batch or even less frequently. This sampling mode often corresponds to the way labels are gathered in practical large-scale processing pipelines.

Batch active learning tries to strike a reasonable balance between the benefits of adaptivity and the costs associated with interaction and re-training. Yet, since the sampling is split into batches, and model updates can only be performed at the end of each batch, a batch active learning algorithm has to prevent to the extent possible the sampling of redundant points. The standard trade-off that arises is then to ensure that the sampled points are informative enough for the model, if taken in isolation, while at the same time being diverse enough so as to avoid sampling redundant labels.

We study batch active learning in the pool-based model, where an unlabeled pool of data is made available to the algorithm beforehand, and the goal is to single out a subset of the data so as to achieve the same statistical performance as if training were carried out on the entire pool. In this setting, we describe and analyze novel algorithms that obtain minimax rates of convergence of their excess risk as a function of the number of requested labels. Interestingly enough, these optimal rates are retained even if we allow the batch size to grow with the pool size, the actual trade-off being ruled by the amount of noise in the data. Another appealing aspect is that our algorithms guarantee a number of re-training rounds which is at worst logarithmic, while being able to automatically adapt to the level of noise.

We operate in specific realizable settings, starting with linear or generalized linear models, and then extending our results to the more general non-linear setting. Unlike what is traditionally done by many algorithmic solutions to active learning available in the literature (e.g., (Balcan et al., 2007; Balcan & Long, 2013; Zhang & Li, 2021)), we do not formulate strong assumptions on the input distribution. We establish careful trade-offs between the informativeness and the diversity of the queried labels, and rigorously quantify the statistical performance on batch active learning in a noisy pool-based setting. To our knowledge, these are the first guarantees of this kind that apply to a noisy (hence

realistic) batch pool-based active learning scenario. See also the related work contained in Section 3.

1.1. Content and contributions

Our contributions can be described as follows.

1. We present an efficient algorithm for pool-based batch active learning for noisy linear models (Algorithm 1). This algorithm generates pseudo-labels by computing sequences of linear classifiers that restrict their attention to exponentially small regions of the margin space, and then trains a single model based on the pseudolabels only. The design inspiring the sampling within each stage is a G-optimal design, computed through a greedy strategy. We show (Theorem 4.1) that under the standard i.i.d. assumption of the (input, label) pairs, the model so trained enjoys an excess risk bound with respect to the Bayes optimal predictor which is best possible, when expressed in terms of the total number of requested labels. The number of re-training stages (that is, the number of linear classifiers computed to generate pseudo-labels) is at most logarithmic in the pool size, and automatically adapts to the noise level without knowing it in advance.
2. Since the above algorithm does not operate on a constant batch size $B$ , we show in Section 4.2 an easy adaptation to the constant batch size, and make the observation that $B$ therein may also scale as $T^{\beta}$ , for some exponent $\beta < 1$ that depends on the amount of noise (see comments surrounding Corollary 4.2), still retaining the above-mentioned optimal rates.
3. We extend in Section 5 our results to the generalized linear case (specifically, the logistic case), and point out that restricting to exponentially small regions of the margin space is also beneficial for obtaining bounds with a milder dependence on the loss curvature.
4. Last but not least, despite we work out the details only for (generalized) linear models, our algorithmic technique can be seen as a skeleton technique that can be applied to more general situations, provided the estimators employed at each stage and the diversity measure guiding the design have matching properties, as briefly discussed in Section 6.

2. Preliminaries and Notation

We denote by $\mathcal{X}$ the input space (e.g., $\mathcal{X} = \mathbb{R}^d$ ), by $\mathcal{Y}$ the output space, and by $\mathcal{D}$ an unknown distribution over $\mathcal{X} \times \mathcal{Y}$ . The corresponding random variables will be denoted by $\mathbf{x}$ and $y$ . We also denote by $\mathcal{D}_{\mathcal{X}}$ the marginal distribution of $\mathcal{D}$ over $\mathcal{X}$ . Given a function $h$ (also called a hypothesis or a

model) mapping $\mathcal{X}$ to $\mathcal{V}$ , the population loss (often referred to as risk) of $h$ is denoted by $\mathcal{L}(h)$ , and defined as $\mathcal{L}(h) = \mathbb{E}_{(\mathbf{x},y)\sim \mathcal{D}}[loss(h(x),y)]$ , where $loss(\cdot ,\cdot):\mathcal{Y}\times \mathcal{Y}\to [0,1]$ is a given loss function. For simplicity of presentation, we restrict ourselves to a binary classification setting with 0-1 loss, so that $\mathcal{Y} = {-1, + 1}$ , and $loss(\hat{y},y) = \mathbb{1}{\hat{y}\neq y} \in {0,1}$ , being $\mathbb{1}{\cdot }$ the indicator function of the predicate at argument. When clear from the surrounding context, we will omit subscripts like “ $(\mathbf{x},y)\sim \mathcal{D}$ ” from probabilities and expectations.

We are given a class of models $\mathcal{F} = {f:\mathcal{X}\to [0,1]}$ and the Bayes optimal predictor $h^* (x) = \operatorname {sgn}(f^* (x) - 1 / 2)$ where

$$f^{\ast }(\mathbf{x})=\mathbb{P}(y=1\mathrm{\mid }\mathbf{x})f\; ^\; \{*\}\; (\backslash mathbf\; \{x\})\; =\; \backslash mathbb\; \{P\}\; (y\; =\; 1\; |\; \backslash mathbf\; \{x\})$$

is assumed to belong to class $\mathcal{F}$ (the so-called realizability assumption). This assumption is reasonable whenever the model class $\mathcal{F}$ we operate on is wide enough. For instance, a realizability (or quasi-realizability) assumption seems natural in overparameterized settings implemented by nowadays' Deep Neural Networks.

As a simple example, we consider a generalized linear model

$$\begin{array}{cccc}& f^{\ast }(\mathbf{x})=\sigma (⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩),& & \text{(1)}\end{array}f\; ^\; \{*\}\; (\backslash mathbf\; \{x\})\; =\; \backslash sigma\; \backslash left(\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\backslash right),\; \backslash tag\; \{1\}$$

where $\sigma : \mathbb{R} \to [0,1]$ is a suitable sigmoidal function, e.g., $\sigma(z) = \frac{e^z}{1 + e^z}$ , $\mathbf{w}^*$ is an unknown vector in $\mathbb{R}^d$ , with bounded (Euclidean) norm $||\mathbf{w}|| \leq R$ for some $R \geq 1$ , and $\langle \cdot, \cdot \rangle$ denotes the usual inner product in $\mathbb{R}^d$ .

Throughout this paper, we adopt the commonly used low-noise condition on the marginal distribution $\mathcal{D}_{\mathcal{X}}$ of Mammen & Tsybakov (1999): there are constant $c > 0$ , $\epsilon_0 \in (0,1]$ and exponent $\alpha \geq 0$ such that for all $\epsilon \in (0,\epsilon_0]$ we have

Notice, in particular, that $\alpha \to \infty$ gives the so-called hard margin condition $\mathbb{P}\big(|f^{*}(\mathbf{x}) - 1 / 2| < \epsilon \big) = 0$ . while, at the opposite end of the spectrum, exponent $\alpha = 0$ (and $c = 1$ ) corresponds to making no assumptions whatsoever on $\mathcal{D}_{\mathcal{X}}$ . For simplicity, we shall assume throughout that the above low-noise condition holds for $c = 1$ . The noise exponent $\alpha$ and range constant $\epsilon_0$ are typically unknown, and our algorithms will not rely on the prior knowledge of them.

We are given a class of models $\mathcal{F}$ , and a pool $\mathcal{P}$ of $T$ unlabeled instances $\mathbf{x}_1, \ldots, \mathbf{x}T \in \mathcal{X}$ , drawn i.i.d. according to a marginal distribution $\mathcal{D}{\mathcal{X}}$ obeying condition (2) (with $c = 1$ ). The associated labels $y_1, \ldots, y_T \in \mathcal{Y}$ are such that the pairs $(\mathbf{x}_t, y_t), t = 1, \ldots, T$ , are drawn i.i.d. according to $\mathcal{D}$ , the labels being generated according to the conditional distribution determined by some $f^* \in \mathcal{F}$ . The labels are not initially revealed to us, and the goal of the active learning algorithm is to come up at the end of training with a model

$\widehat{h}:\mathcal{X}\to \mathcal{Y}$ whose excess risk $\mathcal{L}(\widehat{h}) - \mathcal{L}(h^{*})$ is as small as possible, while querying as few labels as possible in $\mathcal{P}$ .

The way labels are queried follows the standard batch active learning protocol. We are given a batch size $B \geq 1$ . Label acquisition and learning proceeds in a sequence of stages, $\ell = 1,2,\ldots$ . At each stage $\ell$ , the algorithm is allowed to query $B$ -many labels by only relying on labels acquired in the past $\ell - 1$ stages. Notice that each point $\mathbf{x}t$ in pool $\mathcal{P}$ can only be queried once, which is somehow equivalent to assuming that the noise in the corresponding label $y_t$ is persistent. We shall henceforth denote by $N_T(\mathcal{P})$ the total number of labels (sometimes referred to as label complexity) queried by the algorithm at hand on pool $\mathcal{P}$ , and by $N{T,B}(\mathcal{P})$ the same quantity if we want to emphasize the dependence on $B$ .

The analysis of our algorithms hinges upon a suitable measure of diversity, $D(\mathbf{x}, S)$ , that quantifies how far off a data point $\mathbf{x} \in \mathcal{X}$ is from a finite set of points $S \subseteq \mathcal{X}$ . Though many diversity measures may be adopted for practical purposes (e.g., (Wei et al., 2015; Sener & Savarese, 2018; Kirsch et al., 2019; Ash et al., 2020; Killamsetty et al., 2020; Kirsch et al., 2021; Citovsky et al., 2021)), the one enabling tight theoretical analyses for our algorithms is a spectral-like diversity measure defined in the finite dimensional case $\mathcal{X} = \mathbb{R}^d$ as $D(\mathbf{x}, S) = \langle \mathbf{x}, \mathbf{x} \rangle_{A_S^{-1}}^{\frac{1}{2}} = ||\mathbf{x}||{A_S^{-1}} = \sqrt{\mathbf{x}^\top A_S^{-1}\mathbf{x}}$ , that is, the Mahalanobis norm of $\mathbf{x}$ w.r.t. the positive semi-definite matrix $A_S^{-1}$ , where $A_S = I + \sum{\mathbf{z} \in S} \mathbf{z}\mathbf{z}^\top$ , being $I$ the $d \times d$ identity matrix. Notice that $D(\mathbf{x}, S)$ is large when $\mathbf{x}$ is aligned with small eigenvectors of $A_S$ , while it is small if $\mathbf{x}$ is aligned with large eigenvectors of that matrix. In particular, $D(\mathbf{x}, S)$ achieves its maximal value $||\mathbf{x}||^2$ when $\mathbf{x}$ is orthogonal to the space spanned by $S$ . Hence, $\mathbf{x}$ is "very different" from $S$ as measured by $D(\mathbf{x}, S)$ if $\mathbf{x}$ contributes a direction of the input space which is not already spanned by $S$ . We denote by $|A_S|$ the determinant of matrix $A_S$ .

At an intuitive level, since the label requests are batched, and model updates are typically performed only at the end of each stage, a batch active learning algorithm is compelled to operate within each stage by trading off the (predicted) informativeness of the selected labels against the diversity of the data points whose labels are requested. Moreover, the larger the batch size $B$ the less adaptive the algorithm is forced to be, hence we expect $B$ to somehow play a role in the performance of the algorithm.

From a practical standpoint, there are indeed two separate notions of adaptivity to consider. One is the number of interactive rounds with the labeling oracle, the other is the number of times we re-train (or update) a model based on the labels gathered during the interactive rounds. The two notions need not coincide. While the former essentially accounts for the cost of interacting with human labelers, the

latter is more related to the cost of re-training/updating a (potentially very complex) learning system.

3. Related Work

While experimental studies on batch active learning are reported since the early 2000s (see, e.g., (Hoi et al., 2006)), it is only with the deployment at scale of Deep Neural Networks that we have seen a general resurgence of interest in active learning, and batch active learning in particular. The batch pool-based model studied here is the one that has spurred the widest attention, as it corresponds to the way in practice labels are gathered in large-scale processing pipelines. This interest has generated a flurry of recent investigations, mainly of experimental nature, yet containing a lot of interesting and diverse approaches to batch active learning. Among these are (Gu et al., 2012; 2014; Sener & Savarese, 2018; Kirsch et al., 2019; Zhdanov, 2019; Shui et al., 2020; Ash et al., 2020; Kim et al., 2020; Killamsetty et al., 2020; Kirsch et al., 2021; Ghorbani et al., 2021; Citovsky et al., 2021; Kothawade et al., 2021).

On the theoretical side, active learning is a well-studied subfield of statistical learning. General references in pool-based active learning include (Dasgupta, 2004; 2005; Hanneke, 2014; Nowak, 2011; Tosh & Dasgupta, 2017), and specific algorithms for half-spaces under classes of input distributions are contained, e.g., in (Balcan et al., 2007; Balcan & Long, 2013; Zhang & Li, 2021). However, none of these papers tackle the practically relevant scenario of batch active learning. In fact, restricting to theoretical aspects of batch active learning makes the research landscape far less populated. Below we briefly summarize what we think are among the most relevant papers to our work, as directly related to batch active learning, and then mention recent efforts in contiguous fields, like adaptive sampling and subset selection, which may serve as a general reference and inspiration.

Batch active learning in the pool-based scenario is one of the motivating applications in (Chen & Krause, 2013), where the main concern is to investigate general conditions under which a batch greedy policy achieves similar performance as the optimal policy that operates with the same batch size. Yet, the authors consider simple noise free scenarios, while the important observation (Theorem 2 therein) that a batch greedy algorithm is also competitive with respect to an optimal fully sequential policy (batch size one) does not apply to active learning. Chen et al. (2015; 2017) are along similar lines, with the addition of persistent noise, but do not tackle batch active learning problems.

A paper with a similar aim as ours, yet operating in the streaming setting of active learning, is (Amin et al., 2020). The authors show that some classes of fully sequential ac

tive algorithms can be turned into sequential algorithms that query labels in batches and suffer only an additive (times log factors) overhead in the label complexity. This transformation is essentially obtained by freezing the state of the fully sequential algorithm, but it is unclear whether any notion of diversity over the batch is enforced by the resulting batch algorithms.

Very recent stream-based active learning papers that are worth mentioning are (Katz-Samuels et al., 2021; Camilleri et al., 2021b)). These papers share similar methods and modeling assumptions as ours in leveraging optimal design, but they do not deal with batch active learning. The main concern there is essentially to improve the performance of adaptive sampling by reducing the variance of the estimators.

A learning problem similar to pool-based batch active learning is training subset selection (sometimes called dataset summarization), whose goal is to come up with a compressed version of a (big) dataset that offers to a given learning algorithm the same inference capabilities as if applied to the original dataset. The problem can be organized in rounds (as in batch active learning) and bridging one to the other can in practice be done by label hallucination/pseudolabeling. Representative works include (Wei et al., 2015; Killamsetty et al., 2020; Borsos et al., 2021).

4. The Linear Case

We start off by considering a simple linear model of the form $f^{}(\mathbf{x}) = \frac{1 + \langle\mathbf{w}^{},\mathbf{x}\rangle}{2}$ , where both $\mathbf{w}^*$ and $\mathbf{x}$ lie in the $d$ -dimensional Euclidean unit ball (so that $\langle \mathbf{w}^*,\mathbf{x}\rangle \in [-1,1]$ and $f^{*}(\mathbf{x})\in [0,1])$ ). Algorithm 1 contains in a nutshell the main ideas behind our algorithmic solutions, which is to greedily approximate a G-optimal design in the selection of points at each stage. The way it is formulated, Algorithm 1 does not operate with a constant batch size $B$ per stage. We will reduce to the constant batch size case in Section 4.2.

The algorithm takes as input a finite pool of points $\mathcal{P}$ of size $T$ and proceeds across stages $\ell = 1,2,\ldots$ by generating at each stage $\ell$ a (linear-threshold) predictor $\mathrm{sgn}(\langle \mathbf{w}{\ell},\mathbf{x}\rangle)$ , where $\mathbf{w}{\ell}$ is a ridge regression estimator computed only on the labeled pairs $(\mathbf{x}{\ell ,1},y{\ell ,1}),\dots ,(\mathbf{x}{\ell ,T{\ell}},y_{\ell ,T_{\ell}})$ collected during that stage. These predictors are used to trim the current pool $\mathcal{P}{\ell -1}$ by eliminating both the points on which $\mathbf{w}{\ell}$ is itself confident (set $\mathcal{C}{\ell}$ ) and those whose labels have just been queried (set $\mathcal{Q}{\ell}$ ). At each stage $\ell$ , the points $\mathbf{x}{\ell ,t}$ to query are selected in a greedy fashion by maximizing $^1 D(\mathbf{x},\mathcal{Q}\ell) = ||\mathbf{x}||{A{\ell ,t - 1}^{-1}}$ over the current pool $\mathcal{P}{\ell -1}$ (excluding the already selected points $\mathcal{Q}{\ell}$ , which are contained in $A_{\ell ,t - 1}$ ), so as to make $\mathbf{x}{\ell ,t}$ maximally different from $\mathcal{Q}{\ell}$ .

Algorithm 1: Pool-based batch active learning algorithm for linear models.

1 Input: Confidence level $\delta \in (0,1]$ , pool of instances $\mathcal{P} \subseteq \mathbb{R}^d$ of size $|\mathcal{P}| = T$
2 Initialize: $\mathcal{P}_0 = \mathcal{P}$
3 for $\ell = 1,2,\ldots$

4 Initialize within stage $\ell$

$$\begin{array}{c}\bullet {ϵ}_{\mathrm{\ell }}=2^{-\mathrm{\ell }}\mathrm{/}(\sqrt{2\log \frac{2\mathrm{\ell }(\mathrm{\ell }+1)T}{\delta }}+1)\\ \bullet A_{\mathrm{\ell },0}=I,t=0,{\mathcal{Q}}_{\mathrm{\ell }}=\mathrm{\varnothing }\end{array}\backslash begin\{array\}\{l\}\; \backslash bullet\; \backslash epsilon\_\; \{\backslash ell\}\; =\; 2\; ^\; \{-\; \backslash ell\}\; /\; (\backslash sqrt\; \{2\; \backslash log\; \backslash frac\; \{2\; \backslash ell\; (\backslash ell\; +\; 1)\; T\}\{\backslash delta\}\}\; +\; 1)\; \backslash \backslash \; \backslash bullet\; A\; \_\; \{\backslash ell\; ,\; 0\}\; =\; I,\; t\; =\; 0,\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\; =\; \backslash emptyset\; \backslash \backslash \; \backslash end\{array\}$$

while $\mathcal{P}{\ell -1}\backslash \mathcal{Q}\ell \neq \emptyset$ and $\max{\mathbf{x}\in \mathcal{P}{\ell -1}\backslash \mathcal{Q}\ell}| \mathbf{x}|{A_{\ell ,t}^{-1}} > \epsilon_\ell$

$$\begin{array}{c}\bullet t=t+1\\ \bullet \mathrm{Pick}{\mathbf{x}}_{\mathrm{\ell },t}\in \mathrm{\parallel }\mathbf{x}{\mathrm{\parallel }}_{A_{\mathrm{\ell },t-1}^{-1}}\\ \bullet \text{U p d a t e}{A}_{\mathrm{\ell },t}=A_{\mathrm{\ell },t-1}+{\mathbf{x}}_{\mathrm{\ell },t}{\mathbf{x}}_{\mathrm{\ell },t}^{\mathrm{\top }}\\ \bullet {\mathcal{Q}}_{\mathrm{\ell }}={\mathcal{Q}}_{\mathrm{\ell }}\cup \left\{{\mathbf{x}}_{\mathrm{\ell },t}\right\}\end{array}\backslash begin\{array\}\{l\}\; \backslash bullet\; t\; =\; t\; +\; 1\; \backslash \backslash \; \backslash bullet\; \backslash operatorname\; \{P\; i\; c\; k\}\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}\; \backslash in\; \backslash underset\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\; \_\; \{\backslash ell\; -\; 1\}\; \backslash backslash\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\}\; \{\backslash operatorname\; \{a\; r\; g\; m\; a\; x\}\}\; \backslash |\; \backslash mathbf\; \{x\}\; \backslash |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; t\; -\; 1\}\; ^\; \{-\; 1\}\}\; \backslash \backslash \; \backslash bullet\; \backslash text\; \{U\; p\; d\; a\; t\; e\}\; A\; \_\; \{\backslash ell\; ,\; t\}\; =\; A\; \_\; \{\backslash ell\; ,\; t\; -\; 1\}\; +\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}\; ^\; \{\backslash top\}\; \backslash \backslash \; \backslash bullet\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\; =\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\; \backslash cup\; \backslash left\backslash \{\backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}\; \backslash right\backslash \}\; \backslash \backslash \; \backslash end\{array\}$$

Set $T_{\ell} = t$ , the number of queries made in stage $\ell$ if $\mathcal{Q}_{\ell}\neq \emptyset$

• Query the labels $y_{\ell,1}, \ldots, y_{\ell,T_{\ell}}$ associated with the unlabeled data in $\mathcal{Q}_{\ell}$ , and compute

$${\mathbf{w}}_{\mathrm{\ell }}=A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}\sum _{t=1}^{T_{\mathrm{\ell }}}{y}_{\mathrm{\ell },t}{\mathbf{x}}_{\mathrm{\ell },t}\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; =\; A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; y\; \_\; \{\backslash ell\; ,\; t\}\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}$$

• Set $\mathcal{C}{\ell} = {\mathbf{x}\in \mathcal{P}{\ell -1}\backslash \mathcal{Q}{\ell}:|\langle \mathbf{w}{\ell},\mathbf{x}\rangle | > 2^{-\ell}}$
• Compute pseudo-labels on each $\mathbf{x} \in \mathcal{C}{\ell}$ as $\hat{y} = \operatorname{sgn}\langle \mathbf{w}{\ell}, \mathbf{x} \rangle$

else

Set $\mathcal{P}{\ell} = \mathcal{P}{\ell -1}\backslash (\mathcal{C}{\ell}\cup \mathcal{Q}{\ell})$

$$\mathbf{i}\mathbf{f}d\mathrm{/}{2}^{-\mathrm{\ell }+1}>2^{-\mathrm{\ell }+1}\mathrm{\mid }{\mathcal{P}}_{\mathrm{\ell }}\mathrm{\mid }\backslash mathbf\; \{i\; f\}\; d\; /\; 2\; ^\; \{-\; \backslash ell\; +\; 1\}\; >\; 2\; ^\; \{-\; \backslash ell\; +\; 1\}\; |\; \backslash mathcal\; \{P\}\; \_\; \{\backslash ell\}\; |$$

$$\bullet L=\mathrm{\ell }\backslash bullet\; L\; =\; \backslash ell$$

• Exit the for-loop ( $L$ is the total no. of stages)

5 Predict labels in pool $\mathcal{P}$ :

• Train an SVM classifier $\widehat{\mathbf{w}}$ on $\cup_{\ell=1}^{L} \mathcal{C}_{\ell}$ via the generated pseudo-labels $\hat{y}$
• Predict on each $\mathbf{x} \in (\cup_{\ell=1}^{L} \mathcal{Q}{\ell}) \cup \mathcal{P}{L}$ through $\operatorname{sgn}(\langle \widehat{\mathbf{w}}, \mathbf{x} \rangle)$

When stage $\ell$ terminates, we are guaranteed that we have collected a set of points $\mathcal{Q}{\ell}$ such that all remaining points $\mathbf{x}$ in the pool satisfy $D(\mathbf{x},\mathcal{Q}{\ell})\leq \epsilon_{\ell}$ . Threshold $\epsilon_{\ell}$ , defined at the beginning of the stage, is exponentially decaying with $\ell$ . It is this threshold that determines the actual length of the stage, and rules the elimination of un queried points from the pool, along with the corresponding generation of pseudo-labels during the stage.

Algorithm 1 stops generating new stages when the size $|\mathcal{P}{\ell}|$ of pool $\mathcal{P}{\ell}$ triggers the condition $d / 2^{-\ell +1} > 2^{-\ell +1}|\mathcal{P}{\ell}|$ (which is satisfied, in particular, when $\mathcal{P}{\ell}$ becomes empty). In that case, the current stage $\ell$ becomes the final stage $L$

Finally, the algorithm uses the subset of points $\cup_{\ell=1}^{L}\mathcal{C}{\ell}$ and the associated pseudo-labels $\hat{y}$ generated during the $L$ stages to train a linear classifier $\widehat{\mathbf{w}}$ (e.g., an SVM) to zero empirical error on that subset. Our analysis (see Appendix A) shows that with high probability such a consistent linear classifier exists. Each point $\mathbf{x}$ that remains in the pool, that is, each $\mathbf{x} \in (\cup{\ell=1}^{L}\mathcal{Q}{\ell}) \cup \mathcal{P}{L}$ , is assigned label $\mathrm{sgn}(\langle \widehat{\mathbf{w}},\mathbf{x}\rangle)$ . Notice, in particular, that $\widehat{\mathbf{w}}$ is not trying to fit the queried labels of $\cup_{\ell=1}^{L}\mathcal{Q}{\ell}$ , but only the pseudo-labels of $\cup{\ell=1}^{L}\mathcal{C}_{\ell}$ .

The fact that the algorithm only uses pseudo-labels to train its final predictor may look counter-intuitive at first, but this is due to our proof technique that derives an excess risk bound out of weighted empirical risk bounds — see, e.g., the proof sketch of Theorem 4.1. Algorithmically, the queried labels can be noisy and, in general, we do not know whether they are consistent with the Bayes optimal predictor $\mathbf{w}^*$ . In this sense, the process of generating pseudo-labels can be seen as a label denoising process. This is made possible by our algorithm, which guarantees (with high probability) that for the selected data points, pseudo-labels generated by the model are consistent with those of the Bayes optimal predictor, while labels of other data (including those in the training data) may not.

Further, notice that the final predictor $\hat{\mathbf{w}}$ need not be an SVM. Any training algorithm that returns a linear classifier which is consistent with the pseudo-labels will suffice. From our analysis we know that such a linear classifier has to exist (with high probability). Incidentally, this is the main the reason why relying on (denoised) pseudo-labels facilitates our statistical analysis, beyond the involved algorithms.

It is also worth observing how Algorithm 1 resolves the trade-off between informativeness and diversity we alluded to in previous sections. Once we reach stage $\ell$ , what remains in the pool are only the points $\mathbf{x}$ such that $|\langle \mathbf{w}{\ell - 1}, \mathbf{x} \rangle| \leq 2^{-\ell + 1}$ (this is because we have eliminated in stage $\ell - 1$ all the points in $\mathcal{C}{\ell - 1}$ ). Hence, the remaining points which the approximate G-optimal design operates with in stage $\ell$ are those which the previous model $\mathbf{w}_{\ell - 1}$ is not sufficiently confident on. The algorithm then puts all

these low-confident points on the same footing (that is, they are considered equally informative if taken in isolation), and then relies on the approximate G-optimal design scheme to maximize diversity among them. The set-wise diversity measure we end up maximizing is indeed a determinant-like diversity measure. This is easily seen from the fact that $\sum_{t=1}^{T_{\ell}}||\mathbf{x}{\ell,t}||{A_{\ell,t-1}^{-1}}^2 \approx \log |A_{\ell,T_{\ell}}|$ .

On one hand, this careful selection of points contributes to keeping the variance of estimator $\mathbf{w}{\ell}$ under control. On the other hand, the fact that we stop accumulating labels when $\max{\mathbf{x} \in \mathcal{P}{\ell-1} \setminus \mathcal{Q}{\ell}} | \mathbf{x} |{A{\ell, T_{\ell}}^{-1}} \leq \epsilon_{\ell}$ essentially implies that $\operatorname{sgn}(\langle \mathbf{w}_{\ell}, \mathbf{x} \rangle) = \operatorname{sgn}(\langle \mathbf{w}^{}, \mathbf{x} \rangle)$ on all points $\mathbf{x}$ we generate pseudo-labels for, which in turn ensures that these pseudolabels are consistent with $\mathbf{w}^{}$ .

Sequential experimental design has become popular, e.g., in the (contextual) bandits literature, see Ch. 22 in (Lattimore & Szepesvari, 2020), and is explicitly contained in recent works on best arm identification (e.g., (Fiez et al., 2019; Camilleri et al., 2021a)). Notice that in those works a design is a distribution over the set of actions (which would correspond to pool $\mathcal{P}$ in our case), and the algorithm is afforded to sample a given action $\mathbf{x}t$ multiple times, obtaining each time a fresh reward value $y{t}$ such that $\mathbb{E}[y_t\mid \mathbf{x}_t] = \langle \mathbf{w}^*,\mathbf{x}_t\rangle$ . This is not conceivable in a pool-based active learning scenario where label noise is persistent, and each "action" $\mathbf{x}_t$ can only be played once. This explains why the design we rely upon here is necessarily more restrained than in those papers.

4.1. Analysis

The following is the main result of this section.2

Theorem 4.1. Let $T \geq d$ and assume that $| \mathbf{x} |_2 \leq 1$ for all $\mathbf{x} \in \mathcal{P}$ . Then with probability at least $1 - \delta$ over the random draw of $(\mathbf{x}_1, y_1), \ldots, (\mathbf{x}_T, y_T) \sim \mathcal{D}$ the excess risk $\mathcal{L}(\widehat{\mathbf{w}}) - \mathcal{L}(\mathbf{w}^*)$ , the label complexity $N_T(\mathcal{P})$ , and the number of stages $L$ generated by Algorithm 1 are simultaneously upper bounded as follows:

$$\begin{array}{c}\mathcal{L}(\widehat{\mathbf{w}})-\mathcal{L}({\mathbf{w}}^{\ast })\\ \le \bar{C}C(\delta ,T,{ϵ}_0)(\max \{{\left(\frac{d}{T}\right)}^{\frac{\alpha +1}{\alpha +2}},\frac{d}{T{ϵ}_0}\}+\frac{\log \left(\frac{\log T}{\delta }\right)}{T}),\end{array}\backslash begin\{array\}\{l\}\; \backslash mathcal\; \{L\}\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\})\; -\; \backslash mathcal\; \{L\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; \backslash \backslash \; \backslash leq\; \backslash bar\; \{C\}\; C\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; \backslash left(\backslash max\; \backslash left\backslash \{\backslash left(\backslash frac\; \{d\}\{T\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{T\; \backslash epsilon\_\; \{0\}\}\; \backslash right\backslash \}\; +\; \backslash frac\; \{\backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\}\{T\}\backslash right),\; \backslash \backslash \; \backslash end\{array\}$$

$$\begin{array}{c}{N}_T(\mathcal{P})\\ \le \bar{C}C(\delta ,T,{ϵ}_0)(\max \{d^{\frac{\alpha }{\alpha +2}}{T}^{\frac{2}{\alpha +2}},\frac{d}{{ϵ}_0^2}\}+{\log }^2\left(\frac{\log T}{\delta }\right)),\end{array}\backslash begin\{array\}\{l\}\; N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; \backslash \backslash \; \backslash leq\; \backslash bar\; \{C\}\; C\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; \backslash left(\backslash max\; \backslash left\backslash \{d\; ^\; \{\backslash frac\; \{\backslash alpha\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\; ^\; \{2\}\}\; \backslash right\backslash \}\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right),\; \backslash \backslash \; \backslash end\{array\}$$

$$L\le \bar{C}(\max \{\frac{\log \left(\frac{T}{d}\right)}{\alpha +2},\log \left(\frac{4}{{ϵ}_0}\right)\}+\log \left(\frac{\log T}{\delta }\right)),L\; \backslash leq\; \backslash bar\; \{C\}\; \backslash left(\backslash max\; \backslash left\backslash \{\backslash frac\; \{\backslash log\; \backslash left(\backslash frac\; \{T\}\{d\}\backslash right)\}\{\backslash alpha\; +\; 2\},\; \backslash log\; \backslash left(\backslash frac\; \{4\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\; \backslash right\backslash \}\; +\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right),$$

for an absolute constant $\bar{C}$ and

$$C(\delta ,T,{ϵ}_0)={\log }^2\left(\frac{T}{\delta }\right)(1+{\log }^2\left(\frac{1}{{ϵ}_0}\right))\mathrm{.}C\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; =\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; \backslash left(1\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right).$$

Proof sketch. We first derive a high-probability bound on the weighted empirical risk

$$R_T(\mathcal{P})=\sum _{\mathbf{x}\in \mathcal{P}}\mathbb{1}\{\mathrm{s}\mathrm{g}\mathrm{n}⟨\widehat{\mathbf{w}},\mathbf{x}⟩\mathrm{\ne }\mathrm{s}\mathrm{g}\mathrm{n}⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\}\mathrm{\mid }⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\mathrm{\mid },R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; \backslash sum\_\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\}\; \backslash mathbb\; \{1\}\; \backslash \{\backslash mathrm\; \{s\; g\; n\}\; \backslash langle\; \backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash neq\; \backslash mathrm\; \{s\; g\; n\}\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash \}\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |,$$

and then turn it into an excess risk bound through a uniform convergence argument. In order to bound $R_{T}(\mathcal{P})$ , we partition the points in $\mathcal{P}$ into the three subsets

$${\cup }_{\mathrm{\ell }=1}^L{\mathcal{C}}_{\mathrm{\ell }},{\cup }_{\mathrm{\ell }=1}^L{\mathcal{Q}}_{\mathrm{\ell }},{\mathcal{P}}_L,\backslash cup\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{C\}\; \_\; \{\backslash ell\},\; \backslash qquad\; \backslash cup\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\},\; \backslash qquad\; \backslash mathcal\; \{P\}\; \_\; \{L\},$$

and consider the contribution to $R_{T}(\mathcal{P})$ of each subset separately.

When $\mathbf{x} \in \cup_{\ell=1}^{L} \mathcal{C}_{\ell}$ , we show that the pseudo-labels $\hat{y}$ generated by the algorithm are with high probability consistent with those generated by $\mathbf{w}^*$ , that is, $\operatorname{sgn}\langle \widehat{\mathbf{w}}, \mathbf{x} \rangle = \operatorname{sgn}\langle \mathbf{w}^*, \mathbf{x} \rangle$ , hence those $\mathbf{x}$ do not contribute to the weighted empirical risk.

Any $\mathbf{x} \in \mathcal{Q}{\ell}$ , is shown to contribute to $R{T}(\mathcal{P})$ by at most $2^{-\ell}$ , thus the overall contribution of $\cup_{\ell=1}^{L} \mathcal{Q}{\ell}$ can be bounded by $\sum{\ell=1}^{L} T_{\ell} / 2^{\ell}$ . In turn, by the way points are picked, $T_{\ell}$ is roughly bounded by $d / \epsilon_{\ell}^{2}$ , allowing us to conclude that the total contribution of $\cup_{\ell=1}^{L} \mathcal{Q}_{\ell}$ is bounded by

$$\sum _{\mathrm{\ell }=1}^L{2}^{-\mathrm{\ell }}d\mathrm{/}{ϵ}_{\mathrm{\ell }}^2\approx d\mathrm{/}{ϵ}_L\mathrm{.}\backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; 2\; ^\; \{-\; \backslash ell\}\; d\; /\; \backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}\; \backslash approx\; d\; /\; \backslash epsilon\_\; \{L\}.$$

Next, for $\mathbf{x} \in \mathcal{P}_L$ , we show that (with high probability) it must be $|\langle \mathbf{w}^*, \mathbf{x} \rangle| \leq 2^{-L}$ which, combined with the stopping condition defining $L$ implies an overall contribution of the same form $d / \epsilon_L$ .

Finally, since $L$ is itself a random variable, we need to devise high probability upper bounds on it. We rely on the low noise assumption (2) to conclude that $L$ is with high probability of the form

$$\max \{\frac{\log (T\mathrm{/}d)}{\alpha +2},\log \left(\frac{4}{{ϵ}_0}\right)\},\backslash max\; \backslash left\backslash \{\backslash frac\; \{\backslash log\; (T\; /\; d)\}\{\backslash alpha\; +\; 2\},\; \backslash log\; \backslash left(\backslash frac\; \{4\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\; \backslash right\backslash \},$$

which we replace back into the previous bounds yielding a guarantee of the form

$$R_T(\mathcal{P})\lesssim \max \{d^{\frac{\alpha +1}{\alpha +2}}{T}^{\frac{1}{\alpha +2}},\frac{d}{{ϵ}_0}\},R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; \backslash lesssim\; \backslash max\; \backslash left\backslash \{d\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\}\; \backslash right\backslash \},$$

hence an excess risk bound of the form

$$\mathcal{L}(\widehat{\mathbf{w}})-\mathcal{L}({\mathbf{w}}^{\ast })\approx \max \{{\left(\frac{d}{T}\right)}^{\frac{\alpha +1}{\alpha +2}},\frac{d}{T{ϵ}_0}\}\mathrm{.}\backslash mathcal\; \{L\}\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\})\; -\; \backslash mathcal\; \{L\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; \backslash approx\; \backslash max\; \backslash left\backslash \{\backslash left(\backslash frac\; \{d\}\{T\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{T\; \backslash epsilon\_\; \{0\}\}\; \backslash right\backslash \}.$$

The analysis of the label complexity $N_{T}(\mathcal{P}) = \sum_{\ell=1}^{L} T_{\ell}$ follows a similar pattern, but it does not require uniform convergence.

4.2. Constant batch size

We now describe a simple modification to Algorithm 1 that makes it work in the constant batch size case. Let us denote by $T_{\ell}$ the length of stage $\ell$ in Algorithm 1. The modified algorithm simply runs Algorithm 1: If $T_{\ell} < B$ the modified algorithm relies on model $\mathbf{w}{\ell}$ generated by Algorithm 1 without saturating the budget of $B$ labels at that stage. On the contrary, if $T{\ell} \geq B$ , the modified algorithm splits stage $\ell$ of Algorithm 1 into $\lceil T_{\ell} / B \rceil$ stages of size $B$ (except, possibly, for the last one), and then uses the queried set $\mathcal{Q}{\ell}$ generated by Algorithm 1 across all those stages. Hence, in this case, the modified algorithm is not exploiting the potential benefit of updating the model every $B$ queried labels. For instance, if $B = 100$ and $T{\ell} = |\mathcal{Q}{\ell}| = 240$ , the modified algorithm will split this stage into three successive stages of size 100, 100, and 40, respectively, and then rely on the 240 labels queried by Algorithm 1 across the three stages. In particular, the update of the model $\mathbf{w}{\ell}$ , and the associated pseudo-label computation on sets $\mathcal{C}_{\ell}$ is only performed at the end of the third stage.

Notice that the modified algorithm we just described is a legitimate pool-based batch active learning algorithm operating on a constant batch size $B$ , and its analysis is a direct consequence of the one in Theorem 4.1, after we take care of the possible over-counting that may arise in the reduction. Specifically, observe that the final hypothesis $\hat{\mathbf{w}}$ produced by the modified algorithm is the same as the one computed by Algorithm 1, hence the same bound on the excess risk applies. As for label complexity, if we stipulate that a batch algorithm operating on a constant batch size $B$ will be billed $B$ labels at each stage even if it ends up querying less, then the label complexity of the modified algorithm will overcount the number of labels simply due to the rounding off in $\lceil T_{\ell} / B \rceil$ . However, at each of the $L$ stages of Algorithm 1, the over-counting is bounded by $B$ , so that, overall, the label complexity of the constant batch size variant exceeds the one of Algorithm 1 by at most an additive $BL$ term which, due to the bound on $L$ in Theorem 4.1, is of the form $\max \left{\frac{B}{\alpha + 2} \log \left(\frac{T}{d}\right), B \log \left(\frac{1}{\epsilon_0}\right)\right}$ . This is summarized in the following corollary.

Corollary 4.2. With the same assumptions and notation as in Theorem 4.1, with probability at least $1 - \delta$ over the random draw of $(\mathbf{x}_1,y_1),\ldots ,(\mathbf{x}T,y_T)\sim \mathcal{D}$ the label complexity $N{T,B}(\mathcal{P})$ achieved by the modified algorithm operating on a batch of size $B$ is bounded as follows:

$$\begin{array}{c}{N}_{T,B}(\mathcal{P})\\ \le \bar{C}C(\delta ,T,{ϵ}_0)(\max \{d^{\frac{\alpha }{\alpha +2}}{T}^{\frac{2}{\alpha +2}},\frac{d}{{ϵ}_0^2}\}+{\log }^2\left(\frac{\log T}{\delta }\right))\\ +B\bar{C}(\max \{\frac{\log \left(\frac{T}{d}\right)}{\alpha +2},\log \left(\frac{4}{{ϵ}_0}\right)\}+\log \left(\frac{\log T}{\delta }\right)),\end{array}\backslash begin\{array\}\{l\}\; N\; \_\; \{T,\; B\}\; (\backslash mathcal\; \{P\})\; \backslash \backslash \; \backslash leq\; \backslash bar\; \{C\}\; C\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; \backslash left(\backslash max\; \backslash left\backslash \{d\; ^\; \{\backslash frac\; \{\backslash alpha\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\; ^\; \{2\}\}\; \backslash right\backslash \}\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right)\; \backslash \backslash \; +\; B\; \backslash bar\; \{C\}\; \backslash left(\backslash max\; \backslash left\backslash \{\backslash frac\; \{\backslash log\; \backslash left(\backslash frac\; \{T\}\{d\}\backslash right)\}\{\backslash alpha\; +\; 2\},\; \backslash log\; \backslash left(\backslash frac\; \{4\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\; \backslash right\backslash \}\; +\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right),\; \backslash \backslash \; \backslash end\{array\}$$

where $\bar{C}, C(\delta, T, \epsilon_0)$ are the same as in Theorem 4.1.

A few comments are in order.

1. An important practical aspect of this modified algorithm (inherited from Algorithm 1) is the very mild number of re-trainings required to achieve the claimed performance. Despite the total number of labels can be as large as $T^{\frac{2}{\alpha + 2}}$ , the number $L$ of times the model is actually re-trained is not $T^{\frac{2}{\alpha + 2}} / B$ , but only logarithmic in $T$ , irrespective of the noise level $\alpha$ (that is, even when the low-noise assumption (2) is vacuous). On the other hand, it is also important to observe that the bound on $L$ shrinks as $\alpha$ increases, that is, when the problem becomes easier. Overall, these properties make the algorithm attractive in practical learning scenarios where the re-training time turns out to be the main bottleneck in the data acquisition process, and a learning procedure is needed that automatically adapts the re-training effort to the hardness of the problem.

2. Let us disregard lower order terms and only consider the asymptotic behavior as $T \to \infty$ . Comparing the excess risk bound in Theorem 4.1 to the label complexity bound in Corollary 4.2, one can see that when $B = O(T^{\frac{2}{\alpha + 2}})$ we have with high probability

$$\begin{array}{cccc}& \mathcal{L}(\widehat{\mathbf{w}})-\mathcal{L}({\mathbf{w}}^{\ast })\approx \frac{1}{{(N_{T,B}(\mathcal{P}))}^{\frac{1+\alpha }{2}}}& & \text{(3)}\end{array}\backslash mathcal\; \{L\}\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\})\; -\; \backslash mathcal\; \{L\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; \backslash approx\; \backslash frac\; \{1\}\{\backslash left(N\; \_\; \{T\; ,\; B\}\; (\backslash mathcal\; \{P\})\backslash right)\; ^\; \{\backslash frac\; \{1\; +\; \backslash alpha\}\{2\}\}\}\; \backslash tag\; \{3\}$$

which is the minimax rate one can achieve for VC classes $^3$ under the low-noise condition (2) with exponent $\alpha$ (e.g., (Castro & Nowak, 2008; Hanneke, 2009; Koltchinskii, 2010; Dekel et al., 2012)). Hence, in order to achieve high-probability minimax rates, one need not try to make the algorithm more adaptive by having it operate with an even smaller $B$ : any $B$ as small as $T^{\frac{2}{\alpha + 2}}$ will indeed suffice in our learning scenario.

3. Similar minimax bounds on excess risk against label complexity have been shown in the streaming setting in (Dekel et al., 2012; Wang et al., 2021), though their results only hold in the fully sequential case (that is, $B = 1$ ) and only hold in expectation over the random draw of the data, not with high probability.

The fact that a batch greedy algorithm can be competitive with a fully sequential policy has also been observed in problems which are similar in spirit to active learning, like influence maximization (see, in particular, (Chen & Krause,

2013)). More recently, in the context of adaptive sequential decision making, Esfandiari et al. (2021) have proposed an efficient semi-adaptive policy that performs logarithmically many rounds of interaction achieving similar performance as the fully sequential policy. This paper improves on the original ideas contained in (Golovin & Krause, 2017). Yet, when adapted to active learning, these results turn out to apply to very stylized scenarios that assume lack of noise in the labels, and/or disregard the computational aspects associated with maintaining a posterior distribution or a version space (which would be of size $O(T^{d})$ in our case).

5. The Logistic Case

We now discuss how to extend the result of the previous section to the logistic case (the generalized linear model (1) with $\sigma(z) = \frac{e^z}{1 + e^z}$ ).

Algorithm 2 is the adaptation to the logistic case of the algorithm of Section 4, the main difference being that we now assume the comparison vector $\mathbf{w}^*$ to lie in a Euclidean ball of (known) radius $R$ , and compute estimators $\mathbf{w}_{\ell}$ as regularized logistic regressors:

$$\begin{array}{cccc}& \begin{array}{c}{\mathbf{w}}_{\mathrm{\ell }}={\mathrm{argmin}}_{\begin{array}{c}\mathbf{w}:{\max }_{\mathbf{x}\in {\mathcal{Q}}_{\mathrm{\ell }}}\mathrm{\mid }⟨\mathbf{w},\mathbf{x}⟩\mathrm{\mid }\le 2R_{\mathrm{\ell }-1}\end{array}}[{\sum }_{t=1}^{T_{\mathrm{\ell }}}Loss(y_{\mathrm{\ell },t}⟨\mathbf{w},{\mathbf{x}}_{\mathrm{\ell },t}⟩)\\ +\frac{1}{8}{e}^{-4R_{\mathrm{\ell }}}\mathrm{\parallel }\mathbf{w}{\mathrm{\parallel }}^2],\end{array}& & \text{(4)}\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbf\{w\}\_\{\backslash ell\}\; =\; \backslash operatorname\; *\{argmin\}\_\{\backslash substack\{\backslash mathbf\{w\}:\backslash max\_\{\backslash mathbf\{x\}\backslash in\; \backslash mathcal\{Q\}\_\{\backslash ell\}\}|\backslash langle\; \backslash mathbf\{w\},\backslash mathbf\{x\}\backslash rangle\; |\backslash leq\; 2R\_\{\backslash ell\; -1\}\}\}\; \backslash left[\backslash sum\_\{t\; =\; 1\}^\{T\_\{\backslash ell\}\}Loss(y\_\{\backslash ell\; ,t\}\backslash langle\; \backslash mathbf\{w\},\backslash mathbf\{x\}\_\{\backslash ell\; ,t\}\backslash rangle)\backslash right.\; \backslash \backslash \; \backslash left.\; +\; \backslash frac\; \{1\}\{8\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; \backslash |\; \backslash mathbf\; \{w\}\; \backslash |\; ^\; \{2\}\; \backslash right],\; \backslash tag\; \{4\}\; \backslash \backslash \; \backslash end\{array\}$$

where $Loss(\cdot)$ is the logistic function

$$\mathrm{Loss}(a)=\log (1+e^{-a})\mathrm{.}\backslash operatorname\; \{L\; o\; s\; s\}\; (a)\; =\; \backslash log\; \backslash left(1\; +\; e\; ^\; \{-\; a\}\backslash right).$$

One of the main concerns in the logistic case is to investigate how excess risk and label complexity bounds depend on the complexity $R$ of the comparison class. The following is the logistic counterpart to Theorem 4.1.

Theorem 5.1. Let $T \geq d$ and assume that $| \mathbf{x} |_2 \leq 1$ for all $\mathbf{x} \in \mathcal{P}$ . Then with probability at least $1 - \delta$ over the random draw of $(\mathbf{x}_1, y_1), \ldots, (\mathbf{x}_T, y_T) \sim \mathcal{D}$ the excess risk $\mathcal{L}(\widehat{\mathbf{w}}) - \mathcal{L}(\mathbf{w}^*)$ , the label complexity $N_T(\mathcal{P})$ , and the number of stages $L$ generated by Algorithm 2 are simultaneously upper

Algorithm 2: Pool-based batch active learning algorithm for logistic models.

1. Input: Confidence level $\delta \in (0,1]$ , pool of instances $\mathcal{P} \subseteq \mathbb{R}^d$ of size $|\mathcal{P}| = T$ , upper bound $R > 0$ on $||\mathbf{w}^{*}||$
   2 Initialize: $\mathcal{P}_0 = \mathcal{P}$
   3 for $\ell = 1,2,\ldots$
   4 Initialize within stage $\ell$

$R_{\ell} = R2^{-\ell}$

$$\begin{array}{c}\bullet {ϵ}_{\mathrm{\ell }}=R_{\mathrm{\ell }}\mathrm{/}(16e^{8R_{\mathrm{\ell }}}\sqrt{d\log \frac{2d\mathrm{\ell }(\mathrm{\ell }+1)}{\delta }}+4R{e}^{4R_{\mathrm{\ell }}})\\ \bullet A_{\mathrm{\ell },0}=I,t=0,{\mathcal{Q}}_{\mathrm{\ell }}=\mathrm{\varnothing }\end{array}\backslash begin\{array\}\{l\}\; \backslash bullet\; \backslash epsilon\_\; \{\backslash ell\}\; =\; R\; \_\; \{\backslash ell\}\; /\; \backslash left(1\; 6\; e\; ^\; \{8\; R\; \_\; \{\backslash ell\}\}\; \backslash sqrt\; \{d\; \backslash log\; \backslash frac\; \{2\; d\; \backslash ell\; (\backslash ell\; +\; 1)\}\{\backslash delta\}\}\; +\; 4\; R\; e\; ^\; \{4\; R\; \_\; \{\backslash ell\}\}\backslash right)\; \backslash \backslash \; \backslash bullet\; A\; \_\; \{\backslash ell\; ,\; 0\}\; =\; I,\; t\; =\; 0,\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\; =\; \backslash emptyset\; \backslash \backslash \; \backslash end\{array\}$$

while $\mathcal{P}{\ell -1}\backslash \mathcal{Q}\ell \neq \emptyset$ and $\max{\mathbf{x}\in \mathcal{P}{\ell -1}\backslash \mathcal{Q}\ell}| \mathbf{x}|{A_{\ell ,t}^{-1}} > \epsilon_\ell$

$t = t + 1$

$$\begin{array}{c}\bullet \mathrm{Pick}{\mathbf{x}}_{\mathrm{\ell },t}\in \mathrm{\parallel }\mathbf{x}{\mathrm{\parallel }}_{A_{\mathrm{\ell },t-1}^{-1}}\\ \bullet \text{U p d a t e}{A}_{\mathrm{\ell },t}=A_{\mathrm{\ell },t-1}+{\mathbf{x}}_{\mathrm{\ell },t}{\mathbf{x}}_{\mathrm{\ell },t}^{\mathrm{\top }}\\ \bullet {\mathcal{Q}}_{\mathrm{\ell }}={\mathcal{Q}}_{\mathrm{\ell }}\cup \left\{{\mathbf{x}}_{\mathrm{\ell },t}\right\}\end{array}\backslash begin\{array\}\{l\}\; \backslash bullet\; \backslash operatorname\; \{P\; i\; c\; k\}\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}\; \backslash in\; \backslash underset\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\; \_\; \{\backslash ell\; -\; 1\}\; \backslash backslash\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\}\; \{\backslash operatorname\; \{a\; r\; g\; m\; a\; x\}\}\; \backslash |\; \backslash mathbf\; \{x\}\; \backslash |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; t\; -\; 1\}\; ^\; \{-\; 1\}\}\; \backslash \backslash \; \backslash bullet\; \backslash text\; \{U\; p\; d\; a\; t\; e\}\; A\; \_\; \{\backslash ell\; ,\; t\}\; =\; A\; \_\; \{\backslash ell\; ,\; t\; -\; 1\}\; +\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}\; ^\; \{\backslash top\}\; \backslash \backslash \; \backslash bullet\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\; =\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\; \backslash cup\; \backslash left\backslash \{\backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}\; \backslash right\backslash \}\; \backslash \backslash \; \backslash end\{array\}$$

Set $T_{\ell} = t$ , the number of queries made in stage $\ell$ if $\mathcal{Q}_{\ell}\neq \emptyset$

• Query the labels $y_{\ell,1}, \ldots, y_{\ell,T_\ell}$ associated with the unlabeled data in $\mathcal{Q}_\ell$

• Compute $\mathbf{w}_{\ell}$ as in (4)

$$\bullet \mathrm{Set}{\mathcal{C}}_{\mathrm{\ell }}=\{\mathbf{x}\in {\mathcal{P}}_{\mathrm{\ell }-1}\mathrm{\backslash }{\mathcal{Q}}_{\mathrm{\ell }}:\mathrm{\mid }⟨{\mathbf{w}}_{\mathrm{\ell }},\mathbf{x}⟩\mathrm{\mid }>R_{\mathrm{\ell }}\}\backslash bullet\; \backslash operatorname\; \{S\; e\; t\}\; \backslash mathcal\; \{C\}\; \_\; \{\backslash ell\}\; =\; \backslash left\backslash \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\; \_\; \{\backslash ell\; -\; 1\}\; \backslash backslash\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}:\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |\; >\; R\; \_\; \{\backslash ell\}\; \backslash right\backslash \}$$

• Compute pseudo-labels on each $\mathbf{x} \in \mathcal{C}{\ell}$ as $\hat{y} = \operatorname{sgn}\langle \mathbf{w}{\ell}, \mathbf{x} \rangle$

else

$${\mathbf{w}}_{\mathrm{\ell }}=\mathbf{0},{\mathcal{C}}_{\mathrm{\ell }}=\mathrm{\varnothing }\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; =\; \backslash mathbf\; \{0\},\; \backslash mathcal\; \{C\}\; \_\; \{\backslash ell\}\; =\; \backslash emptyset$$

Set $\mathcal{P}{\ell} = \mathcal{P}{\ell -1}\backslash (\mathcal{C}{\ell}\cup \mathcal{Q}{\ell})$

$$\mathbf{i}\mathbf{f}d\mathrm{/}(2R_{\mathrm{\ell }})>2R_{\mathrm{\ell }}\mathrm{\mid }{\mathcal{P}}_{\mathrm{\ell }}\mathrm{\mid }\backslash mathbf\; \{i\; f\}\; d\; /\; (2\; R\; \_\; \{\backslash ell\})\; >\; 2\; R\; \_\; \{\backslash ell\}\; |\; \backslash mathcal\; \{P\}\; \_\; \{\backslash ell\}\; |$$

$L = \ell$

• Exit the for-loop ( $L$ is the total no. of stages)

5 Predict labels in pool $\mathcal{P}$ ..

• Train an SVM classifier $\widehat{\mathbf{w}}$ on $\cup_{\ell=1}^{L} \mathcal{C}_{\ell}$ via the generated pseudo-labels $\hat{y}$
• Predict on each $\mathbf{x} \in (\cup_{\ell=1}^{L} \mathcal{Q}{\ell}) \cup \mathcal{P}{L}$ through $\operatorname{sgn}(\langle \widehat{\mathbf{w}}, \mathbf{x} \rangle)$

bounded as follows:

$$\begin{array}{c}\mathcal{L}(\widehat{\mathbf{w}})-\mathcal{L}({\mathbf{w}}^{\ast })\\ =\bar{C}{C}_{d,R}(\delta ,T,{ϵ}_0)(\max \{{\left(\frac{d}{T}\right)}^{\frac{\alpha +1}{\alpha +2}},\frac{d}{T{ϵ}_0}\}\\ +\frac{\log \left(\frac{\log T}{\bar{\delta }}\right)+d{e}^{8R}\lceil {\log }_{2}R\rceil }{T}),\end{array}\backslash begin\{array\}\{l\}\; \backslash mathcal\; \{L\}\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\})\; -\; \backslash mathcal\; \{L\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; \backslash \backslash \; =\; \backslash bar\; \{C\}\; C\; \_\; \{d,\; R\}\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; \backslash left(\backslash max\; \backslash left\backslash \{\backslash left(\backslash frac\; \{d\}\{T\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{T\; \backslash epsilon\_\; \{0\}\}\; \backslash right\backslash \}\; \backslash right.\; \backslash \backslash \; \backslash left.\; +\; \backslash frac\; \{\backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash bar\; \{\backslash delta\}\}\backslash right)\; +\; d\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\}\{T\}\backslash right),\; \backslash \backslash \; \backslash end\{array\}$$

$$\begin{array}{c}{N}_T(\mathcal{P})\\ =\bar{C}{C}_{d,R}(\delta ,T,{ϵ}_0)(\max \{d^{\frac{\alpha }{\alpha +2}}{T}^{\frac{2}{\alpha +2}},\frac{d}{{ϵ}_0^2}\}\\ +\frac{{\log }^2\left(\frac{\log T}{\delta }\right)+d{e}^{8R}\lceil {\log }_{2}R\rceil }{T}),\end{array}\backslash begin\{array\}\{l\}\; N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; \backslash \backslash \; =\; \backslash bar\; \{C\}\; C\; \_\; \{d,\; R\}\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; \backslash left(\backslash max\; \backslash left\backslash \{d\; ^\; \{\backslash frac\; \{\backslash alpha\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\; ^\; \{2\}\}\; \backslash right\backslash \}\; \backslash right.\; \backslash \backslash \; \backslash left.\; +\; \backslash frac\; \{\backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\; +\; d\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\}\{T\}\backslash right),\; \backslash \backslash \; \backslash end\{array\}$$

$$L\le \bar{C}(\max \{\frac{\log \left(\frac{T}{d}\right)}{\alpha +2},\log \left(\frac{4}{{ϵ}_0}\right)\}+\log \left(\frac{R\log T}{\delta }\right)),L\; \backslash leq\; \backslash bar\; \{C\}\; \backslash left(\backslash max\; \backslash left\backslash \{\backslash frac\; \{\backslash log\; \backslash left(\backslash frac\; \{T\}\{d\}\backslash right)\}\{\backslash alpha\; +\; 2\},\; \backslash log\; \backslash left(\backslash frac\; \{4\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\; \backslash right\backslash \}\; +\; \backslash log\; \backslash left(\backslash frac\; \{R\; \backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right),$$

where $\bar{C}$ is an absolute constant and

$$\begin{array}{c}{C}_{d,R}(\delta ,T,{ϵ}_0)=(1+{\log }^2\left(\frac{1}{{ϵ}_0}\right))(d\log \left(\frac{T}{\delta }\right)+R^2)\\ \times (R+\log \left(\frac{T}{\delta }\right))\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; C\; \_\; \{d,\; R\}\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; =\; \backslash left(1\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; \backslash left(d\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; +\; R\; ^\; \{2\}\backslash right)\; \backslash \backslash \; \backslash times\; \backslash left(R\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\backslash right).\; \backslash \backslash \; \backslash end\{array\}$$

In the above bounds, the complexity term $R$ is meant to be a constant. Notice that the dependence on $e^{R}$ is common to many logistic bounds, specifically in the bandits literature. This is due to the nonlinear shape of $\sigma(\cdot)$ (see, e.g., (Filippi et al., 2010; Gentile & Orabona, 2012; Li et al., 2017; Faury et al., 2020), where it takes the form of an upper bound on $1 / \sigma'(\cdot)$ ). In fact, a closer look at the multiplicative dependence on $e^{R}$ above reveals that this factor multiplies only logarithmic terms in $T$ . This is akin to the more refined self-concordant analysis of logistic models contained in (Faury et al., 2020). Since our algorithm is focusing attention to exponentially shrinking regions of margin values $\langle \mathbf{w}^*, \mathbf{x} \rangle$ around the origin, we have obtained here similar guarantees without resorting to a self-concordant analysis.

A constant batch size version of Algorithm 2 can also be devised, and the associated properties spelled out. The details are very similar to those in Section 4.2, and are therefore omitted.

6. Conclusions and Ongoing Research

We have described and analyzed novel batch active learning algorithms in the pool-based setting that achieve minimax rates of convergence of their excess risk as a function of the number of queried labels. The minimax nature of our results is retained also when the batch size $B$ is allowed to scale polynomially ( $B \leq T^{\beta}$ , for $\beta \leq 1$ ) with the size $T$

of the training set, the allowed exponent $\beta$ depending on the actual level of noise in the data. The algorithms have a number of re-training rounds which is at worst logarithmic, and is able to automatically adapt to the noise level.

Our algorithms generate pseudo-labels by restricting to exponentially small regions of the margin space. In the logistic case, this has the side benefit of delivering performance bounds where the classical exponential dependence on the complexity of the comparator $\mathbf{w}^*$ occurs as a multiplicative factor only in logarithmic terms.

The logistic algorithm we presented in Section 5 has a suboptimal dependence on the input dimension $d$ (notice the extra factor $d$ contained in $C_1$ in the excess risk bound of Theorem 5.1), and we are currently trying to see if it is possible to achieve the same result as in the linear case. For the logistic case, a more computationally efficient algorithm actually exists that is based on the online Newton step-like analysis in (Gentile & Orabona, 2012). Yet, this algorithm will have a similar suboptimal dependence on $d$ .

Related to the above, we are currently investigating to what extent it is possible to improve the logistic analysis so as to turn the constrained minimization problem therein into an unconstrained one. Analyses we are aware of in the contiguous field of contextual bandits in generalized linear scenarios (e.g., (Li et al., 2017)) do not seem to help, given the strong assumptions on the context distribution they formulate to achieve the optimal dependence on $d$ .

The methods we have presented here are instances of a more general approach to batch active learning in realizable settings where, given a diversity measure $D(\mathbf{x}, S)$ , an estimator $\widehat{f} = \widehat{f}(S)$ in fixed design scenarios exists for which we can guarantee $L_{\infty}$ approximation bounds of the form

$$\begin{array}{cccc}& \mid {\widehat{f}}_S(\mathbf{x})-f^{\ast }(x)\mid \le D(\mathbf{x},S)\mathrm{\forall }\mathbf{x}\mathrm{.}& & \text{(5)}\end{array}\backslash left|\; \backslash widehat\; \{f\}\; \_\; \{S\}\; (\backslash mathbf\; \{x\})\; -\; f\; ^\; \{*\}\; (x)\; \backslash right|\; \backslash leq\; D\; (\backslash mathbf\; \{x\},\; S)\; \backslash quad\; \backslash forall\; \backslash mathbf\; \{x\}.\; \backslash tag\; \{5\}$$

For instance, our approach can be seamlessly extended to the case where $f^{*}$ belongs to a RKHS, the algorithmic aspects simply requiring a dual variable formulation of Algorithm 1, and the statistical ones simply resorting to covering number bounds (e.g., (Zhou, 2002)) or empirical versions thereof. As another relevant example, (5) can be shown to hold for known plug-in estimators, like local polynomial estimators (e.g., Sect. 1.6.1. in (Tsybakov, 2009)). Hence our general approach may be extended to those cases as well.

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A. Proofs for Section 4

Consider Algorithm 1, and denote by $T_{\ell}$ the length of stage $\ell$

We denote for any $\epsilon >0$

$${\mathcal{T}}_{ϵ}=\{\mathbf{x}\in \mathcal{P}:\mathrm{\mid }⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\mathrm{\mid }\le ϵ\}\mathrm{.}\backslash mathcal\; \{T\}\; \_\; \{\backslash epsilon\}\; =\; \backslash left\backslash \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}:\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |\; \backslash leq\; \backslash epsilon\; \backslash right\backslash \}.$$

Recall that in Algorithm 1 variable $L$ counts the total number of stages (a random quantity), while the size of the original pool $|\mathcal{P}|$ is denoted by $T$ .

We first show that on the confident sets, that is, on sets $\mathcal{C}_{\ell}$ where pseudo-labels are generated, the learner has with high probability no regret. Before giving our key lemma, it will be useful to define the events

$${\mathcal{E}}_{\mathrm{\ell }}=\{\underset{\mathbf{x}\in {\mathcal{P}}_{\mathrm{\ell }-1}\mathrm{\backslash }{\mathcal{Q}}_{\mathrm{\ell }}}{\max }\mathrm{\mid }⟨{\mathbf{w}}_{\mathrm{\ell }}-{\mathbf{w}}^{\ast },\mathbf{x}⟩\mathrm{\mid }\le 2^{-\mathrm{\ell }}\},\backslash mathcal\; \{E\}\; \_\; \{\backslash ell\}\; =\; \backslash left\backslash \{\backslash max\; \_\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\; \_\; \{\backslash ell\; -\; 1\}\; \backslash backslash\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\}\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; -\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |\; \backslash leq\; 2\; ^\; \{-\; \backslash ell\}\; \backslash right\backslash \},$$

for $\ell = 1,\ldots ,L$

Lemma A.1. For any positive $L$ ,

$$\mathbb{P}\left(\bigcap _{\mathrm{\ell }=1}^L{\mathcal{E}}_{\mathrm{\ell }}\right)>1-\delta \mathrm{.}\backslash mathbb\; \{P\}\; \backslash left(\backslash bigcap\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{E\}\; \_\; \{\backslash ell\}\backslash right)\; >\; 1\; -\; \backslash delta\; .$$

Proof. We assume $\mathcal{P}{\ell -1}\backslash \mathcal{Q}\ell$ is not empty (it could be empty only in the final stage $L$ ). We follow the material contained in Chapters 20 and 21 of Lattimore & Szepesvari (2020). Let $\xi{\ell ,t} = y{\ell ,t} - \langle \mathbf{w}^*,\mathbf{x}{\ell ,t}\rangle$ and notice that $\xi{\ell ,t}$ are independent 1-sub-Gaussian random variables conditioned on $\mathcal{P}{\ell -1}$ . Also, observe that, conditioned on past stages $1,\ldots ,\ell -1$ , we are in a fixed design scenario, where the $\mathbf{x}{\ell ,t}$ are chosen without knowledge of the corresponding labels $y_{\ell ,t}$ . We can write, for any $\mathbf{x}\in \mathcal{P}_{\ell -1}$ ,

$$⟨wℓ−w∗,x⟩=⟨Aℓ,Tℓ−1(∑t=1Tℓyℓ,txℓ,t)−w∗,x⟩=⟨Aℓ,Tℓ−1(∑t=1Tℓxℓ,t⟨w∗,xℓ,t⟩+ξℓ,txℓ,t)−w∗,x⟩=⟨Aℓ,Tℓ−1(Aℓ,Tℓ−I)w∗+Aℓ,Tℓ−1(∑t=1Tℓξℓ,txℓ,t)−w∗,x⟩=−⟨w∗,x⟩Aℓ,Tℓ−1+∑t=1Tℓ⟨xℓ,t,x⟩Aℓ,Tℓ−1ξℓ,t. \begin{array}{l} \left\langle \mathbf {w} _ {\ell} - \mathbf {w} ^ {*}, \mathbf {x} \right\rangle = \left\langle A _ {\ell , T _ {\ell}} ^ {- 1} \left(\sum_ {t = 1} ^ {T _ {\ell}} y _ {\ell , t} \mathbf {x} _ {\ell , t}\right) - \mathbf {w} ^ {*}, \mathbf {x} \right\rangle \\ = \left\langle A _ {\ell , T _ {\ell}} ^ {- 1} \left(\sum_ {t = 1} ^ {T _ {\ell}} \mathbf {x} _ {\ell , t} \left\langle \mathbf {w} ^ {*}, \mathbf {x} _ {\ell , t} \right\rangle + \xi_ {\ell , t} \mathbf {x} _ {\ell , t}\right) - \mathbf {w} ^ {*}, \mathbf {x} \right\rangle \\ = \left\langle A _ {\ell , T _ {\ell}} ^ {- 1} \left(A _ {\ell , T _ {\ell}} - I\right) \mathbf {w} ^ {*} + A _ {\ell , T _ {\ell}} ^ {- 1} \left(\sum_ {t = 1} ^ {T _ {\ell}} \xi_ {\ell , t} \mathbf {x} _ {\ell , t}\right) - \mathbf {w} ^ {*}, \mathbf {x} \right\rangle \\ = - \left\langle \mathbf {w} ^ {*}, \mathbf {x} \right\rangle_ {A _ {\ell , T _ {\ell}} ^ {- 1}} + \sum_ {t = 1} ^ {T _ {\ell}} \left\langle \mathbf {x} _ {\ell , t}, \mathbf {x} \right\rangle_ {A _ {\ell , T _ {\ell}} ^ {- 1}} \xi_ {\ell , t}. \\ \end{array} $$

Since ${\xi_{\ell ,t}}{t = 1}^{T{\ell}}$ are 1-sub-Gaussian and independent conditioned on ${\mathbf{x}{\ell ,t}}$ , the variance term $\sum{t = 1}^{T_{\ell}}\langle \mathbf{x}{\ell ,t},\mathbf{x}\rangle{A_{\ell ,T_{\ell}}^{-1}}\xi_{\ell ,t}$ is $\sqrt{\sum_{t = 1}^{T_{\ell}}\langle\mathbf{x}{\ell,t},\mathbf{x}\rangle{A_{\ell,T_{\ell}}}^2}$ -sub-Gaussian. We apply lemma C.5

$$\mathbb{P}(\mid \sum _{t=1}^{T_{\mathrm{\ell }}}⟨{\mathbf{x}}_{\mathrm{\ell },t},\mathbf{x}{⟩}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}{\xi }_{\mathrm{\ell },t}\mid \ge \sqrt{2\sum _{t=1}^{T_{\mathrm{\ell }}}⟨{\mathbf{x}}_{\mathrm{\ell },t},\mathbf{x}{⟩}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}^2\log \frac{2\mathrm{\ell }(\mathrm{\ell }+1)T}{\delta }})\le \frac{\delta }{\mathrm{\ell }(\mathrm{\ell }+1)T}\mathrm{.}\backslash mathbb\; \{P\}\; \backslash left(\backslash left|\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; \backslash langle\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\},\; \backslash mathbf\; \{x\}\; \backslash rangle\_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; \backslash xi\_\; \{\backslash ell\; ,\; t\}\; \backslash right|\; \backslash geq\; \backslash sqrt\; \{2\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; \backslash langle\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}\; ,\; \backslash mathbf\; \{x\}\; \backslash rangle\_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; ^\; \{2\}\; \backslash log\; \backslash frac\; \{2\; \backslash ell\; (\backslash ell\; +\; 1)\; T\}\{\backslash delta\}\}\backslash right)\; \backslash leq\; \backslash frac\; \{\backslash delta\}\{\backslash ell\; (\backslash ell\; +\; 1)\; T\}.$$

Now observe that

$$\sum _{t=1}^{T_{\mathrm{\ell }}}⟨{\mathbf{x}}_{\mathrm{\ell },t},\mathbf{x}{⟩}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}^2=\mathrm{\parallel }\mathbf{x}{\mathrm{\parallel }}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}^2-\mathrm{\parallel }{A}_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}\mathbf{x}{\mathrm{\parallel }}^2\le \mathrm{\parallel }\mathbf{x}{\mathrm{\parallel }}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}^2\mathrm{.}\backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; \backslash langle\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\},\; \backslash mathbf\; \{x\}\; \backslash rangle\_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; ^\; \{2\}\; =\; \backslash |\; \backslash mathbf\; \{x\}\; \backslash |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; ^\; \{2\}\; -\; \backslash |\; A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\; \backslash mathbf\; \{x\}\; \backslash |\; ^\; \{2\}\; \backslash leq\; \backslash |\; \backslash mathbf\; \{x\}\; \backslash |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; ^\; \{2\}.$$

We plug back into the previous inequality to obtain

$$\mathbb{P}(\mid \sum _{t=1}^{T_{\mathrm{\ell }}}⟨{\mathbf{x}}_{\mathrm{\ell },t},\mathbf{x}{⟩}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}{\xi }_{\mathrm{\ell },t}\mid \ge \sqrt{2\mathrm{\parallel }\mathbf{x}{\mathrm{\parallel }}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}^2\log \frac{2\mathrm{\ell }(\mathrm{\ell }+1)T}{\delta }})\le \frac{\delta }{\mathrm{\ell }(\mathrm{\ell }+1)T}\mathrm{.}\backslash mathbb\; \{P\}\; \backslash left(\backslash left|\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; \backslash langle\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\},\; \backslash mathbf\; \{x\}\; \backslash rangle\_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; \backslash xi\_\; \{\backslash ell\; ,\; t\}\; \backslash right|\; \backslash geq\; \backslash sqrt\; \{2\; \backslash |\; \backslash mathbf\; \{x\}\; \backslash |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; ^\; \{2\}\; \backslash log\; \backslash frac\; \{2\; \backslash ell\; (\backslash ell\; +\; 1)\; T\}\{\backslash delta\}\}\backslash right)\; \backslash leq\; \backslash frac\; \{\backslash delta\}\{\backslash ell\; (\backslash ell\; +\; 1)\; T\}.$$

Using a union bound, we get with probability at least $1 - \frac{\delta}{\ell(\ell + 1)}$

$$\mid \sum _{t=1}^{T_{\mathrm{\ell }}}⟨{\mathbf{x}}_{\mathrm{\ell },t},\mathbf{x}{⟩}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}{\xi }_{\mathrm{\ell },t}\mid \le \sqrt{2\mathrm{\parallel }\mathbf{x}{\mathrm{\parallel }}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}^2\log \frac{2\mathrm{\ell }(\mathrm{\ell }+1)T}{\delta }},\backslash left|\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; \backslash langle\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\},\; \backslash mathbf\; \{x\}\; \backslash rangle\_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; \backslash xi\_\; \{\backslash ell\; ,\; t\}\; \backslash right|\; \backslash leq\; \backslash sqrt\; \{2\; \backslash |\; \backslash mathbf\; \{x\}\; \backslash |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; ^\; \{2\}\; \backslash log\; \backslash frac\; \{2\; \backslash ell\; (\backslash ell\; +\; 1)\; T\}\{\backslash delta\}\},$$

holds uniformly for all $\mathbf{x} \in \mathcal{P}{\ell-1}$ . For the bias term $\langle \mathbf{w}^*, \mathbf{x} \rangle{A_{\ell, T_\ell}^{-1}}$ , notice that $A_{\ell, T_\ell} \succeq I$ implies

$$\mid {⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}\mid \le {\parallel \mathbf{x}\parallel }_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}{\parallel {\mathbf{w}}^{\ast }\parallel }_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}\le {\parallel \mathbf{x}\parallel }_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}}\mathrm{.}\backslash left|\; \backslash left\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash right\backslash rangle\_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; \backslash right|\; \backslash leq\; \backslash left\backslash |\; \backslash mathbf\; \{x\}\; \backslash right\backslash |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; \backslash left\backslash |\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; \backslash right\backslash |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}\; \backslash leq\; \backslash left\backslash |\; \backslash mathbf\; \{x\}\; \backslash right\backslash |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\}.$$

Hence with probability at least $1 - \frac{\delta}{\ell(\ell + 1)}$

$$\mid ⟨{\mathbf{w}}_{\mathrm{\ell }}-{\mathbf{w}}^{\ast },\mathbf{x}⟩\mid \le (\sqrt{2\log \frac{2\mathrm{\ell }(\mathrm{\ell }+1)T}{\delta }}+1)\mathrm{\parallel }\mathbf{x}{\mathrm{\parallel }}_{A_{\mathrm{\ell },T_{\mathrm{\ell }}}^{-1}},\backslash left|\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; -\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash right|\; \backslash leq\; \backslash left(\backslash sqrt\; \{2\; \backslash log\; \backslash frac\; \{2\; \backslash ell\; (\backslash ell\; +\; 1)\; T\}\{\backslash delta\}\}\; +\; 1\backslash right)\; \backslash |\; \backslash mathbf\; \{x\}\; \backslash |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; ^\; \{-\; 1\}\},$$

holds uniformly for all $\mathbf{x} \in \mathcal{P}_{\ell-1}$ .

Notice that by the selection criterion in Algorithm 1, $\max_{\mathbf{x} \in \mathcal{P}{\ell - 1} \setminus \mathcal{Q}\ell} | \mathbf{x} |{A{\ell, T_\ell}^{-1}} \leq \epsilon_\ell^2$ . As a consequence, with probability at least $1 - \frac{\delta}{\ell(\ell + 1)}$ ,

$$\underset{\mathbf{x}\in {\mathcal{P}}_{\mathrm{\ell }-1}\mathrm{\backslash }{\mathcal{Q}}_{\mathrm{\ell }}}{\max }\mathrm{\mid }⟨{\mathbf{w}}_{\mathrm{\ell }}-{\mathbf{w}}^{\ast },\mathbf{x}⟩\mathrm{\mid }\le (\sqrt{2\log \frac{2\mathrm{\ell }(\mathrm{\ell }+1)T}{\delta }}+1){ϵ}_{\mathrm{\ell }}\mathrm{.}\backslash max\; \_\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\; \_\; \{\backslash ell\; -\; 1\}\; \backslash backslash\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\}\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; -\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |\; \backslash leq\; \backslash left(\backslash sqrt\; \{2\; \backslash log\; \backslash frac\; \{2\; \backslash ell\; (\backslash ell\; +\; 1)\; T\}\{\backslash delta\}\}\; +\; 1\backslash right)\; \backslash epsilon\_\; \{\backslash ell\}.$$

Recalling the definition of $\epsilon_{\ell}$ in Algorithm 1 and using an union bound over $\ell$ , we get the desired result.

As a simple consequence, we have the following lemma.

Lemma A.2. Assume $\bigcap_{\ell = 1}^{L}\mathcal{E}{\ell}$ holds. Then Algorithm 1 generates pseudo-labels such that, on all points $\mathbf{x}\in \cup{\ell = 1}^{L}\mathcal{C}_{\ell}$ , $\operatorname {sgn}(\langle \mathbf{w}_\ell ,\mathbf{x}\rangle) = \operatorname {sgn}(\langle \mathbf{w}^*,\mathbf{x}\rangle)$ .

Proof. Simply observe that if $\mathbf{x} \in \cup_{\ell=1}^{L} \mathcal{C}{\ell}$ is such that $\operatorname{sgn}(\langle \mathbf{w}{\ell}, \mathbf{x} \rangle) = 1$ then $\langle \mathbf{w}{\ell}, \mathbf{x} \rangle > 2^{-\ell}$ , which implies $\langle \mathbf{w}^*, \mathbf{x} \rangle > 0$ by the assumption that $\mathcal{E}{\ell}$ holds. Similarly, $\operatorname{sgn}(\langle \mathbf{w}_{\ell}, \mathbf{x} \rangle) = -1$ implies $\langle \mathbf{w}^*, \mathbf{x} \rangle < 0$ .

Lemma A.3. The length $T_{\ell}$ of stage $\ell$ in Algorithm 1 is (deterministically) upper bounded as

$$T_{\mathrm{\ell }}\le \frac{8d}{{ϵ}_{\mathrm{\ell }}^2}\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)\mathrm{.}T\; \_\; \{\backslash ell\}\; \backslash leq\; \backslash frac\; \{8\; d\}\{\backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right).$$

Proof. Since in stage $\ell$ the algorithm terminates at $T_{\ell}$ , any round $t < T_{\ell}$ is such that

$${\mid \mid {\mathbf{x}}_{\mathrm{\ell },t+1}\mid \mid }_{A_{\mathrm{\ell },t}^{-1}}^2>{ϵ}_{\mathrm{\ell }}^2\mathrm{.}\backslash left|\; \backslash left|\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; \backslash right|\; \backslash right|\; \_\; \{A\; \_\; \{\backslash ell\; ,\; t\}\; ^\; \{-\; 1\}\}\; ^\; \{2\}\; >\; \backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}.$$

We denote $|\cdot|$ as the determinant of the matrix at argument and have the known identity

$$\mathrm{\mid }{A}_{\mathrm{\ell },t+1}\mathrm{\mid }=\mathrm{\mid }{A}_{\mathrm{\ell },t}+{\mathbf{x}}_{\mathrm{\ell },t+1}{\mathbf{x}}_{\mathrm{\ell },t+1}^{\mathrm{\top }}\mathrm{\mid }=\mathrm{\mid }{A}_{\mathrm{\ell },t}\mathrm{\mid }\cdot \mathrm{\mid }I+A_{\mathrm{\ell },t}^{-1}{\mathbf{x}}_{\mathrm{\ell },t+1}{\mathbf{x}}_{\mathrm{\ell },t+1}^{\mathrm{\top }}\mathrm{\mid }=(1+\mathrm{\mid }\mathrm{\mid }{\mathbf{x}}_{\mathrm{\ell },t+1}\mathrm{\mid }{\mathrm{\mid }}_{A_{\mathrm{\ell },t}^{-1}}^2)\mathrm{\mid }{A}_{\mathrm{\ell },t}\mathrm{\mid }\le 2\mathrm{\mid }{A}_{\mathrm{\ell },t}\mathrm{\mid },|\; A\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; |\; =\; |\; A\; \_\; \{\backslash ell\; ,\; t\}\; +\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; ^\; \{\backslash top\}\; |\; =\; |\; A\; \_\; \{\backslash ell\; ,\; t\}\; |\; \backslash cdot\; |\; I\; +\; A\; \_\; \{\backslash ell\; ,\; t\}\; ^\; \{-\; 1\}\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; ^\; \{\backslash top\}\; |\; =\; (1\; +\; |\; |\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; |\; |\; \_\; \{A\; \_\; \{\backslash ell\; ,\; t\}\; ^\; \{-\; 1\}\}\; ^\; \{2\})\; |\; A\; \_\; \{\backslash ell\; ,\; t\}\; |\; \backslash leq\; 2\; |\; A\; \_\; \{\backslash ell\; ,\; t\}\; |,$$

where the third equality holds since $I + A_{\ell, t}^{-1/2} \mathbf{x}{\ell, t+1} \mathbf{x}{\ell, t+1}^{\top} A_{\ell, t}^{-1/2}$ has $d - 1$ eigenvalues 1 and one eigenvalue 1 + $||\mathbf{x}{\ell, t+1}||{A_{\ell, t}^{-1}}^{2}$ .

Combining the above equality with the fact that $\log (1 + x) \geq \frac{x}{1 + x} \geq \frac{x}{2}$ for $0 \leq x \leq 1$ , we get

$${\mid \mid {\mathbf{x}}_{\mathrm{\ell },t+1}\mid \mid }_{A_{\mathrm{\ell },t}^{-1}}^2=\frac{\mid A_{\mathrm{\ell },t+1}\mid }{\mid A_{\mathrm{\ell },t}\mid }-1\le 2(\log \mid A_{\mathrm{\ell },t+1}\mid -\log \mid A_{\mathrm{\ell },t}\mid )\mathrm{.}\backslash left|\; \backslash left|\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; \backslash right|\; \backslash right|\; \_\; \{A\; \_\; \{\backslash ell\; ,\; t\}\; ^\; \{-\; 1\}\}\; ^\; \{2\}\; =\; \backslash frac\; \{\backslash left|\; A\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; \backslash right|\}\{\backslash left|\; A\; \_\; \{\backslash ell\; ,\; t\}\; \backslash right|\}\; -\; 1\; \backslash leq\; 2\; \backslash left(\backslash log\; \backslash left|\; A\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; \backslash right|\; -\; \backslash log\; \backslash left|\; A\; \_\; \{\backslash ell\; ,\; t\}\; \backslash right|\backslash right).$$

Therefore,

$$2(\log \mid A_{\mathrm{\ell },t+1}\mid -\log \mid A_{\mathrm{\ell },t}\mid )>{ϵ}_{\mathrm{\ell }}^2\mathrm{.}2\; \backslash left(\backslash log\; \backslash left|\; A\; \_\; \{\backslash ell\; ,\; t\; +\; 1\}\; \backslash right|\; -\; \backslash log\; \backslash left|\; A\; \_\; \{\backslash ell\; ,\; t\}\; \backslash right|\backslash right)\; >\; \backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}.$$

Summing over $t = 0,\dots ,T_{\ell} - 1$ yields,

$$2\log \frac{\mid A_{\mathrm{\ell },T_{\mathrm{\ell }}}\mid }{\mid A_{\mathrm{\ell },0}\mid }\ge {ϵ}_{\mathrm{\ell }}^2{T}_{\mathrm{\ell }}\mathrm{.}2\; \backslash log\; \backslash frac\; \{\backslash left|\; A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; \backslash right|\}\{\backslash left|\; A\; \_\; \{\backslash ell\; ,\; 0\}\; \backslash right|\}\; \backslash geq\; \backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}\; T\; \_\; \{\backslash ell\}.$$

Now, $A_{\ell ,0} = I$ , so that $|A_{\ell ,0}| = 1$ , and

$$\log \mid A_{\mathrm{\ell },T_{\mathrm{\ell }}}\mid \le \log {(\mathrm{trace}\left(A_{\mathrm{\ell },T_{\mathrm{\ell }}}\right)\mathrm{/}d)}^d\le d\log (1+\frac{T_{\mathrm{\ell }}}{d}),\backslash log\; \backslash left|\; A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\; \backslash right|\; \backslash leq\; \backslash log\; \backslash left(\backslash operatorname\; \{t\; r\; a\; c\; e\}\; \backslash left(A\; \_\; \{\backslash ell\; ,\; T\; \_\; \{\backslash ell\}\}\backslash right)\; /\; d\backslash right)\; ^\; \{d\}\; \backslash leq\; d\; \backslash log\; \backslash left(1\; +\; \backslash frac\; \{T\; \_\; \{\backslash ell\}\}\{d\}\backslash right),$$

yields

$$\frac{T_{\mathrm{\ell }}}{d}\le \frac{2}{{ϵ}_{\mathrm{\ell }}^2}\log (1+\frac{T_{\mathrm{\ell }}}{d})\mathrm{.}\backslash frac\; \{T\; \_\; \{\backslash ell\}\}\{d\}\; \backslash leq\; \backslash frac\; \{2\}\{\backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}\}\; \backslash log\; \backslash left(1\; +\; \backslash frac\; \{T\; \_\; \{\backslash ell\}\}\{d\}\backslash right).$$

Let $G(x) = \frac{x}{\log(1 + x)}$ , and notice that $G(x)$ is increasing for $x > 0$ . We have

where the second inequality holds since $\epsilon_{\ell} \leq \epsilon_{1} < \frac{1}{4}$ .

As a consequence,

$$T_{\mathrm{\ell }}\le \frac{8d}{{ϵ}_{\mathrm{\ell }}^2}\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)\mathrm{.}T\; \_\; \{\backslash ell\}\; \backslash leq\; \backslash frac\; \{8\; d\}\{\backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right).$$

The proof then proceeds by bounding two relevant quantities associated with the behavior of Algorithm 1: the label complexity

$$N_T(\mathcal{P})=\sum _{\mathrm{\ell }=1}^L\mid {\mathcal{Q}}_{\mathrm{\ell }}\mid ,N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; \backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash left|\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\; \backslash right|,$$

and the weighted cumulative regret over pool $\mathcal{P}$ of size $T$ , defined as

$$R_T(\mathcal{P})=\sum _{\mathbf{x}\in \mathcal{P}}\mathbb{1}\{\mathrm{s}\mathrm{g}\mathrm{n}⟨\widehat{\mathbf{w}},\mathbf{x}⟩\mathrm{\ne }\mathrm{s}\mathrm{g}\mathrm{n}⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\}\mathrm{\mid }⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\mathrm{\mid }\mathrm{.}R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; \backslash sum\_\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\}\; \backslash mathbb\; \{1\}\; \backslash \{\backslash mathrm\; \{s\; g\; n\}\; \backslash langle\; \backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash neq\; \backslash mathrm\; \{s\; g\; n\}\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash \}\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |.$$

We will first present intermediate bounds on $R_{T}(\mathcal{P})$ and $N_{T}(\mathcal{P})$ as a function of $L$ , and then rely on the properties of the noise (hence the randomness on $\mathcal{P}$ ) to complete the proofs. To simplify the math display we denote

$$K_T(\delta ,\mathrm{\ell })=\sqrt{2\log \frac{2\mathrm{\ell }(\mathrm{\ell }+1)T}{\delta }}+1,K\; \_\; \{T\}\; (\backslash delta\; ,\; \backslash ell)\; =\; \backslash sqrt\; \{2\; \backslash log\; \backslash frac\; \{2\; \backslash ell\; (\backslash ell\; +\; 1)\; T\}\{\backslash delta\}\}\; +\; 1,$$

so that $\epsilon_{\ell} = \frac{1}{2^{\ell}K_{T}(\delta,\ell)}$

Lemma A.3 immediately delivers the following bound on $N_T(\mathcal{P})$ :

Theorem A.4. For any pool realization $\mathcal{P}$ , the label complexity $N_{T}(\mathcal{P})$ of Algorithm 1 operating on a pool $\mathcal{P}$ of size $T$ is bounded deterministically as

$$N_T(\mathcal{P})\le \frac{32}{3}d\log (2^L{K}_T(\delta ,L))K_T^2(\delta ,L)4^{L}N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; \backslash leq\; \backslash frac\; \{3\; 2\}\{3\}\; d\; \backslash log\; \backslash left(2\; ^\; \{L\}\; K\; \_\; \{T\}\; (\backslash delta\; ,\; L)\backslash right)\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; 4\; ^\; \{L\}$$

Proof. By definition

$$\begin{array}{c}{N}_T(\mathcal{P})={\sum }_{\mathrm{\ell }=1}^L{T}_{\mathrm{\ell }}\le {\sum }_{\mathrm{\ell }=1}^L\frac{8d}{{ϵ}_{\mathrm{\ell }}^2}\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)\\ \le 8d\log \left(\frac{1}{{ϵ}_L}\right)K_T^2(\delta ,L){\sum }_{\mathrm{\ell }=1}^L{4}^{\mathrm{\ell }}\\ \le \frac{8}{3}d\log (2^L{K}_T(\delta ,L))K_T^2(\delta ,L)4^{L+1},\end{array}\backslash begin\{array\}\{l\}\; N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; \backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; T\; \_\; \{\backslash ell\}\; \backslash leq\; \backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash frac\; \{8\; d\}\{\backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; \backslash \backslash \; \backslash leq\; 8\; d\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{L\}\}\backslash right)\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; \backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; 4\; ^\; \{\backslash ell\}\; \backslash \backslash \; \backslash leq\; \backslash frac\; \{8\}\{3\}\; d\; \backslash log\; \backslash left(2\; ^\; \{L\}\; K\; \_\; \{T\}\; (\backslash delta\; ,\; L)\backslash right)\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; 4\; ^\; \{L\; +\; 1\},\; \backslash \backslash \; \backslash end\{array\}$$

where the second inequality follows from the fact that both $\frac{1}{\epsilon_{\ell}}$ and $K_{T}(\delta, \ell)$ increase with $\ell$ , and the last inequality follows from $\sum_{\ell=1}^{L} 4^{\ell} < \frac{4}{3} 4^{L}$ .

As for the regret $R_{T}(\mathcal{P})$ , we have the following high probability result.

Theorem A.5. For any pool realization $\mathcal{P}$ , the weighted cumulative regret $R_{T}(\mathcal{P})$ of Algorithm 1 operating on a pool $\mathcal{P}$ of size $T$ is bounded as

$$R_T(\mathcal{P})\le 64d\log (2^L{K}_T(\delta ,L))K_T^2(\delta ,L)2^L+d{2}^{L-1},R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; \backslash leq\; 6\; 4\; d\; \backslash log\; \backslash left(2\; ^\; \{L\}\; K\; \_\; \{T\}\; (\backslash delta\; ,\; L)\backslash right)\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; 2\; ^\; \{L\}\; +\; d\; 2\; ^\; \{L\; -\; 1\},$$

assuming $\bigcap_{\ell = 1}^{L}\mathcal{E}_{\ell}$ holds.

Proof. We decompose the pool $\mathcal{P}$ as the union of following disjoint sets

$$\mathcal{P}=\left({\cup }_{l=1}^L{\mathcal{C}}_{\mathrm{\ell }}\right)\cup \left({\cup }_{l=1}^L{\mathcal{Q}}_{\mathrm{\ell }}\right)\cup {\mathcal{P}}_L\backslash mathcal\; \{P\}\; =\; \backslash left(\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{C\}\; \_\; \{\backslash ell\}\backslash right)\; \backslash cup\; \backslash left(\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\backslash right)\; \backslash cup\; \backslash mathcal\; \{P\}\; \_\; \{L\}$$

and, correspondingly, the weighted cumulative regret $R_{T}(\mathcal{P})$ as the sum of the three components

$$R_T(\mathcal{P})=R\left({\cup }_{l=1}^L{\mathcal{C}}_{\mathrm{\ell }}\right)+R\left({\cup }_{l=1}^L{\mathcal{Q}}_{\mathrm{\ell }}\right)+R\left({\mathcal{P}}_L\right)\mathrm{.}R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; R\; \backslash left(\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{C\}\; \_\; \{\backslash ell\}\backslash right)\; +\; R\; \backslash left(\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\backslash right)\; +\; R\; \backslash left(\backslash mathcal\; \{P\}\; \_\; \{L\}\backslash right).$$

Assume $\bigcap_{\ell = 1}^{L}\mathcal{E}{\ell}$ holds. First, notice that on $\mathcal{C}{\ell}$

$$\mathrm{sgn}⟨\widehat{\mathbf{w}},\mathbf{x}⟩=\mathrm{sgn}⟨{\mathbf{w}}_{\mathrm{\ell }},\mathbf{x}⟩=\mathrm{sgn}⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\backslash operatorname\; \{s\; g\; n\}\; \backslash langle\; \backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; =\; \backslash operatorname\; \{s\; g\; n\}\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; =\; \backslash operatorname\; \{s\; g\; n\}\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle$$

under the assumption that $\mathcal{E}{\ell}$ holds, thus points in $\cup{\ell = 1}^{L}\mathcal{C}_{\ell}$ do not contribute weighted regret for $\widehat{\mathbf{w}}$ , i.e.,

$$R\left({\cup }_{l=1}^L{\mathcal{C}}_{\mathrm{\ell }}\right)=0.R\; \backslash left(\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{C\}\; \_\; \{\backslash ell\}\backslash right)\; =\; 0.$$

Next, on $\mathcal{P}_L$ , we have $|\langle \mathbf{w}_L,\mathbf{x}\rangle |\leq 2^{-L}$ . Combining this with the assumption that $\mathcal{E}_L$ holds, we get $|\langle \mathbf{w}^{*},\mathbf{x}\rangle |\leq 2^{-L + 1}$ which implies that the weighted cumulative regret on $\mathcal{P}_L$ is bounded as

the second inequality deriving from the stopping condition defining $L$ in Algorithm 1.

Finally, on the queried points $\cup_{l=1}^{L}\mathcal{Q}{\ell}$ , it is unclear whether $\operatorname{sgn}\langle \widehat{\mathbf{w}},\mathbf{x}\rangle = \operatorname{sgn}\langle \mathbf{w}^*,\mathbf{x}\rangle$ or not, so we bound the weighted cumulative regret contribution of each data item $\mathbf{x}$ therein by $|\langle \mathbf{w}^*,\mathbf{x}\rangle|$ . Now, by construction, $\mathbf{x} \in \mathcal{Q}{\ell} \subset \mathcal{P}{\ell-1}$ , so that $|\langle \mathbf{w}{\ell-1},\mathbf{x}\rangle| \leq 2^{-\ell+1}$ which, combined with the assumption that $\mathcal{E}{\ell-1}$ holds, yields $|\langle \mathbf{w}^*,\mathbf{x}\rangle| \leq 2^{-\ell+2}$ . Since $|\mathcal{Q}{\ell}| = T_{\ell}$ , we have

$$R\left({\cup }_{\mathrm{\ell }=1}^L{\mathcal{Q}}_{\mathrm{\ell }}\right)\le 4\sum _{\mathrm{\ell }=1}^L{T}_{\mathrm{\ell }}{2}^{-\mathrm{\ell }}R\; \backslash left(\backslash cup\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\backslash right)\; \backslash leq\; 4\; \backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; T\; \_\; \{\backslash ell\}\; 2\; ^\; \{-\; \backslash ell\}$$

and Lemma A.3 allows us to write

$$R\left({\cup }_{l=1}^L{\mathcal{Q}}_{\mathrm{\ell }}\right)\le 32d\sum _{l=1}^L\frac{2^{-\mathrm{\ell }}}{{ϵ}_{\mathrm{\ell }}^2}\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)\le 64d\log (2^L{K}_T(\delta ,L))K_T^2(\delta ,L)2^L,R\; \backslash left(\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\backslash right)\; \backslash leq\; 3\; 2\; d\; \backslash sum\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash frac\; \{2\; ^\; \{-\; \backslash ell\}\}\{\backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; \backslash leq\; 6\; 4\; d\; \backslash log\; \backslash left(2\; ^\; \{L\}\; K\; \_\; \{T\}\; (\backslash delta\; ,\; L)\backslash right)\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; 2\; ^\; \{L\},$$

the last inequality following from a reasoning similar to the one that lead us to theorem A.4.

Given any pool realization $\mathcal{P}$ , both the label complexity and weighted regret are bounded by a function of $L$ . Adding the ingredient of the low noise condition (2) helps us leverage the randomness in $\mathcal{P}$ and further bound from above the number of stages $L$ .

Specifically, assume the low noise condition (2) holds for $f^{}(\mathbf{x}) = \frac{1 + \langle\mathbf{w}^{},\mathbf{x}\rangle}{2}$ , for some unknown exponent $\alpha \geq 0$ and unknown constant $\epsilon_0 \in (0,1]$ . Using a multiplicative Chernoff bound, it is easy to see that for any fixed $\epsilon_{*}$ , with probability at least $1 - \delta$ ,

$$\mid {\mathcal{T}}_{{ϵ}_{\ast }}\mid \le \frac{3}{2}(T{ϵ}_{\ast }^{\alpha }+\log (1\mathrm{/}\delta )),\backslash left|\; \backslash mathcal\; \{T\}\; \_\; \{\backslash epsilon\_\; \{*\}\}\; \backslash right|\; \backslash leq\; \backslash frac\; \{3\}\{2\}\; \backslash left(T\; \backslash epsilon\_\; \{*\}\; ^\; \{\backslash alpha\}\; +\; \backslash log\; (1\; /\; \backslash delta)\backslash right),$$

the probability being over the random draw of the initial pool $\mathcal{P}$ . Now, since $\epsilon_L$ is itself a random variable (since so is $L$ ), we need to resort to a covering argument. For any positive number $M$ , consider the following set of fixed $\epsilon$ values

$${\mathcal{K}}_M=\{\frac{{ϵ}_0}{2^{i\mathrm{/}\alpha }}\text{\hspace{0.17em}⁣}:i=0,\dots ,M\}\mathrm{.}\backslash mathcal\; \{K\}\; \_\; \{M\}\; =\; \backslash left\backslash \{\backslash frac\; \{\backslash epsilon\_\; \{0\}\}\{2\; ^\; \{i\; /\; \backslash alpha\}\}\; \backslash colon\; i\; =\; 0,\; \backslash dots\; ,\; M\; \backslash right\backslash \}.$$

Then with probability at least $1 - \delta$

$$\mid {\mathcal{T}}_{ϵ}\mid \le \frac{3}{2}(T{ϵ}^{\alpha }+\log \left(\frac{M}{\delta }\right)),\backslash left|\; \backslash mathcal\; \{T\}\; \_\; \{\backslash epsilon\}\; \backslash right|\; \backslash leq\; \backslash frac\; \{3\}\{2\}\; \backslash left(T\; \backslash epsilon^\; \{\backslash alpha\}\; +\; \backslash log\; \backslash left(\backslash frac\; \{M\}\{\backslash delta\}\backslash right)\backslash right),$$

holds simultaneously over $\epsilon \in \mathcal{K}M$ . Set $M = \log_2T$ and assume $\epsilon$ is the smallest value in $\mathcal{K}M$ that is bigger than or equal to $\epsilon{}$ . If $\epsilon$ is not the smallest value in $\mathcal{K}M$ , then by construction we have $\epsilon{}^{\alpha} \leq \epsilon^{\alpha} < 2\epsilon_{*}^{\alpha}$ so that, for all $\epsilon_{*} > \frac{\epsilon{0}}{2^{M / \alpha}}$ ,

On the other hand if $\epsilon_{*} \leq \frac{\epsilon_{0}}{2^{M / \alpha}}$ we can write

making Eq. (6) hold in this case as well.

We define the event

Then

$$\begin{array}{cccc}& \mathbb{P}(\bar{\mathcal{E}})\ge 1-\delta ,& & \text{(7)}\end{array}\backslash mathbb\; \{P\}\; (\backslash bar\; \{\backslash mathcal\; \{E\}\})\; \backslash geq\; 1\; -\; \backslash delta\; ,\; \backslash tag\; \{7\}$$

for $M = \log_2T$

We set $\epsilon^{*}$ to be the unique solution of the equation

$$\begin{array}{cccc}& d\mathrm{/}{ϵ}_{\ast }=3(T{ϵ}_{\ast }^{\alpha +1}+{ϵ}_{\ast }\log \left(\frac{M}{\delta }\right))\mathrm{.}& & \text{(8)}\end{array}d\; /\; \backslash epsilon\_\; \{*\}\; =\; 3\; \backslash left(T\; \backslash epsilon\_\; \{*\}\; ^\; \{\backslash alpha\; +\; 1\}\; +\; \backslash epsilon\_\; \{*\}\; \backslash log\; \backslash left(\backslash frac\; \{M\}\{\backslash delta\}\backslash right)\backslash right).\; \backslash tag\; \{8\}$$

Eq. (6) will be applied, in particular, to the margin value $2^{-L + 2}$ when $2^{-L + 2} \leq \epsilon_0$ .

Armed with Eqs. (6) and (8) with $M = \log_2 T$ , we prove a lemma that upper bounds the number of stages $L$ .

Lemma A.6. Let $\epsilon_{*}$ be defined through (8), with $T > \frac{2}{3} d$ . Assume both $\bar{\mathcal{E}}$ and $\bigcap_{\ell = 1}^{L}\mathcal{E}_{\ell}$ hold. Then the number of stages $L$ of Algorithm 1 is upper bounded as

$$\begin{array}{c}L\le \max ({\log }_2\left(\frac{1}{{ϵ}_{\ast }}\right),{\log }_2\left(\frac{1}{{ϵ}_0}\right))+2\\ \le \max ({\log }_2[{\left(\frac{3T}{d}\right)}^{\frac{1}{\alpha +2}}+3{\left(\frac{1}{d}\right)}^{\frac{\alpha +1}{\alpha +2}}{\left(\frac{1}{3T}\right)}^{\frac{1}{\alpha +2}}\log \left(\frac{{\log }_{2}T}{\delta }\right)],{\log }_2\left(\frac{1}{{ϵ}_0}\right))+2\\ =\max (O(\frac{1}{\alpha +2}\log \left(\frac{T}{d}\right)+\log \left(\frac{\log T}{\delta }\right)),\log \left(\frac{4}{{ϵ}_0}\right))\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; L\; \backslash leq\; \backslash max\; \backslash left(\backslash log\_\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{*\}\}\backslash right),\; \backslash log\_\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; +\; 2\; \backslash \backslash \; \backslash leq\; \backslash max\; \backslash left(\backslash log\_\; \{2\}\; \backslash left[\; \backslash left(\backslash frac\; \{3\; T\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; 3\; \backslash left(\backslash frac\; \{1\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; \backslash left(\backslash frac\; \{1\}\{3\; T\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\_\; \{2\}\; T\}\{\backslash delta\}\backslash right)\; \backslash right],\; \backslash log\_\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; +\; 2\; \backslash \backslash \; =\; \backslash max\; \backslash left(O\; \backslash left(\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{d\}\backslash right)\; +\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right),\; \backslash log\; \backslash left(\backslash frac\; \{4\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right).\; \backslash \backslash \; \backslash end\{array\}$$

Here the $O$ -notation only omits absolute constants.

Proof. If at stage $L - 1$ the algorithm has not stopped, then we must have

$$d\mathrm{/}{2}^{-L+2}\le 2^{-L+2}\mid {\mathcal{P}}_{L-1}\mid \mathrm{.}d\; /\; 2\; ^\; \{-\; L\; +\; 2\}\; \backslash leq\; 2\; ^\; \{-\; L\; +\; 2\}\; \backslash left|\; \backslash mathcal\; \{P\}\; \_\; \{L\; -\; 1\}\; \backslash right|.$$

Notice that if $\mathbf{x} \in \mathcal{P}{L-1}$ then $|\langle \mathbf{w}{L-1}, \mathbf{x} \rangle| \leq 2^{-L+1}$ . Combining it with the assumption that $\mathcal{E}{L-1}$ holds, we have $|\langle \mathbf{w}^*, \mathbf{x} \rangle| \leq 2^{-L+2}$ which implies $|\mathcal{P}{L-1}| \leq |\mathcal{T}_{2^{-L+2}}|$ .

We split the analysis into two cases. On one hand, when $2^{-L + 2} > \epsilon_0$ , this condition gives us directly

$$L\le {\log }_2(\frac{1}{{ϵ}_0})+2.L\; \backslash leq\; \backslash log\_\; \{2\}\; (\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\})\; +\; 2.$$

On the other hand if $2^{-L + 2}\leq \epsilon_0$ , then given $\bar{\mathcal{E}}$ holds, $|\mathcal{T}_{2^{-L + 2}}|$ is upper bounded as

$$\mid {\mathcal{T}}_{2^{-L+2}}\mid \le 3(T{2}^{(-L+2)\alpha }+\log \left(\frac{M}{\delta }\right)),\backslash left|\; \backslash mathcal\; \{T\}\; \_\; \{2\; ^\; \{-\; L\; +\; 2\}\}\; \backslash right|\; \backslash leq\; 3\; \backslash left(T\; 2\; ^\; \{(-\; L\; +\; 2)\; \backslash alpha\}\; +\; \backslash log\; \backslash left(\backslash frac\; \{M\}\{\backslash delta\}\backslash right)\backslash right),$$

with $M = \log_2T$ . Plugging into the first display results in

$$d\mathrm{/}{2}^{-L+2}\le 3(T{2}^{(-L+2)(\alpha +1)}+2^{-L+2}\log \left(\frac{M}{\delta }\right)),d\; /\; 2\; ^\; \{-\; L\; +\; 2\}\; \backslash leq\; 3\; \backslash left(T\; 2\; ^\; \{(-\; L\; +\; 2)\; (\backslash alpha\; +\; 1)\}\; +\; 2\; ^\; \{-\; L\; +\; 2\}\; \backslash log\; \backslash left(\backslash frac\; \{M\}\{\backslash delta\}\backslash right)\backslash right),$$

which resembles (8) with $2^{-L + 2}$ here playing the role of $\epsilon^{}$ therein. Then, from the definition of $\epsilon^{}$ in (8) we immediately obtain $2^{-L + 2} \geq \epsilon_{}$ , thus $L \leq \log_2\left(\frac{1}{\epsilon_}\right) + 2$ . Moreover, from (8) we see that $d / \epsilon_{} \geq 3T\epsilon_{}^{\alpha + 1}$ , which is equivalent to $\epsilon_{} \leq \left(\frac{d}{3T}\right)^{\frac{1}{\alpha + 2}}$ . Replacing this upper bound on $\epsilon^{}$ back into the right-hand side of (8) and dividing by $d$ yields

$$\frac{1}{{ϵ}_{\ast }}\le {\left(\frac{3T}{d}\right)}^{\frac{1}{\alpha +2}}+3{\left(\frac{1}{d}\right)}^{\frac{\alpha +1}{\alpha +2}}{\left(\frac{1}{3T}\right)}^{\frac{1}{\alpha +2}}\log (\frac{M}{\delta }),\backslash frac\; \{1\}\{\backslash epsilon\_\; \{*\}\}\; \backslash leq\; \backslash left(\backslash frac\; \{3\; T\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; 3\; \backslash left(\backslash frac\; \{1\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; \backslash left(\backslash frac\; \{1\}\{3\; T\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; \backslash log\; (\backslash frac\; \{M\}\{\backslash delta\}),$$

which gives the claimed upper bound on $L$ through $L \leq \log_2\left(\frac{1}{\epsilon_*}\right) + 2$ .

Corollary A.7. Let $T > d$ . Then with probability at least $1 - 2\delta$ over the random draw of $(\mathbf{x}_1, y_1), \ldots, (\mathbf{x}_T, y_T) \sim \mathcal{D}$ the label complexity $N_T(\mathcal{P})$ and the weighted cumulative regret $R_T(\mathcal{P})$ of Algorithm 1 simultaneously satisfy the following:

$$N_T(\mathcal{P})={\log }^2\left(\frac{T}{\delta }\right)(1+{\log }^2\left(\frac{1}{{ϵ}_0}\right))O(\max \{d^{\frac{\alpha }{\alpha +2}}{T}^{\frac{2}{\alpha +2}},\frac{d}{{ϵ}_0^2}\}+{\log }^2\left(\frac{\log T}{\delta }\right))N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; \backslash left(1\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; O\; \backslash left(\backslash max\; \backslash left\backslash \{d\; ^\; \{\backslash frac\; \{\backslash alpha\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\; ^\; \{2\}\}\; \backslash right\backslash \}\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right)$$

$$R_T(\mathcal{P})={\log }^2\left(\frac{T}{\delta }\right)(1+{\log }^2\left(\frac{1}{{ϵ}_0}\right))O(\max \{d^{\frac{\alpha +1}{\alpha +2}}{T}^{\frac{1}{\alpha +2}},\frac{d}{{ϵ}_0}\}+\log \left(\frac{\log T}{\delta }\right))\mathrm{.}R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; \backslash left(1\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; O\; \backslash left(\backslash max\; \backslash left\backslash \{d\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\}\; \backslash right\backslash \}\; +\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right).$$

where the $O$ -notation only omits absolute constants.

Proof. Assume both $\bar{\mathcal{E}}$ and $\bigcap_{\ell = 1}^{L}\mathcal{E}{\ell}$ hold. Recalling the definition of $K{T}(\delta ,L)$ , we have

$$K_T(\delta ,L)=O\left(\sqrt{\log \left(\frac{T}{\delta }\right)+\log L}\right)=O\left(\sqrt{\log \left(\frac{T}{\delta }\right)+L}\right)\mathrm{.}K\; \_\; \{T\}\; (\backslash delta\; ,\; L)\; =\; O\; \backslash left(\backslash sqrt\; \{\backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; +\; \backslash log\; L\}\backslash right)\; =\; O\; \backslash left(\backslash sqrt\; \{\backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; +\; L\}\backslash right).$$

Similar to lemma A.6, we split the analysis into two cases depending on whether or not $2^{-L + 2}$ is bigger than $\epsilon_0$ . If $2^{-L + 2} \leq \epsilon_0$ , we have

$$L\le {\log }_2[{\left(\frac{3T}{d}\right)}^{\frac{1}{\alpha +2}}+3{\left(\frac{1}{d}\right)}^{\frac{\alpha +1}{\alpha +2}}{\left(\frac{1}{3T}\right)}^{\frac{1}{\alpha +2}}\log (\frac{{\log }_{2}T}{\delta })],L\; \backslash leq\; \backslash log\_\; \{2\}\; \backslash left[\; \backslash left(\backslash frac\; \{3\; T\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; 3\; \backslash left(\backslash frac\; \{1\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; \backslash left(\backslash frac\; \{1\}\{3\; T\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; \backslash log\; (\backslash frac\; \{\backslash log\_\; \{2\}\; T\}\{\backslash delta\})\; \backslash right],$$

therefore,

$$2^L=O({\left(\frac{T}{d}\right)}^{\frac{1}{\alpha +2}}+{\left(\frac{1}{d}\right)}^{\frac{\alpha +1}{\alpha +2}}{\left(\frac{1}{T}\right)}^{\frac{1}{\alpha +2}}\log \left(\frac{\log T}{\delta }\right))\mathrm{.}2\; ^\; \{L\}\; =\; O\; \backslash left(\backslash left(\backslash frac\; \{T\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash left(\backslash frac\; \{1\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; \backslash left(\backslash frac\; \{1\}\{T\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right).$$

Plugging the above bounds into Theorem A.4 gives

$$\begin{array}{c}{N}_T(\mathcal{P})=O(d(L+\log K_T^2(\delta ,L))K_T^2(\delta ,L)4^L)\\ =O((L+K_T^2(\delta ,L))K_T^2(\delta ,L)(d^{\frac{\alpha }{\alpha +2}}{T}^{\frac{2}{\alpha +2}}+{\log }^2\left(\frac{\log T}{\delta }\right)))\\ =O\left({(L+\log \left(\frac{T}{\delta }\right))}^2(d^{\frac{\alpha }{\alpha +2}}{T}^{\frac{2}{\alpha +2}}+{\log }^2\left(\frac{\log T}{\delta }\right))\right)\\ ={\log }^2\left(\frac{T}{\delta }\right)O(d^{\frac{\alpha }{\alpha +2}}{T}^{\frac{2}{\alpha +2}}+{\log }^2\left(\frac{\log T}{\delta }\right))\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; O\; \backslash left(d\; \backslash left(L\; +\; \backslash log\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\backslash right)\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; 4\; ^\; \{L\}\backslash right)\; \backslash \backslash \; =\; O\; \backslash left(\backslash left(L\; +\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\backslash right)\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; \backslash left(d\; ^\; \{\backslash frac\; \{\backslash alpha\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right)\backslash right)\; \backslash \backslash \; =\; O\; \backslash left(\backslash left(L\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\backslash right)\; ^\; \{2\}\; \backslash left(d\; ^\; \{\backslash frac\; \{\backslash alpha\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right)\backslash right)\; \backslash \backslash \; =\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; O\; \backslash left(d\; ^\; \{\backslash frac\; \{\backslash alpha\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right).\; \backslash \backslash \; \backslash end\{array\}$$

Similarly applying them to Theorem A.5,

$$\begin{array}{c}{R}_T(\mathcal{P})=O(d(L+\log K_T^2(\delta ,L))K_T^2(\delta ,L)2^L)\\ =O((L+K_T^2(\delta ,L))K_T^2(\delta ,L)(d^{\frac{\alpha +1}{\alpha +2}}{T}^{\frac{1}{\alpha +2}}+\log \left(\frac{\log T}{\delta }\right)))\\ =O\left({(L+\log \left(\frac{T}{\delta }\right))}^2(d^{\frac{\alpha +1}{\alpha +2}}{T}^{\frac{1}{\alpha +2}}+\log \left(\frac{\log T}{\delta }\right))\right)\\ ={\log }^2\left(\frac{T}{\delta }\right)O(d^{\frac{\alpha +1}{\alpha +2}}{T}^{\frac{1}{\alpha +2}}+\log \left(\frac{\log T}{\delta }\right)),\end{array}\backslash begin\{array\}\{l\}\; R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; O\; \backslash left(d\; \backslash left(L\; +\; \backslash log\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\backslash right)\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; 2\; ^\; \{L\}\backslash right)\; \backslash \backslash \; =\; O\; \backslash left(\backslash left(L\; +\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\backslash right)\; K\; \_\; \{T\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; \backslash left(d\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right)\backslash right)\; \backslash \backslash \; =\; O\; \backslash left(\backslash left(L\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\backslash right)\; ^\; \{2\}\; \backslash left(d\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right)\backslash right)\; \backslash \backslash \; =\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; O\; \backslash left(d\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right),\; \backslash \backslash \; \backslash end\{array\}$$

where in the second equality we used the assumption that $d < T$ .

If $2^{-L + 2} > \epsilon_0$ , then $2^L \leq \frac{4}{\epsilon_0}$ . Plugging these bounds into Theorem A.4 and Theorem A.5 gives

$$N_T(\mathcal{P})=O\left({\log }^2\left(\frac{T}{\delta {ϵ}_0}\right)\frac{d}{{ϵ}_0^2}\right)={\log }^2\left(\frac{T}{\delta }\right)(1+{\log }^2\left(\frac{1}{{ϵ}_0}\right))O\left(\frac{d}{{ϵ}_0^2}\right)N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; O\; \backslash left(\backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\; \backslash epsilon\_\; \{0\}\}\backslash right)\; \backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\; ^\; \{2\}\}\backslash right)\; =\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; \backslash left(1\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; O\; \backslash left(\backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\; ^\; \{2\}\}\backslash right)$$

$$R_T(\mathcal{P})=O\left({\log }^2\left(\frac{T}{\delta {ϵ}_0}\right)\frac{d}{{ϵ}_0}\right)={\log }^2\left(\frac{T}{\delta }\right)(1+{\log }^2\left(\frac{1}{{ϵ}_0}\right))O\left(\frac{d}{{ϵ}_0}\right)R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; O\; \backslash left(\backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\; \backslash epsilon\_\; \{0\}\}\backslash right)\; \backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\; =\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; \backslash left(1\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; O\; \backslash left(\backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\}\backslash right)$$

Lastly, (7) and lemma A.1 together yield

$$\mathbb{P}(\bar{\mathcal{E}}\bigcap \left(\bigcap _{\mathrm{\ell }=1}^L{\mathcal{E}}_{\mathrm{\ell }}\right))\ge 1-2\delta ,\backslash mathbb\; \{P\}\; \backslash left(\backslash bar\; \{\backslash mathcal\; \{E\}\}\; \backslash bigcap\; \backslash left(\backslash bigcap\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{E\}\; \_\; \{\backslash ell\}\backslash right)\backslash right)\; \backslash geq\; 1\; -\; 2\; \backslash delta\; ,$$

which concludes the proof.

We now turn the bound on the weighted cumulative regret $R_{T}(\mathcal{P})$ in the previous corollary into a bound on the excess risk. We can write

$$L(w^)−L(w∗)=E(x,y)∼D[1{y≠sgn⁡(⟨w^,x⟩)}−1{y≠sgn⁡(⟨w∗,x⟩)}]=Ex∼DX[Ey∼DY∣X[1{y≠sgn⁡(⟨w^,x⟩)}−1{y≠sgn⁡(⟨w∗,x⟩)}]]=Ex∼DX[1{sgn⁡(⟨w^,x⟩)≠sgn⁡(⟨w∗,x⟩)}∣⟨w∗,x⟩∣], \begin{array}{l} \mathcal {L} (\widehat {\mathbf {w}}) - \mathcal {L} (\mathbf {w} ^ {*}) = \mathbb {E} _ {(\mathbf {x}, y) \sim \mathcal {D}} \left[ \mathbb {1} \{y \neq \operatorname {s g n} (\langle \widehat {\mathbf {w}}, \mathbf {x} \rangle) \} - \mathbb {1} \{y \neq \operatorname {s g n} (\langle \mathbf {w} ^ {*}, \mathbf {x} \rangle) \} \right] \\ = \mathbb {E} _ {\mathbf {x} \sim \mathcal {D} _ {\mathcal {X}}} \left[ \mathbb {E} _ {y \sim \mathcal {D} _ {\mathcal {Y} | \mathcal {X}}} \left[ \mathbb {1} \{y \neq \operatorname {s g n} (\langle \widehat {\mathbf {w}}, \mathbf {x} \rangle) \} - \mathbb {1} \{y \neq \operatorname {s g n} (\langle \mathbf {w} ^ {*}, \mathbf {x} \rangle) \} \right] \right] \\ = \mathbb {E} _ {\mathbf {x} \sim \mathcal {D} _ {\mathcal {X}}} \left[ \mathbb {1} \{\operatorname {s g n} (\langle \widehat {\mathbf {w}}, \mathbf {x} \rangle) \neq \operatorname {s g n} (\langle \mathbf {w} ^ {*}, \mathbf {x} \rangle) \} | \langle \mathbf {w} ^ {*}, \mathbf {x} \rangle | \right], \\ \end{array} $$

where $\widehat{\mathbf{w}}$ is the hypothesis returned by Algorithm 1. Now, simply observe that

$$\mathbb{1}\{\mathrm{sgn}(⟨\widehat{\mathbf{w}},\mathbf{x}⟩)\mathrm{\ne }\mathrm{sgn}\left(⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\right)\}\mathrm{\mid }⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\mathrm{\mid }\backslash mathbb\; \{1\}\; \backslash left\backslash \{\backslash operatorname\; \{s\; g\; n\}\; \backslash left(\backslash langle\; \backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \backslash rangle\backslash right)\; \backslash neq\; \backslash operatorname\; \{s\; g\; n\}\; \backslash left(\backslash left\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash right\backslash rangle\backslash right)\; \backslash right\backslash \}\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |$$

has the same form as the function $\phi (\widehat{\mathbf{w}},\mathbf{x})$ in Appendix C on which the uniform convergence result of Theorem C.4 applies, with $\widehat{\epsilon} (\delta)$ therein replaced by the bound on $R_{T}(\mathcal{P})$ borrowed from Corollary A.7. This allows us to conclude that with probability at least $1 - \delta$

$$\mathcal{L}(\widehat{\mathbf{w}})-\mathcal{L}({\mathbf{w}}^{\ast })={\log }^2\left(\frac{T}{\delta }\right)(1+{\log }^2\left(\frac{1}{{ϵ}_0}\right))O(\max \{{\left(\frac{d}{T}\right)}^{\frac{\alpha +1}{\alpha +2}},\frac{d}{T{ϵ}_0}\}+\frac{\log \left(\frac{\log T}{\delta }\right)}{T}),\backslash mathcal\; \{L\}\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\})\; -\; \backslash mathcal\; \{L\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; =\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; \backslash left(1\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; O\; \backslash left(\backslash max\; \backslash left\backslash \{\backslash left(\backslash frac\; \{d\}\{T\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{T\; \backslash epsilon\_\; \{0\}\}\; \backslash right\backslash \}\; +\; \backslash frac\; \{\backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\}\{T\}\backslash right),$$

as claimed in Theorem 4.1 in the main body of the paper.

B. Proofs for Section 5

We adopt the same notation as in Section A and follow the same proof structure.

Define the loss function

$$Loss(a)=\log (1+e^{-a}),L\; o\; s\; s\; (a)\; =\; \backslash log\; (1\; +\; e\; ^\; \{-\; a\}),$$

and the sigmoidal function

$$\sigma (a)=\frac{1}{1+e^{-a}}\mathrm{.}\backslash sigma\; (a)\; =\; \backslash frac\; \{1\}\{1\; +\; e\; ^\; \{-\; a\}\}.$$

The noise model in the main body of the paper can be re-formulated as follows: there exists an unknown vector $\mathbf{w}^*$ belonging to a Euclidean ball of radius $R\geq 1$ such that for any instance $\mathbf{x}$ of Euclidean norm at most 1,

$$\mathbb{P}(y=1\mid \mathbf{x})=\sigma (⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩)\mathrm{.}\backslash mathbb\; \{P\}\; (y\; =\; 1\; \backslash mid\; \backslash mathbf\; \{x\})\; =\; \backslash sigma\; (\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle).$$

Therefore we have

$$\mathbb{E}[y\mid \mathbf{x}]=\sigma (⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩)-\sigma (-⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩)=2\sigma (⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩)-1,\backslash mathbb\; \{E\}\; [\; y\; \backslash mid\; \backslash mathbf\; \{x\}\; ]\; =\; \backslash sigma\; (\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle)\; -\; \backslash sigma\; (-\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle)\; =\; 2\; \backslash sigma\; (\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle)\; -\; 1,$$

and the noise variable $\xi$ can be written as

$$\xi :=y-\mathbb{E}[y\mid \mathbf{x}]=\frac{2y}{1+e^{y⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩}}\mathrm{.}\backslash xi\; :=\; y\; -\; \backslash mathbb\; \{E\}\; [\; y\; \backslash mid\; \backslash mathbf\; \{x\}\; ]\; =\; \backslash frac\; \{2\; y\}\{1\; +\; e\; ^\; \{y\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; ,\; \backslash mathbf\; \{x\}\; \backslash rangle\}\}.$$

Similar to the linear case, we denote for any $\epsilon > 0$ ,

$${\mathcal{T}}_{ϵ}^{\sigma }=\{\mathbf{x}\in \mathcal{P}:\mathrm{\mid }2\sigma (⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩)-1\mathrm{\mid }\le ϵ\}\mathrm{.}\backslash mathcal\; \{T\}\; \_\; \{\backslash epsilon\}\; ^\; \{\backslash sigma\}\; =\; \backslash left\backslash \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}:\; |\; 2\; \backslash sigma\; \backslash left(\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\backslash right)\; -\; 1\; |\; \backslash leq\; \backslash epsilon\; \backslash right\backslash \}.$$

Now, recall the notation in Algorithm 2. Similar to $\mathcal{E}_{\ell}$ defined in linear case, it will be useful to define the events

$${\mathcal{E}}_{\mathrm{\ell }}=\{\underset{\mathbf{x}\in {\mathcal{P}}_{\mathrm{\ell }-1}\mathrm{\backslash }{\mathcal{Q}}_{\mathrm{\ell }}}{\max }\mathrm{\mid }⟨{\mathbf{w}}_{\mathrm{\ell }}-{\mathbf{w}}^{\ast },\mathbf{x}⟩\mathrm{\mid }\le R_{\mathrm{\ell }}\},\backslash mathcal\; \{E\}\; \_\; \{\backslash ell\}\; =\; \backslash left\backslash \{\backslash max\; \_\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\; \_\; \{\backslash ell\; -\; 1\}\; \backslash backslash\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\}\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; -\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |\; \backslash leq\; R\; \_\; \{\backslash ell\}\; \backslash right\backslash \},$$

where $R_{\ell} = R2^{-\ell}$ for $\ell = 0,\dots ,L$

Lemma B.1. For any positive $L$ ,

$$\mathbb{P}\left(\bigcap _{\mathrm{\ell }=1}^L{\mathcal{E}}_{\mathrm{\ell }}\right)>1-\delta \mathrm{.}\backslash mathbb\; \{P\}\; \backslash left(\backslash bigcap\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{E\}\; \_\; \{\backslash ell\}\backslash right)\; >\; 1\; -\; \backslash delta\; .$$

Proof. We decompose the above quantity as

$$\mathbb{P}\left(\bigcap _{\mathrm{\ell }=1}^L{\mathcal{E}}_{\mathrm{\ell }}\right)=\mathbb{P}({\mathcal{E}}_L\mid \bigcap _{\mathrm{\ell }=1}^{L-1}{\mathcal{E}}_{\mathrm{\ell }})\mathbb{P}({\mathcal{E}}_{L-1}\mid \bigcap _{\mathrm{\ell }=1}^{L-2}{\mathcal{E}}_{\mathrm{\ell }})\dots \mathbb{P}({\mathcal{E}}_2\mathrm{\mid }{\mathcal{E}}_1)\mathbb{P}({\mathcal{E}}_1),\backslash mathbb\; \{P\}\; \backslash left(\backslash bigcap\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{E\}\; \_\; \{\backslash ell\}\backslash right)\; =\; \backslash mathbb\; \{P\}\; \backslash left(\backslash mathcal\; \{E\}\; \_\; \{L\}\; \backslash Big\; |\; \backslash bigcap\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\; -\; 1\}\; \backslash mathcal\; \{E\}\; \_\; \{\backslash ell\}\backslash right)\; \backslash mathbb\; \{P\}\; \backslash left(\backslash mathcal\; \{E\}\; \_\; \{L\; -\; 1\}\; \backslash Big\; |\; \backslash bigcap\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\; -\; 2\}\; \backslash mathcal\; \{E\}\; \_\; \{\backslash ell\}\backslash right)\; \backslash dots\; \backslash mathbb\; \{P\}\; (\backslash mathcal\; \{E\}\; \_\; \{2\}\; |\; \backslash mathcal\; \{E\}\; \_\; \{1\})\; \backslash mathbb\; \{P\}\; (\backslash mathcal\; \{E\}\; \_\; \{1\}),$$

and bound each factor individually.

At the beginning of the stage $\ell$ , the remaining pool is $\mathcal{P}{\ell - 1}$ , and $\sup{x \in \mathcal{P}{\ell - 1}} |\langle \mathbf{w}{l - 1}, \mathbf{x} \rangle| \leq R_{\ell - 1}$ .

For $\ell \geq 2$ , if $\mathcal{E}_{\ell -1}$ holds then

$$\begin{array}{cccc}& \underset{\mathbf{x}\in {\mathcal{P}}_{\mathrm{\ell }-1}}{\sup }\mathrm{\mid }⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\mathrm{\mid }\le 2R_{\mathrm{\ell }-1}\mathrm{.}& & \text{(9)}\end{array}\backslash sup\; \_\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\; \_\; \{\backslash ell\; -\; 1\}\}\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |\; \backslash leq\; 2\; R\; \_\; \{\backslash ell\; -\; 1\}.\; \backslash tag\; \{9\}$$

Note that (9) also holds for $\ell = 1$ since $| \mathbf{w}^*| \leq R$ and $| \mathbf{x}| \leq 1$ .

Now, for any positive number $b$ , let

$${\mathrm{\Omega }}_{\mathrm{\ell }}(b)=\{\mathbf{w}\in {\mathbb{R}}^d:\underset{\mathbf{x}\in {\mathcal{Q}}_{\mathrm{\ell }}}{\max }\mathrm{\mid }⟨\mathbf{w},\mathbf{x}⟩\mathrm{\mid }\le b\},\backslash Omega\_\{\backslash ell\}(b)\; =\; \backslash \{\backslash mathbf\{w\}\backslash in\; \backslash mathbb\{R\}^\{d\}:\backslash max\_\{\backslash mathbf\{x\}\backslash in\; \backslash mathcal\{Q\}\_\{\backslash ell\}\}|\backslash langle\; \backslash mathbf\{w\},\backslash mathbf\{x\}\backslash rangle\; |\backslash leq\; b\backslash \}\; ,$$

which is a convex compact set of $\mathbf{w}$ 's.

The predictor $\mathbf{w}_{\ell}$ in Eq. (4) in the main body is defined as the solution of the following constraint minimization problem:

$${\mathbf{w}}_{\mathrm{\ell }}=\underset{\mathbf{w}\in {\mathrm{\Omega }}_{\mathrm{\ell }}(2R_{\mathrm{\ell }-1})}{\mathrm{argmin}}[\sum _{t=1}^{T_{\mathrm{\ell }}}Loss(y_{\mathrm{\ell },t}⟨\mathbf{w},{\mathbf{x}}_{\mathrm{\ell },t}⟩)+\frac{1}{8}{e}^{-4R_{\mathrm{\ell }}}\mathrm{\parallel }\mathbf{w}{\mathrm{\parallel }}^2],\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; =\; \backslash operatorname\; *\; \{a\; r\; g\; m\; i\; n\}\; \_\; \{\backslash mathbf\; \{w\}\; \backslash in\; \backslash Omega\_\; \{\backslash ell\}\; (2\; R\; \_\; \{\backslash ell\; -\; 1\})\}\; \backslash left[\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; L\; o\; s\; s\; \backslash left(y\; \_\; \{\backslash ell\; ,\; t\}\; \backslash langle\; \backslash mathbf\; \{w\},\; \backslash mathbf\; \{x\}\; \_\; \{\backslash ell\; ,\; t\}\; \backslash rangle\backslash right)\; +\; \backslash frac\; \{1\}\{8\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; \backslash |\; \backslash mathbf\; \{w\}\; \backslash |\; ^\; \{2\}\; \backslash right],$$

For simplicity, from now on we omit the stage index $\ell$ from the subscripts of $\mathbf{x}{\ell,t}$ and $y{\ell,t}$ and denote $A_{\ell,T_\ell}$ as $A_\ell$ . For $t = 1,\dots,T_\ell$ , denote

$$g_t(\mathbf{w}){\mathbf{x}}_t={\mathrm{\nabla }}_{\mathbf{w}}Loss(y_t⟨\mathbf{w},{\mathbf{x}}_t⟩)=-\frac{y_t}{1+\exp (y_t⟨\mathbf{w},{\mathbf{x}}_t⟩)}{\mathbf{x}}_{t}g\; \_\; \{t\}\; (\backslash mathbf\; \{w\})\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; =\; \backslash nabla\_\; \{\backslash mathbf\; \{w\}\}\; L\; o\; s\; s\; (y\; \_\; \{t\}\; \backslash langle\; \backslash mathbf\; \{w\},\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash rangle)\; =\; -\; \backslash frac\; \{y\; \_\; \{t\}\}\{1\; +\; \backslash exp\; (y\; \_\; \{t\}\; \backslash langle\; \backslash mathbf\; \{w\}\; ,\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash rangle)\}\; \backslash mathbf\; \{x\}\; \_\; \{t\}$$

$$h_t(\mathbf{w}){\mathbf{x}}_t{\mathbf{x}}_t^{\mathrm{\top }}={\mathrm{\nabla }}_{\mathbf{w}}^{2}Loss(y_t⟨\mathbf{w},{\mathbf{x}}_t⟩)=\frac{1}{2(1+\cosh (y_t⟨\mathbf{w},{\mathbf{x}}_t⟩))}{\mathbf{x}}_t{\mathbf{x}}_t^{\mathrm{\top }}\mathrm{.}h\; \_\; \{t\}\; (\backslash mathbf\; \{w\})\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; ^\; \{\backslash top\}\; =\; \backslash nabla\_\; \{\backslash mathbf\; \{w\}\}\; ^\; \{2\}\; L\; o\; s\; s\; (y\; \_\; \{t\}\; \backslash langle\; \backslash mathbf\; \{w\},\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash rangle)\; =\; \backslash frac\; \{1\}\{2\; \backslash big\; (1\; +\; \backslash cosh\; (y\; \_\; \{t\}\; \backslash langle\; \backslash mathbf\; \{w\}\; ,\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash rangle)\; \backslash big)\}\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; ^\; \{\backslash top\}.$$

Notice that by definition

$$g_t\left({\mathbf{w}}^{\ast }\right)=-\frac{1}{2}{\xi }_t,g\; \_\; \{t\}\; \backslash left(\backslash mathbf\; \{w\}\; ^\; \{*\}\backslash right)\; =\; -\; \backslash frac\; \{1\}\{2\}\; \backslash xi\_\; \{t\},$$

where $\xi_{t}$ is the noise term $\xi_{t} = y_{t} - \mathbb{E}[y_{t}|\mathbf{x}_{t}]$ . Since $\cosh (\cdot)$ is an even function,

$$h_t(\mathbf{w})=\frac{1}{2(1+\cosh (⟨\mathbf{w},{\mathbf{x}}_t⟩))}h\; \_\; \{t\}\; (\backslash mathbf\; \{w\})\; =\; \backslash frac\; \{1\}\{2\; \backslash left(1\; +\; \backslash cosh\; \backslash left(\backslash langle\; \backslash mathbf\; \{w\}\; ,\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash rangle\backslash right)\backslash right)\}$$

does not depend on $y_{t}$

Since $\mathbf{w}^{*}\in \Omega_{\ell}(2R_{\ell -1})$ (as a consequence of (9)), the assumption that $\mathcal{E}{\ell -1}$ holds and the optimality of $\mathbf{w}{\ell}$ in $\Omega_{\ell}(2R_{\ell -1})$ allows us to write

$$⟨\mathbf{g}({\mathbf{w}}_{\mathrm{\ell }})+\frac{1}{4}{e}^{-4R_{\mathrm{\ell }}}{\mathbf{w}}_{\mathrm{\ell }},{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}⟩\ge 0,\backslash langle\; \backslash mathbf\; \{g\}\; (\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\})\; +\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\},\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash rangle\; \backslash geq\; 0,$$

where

$$\mathbf{g}(\mathbf{w})=\sum _{t=1}^{T_{\mathrm{\ell }}}{g}_t(\mathbf{w}){\mathbf{x}}_t\mathrm{.}\backslash mathbf\; \{g\}\; (\backslash mathbf\; \{w\})\; =\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; g\; \_\; \{t\}\; (\backslash mathbf\; \{w\})\; \backslash mathbf\; \{x\}\; \_\; \{t\}.$$

It follows that

$$\begin{array}{cccc}& ⟨\mathbf{g}({\mathbf{w}}^{\ast })-\mathbf{g}({\mathbf{w}}_{\mathrm{\ell }}),{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}⟩\le ⟨\mathbf{g}({\mathbf{w}}^{\ast }),{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}⟩+\frac{1}{4}{e}^{-4R_{\mathrm{\ell }}}⟨{\mathbf{w}}_{\mathrm{\ell }},{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}⟩\mathrm{.}& & \text{(10)}\end{array}\backslash langle\; \backslash mathbf\; \{g\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; -\; \backslash mathbf\; \{g\}\; (\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}),\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash rangle\; \backslash leq\; \backslash langle\; \backslash mathbf\; \{g\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\}),\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash rangle\; +\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\},\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash rangle\; .\; \backslash tag\; \{10\}$$

For each $t = 1,\ldots ,T_{\ell}$ , the mean-value theorem insures the existence of a constant $\mu_{\ell}^{t}\in [0,1]$ such that for

$${\mathbf{w}}_{\mathrm{\ell }}^t=(1-{\mu }_{\mathrm{\ell }}^t){\mathbf{w}}_{\mathrm{\ell }}+{\mu }_{\mathrm{\ell }}^t{\mathbf{w}}^{\ast },\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; ^\; \{t\}\; =\; (1\; -\; \backslash mu\_\; \{\backslash ell\}\; ^\; \{t\})\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; +\; \backslash mu\_\; \{\backslash ell\}\; ^\; \{t\}\; \backslash mathbf\; \{w\}\; ^\; \{*\},$$

we have

$$g_t\left({\mathbf{w}}^{\ast }\right)-g_t\left({\mathbf{w}}_{\mathrm{\ell }}\right)=h_t\left({\mathbf{w}}_{\mathrm{\ell }}^t\right)⟨{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }},{\mathbf{x}}_t⟩\mathrm{.}g\; \_\; \{t\}\; \backslash left(\backslash mathbf\; \{w\}\; ^\; \{*\}\backslash right)\; -\; g\; \_\; \{t\}\; \backslash left(\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\backslash right)\; =\; h\; \_\; \{t\}\; \backslash left(\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; ^\; \{t\}\backslash right)\; \backslash left\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\},\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash right\backslash rangle\; .$$

Since

$$\mathrm{\mid }⟨{\mathbf{w}}_{\mathrm{\ell }}^t,{\mathbf{x}}_t⟩\mathrm{\mid }\le (1-{\mu }_{\mathrm{\ell }}^t)\mathrm{\mid }⟨{\mathbf{w}}_{\mathrm{\ell }},{\mathbf{x}}_t⟩\mathrm{\mid }+{\mu }_{\mathrm{\ell }}^t\mathrm{\mid }⟨{\mathbf{w}}^{\ast },{\mathbf{x}}_t⟩\mathrm{\mid }\le 2R_{\mathrm{\ell }-1}=4R_{\mathrm{\ell }},|\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; ^\; \{t\},\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash rangle\; |\; \backslash leq\; (1\; -\; \backslash mu\_\; \{\backslash ell\}\; ^\; \{t\})\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\},\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash rangle\; |\; +\; \backslash mu\_\; \{\backslash ell\}\; ^\; \{t\}\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash rangle\; |\; \backslash leq\; 2\; R\; \_\; \{\backslash ell\; -\; 1\}\; =\; 4\; R\; \_\; \{\backslash ell\},$$

we have

$$h_t({\mathbf{w}}_{\mathrm{\ell }}^t)=\frac{1}{2(1+\cosh (⟨{\mathbf{w}}_{\mathrm{\ell }}^t,{\mathbf{x}}_t⟩))}\ge \frac{1}{4}{e}^{-\mathrm{\mid }⟨{\mathbf{w}}_{\mathrm{\ell }}^t,{\mathbf{x}}_t⟩\mathrm{\mid }}\ge \frac{1}{4}{e}^{-4R_{\mathrm{\ell }}}\mathrm{.}h\; \_\; \{t\}\; (\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; ^\; \{t\})\; =\; \backslash frac\; \{1\}\{2\; \backslash big\; (1\; +\; \backslash cosh\; (\backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; ^\; \{t\}\; ,\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash rangle)\; \backslash big)\}\; \backslash geq\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; ^\; \{t\},\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash rangle\; |\}\; \backslash geq\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}.$$

Introduce now the matrix

$$H_{\mathrm{\ell }}:=\sum _{t=1}^{T_{\mathrm{\ell }}}{h}_t\left({\mathbf{w}}_{\mathrm{\ell }}^t\right){\mathbf{x}}_t{\mathbf{x}}_t^{\mathrm{\top }}+\frac{1}{4}{e}^{-4R_{\mathrm{\ell }}}I,H\; \_\; \{\backslash ell\}\; :=\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; h\; \_\; \{t\}\; \backslash left(\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; ^\; \{t\}\backslash right)\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; ^\; \{\backslash top\}\; +\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; I,$$

where $I$ is the $d\times d$ identity matrix. We can write

$$\mathbf{g}\left({\mathbf{w}}^{\ast }\right)-\mathbf{g}\left({\mathbf{w}}_{\mathrm{\ell }}\right)=(H_{\mathrm{\ell }}-\frac{1}{4}{e}^{-4R_{\mathrm{\ell }}}I)({\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }})\mathrm{.}\backslash mathbf\; \{g\}\; \backslash left(\backslash mathbf\; \{w\}\; ^\; \{*\}\backslash right)\; -\; \backslash mathbf\; \{g\}\; \backslash left(\backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\backslash right)\; =\; \backslash left(H\; \_\; \{\backslash ell\}\; -\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; I\backslash right)\; \backslash left(\backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\backslash right).$$

As a consequence, (10) implies

$$\begin{array}{c}⟨H_{\mathrm{\ell }}({\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}),{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}⟩\le ⟨\mathbf{g}({\mathbf{w}}^{\ast }),{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}⟩+\frac{1}{4}{e}^{-4R_{\mathrm{\ell }}}\mathrm{\parallel }{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}{\mathrm{\parallel }}^2+\frac{1}{4}{e}^{-4R_{\mathrm{\ell }}}⟨{\mathbf{w}}_{\mathrm{\ell }},{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}⟩\\ =⟨\mathbf{g}({\mathbf{w}}^{\ast })+\frac{1}{4}{e}^{-4R_{\mathrm{\ell }}}{\mathbf{w}}^{\ast },{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}⟩\\ \le (\mathrm{\parallel }\mathbf{g}({\mathbf{w}}^{\ast }){\mathrm{\parallel }}_{H_{\mathrm{\ell }}^{-1}}+\frac{1}{4}{e}^{-4R_{\mathrm{\ell }}}\mathrm{\parallel }{\mathbf{w}}^{\ast }{\mathrm{\parallel }}_{H_{\mathrm{\ell }}^{-1}})\mathrm{\parallel }{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}{\mathrm{\parallel }}_{H_{\mathrm{\ell }}}\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash langle\; H\; \_\; \{\backslash ell\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}),\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash rangle\; \backslash leq\; \backslash langle\; \backslash mathbf\; \{g\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\}),\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash rangle\; +\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; \backslash |\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash |\; ^\; \{2\}\; +\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\},\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash rangle\; \backslash \backslash \; =\; \backslash langle\; \backslash mathbf\; \{g\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; +\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash rangle\; \backslash \backslash \; \backslash leq\; \backslash left(\backslash |\; \backslash mathbf\; \{g\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; \backslash |\; \_\; \{H\; \_\; \{\backslash ell\}\; ^\; \{-\; 1\}\}\; +\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; \backslash |\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; \backslash |\; \_\; \{H\; \_\; \{\backslash ell\}\; ^\; \{-\; 1\}\}\backslash right)\; \backslash |\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash |\; \_\; \{H\; \_\; \{\backslash ell\}\}.\; \backslash \backslash \; \backslash end\{array\}$$

We thus obtain

$${\parallel {\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}\parallel }_{H_{\mathrm{\ell }}}\le {\parallel \mathbf{g}\left({\mathbf{w}}^{\ast }\right)\parallel }_{H_{\mathrm{\ell }}^{-1}}+\frac{1}{4}{e}^{-4R_{\mathrm{\ell }}}{\parallel {\mathbf{w}}^{\ast }\parallel }_{H_{\mathrm{\ell }}^{-1}}\le 4e^{4R_{\mathrm{\ell }}}{\parallel \mathbf{g}\left({\mathbf{w}}^{\ast }\right)\parallel }_{A_{\mathrm{\ell }}^{-1}}+R,\backslash left\backslash |\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash right\backslash |\; \_\; \{H\; \_\; \{\backslash ell\}\}\; \backslash leq\; \backslash left\backslash |\; \backslash mathbf\; \{g\}\; \backslash left(\backslash mathbf\; \{w\}\; ^\; \{*\}\backslash right)\; \backslash right\backslash |\; \_\; \{H\; \_\; \{\backslash ell\}\; ^\; \{-\; 1\}\}\; +\; \backslash frac\; \{1\}\{4\}\; e\; ^\; \{-\; 4\; R\; \_\; \{\backslash ell\}\}\; \backslash left\backslash |\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; \backslash right\backslash |\; \_\; \{H\; \_\; \{\backslash ell\}\; ^\; \{-\; 1\}\}\; \backslash leq\; 4\; e\; ^\; \{4\; R\; \_\; \{\backslash ell\}\}\; \backslash left\backslash |\; \backslash mathbf\; \{g\}\; \backslash left(\backslash mathbf\; \{w\}\; ^\; \{*\}\backslash right)\; \backslash right\backslash |\; \_\; \{A\; \_\; \{\backslash ell\}\; ^\; \{-\; 1\}\}\; +\; R,$$

where in the second inequality we used $H_{\ell} \succeq \frac{1}{4} e^{-4R_{\ell}}A_{\ell}$ .

To bound $| \mathbf{g}(\mathbf{w}^{*})|{A{\ell^{-1}}}$ , note that

$$\mathrm{\parallel }\mathbf{g}({\mathbf{w}}^{\ast }){\mathrm{\parallel }}_{A_{\mathrm{\ell }}^{-1}}^2=\mathrm{\parallel }\sum _{t=1}^{T_{\mathrm{\ell }}}{g}_t({\mathbf{w}}^{\ast })A_{\mathrm{\ell }}^{-1\mathrm{/}2}{\mathbf{x}}_t{\mathrm{\parallel }}_2^2=\frac{1}{2}\mathrm{\parallel }\sum _{t=1}^{T_{\mathrm{\ell }}}{\xi }_t{A}_{\mathrm{\ell }}^{-1\mathrm{/}2}{\mathbf{x}}_t{\mathrm{\parallel }}_2^2\mathrm{.}\backslash |\; \backslash mathbf\; \{g\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; \backslash |\; \_\; \{A\; \_\; \{\backslash ell\}\; ^\; \{-\; 1\}\}\; ^\; \{2\}\; =\; \backslash |\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; g\; \_\; \{t\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; A\; \_\; \{\backslash ell\}\; ^\; \{-\; 1\; /\; 2\}\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash |\; \_\; \{2\}\; ^\; \{2\}\; =\; \backslash frac\; \{1\}\{2\}\; \backslash |\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\; \_\; \{\backslash ell\}\}\; \backslash xi\_\; \{t\}\; A\; \_\; \{\backslash ell\}\; ^\; \{-\; 1\; /\; 2\}\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; \backslash |\; \_\; \{2\}\; ^\; \{2\}.$$

We plug in $A = [A_{\ell}^{-1 / 2}\mathbf{x}1,\ldots ,A{\ell}^{-1 / 2}\mathbf{x}{T_\ell}], \xi = (\xi_1,\dots ,\xi{T_\ell})$ into lemma C.6 and get with probability at least $1 - \frac{\delta}{\ell(\ell + 1)}$

Thus for any $\mathbf{x} \in \mathcal{P}{\ell - 1} \backslash \mathcal{Q}{\ell}$ , we obtain that with probability at least $1 - \frac{\delta}{\ell(\ell + 1)}$ :

$$\begin{array}{c}\mid ⟨{\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }},\mathbf{x}⟩\mid \le {\parallel \mathbf{x}\parallel }_{H_{\mathrm{\ell }}^{-1}}{\parallel {\mathbf{w}}^{\ast }-{\mathbf{w}}_{\mathrm{\ell }}\parallel }_{H_{\mathrm{\ell }}}\\ \le 4e^{4R_{\mathrm{\ell }}}{ϵ}_{\mathrm{\ell }}(4e^{4R_{\mathrm{\ell }}}\mathrm{\parallel }\mathbf{g}({\mathbf{w}}^{\ast }){\mathrm{\parallel }}_{A_{\mathrm{\ell }}^{-1}}+R)\\ \le {ϵ}_{\mathrm{\ell }}(16e^{8R_{\mathrm{\ell }}}\sqrt{d\log \frac{2d\mathrm{\ell }(\mathrm{\ell }+1)}{\delta }}+4e^{4R_{\mathrm{\ell }}}R)\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash left|\; \backslash left\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\},\; \backslash mathbf\; \{x\}\; \backslash right\backslash rangle\; \backslash right|\; \backslash leq\; \backslash left\backslash |\; \backslash mathbf\; \{x\}\; \backslash right\backslash |\; \_\; \{H\; \_\; \{\backslash ell\}\; ^\; \{-\; 1\}\}\; \backslash left\backslash |\; \backslash mathbf\; \{w\}\; ^\; \{*\}\; -\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; \backslash right\backslash |\; \_\; \{H\; \_\; \{\backslash ell\}\}\; \backslash \backslash \; \backslash leq\; 4\; e\; ^\; \{4\; R\; \_\; \{\backslash ell\}\}\; \backslash epsilon\_\; \{\backslash ell\}\; \backslash left(4\; e\; ^\; \{4\; R\; \_\; \{\backslash ell\}\}\; \backslash |\; \backslash mathbf\; \{g\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; \backslash |\; \_\; \{A\; \_\; \{\backslash ell\}\; ^\; \{-\; 1\}\}\; +\; R\backslash right)\; \backslash \backslash \; \backslash leq\; \backslash epsilon\_\; \{\backslash ell\}\; \backslash left(1\; 6\; e\; ^\; \{8\; R\; \_\; \{\backslash ell\}\}\; \backslash sqrt\; \{d\; \backslash log\; \backslash frac\; \{2\; d\; \backslash ell\; (\backslash ell\; +\; 1)\}\{\backslash delta\}\}\; +\; 4\; e\; ^\; \{4\; R\; \_\; \{\backslash ell\}\}\; R\backslash right).\; \backslash \backslash \; \backslash end\{array\}$$

Recalling the definition of $\epsilon_{\ell}$ in Algorithm 2, we have, with probability at least $1 - \frac{\delta}{\ell(\ell + 1)}$

$$\underset{\mathbf{x}\in {\mathcal{P}}_{\mathrm{\ell }-1}\mathrm{\backslash }{\mathcal{Q}}_{\mathrm{\ell }}}{\max }\mathrm{\mid }⟨{\mathbf{w}}_{\mathrm{\ell }}-{\mathbf{w}}^{\ast },\mathbf{x}⟩\mathrm{\mid }\le R_{\mathrm{\ell }},\backslash max\; \_\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\; \_\; \{\backslash ell\; -\; 1\}\; \backslash backslash\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\}\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\}\; -\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |\; \backslash leq\; R\; \_\; \{\backslash ell\},$$

that is, $\mathbb{P}(\mathcal{E}{\ell}|\bigcap{s = 1}^{\ell -1}\mathcal{E}_{s}^{\sigma})\geq 1 - \frac{\delta}{\ell(\ell + 1)}$ (for $\ell = 1$ the above analysis gives $\mathbb{P}(\mathcal{E}_1^\sigma)\geq 1 - \frac{\delta}{2})$ . Hence

$$\mathbb{P}\left(\bigcap _{\mathrm{\ell }=1}^L{\mathcal{E}}_{\mathrm{\ell }}\right)\ge \prod _{\mathrm{\ell }=1}^L(1-\frac{\delta }{\mathrm{\ell }(\mathrm{\ell }+1)})\ge 1-\delta \sum _{\mathrm{\ell }=1}^L\frac{1}{\mathrm{\ell }(\mathrm{\ell }+1)}>1-\delta ,\backslash mathbb\; \{P\}\; \backslash left(\backslash bigcap\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{E\}\; \_\; \{\backslash ell\}\backslash right)\; \backslash geq\; \backslash prod\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash left(1\; -\; \backslash frac\; \{\backslash delta\}\{\backslash ell\; (\backslash ell\; +\; 1)\}\backslash right)\; \backslash geq\; 1\; -\; \backslash delta\; \backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash frac\; \{1\}\{\backslash ell\; (\backslash ell\; +\; 1)\}\; >\; 1\; -\; \backslash delta\; ,$$

thereby concluding the proof.

Similar to linear case, Lemma A.2 and Lemma A.3 also hold for logistic case.

We define the weighted cumulative regret for the logistic case as

$$R_T(\mathcal{P})=\sum _{\mathbf{x}\in \mathcal{P}}\mathbb{1}\{\mathrm{sgn}⟨\widehat{\mathbf{w}},\mathbf{x}⟩\mathrm{\ne }\mathrm{sgn}⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\}\mathrm{\mid }2\sigma (⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩)-1\mathrm{\mid },R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; \backslash sum\_\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\}\; \backslash mathbb\; \{1\}\; \backslash \{\backslash operatorname\; \{s\; g\; n\}\; \backslash langle\; \backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash neq\; \backslash operatorname\; \{s\; g\; n\}\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash \}\; |\; 2\; \backslash sigma\; (\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle)\; -\; 1\; |,$$

where $\widehat{\mathbf{w}}$ is the model output by Algorithm 2. Notice that since $|2\sigma (x) - 1|\leq |x| / 2$ for all $x$ , we alternatively upper bound

$$R_T(\mathcal{P})=\sum _{\mathbf{x}\in \mathcal{P}}\mathbb{1}\{\mathrm{s}\mathrm{g}\mathrm{n}⟨\widehat{\mathbf{w}},\mathbf{x}⟩\mathrm{\ne }\mathrm{s}\mathrm{g}\mathrm{n}⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\}\mathrm{\mid }⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\mathrm{\mid }\mathrm{/}2,R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; \backslash sum\_\; \{\backslash mathbf\; \{x\}\; \backslash in\; \backslash mathcal\; \{P\}\}\; \backslash mathbb\; \{1\}\; \backslash \{\backslash mathrm\; \{s\; g\; n\}\; \backslash langle\; \backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash neq\; \backslash mathrm\; \{s\; g\; n\}\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash \}\; |\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; |\; /\; 2,$$

To simplify the math display we denote

$$K_d(\delta ,\mathrm{\ell })=\sqrt{d\log \frac{2d\mathrm{\ell }(\mathrm{\ell }+1)}{\delta }},K\; \_\; \{d\}\; (\backslash delta\; ,\; \backslash ell)\; =\; \backslash sqrt\; \{d\; \backslash log\; \backslash frac\; \{2\; d\; \backslash ell\; (\backslash ell\; +\; 1)\}\{\backslash delta\}\},$$

then $\epsilon_{\ell} = \frac{R_{\ell}}{16e^{8R_{\ell}}K_{d}(\delta,\ell) + 4Re^{4R_{\ell}}}$ . Note that here the factor $K_{d}(\delta ,\ell)$ doesn't depend on $T$ but has a $\sqrt{d}$ dependence.

To bound the number of queries note that lemma A.3 still holds, we use this to prove the following result.

Theorem B.2. For any pool realization $\mathcal{P}$ , the label complexity $N_{T}(\mathcal{P})$ of Algorithm 2 operating on a pool $\mathcal{P}$ of size $T$ is bounded deterministically as

$$N_T(\mathcal{P})=d\underset{\mathrm{\ell }\in [L]}{\max }\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)O(K_d^2(\delta ,L)e^{8R}\lceil {\log }_{2}R\rceil +e^{4R}\lceil {\log }_{2}R\rceil R^2+\frac{K_d^2(\delta ,L)+R^2}{R_L^2}),N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; d\; \backslash max\; \_\; \{\backslash ell\; \backslash in\; [\; L\; ]\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; O\; \backslash left(K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; e\; ^\; \{4\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; R\; ^\; \{2\}\; +\; \backslash frac\; \{K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; +\; R\; ^\; \{2\}\}\{R\; \_\; \{L\}\; ^\; \{2\}\}\backslash right),$$

where the $O$ -notation only omits absolute constants.

Proof. By lemma A.3 and the fact that $K_{d}(\delta, \ell)$ is an increasing function of $\ell$ , we get

$$\begin{array}{c}{T}_{\mathrm{\ell }}\le \frac{8d}{{ϵ}_{\mathrm{\ell }}^2}\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)\\ \le 16d\frac{256e^{16R_{\mathrm{\ell }}}{K}_d^2(\delta ,L)+16R^2{e}^{8R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}^2}{\max }_{\mathrm{\ell }\in [L]}\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)\\ =d{\max }_{\mathrm{\ell }\in [L]}\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)O(\frac{e^{16R_{\mathrm{\ell }}}{K}_d^2(\delta ,L)}{R_{\mathrm{\ell }}^2}+\frac{R^2{e}^{8R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}^2})\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; T\; \_\; \{\backslash ell\}\; \backslash leq\; \backslash frac\; \{8\; d\}\{\backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; \backslash \backslash \; \backslash leq\; 1\; 6\; d\; \backslash frac\; \{2\; 5\; 6\; e\; ^\; \{1\; 6\; R\; \_\; \{\backslash ell\}\}\; K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; +\; 1\; 6\; R\; ^\; \{2\}\; e\; ^\; \{8\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\; ^\; \{2\}\}\; \backslash max\; \_\; \{\backslash ell\; \backslash in\; [\; L\; ]\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; \backslash \backslash \; =\; d\; \backslash max\; \_\; \{\backslash ell\; \backslash in\; [\; L\; ]\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; O\; \backslash left(\backslash frac\; \{e\; ^\; \{1\; 6\; R\; \_\; \{\backslash ell\}\}\; K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\}\{R\; \_\; \{\backslash ell\}\; ^\; \{2\}\}\; +\; \backslash frac\; \{R\; ^\; \{2\}\; e\; ^\; \{8\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\; ^\; \{2\}\}\backslash right).\; \backslash \backslash \; \backslash end\{array\}$$

where the second inequality uses $(a + b)^2 \leq 2a^2 + 2b^2$ .

For the terms within the big-oh, once we sum over $\ell$ we can write

$$\begin{array}{c}{\sum }_{\mathrm{\ell }=1}^L\frac{e^{16R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}^2}={\sum }_{R_{\mathrm{\ell }}>1}\frac{e^{16R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}^2}+{\sum }_{R_{\mathrm{\ell }}\le 1}\frac{e^{16R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}^2}\\ \le e^{8R}\lceil {\log }_{2}R\rceil +\frac{e^{16}}{R_L}{\sum }_{R_{\mathrm{\ell }}\le 1}\frac{1}{R_{\mathrm{\ell }}}\\ \le e^{8R}\lceil {\log }_{2}R\rceil +\frac{2e^{16}}{R_L^2}\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash frac\; \{e\; ^\; \{1\; 6\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\; ^\; \{2\}\}\; =\; \backslash sum\_\; \{R\; \_\; \{\backslash ell\}\; >\; 1\}\; \backslash frac\; \{e\; ^\; \{1\; 6\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\; ^\; \{2\}\}\; +\; \backslash sum\_\; \{R\; \_\; \{\backslash ell\}\; \backslash leq\; 1\}\; \backslash frac\; \{e\; ^\; \{1\; 6\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\; ^\; \{2\}\}\; \backslash \backslash \; \backslash leq\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; \backslash frac\; \{e\; ^\; \{1\; 6\}\}\{R\; \_\; \{L\}\}\; \backslash sum\_\; \{R\; \_\; \{\backslash ell\}\; \backslash leq\; 1\}\; \backslash frac\; \{1\}\{R\; \_\; \{\backslash ell\}\}\; \backslash \backslash \; \backslash leq\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; \backslash frac\; \{2\; e\; ^\; \{1\; 6\}\}\{R\; \_\; \{L\}\; ^\; \{2\}\}.\; \backslash \backslash \; \backslash end\{array\}$$

And similarly

$$\sum _{\mathrm{\ell }=1}^L\frac{e^{8R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}^2}\le e^{4R}\lceil {\log }_{2}R\rceil +\frac{2e^8}{R_L^2}\mathrm{.}\backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash frac\; \{e\; ^\; \{8\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\; ^\; \{2\}\}\; \backslash leq\; e\; ^\; \{4\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; \backslash frac\; \{2\; e\; ^\; \{8\}\}\{R\; \_\; \{L\}\; ^\; \{2\}\}.$$

Putting them together gives

$$N_T(\mathcal{P})=\sum _{\mathrm{\ell }=1}^L{T}_{\mathrm{\ell }}=d\underset{\mathrm{\ell }\in [L]}{\max }\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)O(K_d^2(\delta ,L)e^{8R}\lceil {\log }_{2}R\rceil +e^{4R}\lceil {\log }_{2}R\rceil R^2+\frac{K_d^2(\delta ,L)+R^2}{R_L^2}),N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; \backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; T\; \_\; \{\backslash ell\}\; =\; d\; \backslash max\; \_\; \{\backslash ell\; \backslash in\; [\; L\; ]\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; O\; \backslash left(K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; e\; ^\; \{4\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; R\; ^\; \{2\}\; +\; \backslash frac\; \{K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; +\; R\; ^\; \{2\}\}\{R\; \_\; \{L\}\; ^\; \{2\}\}\backslash right),$$

as claimed.

The following bound on the weighted cumulative regret is the logistic counterpart to Theorem A.5.

Theorem B.3. For any pool realization $\mathcal{P}$ , the weighted cumulative regret $R_{T}(\mathcal{P})$ of Algorithm 2 operating on a pool $\mathcal{P}$ of size $T$ is bounded as

$$R_T(\mathcal{P})=d\underset{\mathrm{\ell }\in [L]}{\max }\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)O(K_d^2(\delta ,L)e^{8R}\lceil {\log }_{2}R\rceil +e^{4R}\lceil {\log }_{2}R\rceil R^2+\frac{K_d^2(\delta ,L)+R^2}{R_L})\mathrm{.}R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; d\; \backslash max\; \_\; \{\backslash ell\; \backslash in\; [\; L\; ]\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; O\; \backslash left(K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; e\; ^\; \{4\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; R\; ^\; \{2\}\; +\; \backslash frac\; \{K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; +\; R\; ^\; \{2\}\}\{R\; \_\; \{L\}\}\backslash right).$$

assuming $\bigcap_{\ell = 1}^{L}\mathcal{E}_{\ell}$ holds.

Proof. We follow the same reasoning as in Theorem A.5. We decompose the pool $\mathcal{P}$ as the union of following disjoint sets

$$\mathcal{P}=\left({\cup }_{l=1}^L{\mathcal{C}}_{\mathrm{\ell }}\right)\cup \left({\cup }_{l=1}^L{\mathcal{Q}}_{\mathrm{\ell }}\right)\cup {\mathcal{P}}_L,\backslash mathcal\; \{P\}\; =\; \backslash left(\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{C\}\; \_\; \{\backslash ell\}\backslash right)\; \backslash cup\; \backslash left(\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\backslash right)\; \backslash cup\; \backslash mathcal\; \{P\}\; \_\; \{L\},$$

and study the weighted cumulative regret components

$$R_T({\cup }_{l=1}^L{\mathcal{C}}_{\mathrm{\ell }}),R_T({\cup }_{l=1}^L{\mathcal{Q}}_{\mathrm{\ell }}),R_T({\mathcal{P}}_L)\mathrm{.}R\; \_\; \{T\}\; (\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{C\}\; \_\; \{\backslash ell\}),\; R\; \_\; \{T\}\; (\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}),\; R\; \_\; \{T\}\; (\backslash mathcal\; \{P\}\; \_\; \{L\}).$$

Assume $\bigcap_{\ell = 1}^{L}\mathcal{E}{\ell}$ holds. First, notice that in $\mathcal{C}{\ell}$

$$\mathrm{sgn}⟨\widehat{\mathbf{w}},\mathbf{x}⟩=\mathrm{sgn}⟨{\mathbf{w}}_{\mathrm{\ell }},\mathbf{x}⟩=\mathrm{sgn}⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\backslash operatorname\; \{s\; g\; n\}\; \backslash langle\; \backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; =\; \backslash operatorname\; \{s\; g\; n\}\; \backslash langle\; \backslash mathbf\; \{w\}\; \_\; \{\backslash ell\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; =\; \backslash operatorname\; \{s\; g\; n\}\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle$$

under the assumption that $\mathcal{E}{\ell}$ holds, thus $\cup{\ell = 1}^{L}\mathcal{C}_{\ell}$ does not contribute weighted regret for $\widehat{\mathbf{w}}$ , i.e.,

$$R_T\left({\cup }_{\mathrm{\ell }=1}^L{\mathcal{C}}_{\mathrm{\ell }}\right)=0.R\; \_\; \{T\}\; \backslash left(\backslash cup\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{C\}\; \_\; \{\backslash ell\}\backslash right)\; =\; 0.$$

Next, on $\mathcal{P}_L$ , we have $|\langle \mathbf{w}_L,\mathbf{x}\rangle |\leq R_L$ . Combining this with the assumption that $\mathcal{E}_L^\sigma$ holds, we get $|\langle \mathbf{w}^*,\mathbf{x}\rangle |\leq 2R_L$ , which implies that the weighted cumulative regret on $\mathcal{P}_L$ is bounded as

the second inequality deriving from the stopping condition defining $L$ in Algorithm 2.

Finally, on the queried points $\cup_{l=1}^{L}\mathcal{Q}{\ell}$ , it is unclear whether $\mathrm{sgn}\langle \widehat{\mathbf{w}},\mathbf{x}\rangle = \mathrm{sgn}\langle \mathbf{w}^*,\mathbf{x}\rangle$ or not, so we bound the weighted cumulative regret contribution of each data item $\mathbf{x}$ therein by $|\langle \mathbf{w}^*,\mathbf{x}\rangle|$ . Now, by construction, $\mathbf{x} \in \mathcal{Q}{\ell} \subset \mathcal{P}{\ell-1}$ , so that $|\langle \mathbf{w}{\ell-1},\mathbf{x}\rangle| \leq R_{\ell-1}$ which, combined with the assumption that $\mathcal{E}{\ell-1}^{\sigma}$ holds, yields $|\langle \mathbf{w}^*,\mathbf{x}\rangle| \leq 2R{\ell-1}$ . Since $|\mathcal{Q}{\ell}| = T{\ell}$ , we have

$$R_T\left({\cup }_{\mathrm{\ell }=1}^L{\mathcal{Q}}_{\mathrm{\ell }}\right)\le 2\sum _{\mathrm{\ell }=1}^L{T}_{\mathrm{\ell }}{R}_{\mathrm{\ell }}R\; \_\; \{T\}\; \backslash left(\backslash cup\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\}\backslash right)\; \backslash leq\; 2\; \backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; T\; \_\; \{\backslash ell\}\; R\; \_\; \{\backslash ell\}$$

and Lemma A.3 allows us to write

$$R_T({\cup }_{l=1}^L{\mathcal{Q}}_{\mathrm{\ell }})\le 16d\sum _{l=1}^L\frac{R_{\mathrm{\ell }}}{{ϵ}_{\mathrm{\ell }}^2}\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)=d\underset{\mathrm{\ell }\in [L]}{\max }\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)O(\frac{e^{16R_{\mathrm{\ell }}}{K}_d^2(\delta ,L)}{R_{\mathrm{\ell }}}+\frac{R^2{e}^{8R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}})\mathrm{.}R\; \_\; \{T\}\; (\backslash cup\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{Q\}\; \_\; \{\backslash ell\})\; \backslash leq\; 1\; 6\; d\; \backslash sum\_\; \{l\; =\; 1\}\; ^\; \{L\}\; \backslash frac\; \{R\; \_\; \{\backslash ell\}\}\{\backslash epsilon\_\; \{\backslash ell\}\; ^\; \{2\}\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; =\; d\; \backslash max\; \_\; \{\backslash ell\; \backslash in\; [\; L\; ]\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; O\; \backslash left(\backslash frac\; \{e\; ^\; \{1\; 6\; R\; \_\; \{\backslash ell\}\}\; K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\}\{R\; \_\; \{\backslash ell\}\}\; +\; \backslash frac\; \{R\; ^\; \{2\}\; e\; ^\; \{8\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\}\backslash right).$$

Similar to the argument in theorem A.4, we have

$$\begin{array}{c}{\sum }_{\mathrm{\ell }=1}^L\frac{e^{16R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}}={\sum }_{R_{\mathrm{\ell }}>1}\frac{e^{16R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}}+{\sum }_{R_{\mathrm{\ell }}\le 1}\frac{e^{16R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}}\\ \le e^{8R}\lceil {\log }_{2}R\rceil +e^{16}{\sum }_{R_{\mathrm{\ell }}\le 1}\frac{1}{R_{\mathrm{\ell }}}\\ \le e^{8R}\lceil {\log }_{2}R\rceil +\frac{2e^{16}}{R_L}\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash frac\; \{e\; ^\; \{1\; 6\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\}\; =\; \backslash sum\_\; \{R\; \_\; \{\backslash ell\}\; >\; 1\}\; \backslash frac\; \{e\; ^\; \{1\; 6\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\}\; +\; \backslash sum\_\; \{R\; \_\; \{\backslash ell\}\; \backslash leq\; 1\}\; \backslash frac\; \{e\; ^\; \{1\; 6\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\}\; \backslash \backslash \; \backslash leq\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; e\; ^\; \{1\; 6\}\; \backslash sum\_\; \{R\; \_\; \{\backslash ell\}\; \backslash leq\; 1\}\; \backslash frac\; \{1\}\{R\; \_\; \{\backslash ell\}\}\; \backslash \backslash \; \backslash leq\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; \backslash frac\; \{2\; e\; ^\; \{1\; 6\}\}\{R\; \_\; \{L\}\}.\; \backslash \backslash \; \backslash end\{array\}$$

and

$$\sum _{\mathrm{\ell }=1}^L\frac{e^{8R_{\mathrm{\ell }}}}{R_{\mathrm{\ell }}}\le e^{4R}\lceil {\log }_{2}R\rceil +\frac{2e^8}{R_L}\mathrm{.}\backslash sum\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash frac\; \{e\; ^\; \{8\; R\; \_\; \{\backslash ell\}\}\}\{R\; \_\; \{\backslash ell\}\}\; \backslash leq\; e\; ^\; \{4\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; \backslash frac\; \{2\; e\; ^\; \{8\}\}\{R\; \_\; \{L\}\}.$$

Piecing together, we conclude that the total regret is bounded as

$$R_T(\mathcal{P})=d\underset{\mathrm{\ell }\in [L]}{\max }\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)O(K_d^2(\delta ,L)e^{8R}\lceil {\log }_{2}R\rceil +e^{4R}\lceil {\log }_{2}R\rceil R^2+\frac{K_d^2(\delta ,L)+R^2}{R_L}),R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; d\; \backslash max\; \_\; \{\backslash ell\; \backslash in\; [\; L\; ]\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; O\; \backslash left(K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; e\; ^\; \{4\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; R\; ^\; \{2\}\; +\; \backslash frac\; \{K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; +\; R\; ^\; \{2\}\}\{R\; \_\; \{L\}\}\backslash right),$$

thereby concluding the proof.

As in the linear case, adding the ingredient of the low noise condition (2) helps us exploit the randomness in $\mathcal{P}$ to further bound from above the number of stages $L$ in the logistic case.

Specifically, assume the low noise condition (2) holds for $f^{}(\mathbf{x}) = \sigma (\langle \mathbf{w}^{},\mathbf{x}\rangle)$ , for some unknown exponent $\alpha \geq 0$ and unknown constant $\epsilon_0\in (0,1]$ . Similar to linear case we define the event

Then

$$\begin{array}{cccc}& \mathbb{P}\left({\bar{\mathcal{E}}}^{\sigma }\right)\ge 1-\delta ,& & \text{(11)}\end{array}\backslash mathbb\; \{P\}\; \backslash left(\backslash bar\; \{\backslash mathcal\; \{E\}\}\; ^\; \{\backslash sigma\}\backslash right)\; \backslash geq\; 1\; -\; \backslash delta\; ,\; \backslash tag\; \{11\}$$

for $M = \log_2T$

Lemma B.4. Let $\epsilon_{*}$ be defined through (8), with $T > \frac{2}{3} d$ . Assume both $\bar{\mathcal{E}}^{\sigma}$ and $\bigcap_{\ell = 1}^{L}\mathcal{E}_{\ell}$ hold. Then the number of stages $L$ of Algorithm 2 is upper bounded as

$$\begin{array}{c}L\le \max ({\log }_2\left(\frac{R}{{ϵ}_{\ast }}\right),{\log }_2\left(\frac{R}{{ϵ}_0}\right))+2\\ \le \max ({\log }_2\left[R({\left(\frac{3T}{d}\right)}^{\frac{1}{\alpha +2}}+3{\left(\frac{1}{d}\right)}^{\frac{\alpha +1}{\alpha +2}}{\left(\frac{1}{3T}\right)}^{\frac{1}{\alpha +2}}\log (\frac{{\log }_{2}T}{\delta }))\right],{\log }_2\left(\frac{R}{{ϵ}_0}\right))+2\\ =\max (O(\frac{1}{\alpha +2}\log \left(\frac{T}{d}\right)+\log \left(\frac{R\log T}{\delta }\right)),{\log }_2\left(\frac{4R}{{ϵ}_0}\right)),\end{array}\backslash begin\{array\}\{l\}\; L\; \backslash leq\; \backslash max\; \backslash left(\backslash log\_\; \{2\}\; \backslash left(\backslash frac\; \{R\}\{\backslash epsilon\_\; \{*\}\}\backslash right),\; \backslash log\_\; \{2\}\; \backslash left(\backslash frac\; \{R\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; +\; 2\; \backslash \backslash \; \backslash leq\; \backslash max\; \backslash left(\backslash log\_\; \{2\}\; \backslash left[\; R\; \backslash left(\backslash left(\backslash frac\; \{3\; T\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; 3\; \backslash left(\backslash frac\; \{1\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; \backslash left(\backslash frac\; \{1\}\{3\; T\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; \backslash log\; (\backslash frac\; \{\backslash log\_\; \{2\}\; T\}\{\backslash delta\})\backslash right)\; \backslash right],\; \backslash log\_\; \{2\}\; \backslash left(\backslash frac\; \{R\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; +\; 2\; \backslash \backslash \; =\; \backslash max\; \backslash left(O\; \backslash left(\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{d\}\backslash right)\; +\; \backslash log\; \backslash left(\backslash frac\; \{R\; \backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right),\; \backslash log\_\; \{2\}\; \backslash left(\backslash frac\; \{4\; R\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right),\; \backslash \backslash \; \backslash end\{array\}$$

where the $O$ -notation only hides absolute constants.

Proof. If at stage $L - 1$ the algorithm has not stopped, then we must have

$$d\mathrm{/}2R_{L-1}\le 2R_{L-1}\mathrm{\mid }{\mathcal{P}}_{L-1}\mathrm{\mid }\mathrm{.}d\; /\; 2\; R\; \_\; \{L\; -\; 1\}\; \backslash leq\; 2\; R\; \_\; \{L\; -\; 1\}\; |\; \backslash mathcal\; \{P\}\; \_\; \{L\; -\; 1\}\; |.$$

Notice that if $\mathbf{x} \in \mathcal{P}{L-1}$ then $|\langle \mathbf{w}{L-1}, \mathbf{x} \rangle| \leq R_{L-1}$ . Combining it with the assumption that $\mathcal{E}{L-1}$ holds, we have $|\langle \mathbf{w}^*, \mathbf{x} \rangle| \leq 2R{L-1}$ , which implies $|\mathcal{P}{L-1}| \leq |\mathcal{T}{\tanh(R_{L-1})}^\sigma| \leq |\mathcal{T}{2R{L-1}}^\sigma|$ .

We split the analysis into two cases. On one hand, when $2R_{L-1} > \epsilon_0$ , this condition gives us directly

$$L\le {\log }_2\left(\frac{R}{{ϵ}_0}\right)+2.L\; \backslash leq\; \backslash log\_\; \{2\}\; \backslash left(\backslash frac\; \{R\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\; +\; 2.$$

On the other hand if $2R_{L - 1}\leq \epsilon_0$ , then given that $\bar{\mathcal{E}}^{\sigma}$ holds, $|\mathcal{T}{2R{L - 1}}^{\sigma}|$ is upper bounded as

$$\mid {\mathcal{T}}_{2R_{L-1}}^{\sigma }\mid \le 3(T(2R_{L-1}{)}^{\alpha }+\log \left(\frac{M}{\delta }\right)),\backslash left|\; \backslash mathcal\; \{T\}\; \_\; \{2\; R\; \_\; \{L\; -\; 1\}\}\; ^\; \{\backslash sigma\}\; \backslash right|\; \backslash leq\; 3\; \backslash left(T\; (2\; R\; \_\; \{L\; -\; 1\})\; ^\; \{\backslash alpha\}\; +\; \backslash log\; \backslash left(\backslash frac\; \{M\}\{\backslash delta\}\backslash right)\backslash right),$$

with $M = \log_2 T$ . Plugging into the first display results in

$$d\mathrm{/}2R_{L-1}\le 3(T(2R_{L-1}{)}^{\alpha +1}+2R_{L-1}\log \left(\frac{M}{\delta }\right)),d\; /\; 2\; R\; \_\; \{L\; -\; 1\}\; \backslash leq\; 3\; \backslash left(T\; (2\; R\; \_\; \{L\; -\; 1\})\; ^\; \{\backslash alpha\; +\; 1\}\; +\; 2\; R\; \_\; \{L\; -\; 1\}\; \backslash log\; \backslash left(\backslash frac\; \{M\}\{\backslash delta\}\backslash right)\backslash right),$$

which resembles (8) with $2R_{L-1}$ here playing the role of $\epsilon^{}$ therein. Then, from the definition of $\epsilon^{}$ in (8) we immediately obtain $2R_{L-1} \geq \epsilon_{}$ , thus $L \leq \log_2\left(\frac{R}{\epsilon_{}}\right) + 2$ . Moreover, from (8) we see that $d / \epsilon_{} \geq 3T\epsilon_{}^{\alpha+1}$ , which is equivalent to $\epsilon_{} \leq \left(\frac{d}{3T}\right)^{\frac{1}{\alpha+2}}$ . Replacing this upper bound on $\epsilon^{}$ back into the right-hand side of (8), dividing by $d$ and multiply by $R$ yields

$$\frac{R}{{ϵ}_{\ast }}\le R({\left(\frac{3T}{d}\right)}^{\frac{1}{\alpha +2}}+3{\left(\frac{1}{d}\right)}^{\frac{\alpha +1}{\alpha +2}}{\left(\frac{1}{3T}\right)}^{\frac{1}{\alpha +2}}\log (\frac{M}{\delta })),\backslash frac\; \{R\}\{\backslash epsilon\_\; \{*\}\}\; \backslash leq\; R\; \backslash left(\backslash left(\backslash frac\; \{3\; T\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; 3\; \backslash left(\backslash frac\; \{1\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; \backslash left(\backslash frac\; \{1\}\{3\; T\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; \backslash log\; (\backslash frac\; \{M\}\{\backslash delta\})\backslash right),$$

which gives the claimed upper bound on $L$ through $L \leq \log_2\left(\frac{R}{\epsilon_*}\right) + 2$ .

Corollary B.5. Let $T > \frac{2}{3} d$ . Then with probability at least $1 - 2\delta$ over the random draw of $(\mathbf{x}_1, y_1), \ldots, (\mathbf{x}_T, y_T) \sim \mathcal{D}$ the label complexity $N_T(\mathcal{P})$ and the weighted cumulative regret $R_T(\mathcal{P})$ of Algorithm 1 simultaneously satisfy the following:

$$N_T(\mathcal{P})=C_{d,R}(\delta ,T,{ϵ}_0)O(\max \{d^{\frac{\alpha }{\alpha +2}}{T}^{\frac{2}{\alpha +2}},\frac{d}{{ϵ}_0^2}\}+{\log }^2\left(\frac{\log T}{\delta }\right)+d{e}^{8R}\lceil {\log }_{2}R\rceil ),N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; C\; \_\; \{d,\; R\}\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; O\; \backslash left(\backslash max\; \backslash left\backslash \{d\; ^\; \{\backslash frac\; \{\backslash alpha\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\; ^\; \{2\}\}\; \backslash right\backslash \}\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\; +\; d\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\backslash right),$$

$$R_T(\mathcal{P})=C_{d,R}(\delta ,T,{ϵ}_0)O(\max \{d^{\frac{\alpha +1}{\alpha +2}}{T}^{\frac{1}{\alpha +2}},\frac{d}{{ϵ}_0}\}+\log \left(\frac{\log T}{\delta }\right)+d{e}^{8R}\lceil {\log }_{2}R\rceil ),R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; C\; \_\; \{d,\; R\}\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; O\; \backslash left(\backslash max\; \backslash left\backslash \{d\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\}\; \backslash right\backslash \}\; +\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\; +\; d\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\backslash right),$$

where the $O$ -notation hides absolute constants and

$$C_{d,R}(\delta ,T,{ϵ}_0)=(1+{\log }^2\left(\frac{1}{{ϵ}_0}\right))(d\log \left(\frac{T}{\delta }\right)+R^2)(R+\log \left(\frac{T}{\delta }\right))\mathrm{.}C\; \_\; \{d,\; R\}\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; =\; \backslash left(1\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; \backslash left(d\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; +\; R\; ^\; \{2\}\backslash right)\; \backslash left(R\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\backslash right).$$

Proof. Assume both $\bar{\mathcal{E}}^{\sigma}$ and $\bigcap_{\ell = 1}^{L}\mathcal{E}{\ell}$ hold. Recalling the definition of $K{d}(\delta ,\ell)$ , we have

$$K_d(\delta ,L)=O\left(\sqrt{d\log \left(\frac{d}{\delta }\right)+L}\right)\mathrm{.}K\; \_\; \{d\}\; (\backslash delta\; ,\; L)\; =\; O\; \backslash left(\backslash sqrt\; \{d\; \backslash log\; \backslash left(\backslash frac\; \{d\}\{\backslash delta\}\backslash right)\; +\; L\}\backslash right).$$

Similar to Lemma B.4, we split the analysis into two cases depending on whether or not $2R_{L-1}$ is bigger than $\epsilon_0$ . If $2R_{L-1} \leq \epsilon_0$ , we have

$$L\le {\log }_2\left[R({\left(\frac{3T}{d}\right)}^{\frac{1}{\alpha +2}}+3{\left(\frac{1}{d}\right)}^{\frac{\alpha +1}{\alpha +2}}{\left(\frac{1}{3T}\right)}^{\frac{1}{\alpha +2}}\log \left(\frac{{\log }_{2}T}{\delta }\right))\right],L\; \backslash leq\; \backslash log\_\; \{2\}\; \backslash left[\; R\; \backslash left(\backslash left(\backslash frac\; \{3\; T\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; 3\; \backslash left(\backslash frac\; \{1\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; \backslash left(\backslash frac\; \{1\}\{3\; T\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\_\; \{2\}\; T\}\{\backslash delta\}\backslash right)\backslash right)\; \backslash right],$$

therefore

$$\frac{1}{R_L}=O({\left(\frac{T}{d}\right)}^{\frac{1}{\alpha +2}}+{\left(\frac{1}{d}\right)}^{\frac{\alpha +1}{\alpha +2}}{\left(\frac{1}{T}\right)}^{\frac{1}{\alpha +2}}\log \left(\frac{\log T}{\delta }\right))\mathrm{.}\backslash frac\; \{1\}\{R\; \_\; \{L\}\}\; =\; O\; \backslash left(\backslash left(\backslash frac\; \{T\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash left(\backslash frac\; \{1\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; \backslash left(\backslash frac\; \{1\}\{T\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right).$$

Moreover, we have

$$K_d(\delta ,L)=O(\sqrt{d\log \left(\frac{d}{\delta }\right)}+\sqrt{\log \left(\frac{T}{d}\right)}+\sqrt{\log \left(\frac{R\log T}{\delta }\right)})\mathrm{.}K\; \_\; \{d\}\; (\backslash delta\; ,\; L)\; =\; O\; \backslash left(\backslash sqrt\; \{d\; \backslash log\; \backslash left(\backslash frac\; \{d\}\{\backslash delta\}\backslash right)\}\; +\; \backslash sqrt\; \{\backslash log\; \backslash left(\backslash frac\; \{T\}\{d\}\backslash right)\}\; +\; \backslash sqrt\; \{\backslash log\; \backslash left(\backslash frac\; \{R\; \backslash log\; T\}\{\backslash delta\}\backslash right)\}\backslash right).$$

and

$$\begin{array}{c}{\max }_{\mathrm{\ell }\in [L]}\log \left(\frac{1}{{ϵ}_{\mathrm{\ell }}}\right)\le \log \left(\frac{16e^{4R}{K}_d(\delta ,L)+4R{e}^{2R}}{R_L}\right)\\ =O(R+\log K_d(\delta ,L)+\log \left(\frac{T}{d}\right)+\log \left(\frac{R\log T}{\delta }\right))\\ =O(R+\log \left(\frac{T}{\delta }\right)),\end{array}\backslash begin\{array\}\{l\}\; \backslash max\; \_\; \{\backslash ell\; \backslash in\; [\; L\; ]\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{\backslash ell\}\}\backslash right)\; \backslash leq\; \backslash log\; \backslash left(\backslash frac\; \{1\; 6\; e\; ^\; \{4\; R\}\; K\; \_\; \{d\}\; (\backslash delta\; ,\; L)\; +\; 4\; R\; e\; ^\; \{2\; R\}\}\{R\; \_\; \{L\}\}\backslash right)\; \backslash \backslash \; =\; O\; \backslash left(R\; +\; \backslash log\; K\; \_\; \{d\}\; (\backslash delta\; ,\; L)\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{d\}\backslash right)\; +\; \backslash log\; \backslash left(\backslash frac\; \{R\; \backslash log\; T\}\{\backslash delta\}\backslash right)\backslash right)\; \backslash \backslash \; =\; O\; \backslash left(R\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\backslash right),\; \backslash \backslash \; \backslash end\{array\}$$

where the last equality is because $T > \frac{2}{3} d$ .

Plugging these bounds together back into factor

$$K_d^2(\delta ,L)e^{8R}\lceil {\log }_{2}R\rceil +e^{4R}\lceil {\log }_{2}R\rceil R^2+\frac{K_d^2(\delta ,L)+R^2}{R_L^2}K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; e\; ^\; \{4\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; R\; ^\; \{2\}\; +\; \backslash frac\; \{K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; +\; R\; ^\; \{2\}\}\{R\; \_\; \{L\}\; ^\; \{2\}\}$$

of Theorem B.2 yields

$$\begin{array}{c}{K}_d^2(\delta ,L)e^{8R}\lceil {\log }_{2}R\rceil +e^{4R}\lceil {\log }_{2}R\rceil R^2+\frac{K_d^2(\delta ,L)+R^2}{R_L^2}\\ =O\left((K_d^2(\delta ,L)+R^2)(e^{8R}\lceil {\log }_{2}R\rceil +\frac{1}{R_L^2})\right)\\ =O\left((d\log \left(\frac{T}{\delta }\right)+R^2)({\left(\frac{T}{d}\right)}^{\frac{2}{\alpha +2}}+{\left(\frac{1}{d}\right)}^{\frac{2\alpha +2}{\alpha +2}}{\left(\frac{1}{T}\right)}^{\frac{2}{\alpha +2}}{\log }^2\left(\frac{\log T}{\delta }\right)+e^{8R}\lceil {\log }_{2}R\rceil )\right),\end{array}\backslash begin\{array\}\{l\}\; K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; e\; ^\; \{4\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; R\; ^\; \{2\}\; +\; \backslash frac\; \{K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; +\; R\; ^\; \{2\}\}\{R\; \_\; \{L\}\; ^\; \{2\}\}\; \backslash \backslash \; =\; O\; \backslash left(\backslash left(K\; \_\; \{d\}\; ^\; \{2\}\; (\backslash delta\; ,\; L)\; +\; R\; ^\; \{2\}\backslash right)\; \backslash left(e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\; +\; \backslash frac\; \{1\}\{R\; \_\; \{L\}\; ^\; \{2\}\}\backslash right)\backslash right)\; \backslash \backslash \; =\; O\; \backslash left(\backslash left(d\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; +\; R\; ^\; \{2\}\backslash right)\; \backslash left(\backslash left(\backslash frac\; \{T\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash left(\backslash frac\; \{1\}\{d\}\backslash right)\; ^\; \{\backslash frac\; \{2\; \backslash alpha\; +\; 2\}\{\backslash alpha\; +\; 2\}\}\; \backslash left(\backslash frac\; \{1\}\{T\}\backslash right)\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\}\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\; +\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\backslash right)\backslash right),\; \backslash \backslash \; \backslash end\{array\}$$

where the last equality is due to the assumption that $T > \frac{2}{3} d$ . Combining the above estimates gives

$$N_T(\mathcal{P})=O\left((d\log \left(\frac{T}{\delta }\right)+R^2)(R+\log \left(\frac{T}{\delta }\right))(d^{\frac{\alpha }{\alpha +2}}{T}^{\frac{2}{\alpha +2}}+{\log }^2\left(\frac{\log T}{\delta }\right)+d{e}^{8R}\lceil {\log }_{2}R\rceil )\right)N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; O\; \backslash left(\backslash left(d\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; +\; R\; ^\; \{2\}\backslash right)\; \backslash left(R\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\backslash right)\; \backslash left(d\; ^\; \{\backslash frac\; \{\backslash alpha\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{2\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\; +\; d\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\backslash right)\backslash right)$$

A similar argument gives

$$R_T(\mathcal{P})=O\left((d\log \left(\frac{T}{\delta }\right)+R^2)(R+\log \left(\frac{T}{\delta }\right))(d^{\frac{\alpha +1}{\alpha +2}}{T}^{\frac{1}{\alpha +2}}+\log \left(\frac{\log T}{\delta }\right)+d{e}^{8R}\lceil {\log }_{2}R\rceil )\right)\mathrm{.}R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; O\; \backslash left(\backslash left(d\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; +\; R\; ^\; \{2\}\backslash right)\; \backslash left(R\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\backslash right)\; \backslash left(d\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\}\; T\; ^\; \{\backslash frac\; \{1\}\{\backslash alpha\; +\; 2\}\}\; +\; \backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\; +\; d\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\backslash right)\backslash right).$$

If $2R_{L-1} > \epsilon_0$ , then $\frac{1}{R_L} \leq \frac{4}{\epsilon_0}$ . Applying these bounds into Theorem B.2 we get

$$\begin{array}{c}{N}_T(\mathcal{P})=O\left((d\log \left(\frac{T}{\delta }\right)+R^2+\log \left(\frac{1}{{ϵ}_0}\right))(R+\log \left(\frac{T}{\delta {ϵ}_0}\right))(\frac{d}{{ϵ}_0^2}+d{e}^{8R}\lceil {\log }_{2}R\rceil )\right)\\ =(d\log \left(\frac{T}{\delta }\right)+R^2)(R+\log \left(\frac{T}{\delta }\right))(1+{\log }^2\left(\frac{1}{{ϵ}_0}\right))O(\frac{d}{{ϵ}_0^2}+d{e}^{8R}\lceil {\log }_{2}R\rceil )\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; N\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; O\; \backslash left(\backslash left(d\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; +\; R\; ^\; \{2\}\; +\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; \backslash left(R\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\; \backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; \backslash left(\backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\; ^\; \{2\}\}\; +\; d\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\backslash right)\backslash right)\; \backslash \backslash \; =\; \backslash left(d\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; +\; R\; ^\; \{2\}\backslash right)\; \backslash left(R\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\backslash right)\; \backslash left(1\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; O\; \backslash left(\backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\; ^\; \{2\}\}\; +\; d\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\backslash right).\; \backslash \backslash \; \backslash end\{array\}$$

Similarly

$$R_T(\mathcal{P})=(d\log \left(\frac{T}{\delta }\right)+R^2)(R+\log \left(\frac{T}{\delta }\right))(1+{\log }^2\left(\frac{1}{{ϵ}_0}\right))O(\frac{d}{{ϵ}_0}+d{e}^{8R}\lceil {\log }_{2}R\rceil )\mathrm{.}R\; \_\; \{T\}\; (\backslash mathcal\; \{P\})\; =\; \backslash left(d\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\; +\; R\; ^\; \{2\}\backslash right)\; \backslash left(R\; +\; \backslash log\; \backslash left(\backslash frac\; \{T\}\{\backslash delta\}\backslash right)\backslash right)\; \backslash left(1\; +\; \backslash log^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\_\; \{0\}\}\backslash right)\backslash right)\; O\; \backslash left(\backslash frac\; \{d\}\{\backslash epsilon\_\; \{0\}\}\; +\; d\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\backslash right).$$

Lastly, (11) and Lemma B.1 together yield

$$\mathbb{P}({\bar{\mathcal{E}}}^{\sigma }\cap \left(\bigcap _{\mathrm{\ell }=1}^L{\mathcal{E}}_{\mathrm{\ell }}\right))\ge 1-2\delta ,\backslash mathbb\; \{P\}\; \backslash left(\backslash bar\; \{\backslash mathcal\; \{E\}\}\; ^\; \{\backslash sigma\}\; \backslash cap\; \backslash left(\backslash bigcap\_\; \{\backslash ell\; =\; 1\}\; ^\; \{L\}\; \backslash mathcal\; \{E\}\; \_\; \{\backslash ell\}\backslash right)\backslash right)\; \backslash geq\; 1\; -\; 2\; \backslash delta\; ,$$

which concludes the proof.

We now turn the bound on the weighted cumulative regret $R_{T}(\mathcal{P})$ in the previous corollary into a bound on the excess risk. As in the linear case, we have

$$L(w^)−L(w∗)=E(x,y)∼D[1{y≠sgn(⟨w^,x⟩)}−1{y≠sgn(⟨w∗,x⟩)}]=Ex∼DX[Ey∼DY∣X[1{y≠sgn(⟨w^,x⟩)}−1{y≠sgn(⟨w∗,x⟩)}]]=Ex∼DX[1{sgn(⟨w^,x⟩)≠sgn(⟨w∗,x⟩)}∣2σ(⟨w∗,x⟩)−1∣], \begin{array}{l} \mathcal {L} (\widehat {\mathbf {w}}) - \mathcal {L} (\mathbf {w} ^ {*}) = \mathbb {E} _ {(\mathbf {x}, y) \sim \mathcal {D}} \Big [ \mathbb {1} \{y \neq \mathrm {s g n} (\langle \widehat {\mathbf {w}}, \mathbf {x} \rangle) \} - \mathbb {1} \{y \neq \mathrm {s g n} (\langle \mathbf {w} ^ {*}, \mathbf {x} \rangle) \} \Big ] \\ = \mathbb {E} _ {\mathbf {x} \sim \mathcal {D} _ {\mathcal {X}}} \left[ \mathbb {E} _ {y \sim \mathcal {D} _ {\mathcal {Y} | \mathcal {X}}} \left[ \mathbb {1} \{y \neq \mathrm {s g n} (\langle \widehat {\mathbf {w}}, \mathbf {x} \rangle) \} - \mathbb {1} \{y \neq \mathrm {s g n} (\langle \mathbf {w} ^ {*}, \mathbf {x} \rangle) \} \right] \right] \\ = \mathbb {E} _ {\mathbf {x} \sim \mathcal {D} _ {\mathcal {X}}} \left[ \mathbb {1} \{\mathrm {s g n} (\langle \widehat {\mathbf {w}}, \mathbf {x} \rangle) \neq \mathrm {s g n} (\langle \mathbf {w} ^ {*}, \mathbf {x} \rangle) \} | 2 \sigma (\langle \mathbf {w} ^ {*}, \mathbf {x} \rangle) - 1 | \right], \\ \end{array} $$

where $\widehat{\mathbf{w}}$ is the hypothesis returned by Algorithm 2. Now, simply observe that

$$\mathbb{1}\{\mathrm{s}\mathrm{g}\mathrm{n}(⟨\widehat{\mathbf{w}},\mathbf{x}⟩)\mathrm{\ne }\mathrm{s}\mathrm{g}\mathrm{n}(⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩)\}\mathrm{\mid }2\sigma (⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩)-1\mathrm{\mid }\backslash mathbb\; \{1\}\; \backslash \{\backslash mathrm\; \{s\; g\; n\}\; (\backslash langle\; \backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \backslash rangle)\; \backslash neq\; \backslash mathrm\; \{s\; g\; n\}\; (\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle)\; \backslash \}\; |\; 2\; \backslash sigma\; (\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle)\; -\; 1\; |$$

has the same form as the function $\phi (\widehat{\mathbf{w}},\mathbf{x})$ in Appendix C on which Theorem C.4 applies, with $\widehat{\epsilon} (\delta)$ therein replaced by the bound on $R_{T}(\mathcal{P})$ deriving from Corollary B.5. This allows us to conclude that with probability at least $1 - \delta$

$$\mathcal{L}(\widehat{\mathbf{w}})-\mathcal{L}({\mathbf{w}}^{\ast })=C_{d,R}(\delta ,T,{ϵ}_0)O(\max \{{\left(\frac{d}{T}\right)}^{\frac{\alpha +1}{\alpha +2}},\frac{d}{T{ϵ}_0}\}+\frac{\log \left(\frac{\log T}{\delta }\right)+d{e}^{8R}\lceil {\log }_{2}R\rceil }{T}),\backslash mathcal\; \{L\}\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\})\; -\; \backslash mathcal\; \{L\}\; (\backslash mathbf\; \{w\}\; ^\; \{*\})\; =\; C\; \_\; \{d,\; R\}\; (\backslash delta\; ,\; T,\; \backslash epsilon\_\; \{0\})\; O\; \backslash left(\backslash max\; \backslash left\backslash \{\backslash left(\backslash frac\; \{d\}\{T\}\backslash right)\; ^\; \{\backslash frac\; \{\backslash alpha\; +\; 1\}\{\backslash alpha\; +\; 2\}\},\; \backslash frac\; \{d\}\{T\; \backslash epsilon\_\; \{0\}\}\; \backslash right\backslash \}\; +\; \backslash frac\; \{\backslash log\; \backslash left(\backslash frac\; \{\backslash log\; T\}\{\backslash delta\}\backslash right)\; +\; d\; e\; ^\; \{8\; R\}\; \backslash lceil\; \backslash log\_\; \{2\}\; R\; \backslash rceil\}\{T\}\backslash right),$$

as claimed in Theorem 5.1 in the main body of the paper.

C. Ancillary Technical Results

This section collects ancillary technical results that are used throughout the appendix.

We first recall the following version of the Hoeffding's bound.

Lemma C.1. Let $a_1, \ldots, a_T$ be $T$ arbitrary real numbers, and ${\sigma_1, \ldots, \sigma_T}$ be $T$ i.i.d. Rademacher variables, each taking values $\pm 1$ with equal probability. Then for any $\epsilon \geq 0$

$$\mathbb{P}(\sum _{t=1}^T{\sigma }_t{a}_t\ge ϵ)\le \exp (-\frac{{ϵ}^2}{2\sum _{t=1}^T{a}_t^2}),\backslash mathbb\; \{P\}\; \backslash left(\backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash sigma\_\; \{t\}\; a\; \_\; \{t\}\; \backslash geq\; \backslash epsilon\backslash right)\; \backslash leq\; \backslash exp\; \backslash left(-\; \backslash frac\; \{\backslash epsilon^\; \{2\}\}\{2\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; a\; \_\; \{t\}\; ^\; \{2\}\}\backslash right),$$

where the probability is with respect to ${\sigma_1,\dots ,\sigma_T}$

Let us consider the linear case first. Define the function $\phi : \mathbb{R}^d \times \mathcal{P} \to [0,1]$ as

$$\varphi (\widehat{\mathbf{w}},\mathbf{x})=\mathbb{1}\{\mathrm{sgn}⟨\widehat{\mathbf{w}},\mathbf{x}⟩\mathrm{\ne }\mathrm{sgn}⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩\}\rho (⟨{\mathbf{w}}^{\ast },\mathbf{x}⟩),\backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\})\; =\; \backslash mathbb\; \{1\}\; \backslash \{\backslash operatorname\; \{s\; g\; n\}\; \backslash langle\; \backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash neq\; \backslash operatorname\; \{s\; g\; n\}\; \backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle\; \backslash \}\; \backslash rho\; (\backslash langle\; \backslash mathbf\; \{w\}\; ^\; \{*\},\; \backslash mathbf\; \{x\}\; \backslash rangle),$$

where $\rho (\cdot)$ has range in [0, 1], and does not depend on $\widehat{\mathbf{w}}$ . We have the following standard covering result, which is a direct consequence of Sauer-Shelah lemma (e.g., (Sauer, 1972)).

Lemma C.2. Consider any given $S_{T} = {\mathbf{x}{1},\dots ,\mathbf{x}{T}} \in \mathbb{R}^{d}$ , and let

$$\mathrm{\Phi }(S_T)=\mid \{[\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_1),\dots ,\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_T)]:\widehat{\mathbf{w}}\in {\mathbb{R}}^d\}\mid \mathrm{.}\backslash Phi\; (S\; \_\; \{T\})\; =\; \backslash left|\; \backslash left\backslash \{\backslash left[\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \_\; \{1\}),\; \backslash dots\; ,\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \_\; \{T\})\; \backslash right]:\; \backslash widehat\; \{\backslash mathbf\; \{w\}\}\; \backslash in\; \backslash mathbb\; \{R\}\; ^\; \{d\}\; \backslash right\backslash \}\; \backslash right|.$$

We have, when $T \geq d$ ,

$$\mathrm{\Phi }(S_T)\le {\left(\frac{eT}{d}\right)}^d\mathrm{.}\backslash Phi\; (S\; \_\; \{T\})\; \backslash leq\; \backslash left(\backslash frac\; \{e\; T\}\{d\}\backslash right)\; ^\; \{d\}.$$

The next result follows from a standard symmetrization argument.

Lemma C.3. Let $\mathcal{X} = \mathbb{R}^d$ , $S_T = (\mathbf{x}_1, \ldots, \mathbf{x}T)$ be a sample drawn i.i.d. according to $\mathcal{D}{\mathcal{X}}$ and $S_T' = (\mathbf{x}_1', \ldots, \mathbf{x}T')$ be another sample drawn according to $\mathcal{D}{\mathcal{X}}$ , with $T \geq d$ . Then with probability at least $1 - \delta$

$$\sum _{t=1}^T\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_t^{\mathrm{\prime }})\le 3\sum _{t=1}^T\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_t)+8\log (1\mathrm{/}\delta )+8d\log (2eT\mathrm{/}d),\backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; ^\; \{\backslash prime\})\; \backslash leq\; 3\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \_\; \{t\})\; +\; 8\; \backslash log\; (1\; /\; \backslash delta)\; +\; 8\; d\; \backslash log\; (2\; e\; T\; /\; d),$$

uniformly over $\widehat{\mathbf{w}}\in \mathbb{R}^d$

Proof. Let ${\sigma_1, \ldots, \sigma_T}$ be independent Rademacher variables as in Lemma C.1. We can write, for any $\epsilon \geq 0$

$$P(∃w^∈Rd:∑t=1Tϕ(w^,xt′)≥3∑t=1Tϕ(w^,xt)+2ϵ)=P(∃w^∈Rd:∑t=1T[ϕ(w^,xt′)−ϕ(w^,xt)]≥12∑t=1T[ϕ(w^,xt)+ϕ(w^,xt′)]+ϵ)=P(∃w^∈Rd:∑t=1Tσt[ϕ(w^,xt′)−ϕ(w^,xt)]≥12∑t=1T[ϕ(w^,xt)+ϕ(w^,xt′)]+ϵ)≤P(∃w^∈Rd:∑t=1Tσtϕ(w^,xt′)≥12∑t=1Tϕ(w^,xt′)+ϵ2)+P(∃w^∈Rd:∑t=1T−σtϕ(w^,xt)≥12∑t=1Tϕ(w^,xt)+ϵ2)≤2sup⁡STP(∃w^∈Rd:∑t=1Tσtϕ(w^,xt)≥12∑t=1Tϕ(w^,xt)+12ϵ∣ST)≤2(eTd)dsup⁡ST,w^∈RdP(∑t=1Tσtϕ(w^,xt)≥12∑t=1Tϕ(w^,xt)+12ϵ∣ST) \begin{array}{l} \mathbb {P} \left(\exists \widehat {\mathbf {w}} \in \mathbb {R} ^ {d}: \sum_ {t = 1} ^ {T} \phi \left(\widehat {\mathbf {w}}, \mathbf {x} _ {t} ^ {\prime}\right) \geq 3 \sum_ {t = 1} ^ {T} \phi \left(\widehat {\mathbf {w}}, \mathbf {x} _ {t}\right) + 2 \epsilon\right) \\ = \mathbb {P} \left(\exists \widehat {\mathbf {w}} \in \mathbb {R} ^ {d}: \sum_ {t = 1} ^ {T} \left[ \phi \left(\widehat {\mathbf {w}}, \mathbf {x} _ {t} ^ {\prime}\right) - \phi \left(\widehat {\mathbf {w}}, \mathbf {x} _ {t}\right) \right] \geq \frac {1}{2} \sum_ {t = 1} ^ {T} \left[ \phi \left(\widehat {\mathbf {w}}, \mathbf {x} _ {t}\right) + \phi \left(\widehat {\mathbf {w}}, \mathbf {x} _ {t} ^ {\prime}\right) \right] + \epsilon\right) \\ = \mathbb {P} \left(\exists \widehat {\mathbf {w}} \in \mathbb {R} ^ {d}: \sum_ {t = 1} ^ {T} \sigma_ {t} \left[ \phi \left(\widehat {\mathbf {w}}, \mathbf {x} _ {t} ^ {\prime}\right) - \phi \left(\widehat {\mathbf {w}}, \mathbf {x} _ {t}\right) \right] \geq \frac {1}{2} \sum_ {t = 1} ^ {T} \left[ \phi \left(\widehat {\mathbf {w}}, \mathbf {x} _ {t}\right) + \phi \left(\widehat {\mathbf {w}}, \mathbf {x} _ {t} ^ {\prime}\right) \right] + \epsilon\right) \\ \leq \mathbb {P} \left(\exists \widehat {\mathbf {w}} \in \mathbb {R} ^ {d}: \sum_ {t = 1} ^ {T} \sigma_ {t} \phi (\widehat {\mathbf {w}}, \mathbf {x} _ {t} ^ {\prime}) \geq \frac {1}{2} \sum_ {t = 1} ^ {T} \phi (\widehat {\mathbf {w}}, \mathbf {x} _ {t} ^ {\prime}) + \frac {\epsilon}{2}\right) \\ + \mathbb {P} \left(\exists \widehat {\mathbf {w}} \in \mathbb {R} ^ {d}: \sum_ {t = 1} ^ {T} - \sigma_ {t} \phi (\widehat {\mathbf {w}}, \mathbf {x} _ {t}) \geq \frac {1}{2} \sum_ {t = 1} ^ {T} \phi (\widehat {\mathbf {w}}, \mathbf {x} _ {t}) + \frac {\epsilon}{2}\right) \\ \leq 2 \sup _ {S _ {T}} \mathbb {P} \left(\exists \widehat {\mathbf {w}} \in \mathbb {R} ^ {d}: \sum_ {t = 1} ^ {T} \sigma_ {t} \phi (\widehat {\mathbf {w}}, \mathbf {x} _ {t}) \geq \frac {1}{2} \sum_ {t = 1} ^ {T} \phi (\widehat {\mathbf {w}}, \mathbf {x} _ {t}) + \frac {1}{2} \epsilon \mid S _ {T}\right) \\ \leq 2 \left(\frac {e T}{d}\right) ^ {d} \sup _ {S _ {T}, \widehat {\mathbf {w}} \in \mathbb {R} ^ {d}} \mathbb {P} \left(\sum_ {t = 1} ^ {T} \sigma_ {t} \phi (\widehat {\mathbf {w}}, \mathbf {x} _ {t}) \geq \frac {1}{2} \sum_ {t = 1} ^ {T} \phi (\widehat {\mathbf {w}}, \mathbf {x} _ {t}) + \frac {1}{2} \epsilon \mid S _ {T}\right) \\ \end{array} $$

(from the union bound and Lemma C.2)

$$\le 2{\left(\frac{eT}{d}\right)}^d\underset{S_T,\widehat{\mathbf{w}}\in {\mathbb{R}}^d}{\sup }\exp (-\frac{{(\sum _{t=1}^T\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_t)+ϵ)}^2}{8\sum _{t=1}^T\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_t{)}^2})\backslash leq\; 2\; \backslash left(\backslash frac\; \{e\; T\}\{d\}\backslash right)\; ^\; \{d\}\; \backslash sup\; \_\; \{S\; \_\; \{T\},\; \backslash widehat\; \{\backslash mathbf\; \{w\}\}\; \backslash in\; \backslash mathbb\; \{R\}\; ^\; \{d\}\}\; \backslash exp\; \backslash left(-\; \backslash frac\; \{\backslash left(\backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\}\; ,\; \backslash mathbf\; \{x\}\; \_\; \{t\})\; +\; \backslash epsilon\backslash right)\; ^\; \{2\}\}\{8\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\}\; ,\; \backslash mathbf\; \{x\}\; \_\; \{t\})\; ^\; \{2\}\}\backslash right)$$

(from Lemma C.1)

$$\le 2{\left(\frac{eT}{d}\right)}^d\exp (-ϵ\mathrm{/}4),\backslash leq\; 2\; \backslash left(\backslash frac\; \{e\; T\}\{d\}\backslash right)\; ^\; \{d\}\; \backslash exp\; (-\; \backslash epsilon\; /\; 4),$$

the last inequality deriving from the fact that, since $\phi (\widehat{\mathbf{w}},\mathbf{x}_t)\in [0,1]$

$$\frac{{(\sum _{t=1}^T\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_t)+ϵ)}^2}{\sum _{t=1}^T\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_t{)}^2}\ge \frac{{(\sum _{t=1}^T\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_t)+ϵ)}^2}{\sum _{t=1}^T\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_t)}\ge 2ϵ\mathrm{.}\backslash frac\; \{\backslash left(\backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\}\; ,\; \backslash mathbf\; \{x\}\; \_\; \{t\})\; +\; \backslash epsilon\backslash right)\; ^\; \{2\}\}\{\backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\}\; ,\; \backslash mathbf\; \{x\}\; \_\; \{t\})\; ^\; \{2\}\}\; \backslash geq\; \backslash frac\; \{\backslash left(\backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\}\; ,\; \backslash mathbf\; \{x\}\; \_\; \{t\})\; +\; \backslash epsilon\backslash right)\; ^\; \{2\}\}\{\backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\}\; ,\; \backslash mathbf\; \{x\}\; \_\; \{t\})\}\; \backslash geq\; 2\; \backslash epsilon\; .$$

Take $\epsilon$ such that $\delta = 2\left(\frac{eT}{d}\right)^{d}\exp (-\epsilon /4)$ , to obtain the claimed bound.

Theorem C.4. With the same notation and assumptions as in Lemma C.3, let $\widehat{\mathbf{w}}\in \mathbb{R}^d$ be a function of $S_{T}$ such that

$$\frac{1}{T}\sum _{t=1}^T\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_t)\le \widehat{ϵ}(\delta )\backslash frac\; \{1\}\{T\}\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \_\; \{t\})\; \backslash leq\; \backslash widehat\; \{\backslash epsilon\}\; (\backslash delta)$$

holds with probability at least $1 - \delta$ , for some $\widehat{\epsilon}(\delta) \in [0, 1]$ . Then with probability at least $1 - 3\delta$ :

$${\mathbb{E}}_{\mathbf{x}\sim \mathcal{D}}\varphi (\widehat{\mathbf{w}},\mathbf{x})\le 4\widehat{ϵ}(\delta )+\frac{22\log \left(\frac{1}{\delta }\right)+11d\log \left(\frac{2eT}{d}\right)}{T}\mathrm{.}\backslash mathbb\; \{E\}\; \_\; \{\backslash mathbf\; \{x\}\; \backslash sim\; \backslash mathcal\; \{D\}\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\})\; \backslash leq\; 4\; \backslash widehat\; \{\backslash epsilon\}\; (\backslash delta)\; +\; \backslash frac\; \{2\; 2\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash delta\}\backslash right)\; +\; 1\; 1\; d\; \backslash log\; \backslash left(\backslash frac\; \{2\; e\; T\}\{d\}\backslash right)\}\{T\}.$$

Proof. Use the multiplicative Chernoff bound

$${\mathbb{E}}_{\mathbf{x}\sim \mathcal{D}}\varphi (\widehat{\mathbf{w}},\mathbf{x})\le \frac{4}{3T}\sum _{t=1}^T\varphi (\widehat{\mathbf{w}},{\mathbf{x}}_t^{\mathrm{\prime }})+\frac{32}{3T}\log (1\mathrm{/}\delta ),\backslash mathbb\; \{E\}\; \_\; \{\backslash mathbf\; \{x\}\; \backslash sim\; \backslash mathcal\; \{D\}\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\})\; \backslash leq\; \backslash frac\; \{4\}\{3\; T\}\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{T\}\; \backslash phi\; (\backslash widehat\; \{\backslash mathbf\; \{w\}\},\; \backslash mathbf\; \{x\}\; \_\; \{t\}\; ^\; \{\backslash prime\})\; +\; \backslash frac\; \{3\; 2\}\{3\; T\}\; \backslash log\; (1\; /\; \backslash delta),$$

and then apply Lemma C.3 to further bound the right-hand side.

To control noise terms, which are 1-subgaussian random variables, we provide the following lemma which is a direct implication of Chernoff bound.

Lemma C.5. Suppose $\xi$ is a $\sigma$ -subgaussian random variable, then for any $\delta > 0$ ,

$$\mathbb{P}(\mathrm{\mid }\xi \mathrm{\mid }\ge \sqrt{2{\sigma }^2\log (2\mathrm{/}\delta )})\le \delta \mathrm{.}\backslash mathbb\; \{P\}\; \backslash left(|\; \backslash xi\; |\; \backslash geq\; \backslash sqrt\; \{2\; \backslash sigma^\; \{2\}\; \backslash log\; (2\; /\; \backslash delta)\}\backslash right)\; \backslash leq\; \backslash delta\; .$$

Lemma C.6. Let $A = [a_{ij}] \in \mathbb{R}^{m \times n}$ be a matrix. Suppose $\xi_1, \ldots, \xi_n$ are independent $\sigma$ -subgaussian random variables. Then for any $\delta > 0$ ,

$$\mathbb{P}(\mathrm{\parallel }A\xi {\mathrm{\parallel }}^2>2{\sigma }^2\log \frac{2m}{\delta }\mathrm{t}\mathrm{r}(A{A}^{\mathrm{\top }}))\le \delta ,\backslash mathbb\; \{P\}\; \backslash left(\backslash |\; A\; \backslash xi\; \backslash |\; ^\; \{2\}\; >\; 2\; \backslash sigma^\; \{2\}\; \backslash log\; \backslash frac\; \{2\; m\}\{\backslash delta\}\; \backslash mathrm\; \{t\; r\}\; (A\; A\; ^\; \{\backslash top\})\backslash right)\; \backslash leq\; \backslash delta\; ,$$

where $\xi = (\xi_1,\dots ,\xi_n)^\top$

Proof. Consider

$$(A\xi {)}_i=\sum _{j=1}^n{a}_{ij}{\xi }_j,(A\; \backslash xi)\; \_\; \{i\}\; =\; \backslash sum\_\; \{j\; =\; 1\}\; ^\; \{n\}\; a\; \_\; \{i\; j\}\; \backslash xi\_\; \{j\},$$

the $i$ th component of vector $A\xi$ . Note that $(A\xi)i$ is a $\sigma \sqrt{\sum{j=1}^{n}a_{ij}^2}$ -subgaussian random variable, by lemma C.5 we have

$$\mathbb{P}(\mid \sum _{j=1}^n{a}_{ij}{\xi }_j\mid \ge \sqrt{2{\sigma }^2\sum _{j=1}^n{a}_{ij}^2\log \frac{2m}{\delta }})\le \frac{\delta }{m}\mathrm{.}\backslash mathbb\; \{P\}\; \backslash left(\backslash left|\; \backslash sum\_\; \{j\; =\; 1\}\; ^\; \{n\}\; a\; \_\; \{i\; j\}\; \backslash xi\_\; \{j\}\; \backslash right|\; \backslash geq\; \backslash sqrt\; \{2\; \backslash sigma^\; \{2\}\; \backslash sum\_\; \{j\; =\; 1\}\; ^\; \{n\}\; a\; \_\; \{i\; j\}\; ^\; \{2\}\; \backslash log\; \backslash frac\; \{2\; m\}\{\backslash delta\}\}\backslash right)\; \backslash leq\; \backslash frac\; \{\backslash delta\}\{m\}.$$

A union bound over $i$ gives, with probability at least $1 - \delta$ ,

$${\left(\sum _{j=1}^n{a}_{ij}{\xi }_j\right)}^2\le 2{\sigma }^2\sum _{j=1}^n{a}_{ij}^2\log \frac{2m}{\delta },\backslash left(\backslash sum\_\; \{j\; =\; 1\}\; ^\; \{n\}\; a\; \_\; \{i\; j\}\; \backslash xi\_\; \{j\}\backslash right)\; ^\; \{2\}\; \backslash leq\; 2\; \backslash sigma^\; \{2\}\; \backslash sum\_\; \{j\; =\; 1\}\; ^\; \{n\}\; a\; \_\; \{i\; j\}\; ^\; \{2\}\; \backslash log\; \backslash frac\; \{2\; m\}\{\backslash delta\},$$

uniformly over $i = 1,\ldots ,m$ . Therefore, with probability at least $1 - \delta$

$$\mathrm{\parallel }A\xi {\mathrm{\parallel }}^2=\sum _{i=1}^m{\left(\sum _{j=1}^n{a}_{ij}{\xi }_j\right)}^2\le 2{\sigma }^2\log \frac{2m}{\delta }\sum _{i,j}{a}_{ij}^2=2{\sigma }^2\log \frac{2m}{\delta }\mathrm{tr}(A{A}^{\mathrm{\top }}),\backslash |\; A\; \backslash xi\; \backslash |\; ^\; \{2\}\; =\; \backslash sum\_\; \{i\; =\; 1\}\; ^\; \{m\}\; \backslash left(\backslash sum\_\; \{j\; =\; 1\}\; ^\; \{n\}\; a\; \_\; \{i\; j\}\; \backslash xi\_\; \{j\}\backslash right)\; ^\; \{2\}\; \backslash leq\; 2\; \backslash sigma^\; \{2\}\; \backslash log\; \backslash frac\; \{2\; m\}\{\backslash delta\}\; \backslash sum\_\; \{i,\; j\}\; a\; \_\; \{i\; j\}\; ^\; \{2\}\; =\; 2\; \backslash sigma^\; \{2\}\; \backslash log\; \backslash frac\; \{2\; m\}\{\backslash delta\}\; \backslash operatorname\; \{t\; r\}\; (A\; A\; ^\; \{\backslash top\}),$$

as claimed.

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