papers/diffusion-convergence-rate/paper.md

# Instance-dependent Convergence Theory for Diffusion Models

Yuchen Jiao ∗ Gen Li ∗

June 13, 2025

# Abstract

Score-based diffusion models have demonstrated outstanding empirical performance in machine learning and artificial intelligence, particularly in generating high-quality new samples from complex probability distributions. Improving the theoretical understanding of diffusion models, with a particular focus on the convergence analysis, has attracted significant attention. In this work, we develop a convergence rate that is adaptive to the smoothness of different target distributions, referred to as instance-dependent bound. Specifically, we establish an iteration complexity of $\operatorname* { m i n } \{ d , d ^ { 2 / 3 } L ^ { 1 / 3 } , d ^ { 1 / 3 } L \} \varepsilon ^ { - 2 / 3 }$ (up to logarithmic factors), where $d$ denotes the data dimension, and $\varepsilon$ quantifies the output accuracy in terms of total variation (TV) distance. In addition, $L$ represents a relaxed Lipschitz constant, which, in the case of Gaussian mixture models, scales only logarithmically with the number of components, the dimension and iteration number, demonstrating broad applicability.

# 1 Introduction

Score-based diffusion models have emerged as a powerful class of generative models capable of synthesizing high-quality data from complex probability distributions (Dhariwal and Nichol, 2021; Ho et al., 2020; Sohl-Dickstein et al., 2015; Song et al., 2021; Song and Ermon, 2019). These models, including Denoising Diffusion Probabilistic Models (DDPM) (Ho et al., 2020) and Denoising Diffusion Implicit Models (DDIM) (Song et al., 2021), transforms pure noise into samples from the data distribution through an iterative denoising process. This process is facilitated by a series of score functions, which approximate the gradient of the data log-density using pretrained neural networks. In practice, diffusion models have achieved remarkable performance in various tasks including image generation (Ramesh et al., 2022; Rombach et al., 2022; Saharia et al., 2022), video generation (Villegas et al., 2022) and so on (Croitoru et al., 2023; Yang et al., 2023; Zhang et al., 2025). For an overview of recent developments, including both empirical and theoretical advancements, readers may refer to Croitoru et al. (2023); Tang and Zhao (2024); Yang et al. (2023).

Diffusion models typically consist of two processes: the forward process and the reverse (or backward) process. The forward process is a simple stochastic process that progressively transforms a data sample $X _ { 0 }$ into nearly pure noise $X _ { T }$ by iteratively adding Gaussian noise:

$$
X _ { 0 } { \stackrel { \mathrm { a d d } } {  } } { \stackrel { \mathrm { n o i s e } } { X _ { 1 } } } { \stackrel { \mathrm { a d d } } {  } } \cdots { \stackrel { \mathrm { a d d } } {  } } \cdots X _ { T } .
$$

Here, $X _ { 0 }$ is a $d$ -dimensional sample from the target data distribution , and $X _ { T }$ approximately follows a $p _ { \mathsf { d a t a } }$ standard Gaussian distribution $\mathcal { N } ( 0 , I _ { d } )$ . The core of diffusion models lies in the reverse process, which aims to learn a process that generates a sample resembling the target data distribution from pure Gaussian noise:

$$
{ Y _ { 0 } } \stackrel { \mathrm { d e n o i s e } } {  } { Y _ { 1 } } \stackrel { \mathrm { d e n o i s e } } {  } \cdot \cdot \cdot \stackrel { \mathrm { d e n o i s e } } {  } { Y _ { T } } ,
$$

Here, $Y _ { 0 }$ is initialized as white Gaussian noise, and the reverse process aims to achieve $Y _ { t } \overset { d } { \approx } X _ { T - t }$ for all $t$ , such that the distribution of the final output $Y _ { T }$ approximates the target distribution $p _ { \mathsf { d a t a } }$ . The most critical component of diffusion models is the efficient construction of this reverse process. To achieve this, diffusion models employ the time-reversal of stochastic differential equations (SDEs) to generate $Y _ { t + 1 }$ from $Y _ { t }$ . This process relies on score functions $( s _ { T - t } ^ { \star } = \nabla \log p _ { X _ { T - t } , }$ ), which are gradients of the log marginal density of the forward process $X _ { T - t }$ . These score functions are typically pretrained using score-matching techniques (Ho et al., 2020; Hyvärinen, 2007; Hyvärinen and Dayan, 2005; Pang et al., 2020; Song and Ermon, 2019; Vincent, 2011).

Due to the impressive empirical success of diffusion models, a lot of efforts in recent years have focused on analyzing their convergence properties. Given the complexity of establishing a comprehensive end-toend theoretical framework, most studies adopt a divide-and-conquer approach, treating the score matching step as a black box and focusing on the data generation process. Following this framework, several studies have analyzed how factors such as data dimension, score estimation error, and the number of iterations affect the accuracy of approximating the target distribution (Benton et al., 2023; Chen et al., 2023, 2024b, 2022; De Bortoli, 2022; Gao et al., 2023; Gupta et al., 2024; Huang et al., 2024a,b,c; Lee et al., 2022, 2023; Li and Cai, 2024; Li et al., 2024a; Li and Jiao, 2024; Li et al., 2023, 2024b; Li and Yan, 2024a,b; Liang et al., 2025). For general data distributions, the best-known results are $\widetilde { O } ( d \varepsilon ^ { - 1 } )$ for DDPM (Li and Yan, 2024a) and $\widetilde { \cal O } ( d ^ { 5 / 4 } \varepsilon ^ { - 1 / 2 } )$ for an accelerated sampler (Li and Cai, 2024), where ${ \widetilde { O } } ( \cdot )$ omits logarithmic factors and $d$ denotes the data dimension. Some works exploited smoothness conditions to further improve the iteration complexity, with the state-of-the-art achieved by Li and Jiao (2024) is $\widetilde { \cal O } ( d ^ { 1 / 3 } L \varepsilon ^ { - 2 / 3 } )$ , where $L$ denotes the Lipschitz constant of score functions. However, this improvement holds only when $L \ \lesssim$ $\operatorname* { m i n } \{ d ^ { 2 / 3 } \varepsilon ^ { - 1 / 3 } , d ^ { 1 1 / 1 2 } \varepsilon ^ { 1 / 6 } \}$ , which is restrictive. This motivates us to develop an $L$ -adaptive convergence bound for diffusion models, with the hope to achieve improvement over the full range of $L$ .

# 1.1 Our contributions

In this paper, we investigate a sampler for SGMs based on the randomized midpoint technique, and establish a convergence rate adaptive to the smoothness of score functions. Specifically, up to logarithmic factors, we achieve the following iteration complexity:

$$
\operatorname* { m i n } \{ d , d ^ { 2 / 3 } L ^ { 1 / 3 } , d ^ { 1 / 3 } L \} \varepsilon ^ { - 2 / 3 } ,
$$

where $L$ , as defined in Definition 2, characterizes the smoothness of score functions. Below, we provide a brief comparison of our results with existing convergence rates, viewed in Figure 1:

• Comparison under smoothness condition. Under a uniform Lipschitz assumption, i.e., $\left\| s _ { t } ^ { \star } ( x ) - \right\|$ $s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } \leq L \| x - x ^ { \prime } \| _ { 2 }$ for all $x , x ^ { \prime } \in \mathbb { R } ^ { d }$ and all $t$ , several works (Chen et al., 2023, 2024b; Gao et al., 2023; Gupta et al., 2024; Lee et al., 2022; Li and Jiao, 2024) have studied the convergence rate of SGMs. In comparison, our result has two advantages:

– The uniform Lipschitz condition used in previous works is significantly more restrictive than the non-uniform condition in Definition 2 adopted here. Specifically, for Gaussian mixture models (GMMs), we show that $L$ scales only logarithmically with the number of components and dimension, whereas the uniform Lipschitz constant can be extremely large. This highlights the superiority of our results (see Example 2 for details). Achieving these advancements requires substantial technical innovations. Detailed comparisons are in Section 3 and Appendix B of supplemental materials.

– Among existing results, the best-known bound under smoothness conditions is $\widetilde { \cal O } ( d ^ { 1 / 3 } L \varepsilon ^ { - 2 / 3 } )$ , established by Li and Jiao (2024). Our convergence rate improves prior theory by a factor of√ $\operatorname* { m a x } \{ d ^ { - 2 / 3 } L , d ^ { - 1 / 3 } L ^ { 2 / 3 } , 1 \}$ , yielding substantial gains when $L \gtrsim \sqrt { d }$ even without considering the additional benefits from relaxing the Lipschitz condition.

• Comparison under general condition. Without smoothness assumption on score functions, Benton et al. (2023) established an iteration complexity at the order of $d \varepsilon ^ { - 2 }$ , which marks the first result with linear dependency on the data dimension $d$ . This was later improved to $\tilde { O } ( d \varepsilon ^ { - 1 } )$ by Li and Yan (2024a) and Li et al. (2024b). In comparison, our analysis further improves these bounds by a factor of $\operatorname* { m a x } \{ 1 , d ^ { 1 / 3 } L ^ { - 1 / 3 } , d ^ { 2 / 3 } L ^ { - 1 } \} \varepsilon ^ { - 1 / 3 }$ , which is significant for the entire range of $L$ . Notably, even in the case of $L = \infty$ , our result achieves a significant improvement, except in the trivial scenario where $\varepsilon \asymp 1$ .

• Comparison with accelerated samplers. Existing accelerated convergence theories often rely on additional assumptions about the target distribution or the estimation error of the higher-order score functions. For example, results in Huang et al. (2024a,b) depend on the bound of the first $p$ -th derivatives of score functions or their estimates, and achieved a convergence rate faster than $\varepsilon ^ { - 2 / 3 }$ when $p \geq 3$ . In addition, Li et al. (2024a) assumed that the estimation errors of the jacobian matrix of score functions are bounded, and developed an improved rate of $\varepsilon ^ { - 1 / 2 }$ . More recently, Li and Cai (2024) achieved the state-of-the-art iteration complexity of $\widetilde { \cal O } ( d ^ { 5 / 4 } \varepsilon ^ { - 1 / 2 } )$ without additional distribution or estimation assumptions. In comparison, our result improves this bound when the number of iterations $T \lesssim \operatorname* { m a x } \{ d ^ { 2 } , d ^ { 3 } L ^ { - 1 } , d ^ { 4 } L ^ { - 3 } \}$ . Notably, even in the case of $L = \infty$ , a significant improvement is achieved as long as $T \lesssim d ^ { 2 }$ .

Organizations: The remaining of this paper is structured as follows. Section 2 provides a brief overview of fundamental concepts on SGMs, and presents the sampling algorithm. In Section 3, we present our assumptions and main results, and make comparison with previous works. The theoretical analysis and proofs are detailed in Section 4. Finally, Section 5 concludes the paper and discusses potential directions for future research.

# 2 Preliminary

In this section, we present some basic concepts of generative diffusion models, introduce the sampler employed in this work, and clarify our goal.

# 2.1 Score-based diffusion models

A diffusion model typically involves two key processes: the forward process and the reverse process, which are explained below.

Forward process. The forward process starts from a random instance $X _ { 0 } \in \mathbb { R } ^ { d }$ sampled from the target data distribution $p _ { \mathsf { d a t a } }$ , and progressively transforms it into pure Gaussian noise by iteratively adding noise at each step:

$$
X _ { t } = \sqrt { \alpha _ { t } } X _ { t - 1 } + \sqrt { 1 - \alpha _ { t } } W _ { t } , \quad 1 \leq t \leq T ,
$$

where $\{ W _ { t } \} _ { 1 \leq t \leq T }$ is a sequence of independent Gaussian vectors drawn from $\mathcal { N } ( 0 , I _ { d } )$ , and $\alpha _ { t } ~ \in ~ ( 0 , 1 )$ represents the step-size. For ease of notations, we define

$$
\overline { { \alpha } } _ { t } : = \prod _ { k = 1 } ^ { t } \alpha _ { k } , \quad 1 \leq t \leq T .
$$

With this notation, $X _ { t }$ can be expressed as a linear combination of the original data instance $X _ { 0 }$ and Gaussian noise with variance $1 - \overline { { \alpha } } _ { t }$ , as below.

$$
X _ { t } = \sqrt { \overline { { \alpha _ { t } } } } X _ { 0 } + \sqrt { 1 - \overline { { \alpha _ { t } } } } \overline { { W } } _ { t } , \quad \overline { { W } } _ { t } \sim \mathcal { N } ( 0 , I _ { d } ) .
$$

As $\overline { { \alpha } } _ { t }$ approaches zero, the distribution of $X _ { t }$ becomes exceedingly close to $\mathcal { N } ( 0 , I _ { d } )$ .

Diffusion models are closely related to stochastic differential equations (SDEs). The continuous-time limit of the forward process can be modeled as:

$$
\mathrm { d } X _ { \tau } = - \frac { 1 } { 2 ( 1 - \tau ) } X _ { \tau } \mathrm { d } \tau + \frac { 1 } { \sqrt { 1 - \tau } } \mathrm { d } B _ { \tau } , \quad \mathrm { f o r } \ 0 \leq \tau < 1 ,
$$

where $B _ { \tau }$ denotes some Brownian motion, $X _ { 0 } \sim p _ { \mathsf { d a t a } }$ , and the distribution of $X _ { \tau }$ approaches Gaussian as $\tau$ gets close to 1.

Reverse process and score functions. The core of diffusion models lies in the reverse (or backward) process, which starts from pure Gaussian noise $Y _ { 0 } \sim { \mathcal { N } } ( 0 , I _ { d } )$ , and aims to generate samples $Y _ { T }$ that resemble

the target data distribution. Many reverse processes are designed based on insights from the probability flow ODE, which is given by

$$
\mathrm { d } Y _ { \tau } = - \frac { 1 } { 2 ( 1 - \tau ) } \big ( Y _ { \tau } + \nabla \log p _ { X _ { \tau } } ( Y ) \big ) \mathrm { d } \tau .
$$

This ODE ensures that $Y _ { \tau }$ follows the same distribution as $X _ { \tau }$ defined in (4), provided that the initial point $Y _ { \tau _ { 0 } } \sim p _ { X _ { \tau _ { 0 } } }$ , where $\tau _ { 0 }$ is close to one and $Y _ { \tau _ { 0 } }$ is approximately Gaussian noise.

In ODE (5), the only additional term except $Y _ { \tau }$ itself is the gradient of the log-density of the forward process, $\nabla \log p _ { X _ { \tau } }$ , known as the score function. Its formal mathematical definition is provided below.

Definition 1. The score function $s _ { t } ^ { \star } : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ for $1 \leq t \leq T$ is defined as:

$$
s _ { t } ^ { \star } ( x ) : = \nabla \log p _ { X _ { t } } ( x ) = - \frac { 1 } { 1 - \overline { { \alpha } } _ { t } } \int _ { x _ { 0 } } p _ { X _ { 0 } \mid X _ { t } } ( x _ { 0 } \mid x ) ( x - \sqrt { \overline { { \alpha } } _ { t } } x _ { 0 } ) \mathrm { d } x _ { 0 } .
$$

For ease of notations, we also define $s _ { \tau } ^ { \star } ( x )$ with a continuous index $0 < \tau < 1$ as follows:

$$
s _ { \tau } ^ { \star } ( x ) : = \nabla \log p _ { X _ { \tau } } ( x ) = - \frac { 1 } { \tau } \int _ { x _ { 0 } } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x ) ( x - \sqrt { 1 - \tau } x _ { 0 } ) \mathrm { d } x _ { 0 } .
$$

It is evident that $s _ { t } ^ { \star } ( \cdot ) = s _ { 1 - \overline { { \alpha } } _ { t } } ^ { \star } ( \cdot )$ . In practice, the true score function $s _ { t } ^ { \star }$ is typically unknown and must be estimated from a training dataset. We assume access to faithful estimates $s _ { t }$ of the score functions $s _ { t } ^ { \star }$ across all steps $t$ . The assumption regarding score estimation errors is formally presented in Assumption 2.

# 2.2 Sampling algorithm

The sampler used is the same as the one employed by Li and Jiao (2024), which is derived from the discretization of a probability flow ODE for $X _ { \tau }$ over $\tau \in ( 0 , 1 )$ . Specifically, we first discretize $\tau$ into intermediate points $^ { \prime } k , n$ within the interval $( 0 , 1 )$ , where $n = 0 , \cdots , N$ and $k = 0 , \cdots , K$ . We then estimate $X _ { \tau _ { k , n } }$ at these intermediate points by approximating the integral of probability flow ODE. The detailed implementation is stated below.

Randomized schedule. We begin by discretizing the interval [0, 1] into a sequence of subintervals $( \widehat { \alpha } _ { t } , \widehat { \alpha } _ { t - 1 } )$ where $\widehat { \alpha } _ { t }$ is defined as

$$
\widehat { \alpha } _ { T + 1 } = \frac { 1 } { T ^ { c _ { 0 } } } , \quad \widehat { \alpha } _ { t - 1 } = \widehat { \alpha } _ { t } + \frac { c _ { 1 } \widehat { \alpha } _ { t } ( 1 - \widehat { \alpha } _ { t } ) \log { T } } { T } , \qquad t = - \frac { N } { 2 } + 1 , \cdots , T + 1 ,
$$

for some sufficiently large constants $c _ { 0 } , c _ { 1 } > 0$ , where the ratio $c _ { 1 } / c _ { 0 }$ is assumed to be sufficiently large. Subsequently, we employ a randomized learning rate schedule by setting $\alpha _ { t }$ in (3) as

$$
\overline { { \alpha } } _ { t } \sim \mathsf { U n i f } ( \widehat { \alpha } _ { t } , \widehat { \alpha } _ { t - 1 } ) , \quad \mathrm { f o r } t = - \frac { N } { 2 } + 1 , \ldots , T + 1 ,
$$

where Unif denotes the uniform distribution.

The algorithm operates over $K$ rounds, each consisting of $\begin{array} { r } { N = \frac { 2 \mathcal { I } } { K } } \end{array}$ steps. We define

$$
\widehat { \tau } _ { k , n } : = 1 - \widehat { \alpha } _ { T - \frac { k N } { 2 } - n } , \qquad \tau _ { k , n } : = 1 - \overline { { { \alpha } } } _ { T - \frac { k N } { 2 } - n + 1 } \qquad \mathrm { f o r } \quad n = - 1 , \dots , N .
$$

It follows that $\tau _ { k , n } \sim \mathsf { U n i f } ( \widehat { \tau } _ { k , n } , \widehat { \tau } _ { k , n - 1 } )$ , which we use as the discretization of $\tau$ bSampling procedure. With $\tau$ discretized, we are now ready to describe the sampling procedure. As aforementioned, the sampler is implemented over $K$ rounds, each consisting of $N$ steps. In the $k$ -th round, the sampler approximates $X _ { \tau _ { k , N } }$ defined by the probability flow ODE with the initial point given by $X _ { \tau _ { k , 0 } }$ , whose probability distribution is denoted as $q _ { k }$ . At the end of each round, a Gaussian noise is injected to convert the total variation distance between the reverse and forward processes into an estimation error in the $\ell _ { 2 }$ norm. The sampling procedure consists of the following steps:

1. Initialization: The sampler begins with an initial sample $Y _ { 0 } \sim { \mathcal { N } } ( 0 , I _ { d } )$ .

2. Iterative update: For each $k$ ranging from $0$ to $K - 1$ , the intermediate variables $Y _ { k , n }$ are iteratively updated for $n = 1 , \cdots , N$ by discretizing the ODE as follows:

$$
\begin{array} { r l } & { \frac { Y _ { k , n } } { \sqrt { 1 - \tau _ { k , n } } } = \frac { Y _ { k , 0 } } { \sqrt { 1 - \tau _ { k , 0 } } } + \frac { s _ { T - \frac { k N } { 2 } + 1 } \left( Y _ { k , 0 } \right) } { 2 \left( 1 - \tau _ { k , 0 } \right) ^ { 3 / 2 } } ( \tau _ { k , 0 } - \widehat { \tau } _ { k , 0 } ) } \\ & { + \displaystyle \sum _ { i = 1 } ^ { n - 1 } \frac { s _ { T - \frac { k N } { 2 } - i + 1 } \left( Y _ { k , i } \right) } { 2 \left( 1 - \tau _ { k , i } \right) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) + \frac { s _ { T - \frac { k N } { 2 } - n + 2 } \left( Y _ { k , n - 1 } \right) } { 2 \left( 1 - \tau _ { k , n - 1 } \right) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , n - 1 } - \tau _ { k , n } ) , } \end{array}
$$

where $Y _ { k , 0 } = Y _ { k }$ and $\begin{array} { r l r } {  { S _ { T - \frac { k N } { 2 } - i + 1 } } } \end{array}$ is an estimation of $s _ { T - \frac { k N } { 2 } - i + 1 } ^ { \star }$ as defined in (6), corresponding to $\overline { { \alpha } } _ { T - \frac { k N } { 2 } - i + 1 } = 1 - \tau _ { k , i }$ .

3. Noise injection: After obtaining $Y _ { k , N }$ , we update $Y _ { k + 1 }$ by injecting stochastic noise as:

$$
Y _ { k + 1 } = \sqrt { \frac { 1 - \tau _ { k + 1 , 0 } } { 1 - \tau _ { k , N } } } Y _ { k , N } + \sqrt { \frac { \tau _ { k + 1 , 0 } - \tau _ { k , N } } { 1 - \tau _ { k , N } } } Z _ { k } ,
$$

where $Z _ { k } \stackrel { \scriptscriptstyle 1 . 1 . 0 . } { \sim } \mathcal { N } ( 0 , I _ { d } )$

Notice that at each step of computing $Y _ { k , n }$ , only one additional score evaluation for $s _ { T - \frac { k N } { 2 } - n + 2 } ( Y _ { k , n - 1 } )$ is required. Therefore, the total iteration complexity of the sampler is $K N = 2 T$ . This sampler can be implemented in parallel, as demonstrated by Li and Jiao (2024). For clarity, we present the details in Appendix E.1 of the supplemental material.

Our goal. The objective of this work is to improve the convergence rate for a broader class of distributions by analyzing the aforementioned sampler. Since $X _ { \tau _ { K , 0 } }$ , which is nearly the starting point of the forward process, is perturbed only by noise with a small variance of $1 - \overline { { \alpha } } _ { 1 }$ , the performance of the sampler is evaluated using the total variation (TV) distance between $p _ { Y _ { K } }$ and $q _ { K }$ , defined as

$$
\mathsf { T V } ( q _ { K } , p _ { Y _ { K } } ) : = \frac { 1 } { 2 } \int | p _ { Y _ { K } } ( x ) - q _ { K } ( x ) | \mathrm { d } x .
$$

# 3 Main results

In this section, we establish an instance-dependent convergence rate for diffusion models under a novel analytical framework that accommodates a relaxed Lipschitz condition, thereby covering a broader class of distributions. Finally, we extend our results to the parallel implementation of the sampler, following an idea similar to that of Li and Jiao (2024).

# 3.1 Assumptions

We first make the following assumption on the target data distribution $p _ { \mathsf { d a t a } }$ , which accommodates a broad class of data distributions.

Assumption 1. We assume that the target distribution $p _ { \mathsf { d a t a } }$ has a bounded second-order moment in the sense that

$$
\begin{array} { r } { \mathbb { E } _ { X _ { 0 } \sim p _ { \mathsf { d a t a } } } [ \| X _ { 0 } \| _ { 2 } ^ { 2 } ] < T ^ { c _ { R } } , } \end{array}
$$

where $c _ { R } > 0$ is an arbitrarily large constant.

The second-order moment of $X _ { 0 }$ is assumed to be polynomial in the number of iterations $T$ . This assumption is mild as the exponent $c _ { R }$ can be arbitrarily large. However, there exist exceptions including extremely heavy-tailed distributions. For example, densities decaying slower than $1 / x ^ { 3 }$ may not align with our theoretical framework.

Our analysis is conducted under a relaxed smoothness condition, which is substantially weaker than the commonly used uniform Lipschitz condition and encompasses a wide range of distributions previously considered non-Lipschitz. Specifically, we define the Lipschitz constant for the normalized score functions $( 1 - \overline { { \alpha } } _ { t } ) s _ { t } ^ { \star }$ as follows.

Definition 2 (Non-uniform Lipschitz property). Let $L$ denote the smallest quantity, which may depend on $T$ and $d$ , such that

$$
\begin{array} { l l } { \displaystyle \mathbb { P } _ { x \sim X _ { t } } \left\{ ( 1 - \overline { { \alpha _ { t } } } ) \| s _ { t } ^ { \star } ( x ^ { \prime } ) - s _ { t } ^ { \star } ( x ) \| _ { 2 } \le L \| x ^ { \prime } - x \| _ { 2 } , \forall \| x ^ { \prime } - x \| _ { 2 } \le \frac { C \sqrt { d ( 1 - \overline { { \alpha } } _ { t } ) \log T } } { L } \right\} } \\ { \displaystyle \ge 1 - \frac { c } { ( T + d ) ^ { 4 } } , } \end{array}
$$

where $C$ and $c$ are some universal constants.

Remark 1. Previous works typically assume a uniform Lipschitz condition, requiring $\big ( 1 - \overline { { \alpha } } _ { t } \big ) \big \| s _ { t } ^ { \star } ( x ) -$ $s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } \leq L \| x - x ^ { \prime } \| _ { 2 }$ (or $\| s _ { t } ^ { \star } ( x ) - s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } \leq L \| x - x ^ { \prime } \| _ { 2 } )$ to hold for all $x$ and $x ^ { \prime }$ . In contrast, this work adopts a non-uniform Lipschitz condition, which is a significant relaxation compared to prior assumptions. In what follows, we present two examples to demonstrate the practical importance and necessity of this relaxation.

Example 1 (Gaussian distribution). Consider Gaussian target distribution $X _ { 0 } ^ { ( i ) } \sim \mathcal { N } ( 0 , \sigma _ { i } ^ { 2 } )$ , where $X _ { 0 } ^ { ( i ) }$ denotes the $i$ -th entry of $X _ { 0 }$ , $i = 1 , \cdots , d$ . The score function $\boldsymbol { s } _ { t } ^ { \star } ( \boldsymbol { x } ) = \nabla \log p _ { X _ { t } } ( \boldsymbol { x } )$ satisfies

$$
\begin{array} { r l } { \forall t > 0 , \ \forall x , x ^ { \prime } \in \mathbb { R } ^ { d } , } & { \quad ( 1 - \overline { { \alpha } } _ { t } ) \| s _ { t } ^ { \star } ( x ) - s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } \leq \| x - x ^ { \prime } \| _ { 2 } , } \\ { \forall t \geq 0 , \ \exists x , x ^ { \prime } \in \mathbb { R } ^ { d } , \quad \mathrm { s u c h ~ t h a t } } & { \quad \| s _ { t } ^ { \star } ( x ) - s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } \geq ( 1 - \overline { { \alpha } } _ { t } ) ^ { - 1 } \| x - x ^ { \prime } \| _ { 2 } , } \end{array}
$$

provided that $\operatorname* { m i n } _ { i } \sigma _ { i } ^ { 2 } = 0$ . This highlights the rationale for defining the Lipschitz condition for $( 1 - \overline { { \alpha } } _ { t } ) s _ { t } ^ { \star }$ The detailed derivations are presented in Appendix C.1.

Examwhere ure M and sider GMM tar. Then for any istribution , we have $\begin{array} { r } { X _ { 0 } \sim \sum _ { h = 1 } ^ { H } \gamma _ { h } \mathcal { N } ( \mu _ { h } , \sigma ^ { 2 } I _ { d } ) } \end{array}$ $\mu _ { h } \in \mathbb { R } ^ { d }$ $\sigma \geq 0$ $\gamma _ { h } \geq 0$ $\textstyle \sum _ { h = 1 } ^ { H } \gamma _ { h } = 1$ $t \geq 0$

$$
\begin{array} { r l } & { \mathbb { P } \left\{ ( 1 - \overline { { \alpha } } _ { t } ) \| s _ { t } ^ { \star } ( x ) - s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } \leq C _ { 1 } \log ( H ( T + d ) ) \| x - x ^ { \prime } \| _ { 2 } , \forall \ \| x - x ^ { \prime } \| _ { 2 } \leq C _ { 2 } \sqrt { d ( 1 - \overline { { \alpha } } _ { t } ) } \right\} } \\ & { \geq 1 - \frac { c } { ( T + d ) ^ { 4 } } , } \end{array}
$$

for some universal constants $C _ { 1 } , C _ { 2 }$ and $c$ . Moreover, consider a simple case that $X _ { 0 } \sim \frac { 1 } { 2 } { \mathcal { N } } ( \mu , \sigma ^ { 2 } I _ { d } ) +$ ${ \scriptstyle { \frac { 1 } { 2 } } } { \mathcal { N } } ( - \mu , \sigma ^ { 2 } I _ { d } )$ , then there exists some $x \in \mathbb { R } ^ { d }$ , such that for $\overline { { \alpha } } _ { t } > 1 / 2$ ,

$$
( 1 - \overline { { \alpha } } _ { t } ) \| \nabla s _ { t } ^ { \star } ( x ) \| _ { \mathsf { o p } } \geq \frac { ( 1 - \overline { { \alpha } } _ { t } ) \| \mu \| _ { 2 } ^ { 2 } } { 4 ( 1 - \overline { { \alpha } } _ { t } + \sigma ^ { 2 } ) ^ { 2 } } .
$$

The above example implies that the non-uniform Lipschitz constant remains relatively small, satisfying $L \leq \log ( H ( T + d ) )$ . In contrast, the uniform Lipschitz constant may be extremely large when $\sigma ^ { 2 }$ is small enough. Here, $\| \mu \| _ { 2 } ^ { 2 } \approx \mathbb { E } [ \| X _ { 0 } \| _ { 2 } ^ { 2 } ]$ is typically large in practice, which often scales on the order of $d$ . The detailed derivations are presented in Appendix C.2.

Finally, we make the following assumption about the estimation error of score functions.

Assumption 2. We assume access to an estimate $s _ { t } ( \cdot )$ for each $s _ { t } ^ { \star } ( \cdot )$ , with the averaged $\ell _ { 2 }$ score estimation error as

$$
\begin{array} { l } { \varepsilon _ { \mathsf { s c o r e } } ^ { 2 } = \displaystyle \frac { 1 } { T } \sum _ { k = 0 } ^ { K - 1 } \sum _ { n = 0 } ^ { N - 1 } \mathbb { E } _ { Y _ { k } \sim q _ { k } } \left[ \| s _ { T - \frac { k N } { 2 } - n + 1 } ( Y _ { k , n } ) - s _ { T - \frac { k N } { 2 } - n + 1 } ^ { \star } ( Y _ { k , n } ) \| _ { 2 } ^ { 2 } \right] } \\ { = : \displaystyle \frac { 1 } { T } \sum _ { k = 0 } ^ { K - 1 } \sum _ { n = 0 } ^ { N - 1 } \varepsilon _ { k , n } ^ { 2 } . } \end{array}
$$

# 3.2 Convergence Analysis

We are now positioned to present the convergence guarantees — measured by the total variation distance between the forward and the reverse processes — for the sampler (10). The proof is postponed to Section 4.

Theorem 1. Suppose that Assumptions 1 and ${ \it \Delta } _ { \it { \phi } } ^ { \it { \Delta } }$ hold true, and $K = c _ { 2 } \operatorname* { m i n } \{ d \log ^ { 2 } T , L \log T \}$ for some constant $c _ { 2 } > 0$ . Then the sampling process (10) with the learning rate schedule (9) satisfies

$$
\mathsf { T V } ( q _ { K } , p _ { Y _ { K } } ) \le \frac { C \operatorname* { m i n } \{ d ^ { 3 / 2 } , d L ^ { 1 / 2 } , d ^ { 1 / 2 } L ^ { 3 / 2 } \} \log ^ { 4 } T } { T ^ { 3 / 2 } } + C \varepsilon _ { \mathrm { s c o r e } } \log ^ { 1 / 2 } T
$$

for some constant $C > 0$ large enough, where $L$ is defined in Definition $\mathcal { Z }$

We now discuss the main implications of Theorem 1.

Relaxation of smoothness. Our result requires only a non-uniform Lipschitz condition, which is substantially weaker than the uniform Lipschitz condition commonly employed in prior studies. As demonstrated in Example 2 for Gaussian mixture models, we have shown that $L$ scales only logarithmically with the number of components, dimension and iteration number, whereas the uniform Lipschitz constant may be extremely large. Given the wide use of GMMs, this demonstrates that the uniform Lipschitz condition employed in previous works is more restrictive compared to the non-uniform condition in Definition 2 used here.

Achieving these improvements requires a lot of technical efforts. For example, the absence of a uniform Lipschitz condition poses significant challenges for controlling error propagation across multiple steps, while a naive stepwise analysis may lead to suboptimal bounds. We address this issue by introducing two auxiliary sequences based on a typical set and quantifying how error propagation affects the probability of $Y _ { k , n }$ outside this set (see Step 2 in Section 4 and Lemma 5). In addition, lacking a uniform condition prevents the logconcavity of $p _ { X _ { \tau } | X _ { \tau + \delta } }$ for small $\delta$ , which plays a crucial rule in controlling the one-step discretization error (see Lemma 1 and (B.2)-(B.4) in Chen et al. (2024b)). We instead directly analyze this derivative based on its definition via a careful decomposition and statistical bounds (see Lemma 10). Further details are provided in Appendix B of supplemental material.

Iteration complexity. For the moment, we focus on the first term in (13), which corresponds to discretization error. To ensure $\mathsf { T V } ( q _ { K } , p _ { Y _ { K } } ) \le \varepsilon$ , it is sufficient to choose

$$
T \gtrsim \frac { \operatorname* { m i n } \{ d , d ^ { 2 / 3 } L ^ { 1 / 3 } , d ^ { 1 / 3 } L \} \log ^ { \frac { 8 } { 3 } } T } { \varepsilon ^ { 2 / 3 } } .
$$

This result is adaptive to the non-uniform Lipschitz constant $L$ of normalized score functions and is thus referred to as instance-dependent. Different from previous works, which only improve iteration complexity under specific conditions on $L$ , our theory improves existing results (Benton et al., 2023; Li and Cai, 2024; Li and Yan, 2024a) over a full range of $L$ , and improves the state-of-the-art (Li and Jiao, 2024) when $L \gtrsim \sqrt { d }$ , even when non-uniform and uniform Lipschitz constants are equal. A detailed comparison of iteration complexity orders has been also provided in Section 1.1.

To illustrate these improvements, we assume the non-uniform and uniform Lipschitz constants are identical and compare iteration complexities for $\varepsilon = O ( 1 )$ across varying values of $L$ , as shown in the left subplot of Figure 1. It indicates that prior works (Benton et al., 2023; Gupta et al., 2024; Li and Cai, 2024; Li and Yan, 2024a) outperform others in specific regimes. In contrast, our instance-dependent result achieves the best result across the full range of $L$ , and improves all previous works when ${ \sqrt { d } } \lesssim L \lesssim d$ . Furthermore, to more clearly illustrate the improvement in the case of $L = \infty$ , we present the iteration complexity as a function of $\varepsilon$ in the right subplot of Figure 1. It demonstrates clear improvement over previous results when $T \lesssim d ^ { 2 }$ . We verify our theoretical result via a numerical simulation, which is provided in Appendix A of supplemental material. More comparisons are provided in Appendix B of supplemental material.

Remark 2. Some studies have explored provably accelerated samplers for diffusion models (Huang et al., 2024a,b), achieving convergence rate faster than the $\varepsilon ^ { - 2 / 3 }$ rate established in this work. However, these results typically rely on the bound of the first $p$ -th derivatives of the score functions or their estimates with $p \geq 3$ .

![](images/figures/diffusion-convergence-rate-fig-0001.jpg)  
Figure 1: Comparison of Theorem 1 with prior results. left: the iteration complexity as a function of $L$ when $\varepsilon = O ( 1 )$ . right: the iteration complexity as a function of $\varepsilon$ when $L = \infty$ .

# 3.3 Extension to parallel sampling

In this section, we present a theoretical guarantee for the parallel implementation described in Appendix E.1 of the supplementary material, following an approach similar to that of Gupta et al. (2024); Li and Jiao (2024). Let $N$ denote the number of parallel processors and $M K$ the total number of parallel rounds. The convergence result is stated below, with a detailed proof provided in Appendix E.2 of the supplementary material.

Theorem 2. Under the same assumptions as Theorem 1, it is sufficient to choose

$$
\begin{array} { r } { N \gtrsim \frac { \left( \operatorname* { m i n } \{ d ^ { 2 / 3 } L ^ { - 2 / 3 } , d ^ { 1 / 3 } \} + 1 \right) \log ^ { 5 / 3 } T } { \varepsilon ^ { 2 / 3 } } , \qquad } \\ { M K \gtrsim \operatorname* { m i n } \{ d \log T , L \} \log ^ { 2 } T , \quad \varepsilon _ { \mathrm { s c o r e } } ^ { 2 } \lesssim \varepsilon ^ { 2 } \log ^ { - 1 } T } \end{array}
$$

to achieve $\mathsf { T V } ( q _ { K } , p _ { Y _ { K } } ) \lesssim \varepsilon$ for the parallel sampler, where $T = K N / 2$ .

Finally, let us briefly compare our theory with the prior works. This theorem states that the parallel sampler achieves $\varepsilon$ -accuracy with respect to total variation distance using $O ( \operatorname* { m i n } \{ L , d \} \log ^ { 2 } ( L d / \varepsilon ) )$ parallel rounds, which is consistent with the results in Chen et al. (2024a); Gupta et al. (2024); Li and Jiao (2024). Moreover, our sampler requires only $\widetilde { \cal O } ( ( \operatorname* { m i n } \{ d ^ { 2 / 3 } L ^ { - 2 / 3 } , d ^ { 1 / 3 } \} + 1 ) \varepsilon ^ { - 2 / 3 } )$ parallel processors, which achieves a significant improvement over previous results.

# 4 Analysis

This section is devoted to establishing Theorem 1. Before proceeding, we rewrite the sampling process with the continuous index as following:

$$
\begin{array} { r l } {  { \frac { Y _ { \tau _ { k , n } } } { \sqrt { 1 - \tau _ { k , n } } } = \frac { Y _ { \tau _ { k , 0 } } } { \sqrt { 1 - \tau _ { k , 0 } } } + \frac { s _ { \tau _ { k , 0 } } ( Y _ { \tau _ { k , 0 } } ) } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } ( \tau _ { k , 0 } - \widehat { \tau } _ { k , 0 } ) } } \\ & { + \sum _ { i = 1 } ^ { n - 1 } \frac { s _ { \tau _ { k , i } } ( Y _ { \tau _ { k , i } } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) + \frac { s _ { \tau _ { k , n - 1 } } ( Y _ { \tau _ { k , n - 1 } } ) } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , n - 1 } - \tau _ { k , n } ) , } \end{array}
$$

and

$$
Y _ { \tau _ { k + 1 , 0 } } = \sqrt { \frac { 1 - \tau _ { k + 1 , 0 } } { 1 - \tau _ { k , N } } } Y _ { \tau _ { k , N } } + \sqrt { \frac { \tau _ { k + 1 , 0 } - \tau _ { k , N } } { 1 - \tau _ { k , N } } } Z _ { k } ,
$$

where $Y _ { \tau _ { 0 , 0 } } \sim \mathcal { N } ( 0 , I _ { d } )$ , $Z _ { k }$ i.i.d. ∼ $\mathcal { N } ( 0 , I _ { d } )$ . Here, $Y _ { \tau _ { k , 0 } }$ corresponds to the $Y _ { k }$ in discrete index, and $s _ { \tau } ( \cdot )$ denotes the estimate of score function $s _ { \tau } ^ { \star } ( \cdot )$ defined in (7). For ease of notations, we denote estimation error at the $( k , n )$ -th step as

$$
\begin{array} { r } { \widetilde { \varepsilon } _ { k , n } = \left\| s _ { \tau _ { k , n } } ( Y _ { \tau _ { k , n } } ) - s _ { \tau _ { k , n } } ^ { \star } ( Y _ { \tau _ { k , n } } ) \right\| _ { 2 } . } \end{array}
$$

Moreover, we shall show that $Y _ { \tau _ { k , n } }$ follows a distribution similar to that of $X _ { \tau _ { k , n } }$ defined in (4), which can also be expressed as

$$
X _ { \tau } \stackrel { \mathrm { d } } { = } \sqrt { 1 - \tau } X _ { 0 } + \sqrt { \tau } Z , \mathrm { w i t h } Z \sim { \mathcal N } ( 0 , I _ { d } ) , \mathrm { f o r } 0 \leq \tau \leq 1 .
$$

# 4.1 Step 1: introduce the auxiliary sequence

We first introduce an auxiliary sequence $\widehat { X } _ { k }$ , $1 \leq k \leq K$ , which has identical distribution with $X _ { \tau _ { k , 0 } }$ . Let $\Phi _ { \tau _ { 1 }  \tau _ { 2 } } ( x ) : = x _ { \tau _ { 2 } } \mid { } _ { x _ { \tau _ { 1 } } = x }$ defined through the following ODE

$$
\mathrm { d } \frac { x _ { \tau } } { \sqrt { 1 - \tau } } = - \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau .
$$

Then, we have the following result, which has been presented in Li and Jiao (2024). For completeness, we present its proof in Appendix F.1 in supplemental material:

Lemma 1. It can be shown that

$$
\Phi _ { \tau _ { 1 }  \tau _ { 2 } } ( X _ { \tau _ { 1 } } ) \stackrel { \mathrm { d } } { = } X _ { \tau _ { 2 } } .
$$

Moreover, assume that $\widehat { X } _ { 0 } \ { \overset { \mathrm { d } } { = } } \ X _ { \tau _ { 0 , 0 } }$ . Then we have for $0 \leq k < K$ ,

$$
\widehat { X } _ { k + 1 } = \sqrt { \frac { 1 - \tau _ { k + 1 , 0 } } { 1 - \tau _ { k , N } } } \Phi _ { \tau _ { k , 0 }  \tau _ { k , N } } ( \widehat { X } _ { k } ) + \sqrt { \frac { \tau _ { k + 1 , 0 } - \tau _ { k , N } } { 1 - \tau _ { k , N } } } Z _ { k } \stackrel { \mathrm { d } } { = } X _ { \tau _ { k + 1 , 0 } } ,
$$

where $Z _ { k } \stackrel { \scriptscriptstyle 1 . 1 . 0 . } { \sim } \mathcal { N } ( 0 , I _ { d } )$

Before introducing other auxiliary sequences, we introduce two notations for ease of presentation. Let

$$
\begin{array} { r l } & { \frac { y _ { \tau _ { k , 0 } } ( x _ { \tau _ { k , 0 } } ) } { \sqrt { 1 - \tau _ { k , n } } } = \frac { x _ { \tau _ { k , 0 } } } { \sqrt { 1 - \tau _ { k , 0 } } } + \frac { s _ { \tau _ { k , 0 } } ( x _ { \tau _ { k , 0 } } ) } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } ( \tau _ { k , 0 } - \widehat { \tau } _ { k , 0 } ) } \\ & { \qquad + \displaystyle \sum _ { i = 1 } ^ { n - 1 } \frac { s _ { \tau _ { k , i } } ( y _ { \tau _ { k , i } } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) + \frac { s _ { \tau _ { k , n - 1 } } ( y _ { \tau _ { k , n - 1 } } ) } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , n - 1 } - \tau _ { k , n } ) , } \\ & { \frac { x _ { \tau _ { k , n } } ( x _ { \tau _ { k , 0 } } ) } { \sqrt { 1 - \tau _ { k , n } } } = \frac { x _ { \tau _ { k , 0 } } } { \sqrt { 1 - \tau _ { k , 0 } } } + \displaystyle \int _ { \tau _ { k , n } } ^ { \tau _ { k , 0 } } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau , } \end{array}
$$

for $n = 1 , \ldots , N$ , which satisfies $x _ { \tau _ { k , n } } ( x _ { \tau _ { k , 0 } } ) = \Phi _ { \tau _ { k , 0 }  \tau _ { k , n } } ( x _ { \tau _ { k , 0 } } )$ . Moreover, we define typical sets $\xi _ { k }$ for $0 \leq k \leq K - 1$ as

$$
\begin{array} { r } { \mathcal { E } _ { k } : = \left\{ \begin{array} { l l } { \{ x _ { \tau _ { k , 0 } } : x _ { \tau _ { k , n } } ( x _ { \tau _ { k , 0 } } ) \in \widetilde { \mathcal { S } } _ { \tau _ { k , n } } \cap \mathcal { L } _ { \tau _ { k , n } } , } \\ { \quad y _ { \tau _ { k , n } } ( x _ { \tau _ { k , 0 } } ) \in \mathcal { S } _ { \tau _ { k , n } } , \forall 0 \leq n \leq N - 1 \} , } & { \mathrm { i f ~ } L > d \log T , } \\ { \emptyset } & { \mathrm { i f ~ } L \leq d \log T , } \end{array} \right. } \end{array}
$$

where $ { \widetilde { \boldsymbol { S } } } _ { \tau }$ , $S _ { \tau }$ and $\scriptstyle { \mathcal { L } } _ { \tau }$ denote high probability sets

$$
\begin{array} { r l } & { \mathcal { S } _ { \tau } : = \{ x : - \log p x _ { \tau } ( x ) \leq \theta d \log T \} , \quad \widetilde { \mathcal { S } } _ { \tau } : = \{ x : - \log p x _ { \tau } ( x ) \leq \theta d \log T - \log 2 \} , } \\ & { \mathcal { L } _ { \tau } : = \left\{ x : \tau \| s _ { \tau } ^ { \star } ( x ^ { \prime } ) - s _ { \tau } ^ { \star } ( x ) \| _ { 2 } \leq L \| x ^ { \prime } - x \| _ { 2 } , \forall \| x ^ { \prime } - x \| _ { 2 } \leq \frac { C \sqrt { d \tau \log T } } { L } \right\} , } \end{array}
$$

with $\theta$ a sufficiently large constant. Moreover, we define $\mathcal { E } _ { K } = \mathbb { R } ^ { d }$ .

Based on the above definitions, we introduce a new auxiliary reverse process $\smash { \widetilde { X } } _ { k }$ for $0 \le k \le K$ , which transforms similar with ODE in Lemma 1, but removes samples out of the typical set $\xi _ { k }$ . Specifically, let $\tilde { X } _ { 0 } = \hat { X } _ { 0 }$ for $\hat { X } _ { 0 } \in \mathcal { E } _ { 0 }$ and $\tilde { X } _ { 0 } = \infty$ otherwise. Then it transits following the probability

$$
\begin{array} { r } { p _ { \widetilde { X } _ { k + 1 } | \widetilde { X } _ { k } } ( x | x _ { k } ) = \left\{ \begin{array} { l l } { p _ { \widehat { X } _ { k + 1 } | \widehat { X } _ { k } } ( x \vert x _ { k } ) \mathbb { 1 } ( x \in \mathcal { E } _ { k + 1 } ) } \\ { \quad + \int _ { \mathcal { E } _ { k + 1 } ^ { c } } p _ { \widehat { X } _ { k + 1 } | \widehat { X } _ { k } } ( x _ { k + 1 } \vert x _ { k } ) \mathrm { d } x _ { k + 1 } \delta _ { \infty } , } & { x _ { k } \neq \infty , } \\ { \delta _ { \infty } , } & { x _ { k } = \infty , } \end{array} \right. } \end{array}
$$

where according to Lemma $^ { 1 }$ , $p _ { \widehat { X } _ { k + 1 } | \widehat { X } _ { k } } ( x | x _ { k } ) = \phi \left( x | x _ { \tau _ { k , N } } ( x _ { k } ) , \sigma _ { k } ^ { 2 } \right)$ , and $\phi ( x | \mu , \sigma ^ { 2 } )$ denotes the probability density function of Gaussian distribution with mean vector $\mu$ and covariance matrix $\sigma ^ { 2 } I _ { d }$ , $\mathcal { E } _ { k } ^ { \mathrm { c } }$ denotes the complementary set of $\xi _ { k }$ $\propto \notin \mathcal { E } _ { k } ^ { \mathrm { c } }$ ), and

$$
\sigma _ { k } ^ { 2 } = \frac { \tau _ { k + 1 , 0 } - \tau _ { k , N } } { 1 - \tau _ { k , N } } .
$$

Moreover, we define another reverse sequence $\widetilde { Y _ { k } }$ similar with $Y _ { k }$ as follows. First, $\widetilde { Y } _ { 0 }$ is initialized as $Y _ { 0 }$ for $Y _ { 0 } \in \mathcal { E } _ { 0 }$ and $\tilde { Y _ { 0 } } = \infty$ otherwise. Then for $k = 0 , \cdots , K - 1$ , the conditional density of $\widetilde { Y } _ { k + 1 }$ given $\tilde { Y _ { k } } = y _ { k }$ is

$$
\begin{array} { r } { p _ { \widetilde { Y } _ { k + 1 } | \widetilde { Y } _ { k } } ( y | y _ { k } ) = \left\{ \begin{array} { l l } { p _ { Y _ { k + 1 } | Y _ { k } } ( y | y _ { k } ) \mathbb { 1 } ( y \in \mathcal { E } _ { k + 1 } ) } \\ { \quad + \int _ { \mathcal { E } _ { k + 1 } ^ { c } } p _ { Y _ { k + 1 } | Y _ { k } } ( y _ { k + 1 } | y _ { k } ) \mathrm { d } y _ { k + 1 } \delta _ { \infty } , } & { y _ { k } \neq \infty , } \\ { \delta _ { \infty } , } & { y _ { k } = \infty , } \end{array} \right. } \end{array}
$$

where according to the algorithm design $P _ { Y _ { k + 1 } | Y _ { k } } ( y | y _ { k } ) = \phi \left( y | y _ { \tau _ { k , N } } ( y _ { k } ) , \sigma _ { k } ^ { 2 } \right)$ , with $\sigma _ { k } ^ { 2 }$ defined in (23). The following lemma states some basic properties about the above auxiliary sequences. The proof is postponed to Appendix F.2 in supplemental material.

Lemma 2. The following properties hold:

$$
\begin{array} { r l } & { p _ { \widetilde { X } _ { k } } ( x ) = 0 , \quad p _ { \widetilde { Y } _ { k } } ( y ) = 0 , \quad \mathrm { f o r } x , y \in \mathcal { E } _ { k } ^ { \mathrm { c } } ; } \\ & { p _ { \widetilde { X } _ { k } } ( x ) \leq p _ { \widehat { X } _ { k } } ( x ) , \quad p _ { \widetilde { Y } _ { k } } ( y ) \leq p _ { Y _ { k } } ( y ) , \quad \mathrm { f o r } x , y \neq \infty . } \end{array}
$$

# 4.2 Step 2: decompose the error terms

Recalling the definition of $\widehat { X } _ { k }$ , and the fact that $p _ { \widetilde { Y } _ { K } } ( x ) \le p _ { Y _ { K } } ( x )$ and $p _ { \widetilde { X } _ { K } } ( x ) \le p _ { \widehat { X } _ { K } } ( x )$ , we have

$$
\begin{array} { r l } { \Gamma \vee ( q _ { K } , p _ { \mathrm { Y K } } ) = \Pi \big ( p _ { \tilde { X } _ { K } } , p _ { \mathrm { Y K } } \big ) = \int ( p _ { \tilde { X } _ { K } } ( x ) - p _ { Y _ { K } } ( x ) ) \mathbf { 1 } \{ p _ { \tilde { X } _ { K } } ( x ) > p _ { Y _ { K } } ( x ) \} \mathrm { d } x } \\ { \leq \int ( p _ { \tilde { X } _ { K } } ( x ) - p _ { \tilde { Y } _ { K } } ( x ) ) \mathbf { 1 } \{ p _ { \tilde { X } _ { K } } ( x ) > p _ { \tilde { Y } _ { K } } ( x ) \} \mathrm { d } x } \\ { = \int ( p _ { \tilde { Y } _ { K } } ( x ) - p _ { \tilde { X } _ { K } } ( x ) ) \mathbf { 1 } \{ p _ { \tilde { Y } _ { K } } ( x ) > p _ { \tilde { X } _ { K } } ( x ) \} \mathrm { d } x + P ( \tilde { Y } _ { K } = \infty ) } \\ { \leq \int ( p _ { \tilde { Y } _ { K } } ( x ) - p _ { \tilde { X } _ { K } } ( x ) ) \mathbf { 1 } \{ p _ { \tilde { Y } _ { K } } ( x ) > p _ { \tilde { X } _ { K } } ( x ) \} \mathrm { d } x + P ( \tilde { Y } _ { K } = \infty ) } \\ { \overset { ( a ) } { \leq } \mathbf { T } \big ( p _ { \tilde { Y } _ { K } } , p _ { \tilde { X } _ { K } } \big ) + P \big ( \tilde { X } _ { K } = \infty \big ) } \\ { \overset { ( b ) } { \leq } \mathbf { T } \big ( p _ { \tilde { Y } _ { K } } , p _ { \tilde { X } _ { K } } \big ) + P \big ( \tilde { X } _ { K } = \infty \big ) } \\ { \overset { ( b ) } { \leq } \mathbf { T } \big ( p _ { \tilde { Y } _ { K } } , p _ { \tilde { X } _ { K } } \big ) + \mathbf { T } \big \nabla ( p _ { \tilde { X } _ { K } } , p _ { \tilde { X } _ { K } } \big ) , }  \end{array}
$$

where (a) uses the fact that

$$
\begin{array} { r l r } {  { \mathsf { T V } ( p _ { \widetilde { Y } _ { K } } , p _ { \widetilde { X } _ { K } } ) = \int ( p _ { \widetilde { Y } _ { K } } ( x ) - p _ { \widetilde { X } _ { K } } ( x ) ) \mathbb { 1 } \{ p _ { \widetilde { Y } _ { K } } ( x ) > p _ { \widetilde { X } _ { K } } ( x ) \} \mathrm { d } x } } \\ & { } & { + \operatorname* { m a x } \{ P ( \widetilde { Y } _ { K } = \infty ) - P ( \widetilde { X } _ { K } = \infty ) , 0 \} , } \end{array}
$$

and (b) uses the fact that

$$
P ( \tilde { X } _ { K } = \infty ) \leq \mathsf { T V } ( p _ { \tilde { X } _ { K } } , p _ { \widehat { X } _ { K } } ) .
$$

Now we intend to bound the two terms in the right-hand-side of (27) separately. For the first term, by using Pinsker’s inequality, we have

$$
\begin{array} { r l } & { \quad \mathsf { T V } ^ { 2 } ( p _ { \widetilde { Y } _ { K } } , p _ { \widetilde { X } _ { K } } ) \le \displaystyle \frac { 1 } { 2 } \mathsf { K L } \left( p _ { \widetilde { X } _ { K } } \| p _ { \widetilde { Y } _ { K } } \right) \le \displaystyle \frac { 1 } { 2 } \mathsf { K L } \left( p _ { \widetilde { X } _ { 0 } , \ldots , \widetilde { X } _ { K } } \| p _ { \widetilde { Y } _ { 0 } , \ldots , \widetilde { Y } _ { K } } \right) } \\ & { = \displaystyle \frac { 1 } { 2 } \mathsf { K L } \left( p _ { \widetilde { X } _ { 0 } } \| p _ { \widetilde { Y } _ { 0 } } \right) + \displaystyle \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } _ { x _ { k } \sim \widetilde { X } _ { k } } \mathsf { K L } ( p _ { \widetilde { X } _ { k + 1 } | \widetilde { X } _ { k } } ( \cdot | x _ { k } ) \| p _ { \widetilde { Y } _ { k + 1 } | \widetilde { Y } _ { k } } ( \cdot | x _ { k } ) ) } \\ & { \overset { \mathrm { ( a ) } } { \le } \displaystyle \frac { 1 } { 2 } \mathsf { K L } \left( p _ { \widehat { X } _ { 0 } } \| p _ { Y _ { 0 } } \right) + \displaystyle \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } _ { x _ { k } \sim \widetilde { X } _ { k } } \mathsf { K L } ( p _ { \widehat { X } _ { k + 1 } | \widehat { X } _ { k } } ( \cdot | x _ { k } ) \| p _ { Y _ { k + 1 } | Y _ { k } } ( \cdot | x _ { k } ) ) , } \end{array}
$$

where (a) is proved in Appendix D.4 of supplemental material. For the second term, considering $x \in \mathcal { E } _ { k }$ , we have

$$
\begin{array} { l } { p _ { \widetilde { X } _ { k } } ( x ) = \displaystyle \int _ { \mathcal { E } _ { k - 1 } } p _ { \widetilde { X } _ { k } | \widetilde { X } _ { k - 1 } } ( x | x _ { k - 1 } ) p _ { \widetilde { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } } \\ { \displaystyle = \int _ { \mathcal { E } _ { k - 1 } } p _ { \widetilde { X } _ { k } | \widetilde { X } _ { k - 1 } } ( x | x _ { k - 1 } ) p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } - \Delta _ { k } ( x ) } \\ { \displaystyle = \int _ { \mathcal { E } _ { k - 1 } } p _ { \widehat { X } _ { k } | \widehat { X } _ { k - 1 } } ( x | x _ { k - 1 } ) p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } - \Delta _ { k } ( x ) } \\ { \displaystyle = p _ { \widehat { X } _ { k } } ( x ) - \displaystyle \int _ { \mathcal { E } _ { k - 1 } ^ { c } } p _ { \widehat { X } _ { k } | \widehat { X } _ { k - 1 } } ( x | x _ { k - 1 } ) p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } - \Delta _ { k } ( x ) , } \end{array}
$$

where

$$
\Delta _ { k } ( x ) = \int _ { \mathscr { E } _ { k - 1 } } p _ { \widetilde { X } _ { k } | } \widetilde { x } _ { k - 1 } ( x \mid x _ { k - 1 } ) \left( p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) - p _ { \widetilde { X } _ { k - 1 } } ( x _ { k - 1 } ) \right) \mathrm { d } x _ { k - 1 } .
$$

Thus we have

$$
\begin{array} { r l } & { \mathsf { T V } ( p _ { \bar { X } _ { k } } , p _ { \hat { X } _ { k } } ) = \displaystyle { \int p _ { \hat { X } _ { k } } ( x ) - p _ { \bar { X } _ { k } } ( x ) \mathrm { d } x } } \\ & { \quad \quad \quad \quad \quad = \displaystyle { \int _ { \mathcal { E } _ { k } ^ { \mathrm { e } } } p _ { \hat { X } _ { k } } ( x ) \mathrm { d } x } + \int _ { \mathcal { E } _ { k } } p _ { \hat { X } _ { k } } ( x ) - p _ { \tilde { X } _ { k } } ( x ) \mathrm { d } x } \\ & { \quad \quad \quad \quad \stackrel { \mathrm { ( a ) } } { = } P ( \widehat { X } _ { k } \in \mathcal { E } _ { k } ^ { \mathrm { c } } ) + \displaystyle { \int _ { \mathcal { E } _ { k } } \int _ { \mathcal { E } _ { k - 1 } ^ { \mathrm { c } } } p _ { \hat { X } _ { k } | \hat { X } _ { k - 1 } } ( x | x _ { k - 1 } ) p _ { \hat { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } \mathrm { d } x } } \\ & { \quad \quad \quad \quad \quad + \displaystyle { \int _ { \mathcal { E } _ { k } } \Delta _ { k } ( x ) \mathrm { d } x } , } \end{array}
$$

where (a) inserts (29). Notice that

$$
\begin{array} { r l } {  { \int _ { \mathcal { E } _ { k } } \int _ { \mathcal { E } _ { k - 1 } ^ { c } } p _ { \widehat { X } _ { k } \mid \widehat { X } _ { k - 1 } } ( x \mid x _ { k - 1 } ) p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } \mathrm { d } x \leq \int _ { \mathcal { E } _ { k - 1 } ^ { c } } p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } } } \\ & { \qquad = P ( \widehat { X } _ { k - 1 } \in \mathcal { E } _ { k - 1 } ^ { c } ) , } \\ & { \int _ { \mathcal { E } _ { k } } \Delta _ { k } ( x ) \mathrm { d } x = \int _ { \mathcal { E } _ { k } } \int _ { \mathcal { E } _ { k - 1 } } p _ { \widetilde { X } _ { k } \mid \widetilde { X } _ { k - 1 } } ( x \mid x _ { k - 1 } ) \Big ( p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) - p _ { \widetilde { X } _ { k - 1 } } ( x _ { k - 1 } ) \Big ) \mathrm { d } x _ { k - 1 } \mathrm { d } x } \\ & { \leq \int _ { \mathcal { E } _ { k - 1 } } p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) - p _ { \widetilde { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } \leq \mathsf { T V } ( p _ { \widetilde { X } _ { k - 1 } } , p _ { \widehat { X } _ { k - 1 } } ) . } \end{array}
$$

Inserting into (30) and by using recursion, we have

$$
\begin{array} { r } { \mathsf { T V } ( p _ { \widetilde { X } _ { K } } , p _ { \widehat { X } _ { K } } ) \le \mathsf { T V } ( p _ { \widetilde { X } _ { K - 1 } } , p _ { \widehat { X } _ { K - 1 } } ) + P ( \widehat { X } _ { K - 1 } \in \mathcal { E } _ { K - 1 } ^ { \mathrm { c } } ) + P ( \widehat { X } _ { K } \in \mathcal { E } _ { K } ^ { \mathrm { c } } ) } \end{array}
$$

$$
\leq 2 \sum _ { k = 0 } ^ { K - 1 } P ( \widehat { X } _ { k } \in \mathcal { E } _ { k } ^ { \mathrm { c } } )
$$

considering that

$$
\mathsf { T V } ( p _ { \widetilde { X } _ { 0 } } , p _ { \widehat { X } _ { 0 } } ) = P ( \widehat { X } _ { 0 } \in \mathcal { E } _ { 0 } ) , \quad P ( \widehat { X } _ { K } \in \mathcal { E } _ { K } ^ { \mathrm { c } } ) = 0 .
$$

Inserting (28) and (31) into (27), we have

$$
\begin{array} { r } { \mathsf { T V } \big ( q _ { K } , p _ { Y _ { K } } \big ) \le \sqrt { \frac { 1 } { 2 } \mathsf { K L } \big ( p _ { \hat { X } _ { 0 } } \| p _ { Y _ { 0 } } \big ) + \frac { 1 } { 2 } \displaystyle \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } _ { x _ { k } \sim p _ { \widetilde X _ { k } } } \left[ \mathsf { K L } \big ( p _ { \hat { X } _ { k + 1 } | \hat { X } _ { k } } \left( \cdot | x _ { k } \right) \| p _ { Y _ { k + 1 } | Y _ { k } } \left( \cdot | x _ { k } \right) \big ) \right] } } \\ { + 2 \displaystyle \sum _ { k = 0 } ^ { K - 1 } P ( \widehat X _ { k } \in \mathcal E _ { k } ^ { \circ } ) . } \end{array}
$$

# 4.3 Step 3: control the KL divergence between $p _ { \widehat { X } _ { k + 1 } | \widehat { X } _ { k } }$ and $p _ { Y _ { k + 1 } | Y _ { k } }$

The KL divergence between conditional distributions $p _ { \widehat { X } _ { k + 1 } | \widehat { X } _ { k } }$ and $p _ { Y _ { k + 1 } | Y _ { k } }$ is

$$
\mathsf { K L } \big ( p _ { \hat { X } _ { k + 1 } \mid \hat { X } _ { k } } \left( \cdot \mid x _ { k } \right) \parallel p _ { Y _ { k + 1 } \mid Y _ { k } } \left( \cdot \mid x _ { k } \right) \big ) = \frac { 1 - \tau _ { k + 1 , 0 } } { 2 ( \tau _ { k + 1 , 0 } - \tau _ { k , N } ) } \lVert y _ { \tau _ { k , N } } \left( x _ { k } \right) - x _ { \tau _ { k , N } } \left( x _ { k } \right) \rVert _ { 2 } ^ { 2 } .
$$

This can be immediately verified by recalling that $\widehat { X } _ { k + 1 } | \widehat { X } _ { k } = x _ { k }$ and $Y _ { k + 1 } | Y _ { k } = x _ { k }$ are both normal distributions with the same variance τk+1,0−τk,N and different means $\sqrt { \frac { 1 - \tau _ { k + 1 , 0 } } { 1 - \tau _ { k , N } } } x _ { \tau _ { k , N } } ( x _ { k } )$ and $\sqrt { \frac { 1 - \tau _ { k + 1 , 0 } } { 1 - \tau _ { k , N } } } y _ { \tau _ { k , N } } ( x _ { k } )$ , respectively.

Thus the core of this step is to control the estimation error. Before proceeding, we introduce additional two sequences of auxiliary variables

$$
\begin{array} { r l } & { \frac { y _ { \tau _ { k , n } } ^ { * } ( x _ { \tau _ { k , 0 } } ) } { \sqrt { 1 - \tau _ { k , n } } } = \frac { x _ { \tau _ { k , 0 } } } { \sqrt { 1 - \tau _ { k , 0 } } } + \frac { s _ { \tau _ { k , 0 } } ^ { * } ( x _ { \tau _ { k , 0 } } ) } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } ( \tau _ { k , 0 } - \widehat { \tau } _ { k , 0 } ) } \\ & { \qquad + \displaystyle { \sum _ { i = 1 } ^ { n - 1 } \frac { s _ { \tau _ { k , i } } ^ { * } ( y _ { \tau _ { k , i } } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) } + \frac { s _ { \tau _ { k , n - 1 } } ^ { * } ( y _ { \tau _ { k , n - 1 } } ) } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , n - 1 } - \tau _ { k , n } ) , } \\ & { \frac { z _ { \tau _ { k , n } } ^ { * } ( x _ { \tau _ { k , 0 } } ) } { \sqrt { 1 - \tau _ { k , n } } } = \frac { x _ { \tau _ { k , 0 } } } { \sqrt { 1 - \tau _ { k , 0 } } } + \frac { s _ { \tau _ { k , 0 } } ^ { * } ( x _ { \tau _ { k , 0 } } ) } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } ( \tau _ { k , 0 } - \widehat { \tau } _ { k , 0 } ) } \\ &  \qquad + \displaystyle { \sum _ { i = 1 } ^ { n - 1 } \frac { s _ { \tau _ { k , i } } ^ { * } ( x _ { \tau _ { k , i } } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) } + \frac \end{array}
$$

and for $n = 1 , \ldots , N$ $z _ { \tau _ { k , n } } ^ { \star }$ on $x _ { \tau _ { k , 0 } }$ . For convenience, in the following proof we shall omit the dependence of without ambiguity. Based on these notations, we define the estimation error $x _ { \tau _ { k , n } }$ , $y _ { \tau _ { k , n } }$ , $y _ { \tau _ { k , n } } ^ { \star }$ ,

$$
\xi _ { k , n } ( x _ { \tau _ { k , 0 } } ) : = \frac { y _ { \tau _ { k , n } } ^ { \star } - y _ { \tau _ { k , n } } } { \sqrt { 1 - \tau _ { k , n } } } + \frac { x _ { \tau _ { k , n } } - z _ { \tau _ { k , n } } ^ { \star } } { \sqrt { 1 - \tau _ { k , n } } } .
$$

Moreover, we define the matrix

$$
\Sigma _ { \tau } ( x ) = \mathsf { C o v } [ Z | \sqrt { 1 - \tau } X _ { 0 } + \sqrt { \tau } Z = x ] ,
$$

where $\mathsf { C o v } [ \cdot ]$ denotes the covariance matrix. The following lemma controls the estimation error $\xi _ { k , n }$ . The proof is postponed to Appendix D.1.

Lemma 3. For any $k$ and $n$ , with probability at least $1 - T ^ { - 1 0 0 }$ ,

$$
\mathbb { E } _ { x _ { \tau _ { k , 0 } } \sim p _ { \widehat { X _ { k } } } } \left[ \| \xi _ { k , n } ( x _ { \tau _ { k , 0 } } ) \| _ { 2 } ^ { 2 } \right] \lesssim \frac { d \log ^ { 4 } T } { T ^ { 3 } } \operatorname* { m i n } \Big \{ \frac { N d \widehat { \tau } _ { k , - 1 } \log T } { T ( 1 - \widehat { \tau } _ { k , - 1 } ) } + \int _ { \tau _ { k , n } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } \left[ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) \right] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau ,
$$

$$
\frac { N L ^ { 2 } \widehat { \tau } _ { k , - 1 } \log T } { T ( 1 - \widehat { \tau } _ { k , - 1 } ) } \biggr \} + \frac { N \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 0 } ^ { n - 1 } \widehat { \tau } _ { k , i } ( 1 - \widehat { \tau } _ { k , i } ) ^ { - 1 } \varepsilon _ { k , i } ^ { 2 } ,
$$

where $\varepsilon _ { k , i } ^ { 2 }$ is defined in (12).

With the above relation, we can bound the divergence as following. The proof is postponed to Appendix D.2 in supplemental material.

Lemma 4. According to Lemma $\boldsymbol { \mathcal { B } }$ , it can be shown that

$$
\begin{array} { r l } & { \displaystyle \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } _ { \boldsymbol { x } _ { \tau _ { k , 0 } } \sim p _ { \widetilde { \boldsymbol X } _ { k } } } \left[ \mathsf { K L } \left( p _ { \widehat { X } _ { k + 1 } | \widehat { X } _ { k } } \left( \cdot | \boldsymbol { x } _ { \tau _ { k , 0 } } \right) \parallel p _ { Y _ { k + 1 } | Y _ { k } } \left( \cdot | \boldsymbol { x } _ { \tau _ { k , 0 } } \right) \right) \right] } \\ & { \lesssim \displaystyle \frac { K d \log ^ { 5 } T } { T ^ { 3 } } \operatorname* { m i n } \left\{ d , L ^ { 2 } \right\} + \varepsilon _ { \mathrm { s c o r e } } ^ { 2 } \log T , } \end{array}
$$

$\varepsilon _ { \mathsf { s c o r e } } ^ { 2 }$

# 4.4 Step 4: putting everything together

The remaining terms in (32) can be bounded through the following lemma, whose proof can be found in Appendix D.3 in supplemental material.

Lemma 5. Under our choice of learning schedule (9), we have

$$
\begin{array} { l } { { \displaystyle { \mathsf { K L } } \big ( p _ { \widehat X _ { 0 } } \| p _ { \widehat Y _ { 0 } } \big ) \le \frac { 1 } { T ^ { 1 0 } } } , \ ~ } \\ { { \displaystyle \sum _ { k = 0 } ^ { K - 1 } P ( \widehat X _ { k } \in \mathcal E _ { k } ^ { \mathrm { c } } ) \lesssim \left( \frac { d ^ { 2 } \log ^ { 5 } T } { T ^ { 2 } } + \varepsilon _ { \mathrm { s c o r e } } ^ { 2 } \log T \right) \mathbb { 1 } ( d \log T < L ) } . } \end{array}
$$

Inserting (36), (37), and (38) into (32) leads to

$$
\begin{array} { r l } & { \mathsf { T V } \left( q _ { K } , p _ { Y _ { K } } \right) \lesssim \sqrt { \frac { 1 } { T ^ { 1 0 } } + \frac { K d \log ^ { 5 } T } { T ^ { 3 } } } \operatorname* { m i n } \Big \{ d , L ^ { 2 } \Big \} + \varepsilon _ { \mathrm { s c o r e } } ^ { 2 } \log T } \\ & { \quad \quad \quad \quad + \left( \frac { d ^ { 2 } \log ^ { 5 } T } { T ^ { 2 } } + \varepsilon _ { \mathrm { s c o r e } } ^ { 2 } \log T \right) \mathbb { 1 } ( d \log T < L ) } \\ & { \quad \quad \quad \lesssim \frac { \operatorname* { m i n } \{ d ^ { 3 / 2 } , d L ^ { 1 / 2 } , d ^ { 1 / 2 } L ^ { 3 / 2 } \} \log ^ { 4 } T } { T ^ { 3 / 2 } } + \varepsilon _ { \mathrm { s c o r e } } \log ^ { 1 / 2 } T , } \end{array}
$$

and we conclude the proof here.

# 5 Discussion

In this paper, we establish a faster convergence rate for generative diffusion models under a relaxed Lipschitz condition, which requires a number of $\operatorname* { m i n } \{ d , d ^ { 2 / 3 } L ^ { 1 / 3 } , d ^ { 1 / 3 } L \} \varepsilon ^ { - 2 / 3 } \log ^ { 8 / 3 } T$ iterations to achieve an $\varepsilon$ accuracy in terms of TV distance, where $L$ denotes the non-uniform Lipschitz constant. As a result, our results demonstrate improvements over the convergence theory without exploiting smoothness across the entire range of $L$ , and accommodates a broader class of distributions, e.g., Gaussian mixture model. In addition, our analysis requires only $\varepsilon$ estimation errors, implying the robustness of the algorithm to imperfect score estimations. Furthermore, we extend the result to the parallel implementation of the sampler. However, several open questions remain. For example, deriving instance-dependent bounds for other settings, such as accelerated samplers or data distributions with low-dimensional structures, which we leave for future research. Moreover, the benefit of the randomized design in our work relies heavily on the deterministic nature of ODE process. It is still unclear for us how to adapt this approach to deal with the inherent stochasticity in SDEs while maintaining similar improvements. This will also be left as future work. In addition, it may be feasible to apply your analysis framework to improve the bound for other variants of samplers such as the Langevin algorithm. Furthermore, estimating the Lipschitz constant in real-world cases would be highly beneficial, potentially broadening the applicability of our results.

![](images/figures/diffusion-convergence-rate-fig-0002.jpg)  
Figure 2: Sampling error of the proposed sampler and fitted rate $T  \Theta ( \log ^ { 4 } T / T ^ { 3 } )$ : (a) $d = 1 0 , k = 1 0$ ; (b) $d = 1 0 0 , k = 1 0$ ; (c) $d = 5 0 0 , k = 1 0 0$ .

# Acknowledgments

Gen Li is supported in part by the Chinese University of Hong Kong Direct Grant for Research and the Hong Kong Research Grants Council ECS 2191363.

# A Numerical experiments

We conduct numerical experiments to validate our theoretical results. For ease of computing, we select a Gaussian distribution as the target distribution. This choice ensures that all $Y _ { k , n }$ in the implementation of the proposed sampler follow a Gaussian distribution and that the KL divergence between $Y _ { k , 0 }$ and $X _ { 1 }$ has a closed-form expression. Moreover, since our primary focus is on the convergence rate, we assume access to the exact score function $s _ { t } ^ { \star } ( \cdot )$ .

The target distribution $p _ { 0 }$ is a $d$ -dimensional Gaussian distribution with zero mean and a diagonal covariance matrix. The first $k$ diagonal entries are uniformly distributed within the interval [0, 10], while the remaining $d - k$ diagonal entries are set to zero. We implement the proposed sampler in Section 2.2 with $K = 1 0$ , and $N = 2 T / K$ . For different number of iterations $T$ , we compute the distribution of output $Y _ { K }$ , and its KL divergence with the distribution $q _ { K }$ of $X _ { \tau _ { K , 0 } }$ , which is approximately the starting point of the forward process.

The results are presented in Figure 2. The blue line represents the empirical results, and the black line corresponds to the theoretical rate $O ( \log ^ { 4 } T / T ^ { 3 } )$ . According to Theorem 1, our theoretical analysis predicts a convergence rate of $O ( \mathsf { p o l y } ( \log T ) / T ^ { 3 } )$ in terms of KL divergence, which is consistent with empirical observations. This further confirms that our sampler achieves a KL divergence convergence rate of $O ( \log ^ { 4 } T / T ^ { 3 } )$ in terms of KL divergence, implying a total variation(TV) distance convergence rate of $O ( \log ^ { 2 } T / T ^ { 3 / 2 } )$ . Finally, we remark that Theorem 1 establishes a convergence rate of $O ( \log ^ { 4 } T / T ^ { 3 / 2 } )$ . Compared to empirical results, this bound is suboptimal in terms of its dependence on logarithmic factors. Refining this dependency requires further effort and is left for future work.

# B Comparison with previous works

To compare Theorem 1 with previous results, we illustrate the TV distance achieved by various theories for a fixed number of iterations $T$ . The corresponding results are presented in Figure 3. Notably, when $T = O ( d )$ , the results from Benton et al. (2023); Li and Cai (2024); Li and Yan (2024a) reduce to a trivial bound $\varepsilon = 1$ . In comparison, for both $T = O ( d )$ and √ $\underline { T } = O ( d ^ { 2 } )$ , our result achieves the best result across a full range of $L$ , and improves previous results when ${ \sqrt { d } } \lesssim L \lesssim d$ .

![](images/figures/diffusion-convergence-rate-fig-0003.jpg)  
Figure 3: TV distance $\varepsilon$ achieved by Theorem 1 and previous results with left: $T = O ( d )$ ; middle: $T =$ $O ( d ^ { 3 / 2 } )$ ; right: $T = O ( d ^ { 2 } )$ .

A lot of technical efforts have been devoted to achieving the improvements stated in Section 3.

a. The absence of a uniform Lipschitz bound (see Definition 2) brings new challenges to the discretization analysis. Prior works (Chen et al., 2024b; Li and Jiao, 2024) leveraged the log-concavity of $p _ { X _ { \tau } | X _ { \tau + \delta } }$ for some small $\delta$ to control the one-step discretization error (see Lemma 1 and (B.2)-(B.4) in Chen et al. (2024b)). However, this approach fails under the high-probability bound employed in this work. To address this issue, we directly handle this derivative based on its definition, decomposing it into two components: one depending on the operator norm of the Jacobian matrix and the other on the data dimension $d$ , and then statistically bounding each term (see Lemma 10). Although the resulting bound is of a similar order, the analytical techniques under the uniform bound assumption and the high-probability bound assumption are fundamentally different.

b. The lack of a uniform Lipschitz condition poses significant challenges for controlling error propagation across multiple steps, while a naive stepwise analysis may result in suboptimal bounds. To overcome this challenge, we introduce two auxiliary sequences, $\smash { \widetilde { X } } _ { k }$ and $\widetilde { Y _ { k } }$ , which constrain $\widehat { X } _ { k }$ and $Y _ { k }$ to lie within a typical set (see Step 1 in Section 4 and Lemma 2). We then relate the TV distance between $\widehat { X } _ { k }$ and $Y _ { k }$ to that between $\smash { \widetilde { X } } _ { k }$ and $\widetilde { Y _ { k } }$ (see Step 2 in Section 4), by analyzing how error propagation affects the probability that $Y _ { k , n } | Y _ { k , 0 } = x _ { \tau _ { k , 0 } }$ falls outside the typical set (see Lemma 5). Within this typical set, we establish a uniform bound for score functions that depends explicitly on the data dimension $d$ (see Lemma 12). As a result, the error at the $( k , n )$ -th step, $\| y _ { \tau _ { k , n } } - x _ { \tau _ { k , n } } \|$ , is effectively controlled by the discretization error $\zeta _ { k , n }$ and estimation error terms related to $\widetilde { \varepsilon } _ { k , i }$ (see (75)), thereby ensuring stable error propagation throughout the process.

c. To make the result adaptive to $L$ and applicable for the minimal condition on the target data distribution (i.e., $L = \infty$ ), new bounds for the discretization error and the number of rounds $K$ are derived. Based on the error propagation analysis discussed earlier, the lower bound for $K$ is $O ( \operatorname* { m i n } \{ d \log T , L \} \log T )$ which remains bounded even when $L = \infty$ . Additionally, to obtain a discretization error adaptive to $L$ , we derive a new bound for the expectation of the product $\lVert J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \rVert _ { 2 } ^ { 2 }$ by carefully analyzing the structure of the Jacobian matrix $J _ { \tau } ( \cdot )$ (see Lemma 9). The new bound $O ( d \operatorname* { m i n } \{ d , L ^ { 2 } \} )$ helps to derive a corresponding discretization error of $\tilde { O } ( d \operatorname* { m i n } \{ d , L ^ { 2 } \} )$ (see Lemma 8), which depends only on $d$ when $L$ is large.

# C Computations of Examples

# C.1 Computation of Example 1

It is easy to check that the score function is

$$
s _ { t } ^ { \star } ( x ) = - \Sigma _ { t } ^ { - 1 } x ,
$$

where $\Sigma _ { t }$ is a diagonal matrix with the $( i , i )$ -th entry equal to

$$
( \Sigma _ { t } ) _ { i , i } = \overline { { { \alpha } } } _ { t } \sigma _ { i } ^ { 2 } + 1 - \overline { { { \alpha } } } _ { t } .
$$

Thus we have

$$
\| s _ { t } ^ { \star } ( x ) - s _ { t } ^ { \star } ( y ) \| _ { 2 } = \| \Sigma _ { t } ^ { - 1 } ( x - y ) \| _ { 2 } ,
$$

and

$$
\| \Sigma _ { t } ^ { - 1 } \| _ { 2 } = \frac { 1 } { \overline { { \alpha } } _ { t } \operatorname* { m i n } \sigma _ { i } ^ { 2 } + 1 - \overline { { \alpha } } _ { t } } = \frac { 1 } { 1 - \overline { { \alpha } } _ { t } } .
$$

Then we complete the proof.

# C.2 Computation of Example 2

It is easy to check that the score function is

$$
s _ { t } ^ { \star } ( x ) = - \frac { 1 } { \sigma _ { t } ^ { 2 } } \sum _ { h = 1 } ^ { H } \pi _ { h } ( x ) ( x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } ) = - \frac { x } { \sigma _ { t } ^ { 2 } } + \frac { \sqrt { \overline { { \alpha } } _ { t } } } { \sigma _ { t } ^ { 2 } } \sum _ { h = 1 } ^ { H } \pi _ { h } ( x ) \mu _ { h } ,
$$

where

$$
\pi _ { h } ( x ) = \frac { \gamma _ { h } \exp { \left( - \frac { 1 } { \sigma _ { t } ^ { 2 } } \| x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } \| ^ { 2 } \right) } } { \sum _ { i = 1 } ^ { H } \gamma _ { i } \exp { \left( - \frac { 1 } { \sigma _ { t } ^ { 2 } } \| x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { i } \| ^ { 2 } \right) } } , \qquad \mathrm { a n d } \qquad \sigma _ { t } ^ { 2 } = \overline { { \alpha } } _ { t } \sigma ^ { 2 } + 1 - \overline { { \alpha } } _ { t } .
$$

# C.2.1 Proof of the upper bound

For ease of notations, we prove that for any $\mu _ { h } , \sigma$ , and $\gamma _ { h }$ , the following inequality holds:

$$
\mathbb { P } \left\{ \exists x ^ { \prime } \in \mathbb { R } ^ { d } , ( 1 - \overline { { \alpha } } _ { t } ) \| s _ { t } ^ { \star } ( x ) - s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } > C \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } \right\} \lesssim \frac { 1 } { T ^ { 4 } } .
$$

The upper bound can be proved by replacing $T$ with $T + d$ .

According to the definition of $s _ { t } ^ { \star } ( x )$ , We have

$$
\begin{array} { l } { \displaystyle ( 1 - \overline { { \alpha } } _ { t } ) \| s _ { t } ^ { \star } ( x ) - s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } } \\ { \displaystyle \overset { \mathrm { ( a ) } } { = } \left\| - \frac { 1 - \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } ( x - x ^ { \prime } ) + \frac { \sqrt { \overline { { \alpha } } _ { t } } ( 1 - \overline { { \alpha } } _ { t } ) } { \sigma _ { t } ^ { 2 } } \displaystyle \sum _ { i = 1 } ^ { H } ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \right\| _ { 2 } } \\ { \displaystyle \leq \frac { 1 - \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \| x - x ^ { \prime } \| _ { 2 } + \frac { \sqrt { \overline { { \alpha } } _ { t } } ( 1 - \overline { { \alpha } } _ { t } ) } { \sigma _ { t } ^ { 2 } } \displaystyle \sum _ { i = 1 } ^ { H } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } } \\ { \displaystyle \overset { \mathrm { ( b ) } } { \leq } \| x - x ^ { \prime } \| _ { 2 } + \sqrt { \overline { { \alpha } } _ { t } } \displaystyle \sum _ { i = 1 } ^ { H } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } , } \end{array}
$$

where (a) uses the fact that $\begin{array} { r } { \sum _ { i = 1 } ^ { H } ( { \pi } _ { i } ( x ) - { \pi } _ { i } ( x ^ { \prime } ) ) ( { \mu } _ { i } - { \mu } _ { h } ) = \sum _ { i = 1 } ^ { H } ( { \pi } _ { i } ( x ) - { \pi } _ { i } ( x ^ { \prime } ) ) { \mu } _ { i } } \end{array}$ , and (b) use the fact that $1 - \overline { { \alpha } } _ { t } \le \sigma _ { t } ^ { 2 }$ . Thus we have

$$
\begin{array} { r l } & { \quad \mathbb { P } \left\{ \exists x ^ { \prime } \in \mathbb { R } ^ { d } , ( 1 - \overline { { \alpha } } _ { t } ) \| s _ { t } ^ { \star } ( x ) - s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } > C \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } \right\} } \\ & { \leq \displaystyle \sum _ { h = 1 } ^ { H } \gamma _ { h } \mathbb { P } _ { h } \left\{ \exists x ^ { \prime } \in \mathbb { R } ^ { d } , ( 1 - \overline { { \alpha } } _ { t } ) \| s _ { t } ^ { \star } ( x ) - s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } > C \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } \right\} } \\ & { \leq \displaystyle \sum _ { h = 1 } ^ { H } \gamma _ { h } \mathbb { P } _ { h } \left\{ \exists x ^ { \prime } \in \mathbb { R } ^ { d } , \sqrt { \overline { { \alpha } } _ { t } } \displaystyle \sum _ { i = 1 } ^ { H } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } > \frac { C \log ( H T ) } { 2 } \| x - x ^ { \prime } \| _ { 2 } \right\} } \end{array}
$$

$$
\begin{array} { r l } & { \le \displaystyle \sum _ { h : \gamma _ { h } \ge \frac { 1 } { H T ^ { \sharp } } } ^ { H } \gamma _ { h } \mathbb { P } _ { h } \left\{ \exists x ^ { \prime } \in \mathbb { R } ^ { d } , \sqrt { \overline { { \alpha } } _ { t } } \displaystyle \sum _ { i = 1 } ^ { H } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } > \frac { C \log ( H T ) } { 2 } \| x - x ^ { \prime } \| _ { 2 } \right\} } \\ & { + \displaystyle \frac { 1 } { T ^ { \sharp } } , } \end{array}
$$

where $\mathbb { P } _ { h } \{ \cdot \}$ denotes the probability of the event when $x$ follows the Gaussian distribution $\mathcal { N } ( \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } , \sigma _ { t } ^ { 2 } I _ { d } )$ Define the set

$$
\mathcal { T } _ { t } ^ { h } = \left\{ x \in \mathbb { R } ^ { d } : \sqrt { \overline { { \alpha _ { t } } } } \lvert ( x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) \rvert \leq C _ { 2 } \sigma _ { t } \sqrt { \overline { { \alpha _ { t } } } \log ( H T ) } \lVert \mu _ { i } - \mu _ { h } \rVert , \forall i \right\} .
$$

According to the concentration inequality of Gaussian distribution, it is easy to check that for sufficiently large $C _ { 2 }$ , we have

$$
\mathbb { P } \left\{ x \sim \mathcal { N } ( \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } , \sigma _ { t } ^ { 2 } I _ { d } ) , x \notin \mathcal { T } _ { t } ^ { h } \right\} \lesssim \frac { 1 } { H T ^ { 4 } } , \qquad \forall h .
$$

Below we shall prove that for $h$ satisfying $\begin{array} { r } { \gamma _ { h } \ge \frac { 1 } { H T ^ { 4 } } } \end{array}$ ,

$$
T _ { t } ^ { h } \subset \left\{ x \in \mathbb { R } ^ { d } : \forall x ^ { \prime } \in \mathbb { R } ^ { d } , \sqrt { \alpha _ { t } } \sum _ { i = 1 } ^ { H } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } \leq \frac { C \log ( H T ) } { 2 } \| x - x ^ { \prime } \| _ { 2 } \right\} .
$$

Inserting it into (39), we have

$$
\begin{array} { l } { \displaystyle \mathbb { P } \left\{ \exists \boldsymbol { x } ^ { \prime } \in \mathbb { R } ^ { d } , ( 1 - \overline { { \alpha } } _ { t } ) \| \boldsymbol { s } _ { t } ^ { \star } ( \boldsymbol { x } ) - \boldsymbol { s } _ { t } ^ { \star } ( \boldsymbol { x } ^ { \prime } ) \| _ { 2 } > C \log ( H T ) \| \boldsymbol { x } - \boldsymbol { x } ^ { \prime } \| _ { 2 } \right\} } \\ { \displaystyle \le \sum _ { h = 1 } ^ { H } \gamma _ { h } \mathbb { P } _ { h } \left\{ \boldsymbol { x } \notin \mathcal { T } _ { t } ^ { h } \right\} \mathbb { 1 } \left( \gamma _ { h } \ge \frac { 1 } { H T ^ { 4 } } \right) + \frac { 1 } { T ^ { 4 } } \lesssim \frac { 1 } { T ^ { 4 } } , } \end{array}
$$

and complete the proof.

Proof of (40). To this end, we introduce an auxiliary set

$$
\begin{array} { r } { \mathcal { F } _ { h } = \left. i : \| x - x ^ { \prime } \| _ { 2 } \leq c \sqrt { \overline { { \alpha } } _ { t } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } \right. , } \end{array}
$$

where $c$ is a sufficiently small constant. Then we have

$$
\begin{array} { r l } & { \sqrt { \overline { { \alpha _ { t } } } } \displaystyle \sum _ { i = 1 } ^ { H } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } } \\ & { \leq \sqrt { \overline { { \alpha _ { t } } } } \displaystyle \sum _ { i \in \mathcal { F } _ { h } } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } + \sqrt { \overline { { \alpha _ { t } } } } \displaystyle \sum _ { i \in \mathcal { F } _ { h } ^ { c } } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } } \\ & { \leq \sqrt { \overline { { \alpha _ { t } } } } \displaystyle \sum _ { i \in \mathcal { F } _ { h } } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } + \displaystyle \frac { 2 } { c } \| x - x ^ { \prime } \| _ { 2 } . } \end{array}
$$

For $i \in \mathcal { F } _ { h }$ , we make a decomposition on the term √ √ $| \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) |$ . Define √ $\widetilde { x } _ { i } \ = \ x \mathbb { 1 } ( \| x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { i } \| _ { 2 } \ \leq$ $\| x ^ { \prime } - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } ) + x ^ { \prime } \mathbb { 1 } ( \| x ^ { \prime } - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } \leq \| x - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } )$ and $\widetilde { x } _ { i } ^ { \prime } = x ^ { \prime } \mathbb { 1 } ( \| x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { i } \| _ { 2 } \leq \| x ^ { \prime } - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { i } \| _ { 2 } ) \ +$ $x \mathbf { 1 } [ \left\| x ^ { \prime } - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { i } \right\| _ { 2 } \leq \left\| x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { i } \right\| _ { 2 } )$

$$
\begin{array} { r l } & { | \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) | \leq \pi _ { i } ( \widetilde { x } _ { i } ) \left| 1 - \exp \left( - \frac { \| \widetilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } + \frac { \| \widetilde { x } _ { i } - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) \right| } \\ & { \qquad + \displaystyle \sum _ { \ell = 1 } ^ { H } \frac { \gamma _ { \ell } \gamma _ { i } \exp \left( - \frac { \| \widetilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) \left| \exp \left( - \frac { \| \widetilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { t } } \mu _ { \ell } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) - \exp \left( - \frac { \| \widetilde { x } _ { i } - \sqrt { \alpha _ { t } } \mu _ { \ell } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) \right| } { \left( \sum _ { \ell = 1 } ^ { H } \gamma _ { \ell } \exp \left( - \frac { \| \widetilde { x } _ { i } - \sqrt { \alpha _ { t } } \mu _ { \ell } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) \right) \left( \sum _ { \ell = 1 } ^ { H } \gamma _ { \ell } \exp \left( - \frac { \| \widetilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { t } } \mu _ { \ell } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) \right) } . } \end{array}
$$

For the first term, we have

$$
\pi _ { i } ( \widetilde { x } _ { i } ) \left| 1 - \exp \left( - \frac { \| \widetilde { x } _ { i } ^ { \prime } - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } + \frac { \| \widetilde { x } _ { i } - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) \right|
$$

$$
\begin{array} { r l } & { \le \displaystyle \frac { \pi _ { i } ( \widetilde { x } _ { i } ) } { \sigma _ { t } ^ { 2 } } \left( | ( x - \sqrt { \alpha _ { t } } \mu _ { h } ) ^ { \top } ( \widetilde { x } _ { i } ^ { \prime } - \widetilde { x } _ { i } ) | + \sqrt { \alpha _ { t } } | ( \widetilde { x } _ { i } ^ { \prime } - \widetilde { x } _ { i } ) ^ { \top } ( \mu _ { h } - \mu _ { i } ) | + \frac { \| x - x ^ { \prime } \| _ { 2 } ^ { 2 } } { 2 } \right) } \\ & { \overset { \mathrm { ( a ) } } { \le } \displaystyle \frac { \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} } { \sigma _ { t } ^ { 2 } } \left( | ( x - \sqrt { \alpha _ { t } } \mu _ { h } ) ^ { \top } ( x ^ { \prime } - x ) | + \left( 1 + \frac { c } { 2 } \right) \sqrt { \alpha _ { t } } \| x ^ { \prime } - x \| _ { 2 } \| \mu _ { h } - \mu _ { i } \| _ { 2 } \right) , } \end{array}
$$

where (a) uses the fact that $| ( \widetilde { \boldsymbol { x } } _ { i } ^ { \prime } - \widetilde { \boldsymbol { x } } _ { i } ) ^ { \top } ( \boldsymbol { \mu } _ { k } - \boldsymbol { \mu } _ { i } ) | \leq \| \boldsymbol { x } ^ { \prime } - \boldsymbol { x } \| _ { 2 } \| \boldsymbol { \mu } _ { k } - \boldsymbol { \mu } _ { i } \| _ { 2 }$ and $\| x - x ^ { \prime } \| _ { 2 } \leq c \sqrt { \overline { { \alpha } } _ { t } } \| \mu _ { i } - \mu _ { h } \| _ { 2 }$ . For the second term, we have

$$
\begin{array} { r l } & { \quad \frac { \gamma _ { \ell } \gamma _ { i } \exp \big ( - \frac { \| \tilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { i } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { i } ^ { 2 } } \big ) } { ( \sum _ { \ell = 1 } ^ { H } \gamma _ { \ell } \exp \big ( - \frac { \| \tilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { i } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { i } ^ { 2 } } \big ) ) } ( \exp ( - \frac { \| \tilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { i } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { i } ^ { 2 } } ) - \exp ( - \frac { \| \tilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { i } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { i } ^ { 2 } } ) ) } \\ & { \qquad ( \sum _ { \ell = 1 } ^ { H } \gamma _ { \ell } \exp ( - \frac { \| \tilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { i } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { i } ^ { 2 } } ) ) ( \sum _ { \ell = 1 } ^ { H } \gamma _ { \ell } \exp ( - \frac { \| \tilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { i } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { i } ^ { 2 } } ) ) } \\ &  \leq \frac { \gamma _ { \ell } \exp \big ( - \frac { \| \tilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { i } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { i } ^ { 2 } } \big ) } { \operatorname* { m i n } \{ s ( x ) , s ( x ^ { \prime } ) \} } \gamma _ { \ell } | \exp ( - \frac  \| \tilde  \end{array}
$$

where

$$
s ( \boldsymbol { x } ) : = \sum _ { \ell = 1 } ^ { H } \gamma _ { \ell } \exp \left( - \frac { \| \boldsymbol { x } - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { \ell } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) ,
$$

and (a) comes from the fact that

$$
\begin{array} { r l } & { \frac { \gamma _ { i } \exp { \left( - \frac { \| \widetilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) } } { \operatorname* { m i n } \{ s ( x ) , s ( x ^ { \prime } ) \} } \leq \operatorname* { m a x } \left\{ \frac { \gamma _ { i } \exp { \left( - \frac { \| \widetilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) } } { s ( \widetilde { x } _ { i } ) } , \frac { \gamma _ { i } \exp { \left( - \frac { \| \widetilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) } } { s ( \widetilde { x } _ { i } ^ { \prime } ) } \right\} } \\ & { \qquad \leq \operatorname* { m a x } \left\{ \frac { \gamma _ { i } \exp { \left( - \frac { \| \widetilde { x } _ { i } - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) } } { s ( \widetilde { x } _ { i } ) } , \frac { \gamma _ { i } \exp { \left( - \frac { \| \widetilde { x } _ { i } ^ { \prime } - \sqrt { \alpha _ { t } } \mu _ { i } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \right) } } { s ( \widetilde { x } _ { i } ^ { \prime } ) } \right\} } \\ & { \qquad = \operatorname* { m a x } \left\{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \right\} , } \end{array}
$$

and the following inequality holds according to (43):

$$
\begin{array} { r l } & { \quad \frac { \gamma _ { \ell } \left| \exp \big ( - \frac { \| \tilde { x } _ { \ell } ^ { \prime } - \sqrt { \alpha _ { t } } \mu \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \big ) - \exp \big ( - \frac { \| \tilde { x } _ { \ell } - \sqrt { \alpha _ { t } } \mu \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } \big ) \right| } { \operatorname* { m a x } \{ s ( x ) , s ( x ^ { \prime } ) \} } } \\ & { \leq \frac { \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} } { \sigma _ { t } ^ { 2 } } \bigg ( | ( x - \sqrt { \alpha _ { t } } \mu _ { h } ) ^ { \top } ( x ^ { \prime } - x ) | + \sqrt { \alpha _ { t } } | ( x ^ { \prime } - x ) ^ { \top } ( \mu _ { h } - \mu _ { \ell } ) | + \frac { \| x - x ^ { \prime } \| _ { 2 } ^ { 2 } } { 2 } \bigg ) } \\ & { \leq \frac { \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} } { \sigma _ { t } ^ { 2 } } \Big ( | ( x - \sqrt { \alpha _ { t } } \mu _ { h } ) ^ { \top } ( x ^ { \prime } - x ) | + \sqrt { \alpha _ { t } } \| x ^ { \prime } - x \| _ { 2 } \| \mu _ { h } - \mu _ { \ell } \| _ { 2 } } \\ & { \quad + \frac { c \sqrt { \alpha _ { t } } \| \mu _ { i } - \mu _ { h } \| \| x - x ^ { \prime } \| _ { 2 } } { 2 } \Big ) . } \end{array}
$$

Inserting (44) and (45) into (42), for $c \leq 2$ , we have

$$
\sqrt { \overline { { \alpha } } _ { t } } \lvert \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) \rvert \lVert \mu _ { i } - \mu _ { h } \rVert _ { 2 } \leq \epsilon _ { i , 1 } + \sum _ { \ell = 1 } ^ { H } \epsilon _ { i , 2 } ^ { ( \ell ) } ,
$$

where

$$
\epsilon _ { i , 1 } : = \frac { \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} } { \sigma _ { t } ^ { 2 } } \left( \sqrt { \overline { { \alpha _ { t } } } } | ( x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | + 2 \overline { { \alpha } } _ { t } \| \mu _ { h } - \mu _ { i } \| _ { 2 } ^ { 2 } \right) \| x ^ { \prime } - x \| _ { 2 } ,
$$

$$
\begin{array} { r l r } & { } & { \epsilon _ { i , 2 } ^ { ( \ell ) } : = \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \frac { \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} } { \sigma _ { t } ^ { 2 } } \Biggl ( \sqrt { \overline { { \alpha } } _ { t } } | ( x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | } \\ & { } & { + \overline { { \alpha } } _ { t } \| \mu _ { h } - \mu _ { \ell } \| _ { 2 } \| \mu _ { h } - \mu _ { i } \| _ { 2 } + \overline { { \alpha } } _ { t } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } \Biggr ) \| x ^ { \prime } - x \| _ { 2 } . } \end{array}
$$

We use the following lemma to continue the proof.

Lemma 6. For any $i$ and $h$ , suppose that $x \in \mathcal { T } _ { t } ^ { h }$ , $\| x - x ^ { \prime } \| _ { 2 } \leq c \sqrt { \overline { { \alpha } } _ { t } } \| \mu _ { i } - \mu _ { h } \| _ { 2 }$ , and $\gamma _ { h } \geq \frac { 1 } { H T ^ { 4 } }$ , where c is a sufficiently small constant. Then we have

$$
\begin{array} { r l r } & { } & { \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \Vert \mu _ { i } - \mu _ { h } \Vert _ { 2 } ^ { 2 } \lesssim \operatorname* { m a x } \{ \gamma _ { i } , \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \log ( H T ) , } \\ & { } & { \frac { \sqrt { \alpha _ { t } } } { \sigma _ { t } ^ { 2 } } \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} | ( x - \sqrt { \alpha _ { t } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | \lesssim \operatorname* { m a x } \{ \gamma _ { i } , \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \log ( H T ) . } \end{array}
$$

By using Lemma 6, we have

$$
\begin{array} { r l } & { \epsilon _ { i , 1 } \lesssim \operatorname* { m a x } \{ \gamma _ { i } , \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } , } \\ & { \epsilon _ { i , 2 } ^ { ( \ell ) } \lesssim \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \operatorname* { m a x } \{ \gamma _ { i } , \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } } \\ & { \qquad + \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \| \mu _ { h } - \mu _ { \ell } \| _ { 2 } \| \mu _ { h } - \mu _ { i } \| _ { 2 } \| x - x ^ { \prime } \| _ { 2 } . } \end{array}
$$

If $\| x - x ^ { \prime } \| _ { 2 } \leq c \sqrt { \overline { { \alpha } } _ { t } } \| \mu _ { \ell } - \mu _ { h } \| _ { 2 }$ , then we have

$$
\begin{array} { r l } & { \quad \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \| \mu _ { h } - \mu _ { \ell } \| _ { 2 } \| \mu _ { h } - \mu _ { i } \| _ { 2 } \| x - x ^ { \prime } \| _ { 2 } } \\ & { \le \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \operatorname* { m a x } \{ \| \mu _ { h } - \mu _ { \ell } \| _ { 2 } ^ { 2 } , \| \mu _ { h } - \mu _ { i } \| _ { 2 } ^ { 2 } \} \| x - x ^ { \prime } \| _ { 2 } } \\ & { \le \operatorname* { m a x } \{ \gamma _ { \ell } , \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \operatorname* { m a x } \{ \gamma _ { i } , \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } . } \end{array}
$$

Otherwise, that is $\| x - x ^ { \prime } \| _ { 2 } > c \sqrt { \overline { { \alpha } } _ { t } } \| \mu _ { \ell } - \mu _ { h } \| _ { 2 }$ , then we have

$$
\begin{array} { r l } & { \quad \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \| \mu _ { h } - \mu _ { \ell } \| _ { 2 } \| \mu _ { h } - \mu _ { i } \| _ { 2 } \| x - x ^ { \prime } \| _ { 2 } } \\ & { \le \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \frac { \| x - x ^ { \prime } \| _ { 2 } } { c \sqrt { \overline { { \alpha } } _ { t } } } \| \mu _ { h } - \mu _ { i } \| _ { 2 } c \sqrt { \overline { { \alpha } } _ { t } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } } \\ & { = \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \| x - x ^ { \prime } \| _ { 2 } \| \mu _ { h } - \mu _ { i } \| _ { 2 } ^ { 2 } } \\ & { \le \operatorname* { m a x } \{ \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \operatorname* { m a x } \{ \gamma _ { i } , \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } . } \end{array}
$$

Thus we have

$$
\epsilon _ { i , 2 } ^ { ( \ell ) } \lesssim \operatorname* { m a x } \{ \gamma _ { \ell } , \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \operatorname* { m a x } \{ \gamma _ { i } , \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } .
$$

Inserting into (46), we have

$$
\begin{array} { r l } & { \sqrt { \alpha _ { t } } | \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) | \| \mu _ { i } - \mu _ { h } \| _ { 2 } } \\ & { \lesssim \operatorname* { m a x } \{ \gamma _ { i } , \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } \left( 1 + \displaystyle \sum _ { \ell = 1 } ^ { H } \operatorname* { m a x } \{ \gamma _ { \ell } , \pi _ { \ell } ( x ) , \pi _ { \ell } ( x ^ { \prime } ) \} \right) } \\ & { \lesssim \operatorname* { m a x } \{ \gamma _ { i } , \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } . } \end{array}
$$

Inserting it into (41) we have

$$
\sqrt { \overline { { \alpha } } _ { t } } \sum _ { i = 1 } ^ { H } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 }
$$

$$
\begin{array} { r l } {  { \lesssim \sum _ { i \in \mathcal { F } _ { h } } \operatorname* { m a x } \{ \gamma _ { i } , \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } + \frac { 2 } { c } \| x - x ^ { \prime } \| _ { 2 } } } \\ & { \lesssim \log ( H T ) \| x - x ^ { \prime } \| _ { 2 } , } \end{array}
$$

and complete the proof.

Proof of Lemma 6. We first bound $\pi _ { i } ( x )$ and $\pi _ { i } ( x ^ { \prime } )$ . Noticing that for any $h$ such that $\begin{array} { r } { \gamma _ { h } \ge \frac { 1 } { H T ^ { 4 } } } \end{array}$ , any $i$ , and any $x \in \mathcal { T } _ { t } ^ { h }$ , we have

$$
\begin{array} { r l } & { \pi _ { i } ( x ) \leq \displaystyle \frac { \pi _ { i } ( x ) } { \pi _ { h } ( x ) } = \frac { \gamma _ { i } } { \gamma _ { h } } \exp \left( - \frac { 1 } { 2 \sigma _ { t } ^ { 2 } } \| x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { i } \| ^ { 2 } + \displaystyle \frac { 1 } { 2 \sigma _ { t } ^ { 2 } } \| x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { h } \| ^ { 2 } \right) } \\ & { \quad \quad = \displaystyle \frac { \gamma _ { i } } { \gamma _ { h } } \exp \left( - \frac { \overline { { \alpha _ { t } } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } + \frac { \sqrt { \overline { { \alpha _ { t } } } } } { \sigma _ { t } ^ { 2 } } ( x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) \right) } \\ & { \overset { ( \mathrm { a } ) } { \leq } \gamma _ { i } \exp \left( - \frac { \overline { { \alpha _ { t } } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } + \frac { \sqrt { \overline { { \alpha _ { t } } } } } { \sigma _ { t } ^ { 2 } } | ( x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | + 4 \log ( H T ) \right) , } \end{array}
$$

where (a) uses the fact that $\gamma _ { h } \geq \frac { 1 } { H T ^ { 4 } }$ . Similarly, we bound the $\pi _ { i } ( x ^ { \prime } )$ as

$$
\begin{array} { r l } & { \pi _ { i } ( x ^ { \prime } ) \leq \displaystyle \frac { \pi _ { i } ( x ^ { \prime } ) } { \pi _ { h } ( x ^ { \prime } ) } = \frac { \gamma _ { i } } { \gamma _ { h } } \exp \Big ( - \frac { \overline { { \alpha } } _ { t } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } } + \frac { \sqrt { \overline { { \alpha } } _ { t } } } { \sigma _ { t } ^ { 2 } } ( x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) } \\ & { \qquad + \frac { \sqrt { \overline { { \alpha } } _ { t } } } { \sigma _ { t } ^ { 2 } } ( x ^ { \prime } - x ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) \Big ) } \\ & { \overset { ( \mathrm { a } ) } { \leq } \gamma _ { i } \exp \left( - \frac { 1 1 \overline { { \alpha } } _ { t } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 2 4 \sigma _ { t } ^ { 2 } } + \frac { \sqrt { \overline { { \alpha } } _ { t } } } { \sigma _ { t } ^ { 2 } } | ( x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | + 4 \log ( H T ) \right) , } \end{array}
$$

where (a) uses the fact that for $c \leq 1 / 2 4$ ,

$$
\frac { \sqrt { \overline { { \alpha _ { t } } } } } { \sigma _ { t } ^ { 2 } } \big ( x ^ { \prime } - x \big ) ^ { \top } \big ( \mu _ { i } - \mu _ { h } \big ) \leq \frac { \sqrt { \overline { { \alpha _ { t } } } } } { \sigma _ { t } ^ { 2 } } \| x ^ { \prime } - x \| _ { 2 } \| \mu _ { i } - \mu _ { h } \| _ { 2 } \leq \frac { c \overline { { \alpha _ { t } } } } { \sigma _ { t } ^ { 2 } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } \leq \frac { \overline { { \alpha _ { t } } } } { 2 4 \sigma _ { t } ^ { 2 } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } .
$$

Let us discuss two cases:

• If $\sqrt { \overline { { \alpha } } _ { t } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } \geq 6 \sigma _ { t } C _ { 2 } \sqrt { \log ( H T ) }$ , then

$$
\begin{array} { r l r } {  { \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \leq \gamma _ { i } \exp \Big ( - \frac { 1 1 \overline { { \alpha _ { t } } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 2 4 \sigma _ { t } ^ { 2 } } + \frac { \sqrt { \overline { { \alpha _ { t } } } } } { \sigma _ { t } ^ { 2 } } | ( x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | } } \\ & { } & { + \ 4 \log ( H T ) \Big ) } \\ & { } & { \overset { \mathrm { ( a ) } } { \leq } \gamma _ { i } \exp \Big ( - \frac { 5 \overline { { \alpha _ { t } } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 1 2 \sigma _ { t } ^ { 2 } } + \frac { \sqrt { \overline { { \alpha _ { t } } } } } { \sigma _ { t } ^ { 2 } } | ( x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | \Big ) , } \end{array}
$$

where (a) holds as long as $C _ { 2 } \geq \sqrt { 8 / 3 }$ and

$$
4 \log ( H T ) \leq \frac { \overline { { \alpha } } _ { t } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 9 C _ { 2 } ^ { 2 } \sigma _ { t } ^ { 2 } } \leq \frac { \overline { { \alpha } } _ { t } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 2 4 \sigma _ { t } ^ { 2 } } .
$$

Furthermore, we have

$$
\begin{array} { r l } & { \quad \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } \\ & { \le 2 4 \gamma _ { i } \exp \bigg ( - \frac { 5 \overline { { \alpha } } _ { t } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 1 2 \sigma _ { t } ^ { 2 } } + \frac { \sqrt { \overline { { \alpha } } _ { t } } } { \sigma _ { t } ^ { 2 } } | ( x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | + \frac { \overline { { \alpha } } _ { t } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 2 4 \sigma _ { t } ^ { 2 } } \bigg ) } \\ & { \lesssim \gamma _ { i } \exp \bigg ( - \frac { 3 \overline { { \alpha } } _ { t } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 8 \sigma _ { t } ^ { 2 } } + \frac { \sqrt { \overline { { \alpha } } _ { t } } } { \sigma _ { t } ^ { 2 } } | ( x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | \bigg ) } \end{array}
$$

and

$$
\begin{array} { l } { \displaystyle \frac { \sqrt { \overline { { \alpha _ { t } } } } } { \sigma _ { t } ^ { 2 } } \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \vert ( x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) \vert } \\ { \displaystyle \leq \gamma _ { i } \exp \left( - \frac { 5 \overline { { \alpha _ { t } } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 1 2 \sigma _ { t } ^ { 2 } } + \frac { 2 \sqrt { \overline { { \alpha _ { t } } } } } { \sigma _ { t } ^ { 2 } } \vert ( x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) \vert \right) } \end{array}
$$

Thus to establish Lemma 6, it suffices to prove that

$$
\exp \left( - \frac { 3 \overline { { \alpha } } _ { t } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } } { 8 \sigma _ { t } ^ { 2 } } + \frac { 2 \sqrt { \overline { { \alpha } } _ { t } } } { \sigma _ { t } ^ { 2 } } | ( x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | \right) \leq 1 .
$$

Recalling that $x \in \mathcal { T } _ { t } ^ { h }$ , which implies that $\begin{array} { r } { \sqrt { \overline { { \alpha _ { t } } } } | ( x - \sqrt { \overline { { \alpha _ { t } } } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | \leq C _ { 2 } \sigma _ { t } \sqrt { \overline { { \alpha } } _ { t } \log ( H T ) } \| \mu _ { i } - \mu _ { h } \| _ { 2 } } \end{array}$ , we have

$$
\frac { 2 \sqrt { \alpha _ { t } } } { \sigma _ { t } ^ { 2 } } | ( x - \sqrt { \alpha _ { t } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | \leq \frac { 2 C _ { 2 } } { \sigma _ { t } } \sqrt { \overline { { \alpha } } _ { t } \log ( H T ) } \| \mu _ { i } - \mu _ { h } \| _ { 2 } \leq \frac { \overline { { \alpha } } _ { t } } { 3 \sigma _ { t } ^ { 2 } } \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } ,
$$

and complete the proof.

• If $\sqrt { \overline { { \alpha } } _ { t } } \| \mu _ { i } - \mu _ { h } \| < 6 C _ { 2 } \sigma _ { t } \sqrt { \log ( H T ) }$ , then we have

$$
\begin{array} { r l } & { \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ) \} \| \mu _ { i } - \mu _ { h } \| _ { 2 } ^ { 2 } \leq 3 6 C _ { 2 } ^ { 2 } \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ) \} \log ( H T ) , } \\ & { \frac { \sqrt { \overline { { \alpha } } _ { t } } } { \sigma _ { t } ^ { 2 } } \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} | ( x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { h } ) ^ { \top } ( \mu _ { i } - \mu _ { h } ) | } \\ & { \leq \frac { C _ { 2 } \sqrt { \overline { { \alpha } } _ { t } } \log ( H T ) } { \sigma _ { t } } \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} \| \mu _ { i } - \mu _ { h } \| _ { 2 } } \\ & { \leq 6 C _ { 2 } ^ { 2 } \log ( H T ) \operatorname* { m a x } \{ \pi _ { i } ( x ) , \pi _ { i } ( x ^ { \prime } ) \} . } \end{array}
$$

# C.2.2 Proof of the lower bound

By calculation, we have

$$
{ \cal J } _ { t } ( x ) = \frac { 1 } { \sigma _ { t } ^ { 2 } } \left[ - I _ { d } + \frac { \overline { { { \alpha } } } _ { t } } { \sigma _ { t } ^ { 2 } } \left( \sum _ { h = 1 } ^ { H } \gamma _ { h } \mu _ { h } \mu _ { h } ^ { \top } - \overline { { { \mu } \mu } } ^ { \top } \right) \right] , \qquad \mathrm { w h e r e } \qquad \overline { { { \mu } } } = \sum _ { h = 1 } ^ { H } \gamma _ { h } \mu _ { h } .
$$

In the case of $X _ { 0 } \sim \textstyle { \frac { 1 } { 2 } } { \mathcal { N } } ( \mu , \sigma ^ { 2 } I _ { d } ) + \textstyle { \frac { 1 } { 2 } } { \mathcal { N } } ( - \mu , \sigma ^ { 2 } I _ { d } )$ , we have

$$
( 1 - \overline { { \alpha } } _ { t } ) \nabla s _ { t } ^ { \star } ( x ) = ( 1 - \overline { { \alpha } } _ { t } ) J _ { t } ( x ) = \frac { \left( 1 - \overline { { \alpha } } _ { t } \right) } { \sigma _ { t } ^ { 2 } } \left[ - I _ { d } + \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \mu \mu ^ { \top } \right] .
$$

Thus we have

$$
\begin{array} { r l } & { \| ( 1 - \overline { { \alpha } } _ { t } ) \nabla s _ { t } ^ { \star } ( x ) \| _ { 2 } = \frac { ( 1 - \overline { { \alpha } } _ { t } ) } { \sigma _ { t } ^ { 2 } } \left\| - I _ { d } + \frac { \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \mu \mu ^ { \top } \right\| _ { \mathrm { o p } } } \\ & { \quad \quad \quad \quad = \frac { ( 1 - \overline { { \alpha } } _ { t } ) } { \sigma _ { t } ^ { 2 } } \operatorname* { m a x } \left\{ \frac { \overline { { \alpha } } _ { t } \| \mu \| _ { 2 } ^ { 2 } } { \sigma _ { t } ^ { 2 } } - 1 , 1 \right\} \overset { \mathrm { ( a ) } } { \geq } \frac { \overline { { \alpha } } _ { t } ( 1 - \overline { { \alpha } } _ { t } ) \| \mu \| _ { 2 } ^ { 2 } } { 2 \sigma _ { t } ^ { 4 } } } \\ & { \quad \quad \geq \frac { ( 1 - \overline { { \alpha } } _ { t } ) \| \mu \| _ { 2 } ^ { 2 } } { 4 ( 1 - \overline { { \alpha } } _ { t } + \sigma ^ { 2 } ) ^ { 2 } } , } \end{array}
$$

where the last inequality uses the fact that $\begin{array} { r } { \overline { { \alpha } } _ { t } \geq \frac { 1 } { 2 } } \end{array}$

# D Proof of key lemmas in Theorem 1

Before diving into the proof details, we first present a preliminary lemma about the learning schedule $\tau _ { k , n }$ . It has been stated in Li and Jiao (2024). For completeness, we rewrite its proof in Appendix F.4 in supplemental material.

Lemma 7. Our choice of learning schedules (9) satisfies

$$
1 - \tau _ { 0 , 0 } \leq \widehat { \alpha } _ { T } \leq \frac { 2 } { T ^ { c _ { 0 } } } , \qquad \tau _ { K , 0 } \leq 1 - \widehat { \alpha } _ { 1 } \leq \frac { 1 } { T ^ { c _ { 0 } } } , \quad a n d \ \frac { \widehat { \tau } _ { k , n - 1 } - \widehat { \tau } _ { k , n } } { \widehat { \tau } _ { k , n - 1 } \left( 1 - \widehat { \tau } _ { k , n - 1 } \right) } = \frac { c _ { 1 } \log T } { T } .
$$

Moreover, we have

$$
\frac { \widehat { \tau } _ { k , i - 1 } } { \widehat { \tau } _ { k , i } } \leq \frac { \widehat { \tau } _ { k , - 1 } } { \widehat { \tau } _ { k , N } } \lesssim 1 , \qquad \frac { 1 - \widehat { \tau } _ { k , i } } { 1 - \widehat { \tau } _ { k , i - 1 } } \leq \frac { 1 - \widehat { \tau } _ { k , N } } { 1 - \widehat { \tau } _ { k , - 1 } } \lesssim 1 .
$$

# D.1 Proof of Lemma 3

We present a more preliminary conclusion stated as below.

Lemma 8. For any $k$ and $n$ , with probability at least $1 - T ^ { - 1 0 0 }$ ,

$$
\begin{array} { r l } & { \frac { \Vert y _ { T _ { k , n } } ^ { \star } ( x _ { k } ) - y _ { \tau _ { k , n } } ( x _ { k } ) \Vert _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } \lesssim \frac { N \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 0 } ^ { n - 1 } \frac { \widehat \tau _ { k , i } \widehat \tau _ { k , i } ^ { 2 } ( x _ { k } ) } { 1 - \widehat \tau _ { k , i } } , } \\ & { \frac { \mathbb { E } _ { x _ { k } \sim \widehat { x } _ { k } } \Vert x _ { \tau _ { k , n } } ( x _ { k } ) - z _ { \tau _ { k , n } } ^ { \star } ( x _ { k } ) \Vert _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } } \\ & { \lesssim \frac { d \log ^ { 5 } T } { T ^ { 3 } } \operatorname* { m i n } \Big \{ \frac { N d \widehat \tau _ { k , - 1 } \log T } { T ( 1 - \widehat \tau _ { k , - 1 } ) } + \int _ { \tau _ { k , n } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , \frac { N L ^ { 2 } \widehat \tau _ { k , - 1 } \log T } { T ( 1 - \widehat \tau _ { k , - 1 } ) } \Big \} , } \\ & { = \Vert s _ { \tau _ { k , i } } ( y _ { \tau _ { k , i } } ( x _ { k } ) ) - s _ { \tau _ { k , i } } ^ { \star } ( y _ { \tau _ { k , i } } ( x _ { k } ) ) \Vert _ { 2 } . } \end{array}
$$

where

Then according to the definition of $\xi _ { k , n }$ (cf. (34)) and $\varepsilon _ { k , i } ^ { 2 }$ (cf. (12)), we could prove Lemma 3. The remaining proof focuses on establishing Lemma 8. According to definitions of $y _ { \tau _ { k , n } }$ and $y _ { \tau _ { k , n } } ^ { \star }$ , we have

$$
\begin{array} { r l r } {  { \frac { \| y _ { \tau _ { k , n } } ( x _ { k } ) - y _ { \tau _ { k , n } } ^ { \star } ( x _ { k } ) \| _ { 2 } } { \sqrt { 1 - \tau _ { k , n } } } \leq \frac { \widetilde { \varepsilon } _ { k , 0 } ( x _ { k } ) ( \tau _ { k , 0 } - \widehat { \tau } _ { k , 0 } ) } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } + \displaystyle \sum _ { i = 1 } ^ { n - 1 } \frac { \widetilde { \varepsilon } _ { k , i } ( x _ { k } ) ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } } } \\ & { } & { \quad + \frac { \widetilde { \varepsilon } _ { k , n - 1 } ( x _ { k } ) ( \widehat { \tau } _ { k , n - 1 } - \tau _ { k , n } ) } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } . } \end{array}
$$

Recalling Lemma 7, we have

$$
\begin{array} { r l } & { \frac { \bar { \tau } _ { k , 0 } - \hat { \tau } _ { k , 0 } } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } \leq \frac { \hat { \tau } _ { k , - 1 } - \hat { \tau } _ { k , 0 } } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } \leq \frac { \hat { \tau } _ { k , - 1 } \log T } { 2 T ( 1 - \tau _ { k , 0 } ) ^ { 1 / 2 } } \lesssim \frac { \hat { \tau } _ { k , - 1 } \log T } { T ( 1 - \hat { \tau } _ { k , - 1 } ) ^ { 1 / 2 } } } \\ & { \frac { \hat { \tau } _ { k , i - 1 } - \hat { \tau } _ { k , i } } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } \leq \frac { \hat { \tau } _ { k , i - 1 } \log T } { 2 T ( 1 - \tau _ { k , i } ) ^ { 1 / 2 } } \lesssim \frac { \hat { \tau } _ { k , i - 1 } \log T } { T ( 1 - \hat { \tau } _ { k , i - 1 } ) ^ { 1 / 2 } } , } \\ & { \frac { \hat { \tau } _ { k , n - 1 } - \tau _ { k , n } } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } \leq \frac { \hat { \tau } _ { k , n - 1 } - \hat { \tau } _ { k , 0 } } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } \leq \frac { \hat { \tau } _ { k , n - 1 } \log T } { 2 T ( 1 - \tau _ { k , n - 1 } ) ^ { 1 / 2 } } } \\ & { \qquad \lesssim \frac { \hat { \tau } _ { k , n - 1 } \log T } { T ( 1 - \hat { \tau } _ { k , n - 1 } ) ^ { 1 / 2 } } \lesssim \frac { \hat { \tau } _ { k , n - 2 } \log T } { T ( 1 - \hat { \tau } _ { k , n - 2 } ) ^ { 1 / 2 } } . } \end{array}
$$

Inserting into (54), we have that

$$
\frac { \| y _ { \tau _ { k , n } } ( x _ { k } ) - y _ { \tau _ { k , n } } ^ { \star } ( x _ { k } ) \| _ { 2 } } { \sqrt { 1 - \tau _ { k , n } } } \lesssim \sum _ { i = 0 } ^ { n - 1 } \frac { \widetilde { \varepsilon } _ { k , i } ( x _ { k } ) \widehat { \tau } _ { k , i - 1 } \log T } { T ( 1 - \widehat { \tau } _ { k , i - 1 } ) ^ { 1 / 2 } } .
$$

Thus we have

$$
\frac { \Vert y _ { \tau _ { k , n } } ^ { \star } ( x _ { k } ) - y _ { \tau _ { k , n } } ( x _ { k } ) \Vert _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } \lesssim \frac { N \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 0 } ^ { n - 1 } \frac { \tilde { \varepsilon } _ { k , i } ^ { 2 } ( x _ { k } ) \hat { \tau } _ { k , i - 1 } } { 1 - \hat { \tau } _ { k , i - 1 } } \lesssim \frac { N \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 0 } ^ { n - 1 } \frac { \tilde { \varepsilon } _ { k , i } ^ { 2 } ( x _ { k } ) \hat { \tau } _ { k , i } } { 1 - \hat { \tau } _ { k , i } }
$$

and complete the proof of (52), where the last inequality uses (51).

Now we are ready to prove (53). The definition of $x _ { \tau _ { k , n } }$ (cf. (19)) can be decomposed as

$$
\begin{array} { l } { \displaystyle \frac { x _ { \tau _ { k , n } } } { \sqrt { 1 - \tau _ { k , n } } } = \displaystyle \frac { x _ { \tau _ { k , 0 } } } { \sqrt { 1 - \tau _ { k , 0 } } } + \int _ { \hat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } { \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau } + \displaystyle \sum _ { i = 1 } ^ { n - 1 } \int _ { \hat { \tau } _ { k , i } } ^ { \hat { \tau } _ { k , i - 1 } } { \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau } } \\ { \displaystyle \quad \quad + \int _ { \tau _ { k , n } } ^ { \hat { \tau } _ { k , n - 1 } } { \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau } . } \end{array}
$$

Based on the definitions of $x _ { \tau _ { k , n } }$ and $z _ { \tau _ { k , n } } ^ { \star }$ , we first make the following decomposition as

$$
\frac { x _ { \tau _ { k , n } } ( x _ { k } ) - z _ { \tau _ { k , n } } ^ { \star } ( x _ { k } ) } { \sqrt { 1 - \tau _ { k , n } } } = \mathcal { E } _ { 1 , 1 } ( x _ { \tau _ { k , 0 } } ) + \mathcal { E } _ { 1 , 0 } ( x _ { \tau _ { k , 0 } } ) - \mathcal { E } _ { 2 } ( x _ { \tau _ { k , 0 } } ) ,
$$

where

$$
\begin{array} { r l } & { \mathcal { E } _ { 1 , 1 } ( x _ { \tau _ { k , 0 } } ) : = \displaystyle \int _ { \tau _ { k , n } } ^ { \tilde { \tau } _ { k , n - 1 } } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau - \frac { s _ { \tau _ { k , n - 1 } } ^ { \star } ( x _ { \tau _ { k , n - 1 } } ) } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , n - 1 } - \tau _ { k , n } ) ; } \\ & { \mathcal { E } _ { 1 , 0 } ( x _ { \tau _ { k , 0 } } ) : = \displaystyle \int _ { \widehat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau - \frac { s _ { \tau _ { k , 0 } } ^ { \star } ( x _ { \tau _ { k , 0 } } ) } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } ( \tau _ { k , 0 } - \widehat { \tau } _ { k , 0 } ) ; } \\ & { \mathcal { E } _ { 2 } ( x _ { \tau _ { k , 0 } } ) : = \displaystyle \sum _ { i = 1 } ^ { n - 1 } \left[ \frac { s _ { \tau _ { k , i } } ^ { \star } ( x _ { \tau _ { k , i } } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) - \displaystyle \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau \right] . } \end{array}
$$

In the following, we will control these three terms separately, and then combine them. The remaining proof is divided into four steps.

1. Analysis of $\mathcal { E } _ { 1 , 1 } ( x _ { \tau _ { k , 0 } } )$ . This term can be calculated as

$$
\begin{array} { r l } & { \mathcal { E } _ { 1 , 1 } ( x _ { \tau _ { k , 0 } } ) = \int _ { \tau _ { k , 0 } } ^ { \tilde { \tau } _ { k , n - 1 } } \bigg ( \frac { \delta _ { \tau } ^ { s } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } - \frac { \delta _ { \tau _ { k , n - 1 } } ^ { s } ( x _ { \tau _ { k , n - 1 } } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \bigg ) \mathrm { d } \tau } \\ & { \stackrel { ( a ) } { = } - \int _ { \tau _ { k , n } } ^ { \tilde { \tau } _ { k , n - 1 } } \bigg ( \int _ { \tau ^ { \prime } } ^ { \tau _ { k , n - 1 } } \frac { \delta } { \partial \tau } \frac { \delta _ { \tau } ^ { s } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau \bigg ) \mathrm { d } \tau ^ { \prime } } \\ & { \stackrel { ( b ) } { = } - \int _ { \tau _ { k , n } } ^ { \tau _ { k , n - 1 } } \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { s } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \bigg ( \int _ { \tau _ { k , n } } ^ { \tilde { \tau } _ { k , n - 1 } } 1 ( \tau ^ { \prime } < \tau ) \mathrm { d } \tau ^ { \prime } \bigg ) \mathrm { d } \tau } \\ & { = - \int _ { \tau _ { k , n } } ^ { \tilde { \tau } _ { k , n - 1 } } ( \tau - \tau _ { k , n } ) \frac { \delta } { \partial \tau } \frac { s _ { \tau } ^ { s } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau - \int _ { \tilde { \tau } _ { k , n - 1 } } ^ { \tilde { \tau } _ { k , n - 1 } } ( \hat { \tau } _ { k , n - 1 } } \\ &  \quad \quad - \tau _ { k , n } ) \frac { \partial }  \end{array}
$$

where (a) comes from the fact that

$$
\frac { s _ { \tau ^ { \prime } } ^ { \star } ( x _ { \tau ^ { \prime } } ) } { 2 ( 1 - \tau ^ { \prime } ) ^ { 3 / 2 } } - \frac { s _ { \tau _ { k , n - 1 } } ^ { \star } ( x _ { \tau _ { k , n - 1 } } ) } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } = \int _ { \tau _ { k , n - 1 } } ^ { \tau ^ { \prime } } \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau
$$

and the $\Im ( \tau ^ { \prime } < \tau )$ in (b) denotes an indicator function. We further have

$$
\lVert \mathcal { E } _ { 1 , 1 } ( x _ { \tau _ { k , 0 } } ) \rVert _ { 2 } ^ { 2 } \leq ( \widehat { \tau } _ { k , n - 1 } - \tau _ { k , n } ) ^ { 2 } \left( \int _ { \tau _ { k , n } } ^ { \tau _ { k , n - 1 } } \left. \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \right. _ { 2 } \mathrm { d } \tau \right) ^ { 2 }
$$

$$
\leq ( \widehat { \tau } _ { k , n - 1 } - \widehat { \tau } _ { k , n } ) ^ { 2 } ( \widehat { \tau } _ { k , n - 2 } - \widehat { \tau } _ { k , n } ) \int _ { \tau _ { k , n } } ^ { \widehat { \tau } _ { k , n - 2 } } \left\| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \right\| _ { 2 } ^ { 2 } \mathrm { d } \tau .
$$

By using (50) and (51), we have

$$
\begin{array} { r l } & { \| \mathcal { E } _ { 1 , 1 } ( x _ { \gamma _ { k , 0 } } ) \| _ { 2 } ^ { 2 } \leq \bigg ( 1 + \frac { \widehat \gamma _ { k , n - 2 } - \widehat \gamma _ { k , n - 1 } } { \widehat \gamma _ { k , n - 1 } - \widehat \gamma _ { k , n - 1 } } \bigg ) ( \widehat \gamma _ { k , n - 1 } - \widehat \gamma _ { k , n } ) ^ { 3 } \int _ { \mathbb { R } _ { n , \star } } ^ { \widehat \gamma _ { k , n - 1 } } \bigg \| \frac { \partial } { \partial \tau } \frac { s _ { \star } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \bigg \| _ { 2 } ^ { 2 } \mathrm { d } \tau } \\ & { \qquad + \bigg ( \frac { ( \widehat \gamma _ { k , n - 1 } - \widehat \gamma _ { k , n } ) ^ { 3 } } { ( \widehat \gamma _ { k , n - 2 } - \widehat \gamma _ { k , n - 1 } ) ^ { 3 } } + \frac { ( \widehat \gamma _ { k , n - 1 } - \widehat \gamma _ { k , n } ) ^ { 2 } } { ( \widehat \gamma _ { k , n - 2 } - \widehat \gamma _ { k , n - 1 } ) ^ { 2 } } \bigg ) ( \widehat \gamma _ { k , n - 2 } - \widehat \gamma _ { k , n - 1 } ) ^ { 3 } } \\ & { \qquad \cdot \int _ { \widehat \gamma _ { k , n - 1 } } ^ { \widehat \gamma _ { k , n - 2 } } \bigg \| \frac { \partial } { \partial \tau } \frac { s _ { \star } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \bigg \| _ { 2 } ^ { 2 } } \\ &  \overset { ( a ) } { \lesssim } ( \widehat \gamma _ { k , n - 1 } - \widehat \gamma _ { k , n } ) ^ { 3 } \int _ { \gamma _ { k , n } } ^ { \widehat \gamma _ { k , n - 1 } } \bigg \| \frac { \partial } { \partial \tau } \frac { s _ { \star } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \bigg \| _   \end{array}
$$

where (a) uses (51). By taking expectation, we have,

$$
\begin{array} { r l } & { \mathbb { E } _ { \boldsymbol { x } _ { k } \sim \widehat { X } _ { k } } \left[ \| \boldsymbol { \mathcal { E } } _ { 1 , 1 } ( \boldsymbol { x } _ { k } ) \| _ { 2 } ^ { 2 } \right] \lesssim ( \widehat { \tau } _ { k , n - 1 } - \widehat { \tau } _ { k , n } ) ^ { 3 } \displaystyle \int _ { \tau _ { k , n } } ^ { \widehat { \tau } _ { k , n - 1 } } \mathbb { E } _ { \boldsymbol { x } _ { \tau } \sim \boldsymbol { X } _ { \tau } } \left[ \left\| \displaystyle \frac { \partial } { \partial \tau } \displaystyle \frac { s _ { \tau } ^ { \star } ( \boldsymbol { x } _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \right\| _ { 2 } ^ { 2 } \right] \mathrm { d } \tau } \\ & { \qquad + \left( \widehat { \tau } _ { k , n - 2 } - \widehat { \tau } _ { k , n - 1 } \right) ^ { 3 } \displaystyle \int _ { \widehat { \tau } _ { k , n - 1 } } ^ { \widehat { \tau } _ { k , n - 2 } } \mathbb { E } _ { \boldsymbol { x } _ { \tau } \sim \boldsymbol { X } _ { \tau } } \left[ \left\| \displaystyle \frac { \partial } { \partial \tau } \displaystyle \frac { s _ { \tau } ^ { \star } ( \boldsymbol { x } _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \right\| _ { 2 } ^ { 2 } \right] \mathrm { d } \tau . } \end{array}
$$

2. Analysis of $\mathcal { E } _ { 1 , 0 } ( x _ { \tau _ { k , 0 } } )$ . This term can be calculated in a similar way as

$$
\begin{array} { r l } & { \mathcal { E } _ { 1 , 0 } ( x _ { \tau _ { k , 0 } } ) = \displaystyle { \int _ { \hat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } \left( \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } - \frac { s _ { \tau _ { k , 0 } } ^ { \star } ( x _ { \tau _ { k , 0 } } ) } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } \right) } \mathrm { d } \tau } \\ & { \quad \quad \quad \quad = \displaystyle { - \int _ { \hat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } \left( \int _ { \tau ^ { \prime } } ^ { \tau _ { k , 0 } } \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau \right) } \mathrm { d } \tau ^ { \prime } } \\ & { \quad \quad \quad \quad = \displaystyle { - \int _ { \hat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \left( \int _ { \hat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } 1 ( \tau ^ { \prime } < \tau ) \mathrm { d } \tau ^ { \prime } \right) } \mathrm { d } \tau } \\ & { \quad \quad \quad \quad = \displaystyle { - \int _ { \hat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } ( \tau - \hat { \tau } _ { k , 0 } ) \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau } . } \end{array}
$$

We further have

$$
\| \mathcal { E } _ { 1 , 0 } ( x _ { \tau _ { k , 0 } } ) \| _ { 2 } ^ { 2 } \leq ( \tau _ { k , 0 } - \widehat { \tau } _ { k , 0 } ) ^ { 3 } \int _ { \widehat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } \bigg \| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \bigg \| _ { 2 } ^ { 2 } \mathrm { d } \tau .
$$

By taking expectation, we have

$$
\begin{array} { r l r } {  { \mathbb { E } _ { x _ { \tau _ { k , 0 } } \sim \hat { X } _ { k } } [ \| \mathcal { E } _ { 1 , 0 } ( x _ { \tau _ { k , 0 } } ) \| _ { 2 } ^ { 2 } ] \lesssim ( \tau _ { k , 0 } - \widehat { \tau } _ { k , 0 } ) ^ { 3 } \int _ { \widehat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } [ \| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \| _ { 2 } ^ { 2 } ] \mathrm { d } \tau } } \\ & { } & { \lesssim ( \widehat { \tau } _ { k , - 1 } - \widehat { \tau } _ { k , 0 } ) ^ { 3 } \int _ { \widehat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } [ \| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \| _ { 2 } ^ { 2 } ] \mathrm { d } \tau . } \end{array}
$$

3. Analysis of $\mathcal { E } _ { 2 } ( x _ { \tau _ { k , 0 } } )$ . This term is a random variable about $\tau = \{ \tau _ { k , n } \} _ { n = 1 } ^ { N - 1 }$ . To bound it with highr probability, we intend to calculate its $r$ -th order moment $\mathbb { E } \big [ \big ( \mathbb { E } _ { \boldsymbol { x } _ { \tau _ { k , 0 } } \sim p _ { \widehat { X } _ { k } } } \big [ \| \mathcal { E } _ { 2 } ( \boldsymbol { x } _ { \tau _ { k , 0 } } ) \| _ { 2 } ^ { 2 } | \tau \big ] \big ) ^ { r } \big ]$ . Towards this, we introduce $r$ independent random variables $x _ { k } ^ { ( j ) } \sim p _ { \widehat { X } _ { k } }$ , $j = 1 , \cdots , r$ . For any integer $r > 0$ , considering that $\mathbb { E } _ { \boldsymbol { x } _ { \tau _ { k , 0 } } \sim p _ { \widehat { X } _ { k } } } \left\lfloor \left\| \mathcal { E } _ { 2 } ( \boldsymbol { x } _ { \tau _ { k , 0 } } ) \right\| _ { 2 } ^ { 2 } \big | \tau \right\rfloor$ is a random variable independent on $x _ { \tau _ { k , 0 } }$ , we get

$$
\mathbb { E } \big [ \big ( \mathbb { E } _ { \boldsymbol { x } _ { \tau _ { k , 0 } } \sim p _ { \widehat { X _ { k } } } } \big [ \| \mathcal { E } _ { 2 } ( \boldsymbol { x } _ { \tau _ { k , 0 } } ) \| _ { 2 } ^ { 2 } \mid \tau \big ] \big ) ^ { r } \big ] = \mathbb { E } \Big [ \mathbb { E } _ { \boldsymbol { x } _ { k } ^ { ( j ) } \sim p _ { \widehat { X _ { k } } } } \Big [ \prod _ { 1 \leq j \leq r } \| \mathcal { E } _ { 2 } ( \boldsymbol { x } _ { k } ^ { ( j ) } ) \| _ { 2 } ^ { 2 } \mid \tau \Big ] \Big ]
$$

$$
= \mathbb { E } _ { \boldsymbol { x } _ { k } ^ { ( j ) } \sim p _ { \widehat { X } _ { k } } } \left[ \mathbb { E } \Big [ \prod _ { 1 \leq j \leq r } \| \mathcal { E } _ { 2 } ( \boldsymbol { x } _ { k } ^ { ( j ) } ) \| _ { 2 } ^ { 2 } | \boldsymbol { x } _ { k } ^ { ( j ) } \Big ] \right] .
$$

To control the above term, we first make the following observations about the expectation and bias of random variable $\frac { s _ { \tau _ { k , i } } ^ { \star } ( x _ { \tau _ { k , i } } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } }$ conditioned on $x _ { \tau _ { k , 0 } } = x _ { k } ^ { ( j ) }$ :

$$
\begin{array} { r l } & { \mathbb { E } \bigg [ \frac { \delta _ { \tau _ { k , i } } ^ { \star } \big ( x _ { \tau _ { k , i } } \big ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } \big ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } \big ) | x _ { k } ^ { ( j ) } \bigg ] \overset { \mathrm { ( a ) } } { = } \displaystyle \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \big ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } \big ) \frac { 1 } { \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } } \mathrm { d } \tau } \\ & { \qquad = \displaystyle \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau , } \end{array}
$$

where (a) comes from the fact that $\tau _ { k , i }$ is uniformly distributed within $[ \widehat { \tau } _ { k , i } , \widehat { \tau } _ { k , i - 1 } ]$ , and

$$
\begin{array} { r l } & { \int _ { \hat { \mathcal { T } } _ { k , \delta } } ^ { \hat { \mathcal { T } } _ { k , \delta - 1 } } \frac { s _ { \mathrm { F } } ^ { \star } ( x _ { T } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau - \frac { s _ { \mathrm { F } _ { k , \delta } } ^ { \star } ( x _ { T , \delta , 1 } ) } { 2 ( 1 - \tau ) _ { k + 1 } \lambda ^ { 2 / 2 } } ( \hat { \mathcal { T } } _ { k , \delta - 1 } - \hat { \mathcal { T } } _ { k , \delta } ) } \\ & { = \int _ { \hat { \mathcal { T } } _ { k , \delta } } ^ { \hat { \mathcal { T } } _ { k , \delta } } 1 [ \frac { s _ { \mathrm { F } } ^ { \star } ( x _ { T } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } - \frac { s _ { \mathrm { F } _ { k , \delta } } ^ { \star } ( x _ { T , \delta , 1 } ) } { 2 ( 1 - \tau _ { k , \delta } ) ^ { 3 / 2 } } ] \mathrm { d } \tau } \\ & { = \int _ { \hat { \mathcal { T } } _ { k , \delta } } ^ { \hat { \mathcal { T } } _ { k , \delta } } 1 \int _ { \tau _ { k , \delta } } ^ { \tau ^ { \prime } } \frac { \partial } { \partial \tau } \frac { s _ { \mathrm { F } } ^ { \star } ( x _ { T } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau \mathrm { d } \tau ^ { \prime } } \\ &  = ( \int _ { \hat { \mathcal { T } } _ { k , \delta } } ^ { \hat { \mathcal { T } } _ { k , \delta - 1 } } \int _ { \tau _ { k } } ^ { \hat { \mathcal { T } } _ { k , \delta - 1 } } + \int _ { \hat { \mathcal { T } } _ { k , \delta } } ^ { \tau _ { k , \delta } } \int _ { \hat { \mathcal { T } } _ { k , \delta } } ^ { \frac { \partial } { \partial \tau } } \frac  s _  \mathrm { F } \end{array}
$$

Furthermore, we have

$$
\begin{array} { r l } & { \quad \left\| \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \mathrm { d } \tau - \frac { s _ { \tau _ { k , i } } ^ { \star } ( x _ { \tau _ { k , i } } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) \right\| _ { 2 } } \\ & { \leq ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) \int _ { \tau _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \left\| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \right\| _ { 2 } \mathrm { d } \tau + ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) \int _ { \widehat { \tau } _ { k , i } } ^ { \tau _ { k , i } } \left\| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \right\| _ { 2 } \mathrm { d } \tau } \\ & { \leq ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \left\| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \right\| _ { 2 } \mathrm { d } \tau . } \end{array}
$$

$x _ { k } ^ { ( j ) }$ , according to Bernstein inequality, we have with probability at least $1 - \delta$ , for all $1 \le j \le r$

$$
\begin{array} { r } { \lVert \mathcal { E } _ { 2 } ( x _ { k } ^ { ( j ) } ) \rVert _ { 2 } ^ { 2 } \lesssim \log { \frac { r } { \delta } } \displaystyle \sum _ { i = 1 } ^ { n - 1 } \left[ ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \Big \lVert \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big \rVert _ { 2 } \mathrm { d } \tau \right] ^ { 2 } } \\ { \stackrel { \mathrm { ( a ) } } { \lesssim } \log { \frac { r } { \delta } } \displaystyle \sum _ { i = 1 } ^ { n - 1 } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) ^ { 3 } \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \Big \lVert \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big \rVert _ { 2 } ^ { 2 } \mathrm { d } \tau , } \end{array}
$$

where (a) uses the cauchy-schwarz inequality.

In addition, notice the relation that for $\begin{array} { r } { \mathbb { P } ( X > \log \frac { r } { \delta } ) < \frac { \delta } { r } } \end{array}$ and $Y \sim \mathsf { E x p } ( 1 )$ , the following inequality holds.

$$
\mathbb { E } [ X ^ { r } ] \leq \mathbb { E } [ Y ^ { r } ] = r ! .
$$

Then the (56) implies that

$$
\mathbb { E } \Big [ \prod _ { 1 \le j \le r } \| \mathcal { E } _ { 2 } ( x _ { k } ^ { ( j ) } ) \| _ { 2 } ^ { 2 } | x _ { k } ^ { ( j ) } \Big ] \lesssim r ! \left( \sum _ { i = 1 } ^ { n - 1 } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) ^ { 3 } \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \Big \| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big \| _ { 2 } ^ { 2 } \mathrm { d } \tau \right) ^ { r } .
$$

Inserting (57) into (55), we have

$$
\begin{array} { r l } & { \quad \mathbb { E } \left[ \left( \mathbb { E } _ { x _ { k } \sim p _ { \widehat { X } _ { k } } } \left[ \| \mathcal { E } _ { 2 } ( x _ { k } ) \| _ { 2 } ^ { 2 } | \tau \right] \right) ^ { r } \right] } \\ & { \lesssim r ! \left( \displaystyle \sum _ { i = 1 } ^ { n - 1 } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) ^ { 3 } \displaystyle \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } \left[ \left\| \displaystyle \frac { \partial } { \partial \tau } \displaystyle \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ^ { ( j ) } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \right\| _ { 2 } ^ { 2 } \right] \mathrm { d } \tau \right) ^ { r } , } \end{array}
$$

which tells us that with probability at least $1 - T ^ { - 1 0 0 }$ ,

$$
\mathbb { E } _ { \boldsymbol { x } _ { k } \sim p _ { \hat { X } _ { k } } } \big [ \| \mathcal { E } _ { 2 } ( \boldsymbol { x } _ { k } ) \| _ { 2 } ^ { 2 } \big ] \lesssim \log T \sum _ { i = 1 } ^ { n - 1 } ( \hat { \tau } _ { k , i - 1 } - \hat { \tau } _ { k , i } ) ^ { 3 } \int _ { \hat { \tau } _ { k , i } } ^ { \hat { \tau } _ { k , i - 1 } } \mathbb { E } _ { \boldsymbol { x } _ { \tau } \sim \boldsymbol { X } _ { \tau } } \bigg [ \Big \| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( \boldsymbol { x } _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big \| _ { 2 } ^ { 2 } \bigg ] \mathrm { d } \tau .
$$

4. Combining the above results. Combining these together, we have

$$
\begin{array} { r l } & { \quad \quad \frac { \mathbb { E } _ { x _ { k } \sim \mathcal { P } _ { \widehat { X } _ { k } } } [ \| \boldsymbol { 1 } \boldsymbol { \tau } _ { { \widehat { \tau } } _ { k , n } } ( x _ { k } ) - \boldsymbol { z } _ { { \tau } _ { k , n } } ^ { * } ( x _ { k } ) \| _ { 2 } ^ { 2 } ] } { 1 - \tau _ { k , n } } } \\ & { \lesssim \log T \displaystyle \sum _ { i = 1 } ^ { n - 1 } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) ^ { 3 } \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } [ \| \displaystyle \frac { \partial } { \partial \tau } \displaystyle \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \| _ { 2 } ^ { 2 } ] \mathrm { d } \tau } \\ & { \quad \quad + ( \widehat { \tau } _ { k , - 1 } - \widehat { \tau } _ { k , 0 } ) ^ { 3 } \displaystyle \int _ { \widehat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } [ \| \displaystyle \frac { \partial } { \partial \tau } \displaystyle \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \| _ { 2 } ^ { 2 } ] \mathrm { d } \tau } \\ &  \quad \quad + ( \widehat { \tau } _ { k , n - 1 } - \widehat { \tau } _ { k , n } ) ^ { 3 } \displaystyle \int _ { \tau _ { k , n } } ^ { \widehat { \tau } _ { k , n - 1 } } \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } [ \| \displaystyle \frac { \partial } { \partial \tau } \displaystyle \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) }  2 ( 1 - \tau  \end{array}
$$

We claim that for any $0 < \tau < 1$ ,

$$
\mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } \bigg [ \Big \lVert \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big \rVert _ { 2 } ^ { 2 } \bigg ] \lesssim \frac { d \log T } { ( 1 - \tau ) ^ { 5 } \tau ^ { 3 } } \operatorname* { m i n } \left\{ d + \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] , L ^ { 2 } \right\} .
$$

The proof is deferred to the end of this section. Inserting into (58), we have

$$
\begin{array} { r l } & { ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) ^ { 3 } \displaystyle \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } [ \| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { * } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \| _ { 2 } ^ { 2 } ] \mathrm { d } \tau } \\ & { \displaystyle \cdot \frac { ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) ^ { 3 } d \log T } { ( 1 - \widehat { \tau } _ { k , i - 1 } ) ^ { 3 } \widehat { \tau } _ { k , i } ^ { 3 } } \operatorname* { m i n } \{ \frac { d ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) } { ( 1 - \widehat { \tau } _ { k , i - 1 } ) ^ { 2 } } + \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \frac { \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , \frac { L ^ { 2 } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) } { ( 1 - \widehat { \tau } _ { k , i - 1 } ) ^ { 2 } } \} } \\ &  \displaystyle \overset { ! ) } { \underset { T ^ { 3 } } { \longrightarrow } } \frac { d \log ^ { 4 } T } { T ^ { 3 } } \operatorname* { m i n } \{ \frac { d \widehat { \tau } _ { k , i - 1 } \log T } { ( 1 - \widehat { \tau } _ { k , i - 1 } ) T } + \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \frac  \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } \end{array}
$$

where (a) uses (50) and (51). Similarly, we have

$$
\begin{array} { r l } & { \quad ( \widehat { \tau } _ { k , - 1 } - \widehat { \tau } _ { k , 0 } ) ^ { 3 } \displaystyle \int _ { \tau _ { k , 0 } } ^ { \tau _ { k , 0 } } \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } [ \| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { * } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \| _ { 2 } ^ { 2 } ] \mathrm { d } \tau } \\ & { \lesssim \displaystyle \frac { d \log ^ { 4 } T } { T ^ { 3 } } \operatorname* { m i n } \{ \frac { d \widehat { \tau } _ { k , - 1 } \log T } { ( 1 - \widehat { \tau } _ { k , - 1 } ) T } + \int _ { \widehat { \tau } _ { k , 0 } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , \frac { L ^ { 2 } \widehat { \tau } _ { k , - 1 } \log T } { ( 1 - \widehat { \tau } _ { k , - 1 } ) T } \} , } \\ & { \quad ( \widehat { \tau } _ { k , n - 1 } - \widehat { \tau } _ { k , n } ) ^ { 3 } \displaystyle \int _ { \tau _ { k , n } } ^ { \widehat { \tau } _ { k , n - 1 } } \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } [ \| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { * } ( x _ { \tau } ) } { 2 ( 1 - \tau ) ^ { 3 / 2 } } \| _ { 2 } ^ { 2 } ] \mathrm { d } \tau } \\ &  \lesssim \displaystyle \frac { d \log ^ { 4 } T } { T ^ { 3 } } \operatorname* { m i n } \{ \frac { d \widehat { \tau } _ { k , n - 1 } \log T } { ( 1 - \widehat { \tau } _ { k , n - 1 } ) T } + \int _ { \tau _ { k , n } } ^  \widehat { \tau } _ { k , n - 1 }  \end{array}
$$

Thus we have

$$
\frac { \mathbb { E } _ { \boldsymbol { x } _ { k } \sim p _ { \widehat { X } _ { k } } } \left[ \| \boldsymbol { x } _ { \tau _ { k , n } } ( \boldsymbol { x } _ { k } ) - \boldsymbol { z } _ { \tau _ { k , n } } ^ { \star } ( \boldsymbol { x } _ { k } ) \| _ { 2 } ^ { 2 } \right] } { 1 - \tau _ { k , n } }
$$

$$
\lesssim \operatorname* { m i n } \Big \{ \frac { N d ^ { 2 } \widehat \tau _ { k , - 1 } \log ^ { 6 } T } { T ^ { 4 } ( 1 - \widehat \tau _ { k , - 1 } ) } + \frac { d \log ^ { 5 } T } { T ^ { 3 } } \int _ { \tau _ { k , n } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , \frac { N d L ^ { 2 } \widehat \tau _ { k , - 1 } \log ^ { 6 } T } { ( 1 - \widehat \tau _ { k , - 1 } ) T ^ { 4 } } \Big \}
$$

and complete the proof.

Proof of (59). Notice that

$$
\frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } = \frac { \partial } { \partial y } \frac { s _ { \tau } ^ { \star } ( \sqrt { 1 - \tau } y ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big | _ { y = \frac { x _ { \tau } } { \sqrt { 1 - \tau } } } \frac { \partial } { \partial \tau } \frac { x _ { \tau } } { \sqrt { 1 - \tau } } + \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( \sqrt { 1 - \tau } y ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big | _ { y = \frac { x _ { \tau } } { \sqrt { 1 - \tau } } } .
$$

For the first term, we have

$$
\begin{array} { l } { { \displaystyle \frac { \partial } { \partial y } \frac { s _ { \tau } ^ { \star } ( \sqrt { 1 - \tau } y ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big \vert _ { y = \frac { x _ { \tau } } { \sqrt { 1 - \tau } } } \frac { \partial } { \partial \tau } \frac { x _ { \tau } } { \sqrt { 1 - \tau } } = - \frac { \sqrt { 1 - \tau } } { ( 1 - \tau ) ^ { 3 / 2 } } J _ { \tau } ( x _ { \tau } ) \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } } } \\ { { = - J _ { \tau } ( x _ { \tau } ) \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 5 / 2 } } , } } \end{array}
$$

where

$$
J _ { \tau } ( x ) : = \frac { \partial s _ { \tau } ^ { \star } ( x ) } { \partial x } .
$$

According to Definition 2, we have $\tau \| J _ { \tau } ( x ) \| \leq L$ for $x \in { \mathcal { L } } _ { \tau }$ . Moreover, we use the following lemma, the proof of which is postponed to the end of this section.

Lemma 9. Suppose that $x _ { \tau } \in S _ { \tau }$ . Then we have

$$
\| s _ { \tau } ^ { \star } ( x _ { \tau } ) \| _ { 2 } ^ { 2 } \leq \frac { 1 } { \tau ^ { 2 } } \int \left\| x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } ^ { 2 } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x _ { \tau } ) \mathrm { d } x _ { 0 } \leq \frac { 2 5 ( \theta + c _ { 0 } ) d \log T } { \tau } .
$$

Moreover, we have

$$
\int _ { x _ { \tau } } \Vert J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \Vert _ { 2 } ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } \lesssim \frac { d \log T } { \tau ^ { 3 } } \operatorname* { m i n } \left\{ d + \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] , L ^ { 2 } \right\} ,
$$

where $C _ { 0 }$ is a universal constant.

By using Lemma 9, we have

$$
\begin{array} { r l } & { \displaystyle \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } \left\| \frac { \partial } { \partial y } \frac { s _ { \tau } ^ { \star } ( \sqrt { 1 - \tau } y ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big | _ { y = \frac { x _ { \tau } } { \sqrt { 1 - \tau } } } \frac { \partial } { \partial \tau } \frac { x _ { \tau } } { \sqrt { 1 - \tau } } \right\| _ { 2 } ^ { 2 } } \\ & { \displaystyle \leq \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } \Big \| J _ { \tau } ( x _ { \tau } ) \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 5 / 2 } } \Big \| _ { 2 } ^ { 2 } \lesssim \frac { d \log T } { \tau ^ { 3 } ( 1 - \tau ) ^ { 5 } } \operatorname* { m i n } \big \{ d + \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] , L ^ { 2 } \big \} . } \end{array}
$$

For the second term, we have

$$
\begin{array} { r l r } {  { \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { * } ( \sqrt { 1 - \tau } y ) } { ( 1 - \tau ) ^ { 3 / 2 } } = - \frac { \partial } { \partial \tau } \frac { 1 } { ( 1 - \tau ) \tau } \int _ { x _ { 0 } } ( y - x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | \sqrt { 1 - \tau } y ) \mathrm { d } x _ { 0 } } } \\ & { } & { = \frac { 1 - 2 \tau } { ( 1 - \tau ) ^ { 2 } \tau ^ { 2 } } \int _ { x _ { 0 } } ( y - x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | \sqrt { 1 - \tau } y ) \mathrm { d } x _ { 0 } } \\ & { } & { - \frac { 1 } { ( 1 - \tau ) \tau } \frac { \partial } { \partial \tau } \int _ { x _ { 0 } } ( y - x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | \sqrt { 1 - \tau } y ) \mathrm { d } x _ { 0 } . } \end{array}
$$

Thus we have

$$
\begin{array} { r l } & { \displaystyle \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } \left\| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( \sqrt { 1 - \tau } y ) } { ( 1 - \tau ) ^ { 3 / 2 } } \right\| _ { 2 } ^ { 2 } } \\ & { \displaystyle \leq \frac { 2 ( 1 - 2 \tau ) ^ { 2 } } { ( 1 - \tau ) ^ { 4 } \tau ^ { 4 } } \int _ { x _ { 0 } } \mathbb { E } _ { y \sim X _ { \tau } / \sqrt { 1 - \tau } } \| y - x _ { 0 } \| _ { 2 } ^ { 2 } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | \sqrt { 1 - \tau } y ) \mathrm { d } x _ { 0 } } \end{array}
$$

$$
\begin{array} { r l } & { \quad + \displaystyle \frac { 2 } { ( 1 - \tau ) ^ { 2 } \tau ^ { 2 } } \mathbb { E } \left\| \displaystyle \frac { \partial } { \partial \tau } \int _ { x _ { 0 } } ( y - x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | \sqrt { 1 - \tau } y ) \mathrm { d } x _ { 0 } \right\| _ { 2 } ^ { 2 } } \\ & { \stackrel { ( i ) } { \leq } \displaystyle \frac { 2 ( 1 - 2 \tau ) ^ { 2 } } { ( 1 - \tau ) ^ { 4 } \tau ^ { 4 } } \displaystyle \frac { \tau d } { 1 - \tau } + \displaystyle \frac { 2 } { ( 1 - \tau ) ^ { 2 } \tau ^ { 2 } } \displaystyle \frac { d \mathrm { m i n } \{ d \log T , L ^ { 2 } \} } { \tau ( 1 - \tau ) ^ { 3 } } } \\ & { \lesssim \displaystyle \frac { ( 1 - 2 \tau ) ^ { 2 } d } { ( 1 - \tau ) ^ { 5 } \tau ^ { 3 } } + \displaystyle \frac { d \mathrm { m i n } \{ d \log T , L ^ { 2 } \} } { ( 1 - \tau ) ^ { 5 } \tau ^ { 3 } } \lesssim \displaystyle \frac { d \mathrm { m i n } \{ d \log T , L ^ { 2 } \} } { ( 1 - \tau ) ^ { 5 } \tau ^ { 3 } } , } \end{array}
$$

where (i) comes from the following Lemma.

Lemma 10. For $y \sim p _ { X _ { \tau } / \sqrt { 1 - \tau } }$ , the following inequality holds.

$$
\mathbb { E } \left\| \frac { \partial } { \partial \tau } \int _ { x _ { 0 } } ( y - x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | \sqrt { 1 - \tau } y ) \mathrm { d } x _ { 0 } \right\| _ { 2 } ^ { 2 } \leq \frac { d \operatorname* { m i n } \{ d \log T , L \} } { \tau ( 1 - \tau ) ^ { 3 } } .
$$

Inserting (63) and (64) into (60), we have

$$
\mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } \left\| \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \right\| ^ { 2 } \lesssim \frac { d \log T } { \tau ^ { 3 } ( 1 - \tau ) ^ { 5 } } \operatorname* { m i n } \left. d + \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] , L ^ { 2 } \right. .
$$

Proof of Lemma 9. Inequality (61) can be proved in a similar way as Li and Yan (2024a) (cf. Lemma 1). Specifically, we have

$$
\begin{array} { r l } & { \quad \displaystyle \int \| x _ { \tau } - \sqrt { 1 - \tau x _ { 0 } } \| _ { 2 } ^ { 2 } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x _ { \tau } ) \mathrm { d } x _ { 0 } } \\ & { \le \displaystyle \int \| x _ { \tau } - \sqrt { 1 - \tau x _ { 0 } } \| _ { 2 } ^ { 2 } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x _ { \tau } ) \mathbb { 1 } ( \left\| x _ { \tau } - \sqrt { 1 - \tau x _ { 0 } } \right\| _ { 2 } \le R ) \mathrm { d } x _ { 0 } } \\ & { \quad \displaystyle + \int \| x _ { \tau } - \sqrt { 1 - \tau x _ { 0 } } \| _ { 2 } ^ { 2 } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x _ { \tau } ) \mathbb { 1 } ( \left\| x _ { \tau } - \sqrt { 1 - \tau x _ { 0 } } \right\| _ { 2 } > R ) \mathrm { d } x _ { 0 } } \\ & { \stackrel { ( a ) } { \le } R ^ { 2 } + \displaystyle \int p _ { X _ { 0 } } ( x _ { 0 } ) \exp \left( - \frac { \| x _ { \tau } - \sqrt { 1 - \tau x _ { 0 } } \| _ { 2 } ^ { 2 } } { 3 T } \right) \left\| x _ { \tau } - \sqrt { 1 - \tau x _ { 0 } } \right\| _ { 2 } ^ { 2 } } \\ & { \quad \quad \quad \cdot \mathbb { 1 } ( \left\| x _ { \tau } - \sqrt { 1 - \tau x _ { 0 } } \right\| _ { 2 } > R ) \mathrm { d } x _ { 0 } } \\ & { \stackrel { ( b ) } { \le } R ^ { 2 } + 3 \tau \displaystyle \int p _ { X _ { 0 } } ( x _ { 0 } ) \frac { R ^ { 2 } } { 3 T } \exp \left( - \frac { R ^ { 2 } } { 3 T } \right) \mathrm { d } x _ { 0 } \le 2 R ^ { 2 } , } \end{array}
$$

$R ^ { 2 } = ( 6 \theta + 3 c _ { 0 } ) d \tau \log T$ $\begin{array} { r } { p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x _ { \tau } ) \leq p _ { X _ { 0 } } ( x _ { 0 } ) \exp ( - \frac { \| x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \| _ { 2 } ^ { 2 } } { 3 \tau } ) } \end{array}$ $\left. x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \right. _ { 2 } > R$ $\begin{array} { r } { z \exp ( - z ) \le \frac { R ^ { 2 } } { 3 \tau } \exp \bigl ( - \frac { R ^ { 2 } } { 3 \tau } \bigr ) } \end{array}$ $\begin{array} { r } { z \ge \frac { R ^ { 2 } } { 3 \tau } \ge 1 } \end{array}$ $( 2 \theta + c _ { 0 } ) d \log T \geq$ $^ { 1 }$ . The remaining proof focuses on (62). For convenience, we denote $p _ { \tau }$ as the probability density function of $X _ { \tau }$ in this section. For $d < L ^ { 2 }$ , we have

$$
\begin{array} { r l } & { \quad \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } \Big \lVert J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \Big \rVert _ { 2 } ^ { 2 } } \\ & { \le \displaystyle \int _ { S _ { \tau } } \Big \lVert J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \Big \rVert _ { 2 } ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } + \displaystyle \int _ { S _ { \tau } ^ { \mathrm { c } } } \Big \lVert J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \Big \rVert _ { 2 } ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } } \\ & { \le \frac { 2 5 ( \theta + c _ { 0 } ) d \log T } { \tau } \int \Big \lVert J _ { \tau } ( x _ { \tau } ) \Big \rVert _ { 2 } ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } + \displaystyle \int _ { S _ { \tau } ^ { \mathrm { c } } } \Big \lVert J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \Big \rVert _ { 2 } ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } . } \end{array}
$$

Noticing that

$$
J _ { \tau } ( x _ { \tau } ) = - \frac 1 \tau I _ { d } + \frac 1 \tau \Sigma _ { \tau } ( x _ { \tau } ) ,
$$

we have

$$
\| J _ { \tau } ( x _ { \tau } ) \| _ { 2 } ^ { 2 } \leq \| J _ { \tau } ( x _ { \tau } ) \| _ { F } ^ { 2 } = { \mathsf { T r } } ( J _ { \tau } ^ { 2 } ( x _ { \tau } ) ) = { \frac { 1 } { \tau ^ { 2 } } } { \mathsf { T r } } ( I _ { d } - 2 \Sigma _ { \tau } ( x _ { \tau } ) + \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) )
$$

$$
\leq \frac { 1 } { \tau ^ { 2 } } \mathsf { T r } ( I _ { d } + \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) \leq \frac { d + \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) } { \tau ^ { 2 } } .
$$

Thus we have

$$
\int \Big \| J _ { \tau } ( x _ { \tau } ) \Big \| _ { 2 } ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } \leq \frac { d + \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { \tau ^ { 2 } } .
$$

To bound the second term, we notice that

$$
\begin{array} { r l } & { \| J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \| _ { 2 } \leq \frac { 1 } { \tau ^ { 3 / 2 } } ( \| \sqrt { \tau } s _ { \tau } ^ { \star } ( x _ { \tau } ) \| _ { 2 } + \| \Sigma _ { \tau } ( x _ { \tau } ) \sqrt { \tau } s _ { \tau } ^ { \star } ( x _ { \tau } ) \| _ { 2 } ) } \\ & { \qquad \leq \frac { 1 } { \tau ^ { 3 / 2 } } ( \| \sqrt { \tau } s _ { \tau } ^ { \star } ( x _ { \tau } ) \| _ { 2 } + \| \Sigma _ { \tau } ( x _ { \tau } ) \| _ { 2 } \| \sqrt { \tau } s _ { \tau } ^ { \star } ( x _ { \tau } ) \| _ { 2 } ) } \\ & { \qquad \leq \frac { \| \sqrt { \tau } s _ { \tau } ^ { \star } ( x _ { \tau } ) \| _ { 2 } } { \tau ^ { 3 / 2 } } ( 1 + \| \Sigma _ { \tau } ( x _ { \tau } ) \| _ { 2 } ) . } \end{array}
$$

Then according to Jensen’s inequality, we have

$$
\begin{array} { l } { \displaystyle \int _ { S _ { \tau } ^ { c } } \Big \| J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \Big \| _ { 2 } ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } \leq \frac { 1 } { \tau ^ { 3 } } \int _ { S _ { \tau } ^ { c } } \Big \| \sqrt { \tau } s _ { \tau } ^ { \star } ( x _ { \tau } ) \Big \| _ { 2 } ^ { 2 } \big ( 1 + \| \Sigma _ { \tau } ( x _ { \tau } ) \| _ { 2 } \big ) ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } } \\ { \displaystyle \lesssim \frac { 1 } { \tau ^ { 3 } } \left( \int \| \sqrt { \tau } s _ { \tau } ^ { \star } ( x _ { \tau } ) \| _ { 2 } ^ { 4 } ( 1 + \| \Sigma _ { \tau } ( x _ { \tau } ) \| _ { 2 } ) ^ { 4 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } \right) ^ { 1 / 2 } \left( \int _ { S _ { \tau } ^ { c } } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } \right) ^ { 1 / 2 } . } \end{array}
$$

Noticing that

$$
\begin{array} { r l } & { \displaystyle \| s _ { \tau } ^ { \star } ( x _ { \tau } ) \| _ { 2 } \leq \frac { 1 } { \tau } \mathbb { E } _ { x _ { 0 } } \left[ \left\| x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } | x _ { \tau } \right] , } \\ & { \displaystyle \| \Sigma _ { \tau } ( x _ { \tau } ) \| _ { 2 } \leq \frac { 1 } { \tau } \mathbb { E } _ { x _ { 0 } } \left[ \left\| x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } ^ { 2 } | x _ { \tau } \right] , } \end{array}
$$

we have

$$
\begin{array} { r l } & { \quad \displaystyle \int \| \sqrt { \tau } s _ { \tau } ^ { \star } ( x _ { \tau } ) \| _ { 2 } ^ { 4 } \big ( 1 + \| \Sigma _ { \tau } ( x _ { \tau } ) \| _ { 2 } \big ) ^ { 4 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } } \\ & { \lesssim \displaystyle \int \frac { 1 } { \tau ^ { 2 } } \mathbb { E } _ { x _ { 0 } } \left[ \left\| x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } ^ { 4 } | x _ { \tau } \right] \left( 1 + \frac { 1 } { \tau ^ { 4 } } \mathbb { E } _ { x _ { 0 } } \left[ \left\| x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } ^ { 8 } | x _ { \tau } \right] \right) p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } } \\ & { \lesssim \displaystyle \int \frac { 1 } { \tau ^ { 2 } } \left\| x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } ^ { 4 } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x _ { \tau } ) p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { 0 } \mathrm { d } x _ { \tau } } \\ & { \quad + \displaystyle \int \frac { 1 } { \tau ^ { 6 } } \| x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \| _ { 2 } ^ { 1 2 } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x _ { \tau } ) p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { 0 } \mathrm { d } x _ { \tau } \lesssim d ^ { 6 } . } \end{array}
$$

Moreover, we have

$$
\begin{array} { r l } { \displaystyle \int _ { S _ { \tau } ^ { \infty } } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } \le \int _ { S _ { \tau } ^ { \infty } \cap \{ x _ { \tau } : \| x _ { \tau } \| \le \sqrt { 1 - \tau } T ^ { c n + 1 } + \sqrt { \tau d \log T } \} } p _ { \tau } \mathrm { d } x _ { \tau } } & { } \\ { \displaystyle } & { \quad + \int _ { \{ x _ { \tau } : \| x _ { \tau } \| > \sqrt { 1 - \tau } T ^ { c n + 4 } + \sqrt { \tau d \log T } \} } p _ { \tau } \mathrm { d } x _ { \tau } } \\ { \displaystyle } & { \le \left( 2 \sqrt { 1 - \tau } T ^ { c \kappa + 4 } + 2 \sqrt { \tau d \log T } \right) ^ { d } \exp ( - \theta d \log T ) } \\ { \displaystyle } & { \quad + \mathbb { P } \left( \| x _ { \tau } \| > \sqrt { 1 - \tau } T ^ { c \kappa + 4 } + \sqrt { \tau d \log T } \right) } \\ { \displaystyle } & { \le \exp ( - ( \theta - c _ { n } - 6 ) d \log T ) + \mathbb { P } \left( \| X _ { 0 } \| > T ^ { c n + 4 } \right) } \\ { \displaystyle } & { \quad + P \left( \| W _ { \tau } \| \ge \sqrt { d \log T } \right) } \end{array}
$$

$$
\leq \frac { 1 } { T ^ { 4 } } + \frac { \mathbb { E } \| X _ { 0 } \| } { T ^ { c _ { R } + 4 } } + \frac { 1 } { T ^ { 4 } } \lesssim \frac { 1 } { T ^ { 4 } } ,
$$

as long as $\theta \geq c _ { R } + 1 0 $ . Thus we have

$$
\int _ { S _ { \tau } ^ { \mathrm { c } } } \Big \| J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \Big \| ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } \leq \frac { d ^ { 3 } } { \tau ^ { 3 } T ^ { 2 } } \lesssim \frac { d ^ { 2 } } { \tau ^ { 3 } }
$$

for $T \geq K \gtrsim \sqrt { d }$ . Inserting (66) and (67) into (65), we have

$$
\begin{array} { r l } & { \mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } \Big \| J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \Big \| ^ { 2 } \lesssim \frac { d \log T } { \tau ^ { 3 } } \left( d + \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ( x _ { \tau } ) ) ] \right) + \frac { d ^ { 2 } } { \tau ^ { 3 } } } \\ & { \qquad \lesssim \frac { d \log T } { \tau ^ { 3 } } \left( d + \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ( x _ { \tau } ) ) ] \right) . } \end{array}
$$

Otherwise, for $d \geq L ^ { 2 }$ , we have

$$
\begin{array} { r l } & { \mathbb { E } _ { \boldsymbol { x } _ { \tau } \sim X _ { \tau } } \Big \lVert J _ { \tau } ( \boldsymbol { x } _ { \tau } ) s _ { \tau } ^ { \star } ( \boldsymbol { x } _ { \tau } ) \Big \rVert _ { 2 } ^ { 2 } \le \frac { L ^ { 2 } } { \tau ^ { 2 } } \int _ { \mathcal { L } _ { \tau } } \Big \lVert s _ { \tau } ^ { \star } ( \boldsymbol { x } _ { \tau } ) \Big \rVert _ { 2 } ^ { 2 } p _ { \tau } ( \boldsymbol { x } _ { \tau } ) \mathrm { d } \boldsymbol { x } _ { \tau } } \\ & { \qquad + \int _ { \mathcal { L } _ { \tau } ^ { \mathrm { c } } } \Big \lVert J _ { \tau } ( \boldsymbol { x } _ { \tau } ) s _ { \tau } ^ { \star } ( \boldsymbol { x } _ { \tau } ) \Big \rVert _ { 2 } ^ { 2 } p _ { \tau } ( \boldsymbol { x } _ { \tau } ) \mathrm { d } \boldsymbol { x } _ { \tau } . } \end{array}
$$

For the first term, we have

$$
\int _ { \mathcal { L } _ { \tau } } \left\| s _ { \tau } ^ { \star } ( x _ { \tau } ) \right\| _ { 2 } ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } \leq \frac { 1 } { \tau ^ { 2 } } \int \left\| x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } ^ { 2 } p _ { X _ { 0 } } | _ { X _ { \tau } } ( x _ { 0 } | x _ { \tau } ) p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { 0 } \mathrm { d } x _ { \tau } \leq \frac { d } { \tau } .
$$

For the second term, we have

$$
\begin{array} { r l } & { \displaystyle \int _ { \mathcal { L } _ { \tau } ^ { \mathrm { c } } } \left\| J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \right\| _ { 2 } ^ { 2 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } } \\ & { \displaystyle \leq \left( \int \left\| J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \right\| _ { 2 } ^ { 4 } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } \right) ^ { 1 / 2 } \left( \displaystyle \int _ { \mathcal { L } _ { \tau } ^ { \mathrm { c } } } p _ { \tau } ( x _ { \tau } ) \mathrm { d } x _ { \tau } \right) ^ { 1 / 2 } \lesssim \frac { d ^ { 3 } } { \tau ^ { 3 } } \frac { 1 } { d ^ { 2 } } \lesssim \frac { d } { \tau ^ { 3 } } . } \end{array}
$$

Thus we have

$$
\mathbb { E } _ { x _ { \tau } \sim X _ { \tau } } \Big \| J _ { \tau } ( x _ { \tau } ) s _ { \tau } ^ { \star } ( x _ { \tau } ) \Big \| _ { 2 } ^ { 2 } \lesssim \frac { d L ^ { 2 } } { \tau ^ { 3 } } + \frac { d } { \tau ^ { 3 } } \lesssim \frac { d L ^ { 2 } } { \tau ^ { 3 } } .
$$

Combining (68) and (69), we could complete the proof.

Proof of Lemma 10. According to the definition, we have

$$
\begin{array} { r l } & { \displaystyle \int _ { x _ { 0 } } ( y - x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | \sqrt { 1 - \tau } y ) \mathrm { d } x _ { 0 } } \\ & { = \frac { \displaystyle \int ( y - x _ { 0 } ) \phi ( \sqrt { 1 - \tau } y | \sqrt { 1 - \tau } x _ { 0 } , \tau I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } { \displaystyle \int \phi ( \sqrt { 1 - \tau } y | \sqrt { 1 - \tau } x _ { 0 } , \tau I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } } \\ & { = \frac { \displaystyle \int ( y - x _ { 0 } ) \phi ( y | x _ { 0 } , \frac { \tau } { 1 - \tau } I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } { \displaystyle \int \phi ( y | x _ { 0 } , \frac { \tau } { 1 - \tau } I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } , } \end{array}
$$

where $\phi ( \cdot | \boldsymbol { \mu } , \Sigma )$ denotes the probability density function of Gaussian distribution with mean vector $\mu$ and covariance matrix $\Sigma$ . Define $\begin{array} { r } { t = \frac { \tau } { 1 - \tau } } \end{array}$ . We have

$$
\begin{array} { l } { { \displaystyle { \frac { \partial } { \partial \tau } \int _ { x _ { 0 } } ( y - x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | \sqrt { 1 - \tau } y ) \mathrm { d } x _ { 0 } } } } \\ { { \displaystyle { = \frac { \partial } { \partial t } \frac { \int ( y - x _ { 0 } ) \phi ( y | x _ { 0 } , t I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } { \int \phi ( y | x _ { 0 } , t I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } \frac { \partial \frac { \tau } { 1 - \tau } } { \partial \tau } } } } \end{array}
$$

$$
\begin{array} { r l } & { = - \displaystyle \frac { 1 } { ( 1 - \tau ) ^ { 2 } } \Big ( \frac { \int ( y - x _ { 0 } ) \frac { \| y - x \| _ { 2 } ^ { 2 } } { 2 t ^ { 2 } } \phi ( y | x _ { 0 } , t I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } { \int \phi ( y | x _ { 0 } , t I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } } \\ & { \phantom { = \ } - \frac { \int ( y - x _ { 0 } ) \phi ( y | x _ { 0 } , t I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } { \int \phi ( y | x _ { 0 } , t I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } \frac { \int \frac { \| y - x \| _ { 2 } ^ { 2 } } { 2 t ^ { 2 } } \phi ( y | x _ { 0 } , t I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } { \int \phi ( y | x _ { 0 } , t I _ { d } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } \Big ) } \\ & { = - \frac { 1 } { ( 1 - \tau ) ^ { 2 } } \left( \frac { \tau } { 1 - \tau } \right) ^ { 3 / 2 } \frac { 1 } { 2 t ^ { 2 } } \left( \mathbb { E } _ { Z | y } \left[ \| Z \| _ { 2 } ^ { 2 } Z - \mathbb { E } _ { Z | y } Z \mathbb { E } _ { Z | y } \| Z \| _ { 2 } ^ { 2 } \right] \right) , } \end{array}
$$

where $\mathbb { E } _ { Z \mid y } [ \cdot ]$ denotes the expectation conditioned on $\begin{array} { r } { X _ { 0 } + \sqrt { \frac { \tau } { 1 - \tau } } Z = y } \end{array}$ . Notice that

$$
\begin{array} { r } { \mathbb { E } _ { Z | y } \left[ \| Z \| _ { 2 } ^ { 2 } Z - \mathbb { E } _ { Z | y } Z \mathbb { E } _ { Z | y } { \| Z \| _ { 2 } ^ { 2 } } \right] = \mathbb { E } _ { Z | y } \left[ \left( Z - \mathbb { E } _ { Z | y } Z \right) \left( \| Z \| _ { 2 } ^ { 2 } - \mathbb { E } _ { Z | y } { \| Z \| _ { 2 } ^ { 2 } } \right) \right] . } \end{array}
$$

According to Cauchy-Switch inequality, we have

$$
\begin{array} { r l } & { \quad \left\| \mathbb { E } _ { Z | y } \left[ \left( Z - \mathbb { E } _ { Z | y } Z \right) \left( \| Z \| _ { 2 } ^ { 2 } - \mathbb { E } _ { Z | y } \| Z \| _ { 2 } ^ { 2 } \right) \right] \right\| ^ { 2 } } \\ & { = \underset { u \in \mathbb { R } ^ { d } , \| u \| = 1 } { \operatorname* { m a x } } \left\| \mathbb { E } _ { Z | y } \left[ u ^ { \top } \left( Z - \mathbb { E } _ { Z | y } Z \right) \left( \| Z \| _ { 2 } ^ { 2 } - \mathbb { E } _ { Z | y } \| Z \| _ { 2 } ^ { 2 } \right) \right] \right\| ^ { 2 } } \\ & { \leq \underset { u \in \mathbb { R } ^ { d } , \| u \| = 1 } { \operatorname* { m a x } } \mathbb { E } _ { Z | y } \left| u ^ { \top } \left( Z - \mathbb { E } _ { Z | y } Z \right) \right| ^ { 2 } \mathbb { E } _ { Z | y } \left[ \left( \| Z \| _ { 2 } ^ { 2 } - \mathbb { E } _ { Z | y } \| Z \| _ { 2 } ^ { 2 } \right) ^ { 2 } \right] . } \end{array}
$$

Notice that according to the definition of Jacobian matrix $J _ { \tau }$ , the first factor is bounded by

$$
\begin{array} { r } { \mathbb { E } _ { Z | y } \left| u ^ { \top } \left( Z - \mathbb { E } _ { Z | y } Z \right) \right| ^ { 2 } = u ^ { \top } \mathbb { E } _ { Z | y } \left( Z - \mathbb { E } _ { Z | y } Z \right) \left( Z - \mathbb { E } _ { Z | y } Z \right) ^ { \top } u } \\ { = u ^ { \top } \left( I _ { d } + \tau J _ { \tau } ( X _ { \tau } ) \right) u \le \| I _ { d } + \tau J _ { \tau } ( X _ { \tau } ) \| _ { 2 } . } \end{array}
$$

For the second factor, we have

$$
\begin{array} { r l } & { \quad \displaystyle \int \mathbb E _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \left( \| Z \| _ { 2 } ^ { 2 } - \mathbb E _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } \right) ^ { 2 } p _ { X _ { \tau } } ( x ) \mathrm d x } \\ & { \le \displaystyle \int \left( \mathbb E _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 4 } - \left( \mathbb E _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } \right) ^ { 2 } \right) p _ { X _ { \tau } } ( x ) \mathrm d x } \\ & { = \displaystyle \int \| z \| _ { 2 } ^ { 4 } p _ { Z } ( z ) \mathrm d z - \frac { 1 } { \tau ^ { 2 } } \int \left( \int \| x - \sqrt { 1 - \tau } x _ { 0 } \| _ { 2 } ^ { 2 } p _ { X _ { \tau } | X _ { \tau } } ( x _ { 0 } | x ) \mathrm d z \right) ^ { 2 } p _ { X _ { \tau } } ( x ) \mathrm d x } \\ & { \stackrel { \mathrm { ( a ) } } { \le } \displaystyle \int \| z \| _ { 2 } ^ { 4 } p _ { Z } ( z ) \mathrm d z - \left( \int \| z \| _ { 2 } ^ { 2 } p _ { Z } ( z ) \mathrm d z \right) ^ { 2 } } \\ & { = \left( d ^ { 2 } + 2 d - d ^ { 2 } \right) \lesssim d , } \end{array}
$$

where (a) uses the fact that $\begin{array} { r } { \mathbb { E } \left( \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } \right) ^ { 2 } \ge \left( \mathbb { E } \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } \right) ^ { 2 } } \end{array}$

In addition, for each set $A _ { \tau }$ holding with probability at least $1 - O ( 1 / d ^ { 4 } )$ , we have

$$
\begin{array} { r l } & { \quad \displaystyle \int _ { A _ { 2 } ^ { c } } ( \| I _ { d } + \tau J _ { \tau } ( x ) \| _ { 2 } ) \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } ( \| Z \| _ { 2 } ^ { 2 } - \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } ) ^ { 2 } p _ { X _ { \tau } } ( x ) \mathrm { d } x } \\ & { \lesssim \displaystyle \int _ { A _ { \tau } ^ { c } } \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z - \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } Z \| _ { 2 } ^ { 2 } \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } ( \| Z \| _ { 2 } ^ { 2 } - \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } ) ^ { 2 } p _ { X _ { \tau } } ( x ) \mathrm { d } x } \\ & { \lesssim \displaystyle \int _ { A _ { \tau } ^ { c } } \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 4 } p _ { X _ { \tau } } ( x ) \mathrm { d } x } \\ & { \lesssim ( \int \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 4 } \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 8 } p _ { X _ { \tau } } ( x ) \mathrm { d } x ) ^ { 1 / 2 } ( \displaystyle \int _ { A _ { \tau } ^ { c } } p _ { X _ { \tau } } ( x ) \mathrm { d } x ) ^ { 1 / 2 } } \\ &  \lesssim \frac { 1 } { d ^ { 2 } } ( \mathbb { E } \| Z \| _   \end{array}
$$

Armed with the above observations, we decompose the expectation as

$$
\begin{array} { r l } & { \mathbb { E } \bigg \| \frac { \partial } { \partial \tau } \int _ { \boldsymbol { \mathcal { X } } _ { \sigma } } \langle y - x _ { 0 } \rangle p _ { X _ { \sigma } | X _ { \epsilon } } \langle x _ { 0 } | \sqrt { 1 - \tau y } y \rangle \mathrm { d } x _ { 0 } \bigg \| _ { 2 } ^ { 2 } } \\ & { \leq \frac { 1 } { \tau ( 1 - \tau ) ^ { 3 } } \mathrm { E } \Big \{ \big \| [ d _ { \boldsymbol { \mathcal { X } } } + \tau J _ { \boldsymbol { \mathcal { Y } } } ( X _ { \tau } ) ] _ { \boldsymbol { 2 } } \mathbb { E } _ { \boldsymbol { \mathcal { Z } } | \boldsymbol { \mathcal { X } } } \big ( \| \boldsymbol { \mathcal { Z } } \| _ { 2 } ^ { 2 } - \mathbb { E } _ { \boldsymbol { \mathcal { Z } } | \boldsymbol { \mathcal { Y } } } \| \boldsymbol { \mathcal { Z } } \| _ { 2 } ^ { 2 } \big ) \Big ^ { 2 } \Big \} } \\ & { = \frac { 1 } { \tau ( 1 - \tau ) ^ { 3 } } \int _ { \boldsymbol { \mathcal { X } } _ { \tau } } \| I _ { d } + \tau J _ { \boldsymbol { \mathcal { Y } } } ( x ) \| _ { 2 } \mathbb { E } _ { \boldsymbol { \mathcal { Z } } | \boldsymbol { \mathcal { X } } ^ { \frac { \epsilon } { 3 } } } \bigg ( \| \boldsymbol { \mathcal { Z } } \| _ { 2 } ^ { 2 } - \mathbb { E } _ { \boldsymbol { \mathcal { Z } } | \boldsymbol { \mathcal { Y } } _ { \overline { { \mathcal { X } } } ^ { \frac { \epsilon } { 3 } } } } \| \boldsymbol { \mathcal { Z } } \| _ { 2 } ^ { 2 } \bigg ) ^ { 2 } p _ { X _ { \tau } } \langle x \rangle \mathrm { d } x } \\ &  \quad + \frac { 1 } { \tau ( 1 - \tau ) ^ { 3 } } \int _ { \boldsymbol { \mathcal { Z } } _ { \tau } } \| I _ { d } + \tau J _ { \tau } ( x ) \| _ { 2 } \mathbb { E } _  \boldsymbol { \mathcal { Z } } | \boldsymbol { \mathcal { Y } } _ { \tau } ^  \frac  \ \end{array}
$$

where (a) uses the fact that $\mathbb { P } ( X _ { \tau } \in \mathcal { L } _ { \tau } ^ { \mathrm { c } } ) \lesssim 1 / d ^ { 4 }$ and (72), and (b) uses (71).

In addition, recalling that for $x _ { \tau } \in S _ { \tau }$ , we have

$$
\begin{array} { r l } & { \displaystyle \| I _ { d } + \tau J _ { \tau } ( x _ { \tau } ) \| _ { 2 } \leq \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \left\| Z - \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } Z \right\| _ { 2 } ^ { 2 } \leq \mathbb { E } _ { Z | \frac { x } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } } \\ & { \quad \quad \quad \quad \quad = \frac { 1 } { \tau } \displaystyle \int \| x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \| _ { 2 } ^ { 2 } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x _ { \tau } ) \mathrm { d } x _ { 0 } \stackrel { ( \mathrm { a } ) } { \leq } 2 5 ( \theta + c _ { 0 } ) d \log T , } \end{array}
$$

where (a) uses Lemma 9. Thus for $d < L$ , we have

$$
\begin{array} { r l } & { \mathbb { E } \left\| \displaystyle \frac { \partial } { \partial \tau } \int _ { x _ { 0 } } ( y - x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | \sqrt { 1 - \tau } y ) \mathrm { d } x _ { 0 } \right\| _ { 2 } ^ { 2 } } \\ & { = \frac { 1 } { \tau ( 1 - \tau ) ^ { 3 } } \int _ { S _ { \tau } } \| I _ { d } + \tau J _ { \tau } ( X _ { \tau } ) \| _ { 2 } \mathbb { E } _ { Z | \frac { - \tau } { \sqrt { 1 - \tau } } } \left( \| Z \| _ { 2 } ^ { 2 } - \mathbb { E } _ { Z | \frac { 1 } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } \right) ^ { 2 } p _ { X _ { \tau } } ( x ) \mathrm { d } x } \\ & { \quad + \frac { 1 } { \tau ( 1 - \tau ) ^ { 3 } } \int _ { S _ { \tau } } \| I _ { d } + \tau J _ { \tau } ( X _ { \tau } ) \| _ { 2 } \mathbb { E } _ { Z | \frac { - \tau } { \sqrt { 1 - \tau } } } \left( \| Z \| _ { 2 } ^ { 2 } - \mathbb { E } _ { Z | \frac { 1 } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } \right) ^ { 2 } p _ { X _ { \tau } } ( x ) \mathrm { d } x } \\ & { \lesssim \frac { d \log T } { \tau ( 1 - \tau ) ^ { 3 } } \int _ { S _ { \tau } } \mathbb { E } _ { Z | \frac { 1 } { \sqrt { 1 - \tau } } } \left( \| Z \| _ { 2 } ^ { 2 } - \mathbb { E } _ { Z | \frac { 1 } { \sqrt { 1 - \tau } } } \| Z \| _ { 2 } ^ { 2 } \right) ^ { 2 } p _ { X _ { \tau } } ( x ) \mathrm { d } x + \frac { d } { \tau ( 1 - \tau ) ^ { 3 } } } \\ &  \lesssim \frac { d ^ { 2 } \log T } { \tau ( 1 - \tau ) ^ { 3 } } + \frac { d }  \tau ( 1 - \end{array}
$$

where (a) uses (72) and the fact the $\mathbb { P } ( X _ { \tau } \in { \mathcal { S } } _ { \tau } ^ { \mathrm { c } } ) \lesssim 1 / T ^ { 4 } \lesssim 1 / d ^ { 4 }$ for $T \gtrsim K \gtrsim \operatorname* { m i n } \{ d \log T , L \} \log T \gtrsim d$ , and (b) uses (71). Combining (73) and (74), we have

$$
\mathbb { E } \left\| \frac { \partial } { \partial \tau } \int _ { x _ { 0 } } ( y - x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | \sqrt { 1 - \tau } y ) \mathrm { d } x _ { 0 } \right\| _ { 2 } ^ { 2 } \leq \frac { d \operatorname* { m i n } \{ d \log T , L \} } { \tau ( 1 - \tau ) ^ { 3 } } .
$$

and complete the proof.

# D.2 Proof of Lemma 4

We introduce a more preliminary lemma which leads to Lemma 4 immediately.

Lemma 11. According to Lemma 3, it can be shown that for any $0 \leq n \leq N$ ,

$$
\| y _ { \tau _ { k , n } } ( x _ { k } ) - x _ { \tau _ { k , n } } ( x _ { k } ) \| _ { 2 } ^ { 2 } \leq \zeta _ { k , n } ( x _ { k } ) + \frac { N \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 0 } ^ { n - 1 } \widehat { \tau } _ { k , i - 1 } \widehat { \varepsilon } _ { k , i } ^ { 2 } ( x _ { k } ) , x _ { k } \in \mathcal { E } _ { k } ,
$$

$$
\begin{array} { r l } & { \displaystyle \int _ { \mathcal { E } _ { k , n } } \zeta _ { k , n } ( x _ { k } ) p _ { \widehat { X } _ { k } } ( x _ { k } ) \mathrm { d } x _ { k } \lesssim \frac { d \log ^ { 5 } T } { T ^ { 3 } } \operatorname* { m i n } \Big \{ \frac { N d \widehat { \tau } _ { k , - 1 } \log T } { T } } \\ & { \qquad + ( 1 - \widehat { \tau } _ { k , n } ) \displaystyle \int _ { \tau _ { k , n } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } [ \mathsf { T } \mathsf { r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , \frac { N L ^ { 2 } \widehat { \tau } _ { k , - 1 } \log T } { T } \Big \} , } \end{array}
$$

where $\widetilde { \varepsilon } _ { k , i } ^ { 2 } ( \boldsymbol { x } _ { k } )$ is defined in Lemma $\delta$ , and

$$
\mathcal { E } _ { k , n } : = \left\{ \begin{array} { l l } { \{ x _ { \tau _ { k , 0 } } : x _ { \tau _ { k , i } } ( x _ { \tau _ { k , 0 } } ) \in \widetilde { \mathcal { S } } _ { \tau _ { k , i } } \cap \mathcal { L } _ { \tau _ { k , i } } , y _ { \tau _ { k , i } } ( x _ { \tau _ { k , 0 } } ) \in \mathcal { S } _ { \tau _ { k , i } } , \forall 0 \leq i \leq n - 1 \} , } & { \mathrm { i f ~ } L > d \log T , } \\ { \emptyset } & { \mathrm { i f ~ } L \leq d \log T . } \end{array} \right.
$$

According to the definition of $\varepsilon _ { k , i } ^ { 2 }$ in (12), we have

$$
\begin{array} { r l } & { \mathbb { E } _ { \boldsymbol { x } _ { \tau _ { k , 0 } } \sim p _ { \tilde { X } _ { k } } } \left\| y _ { \tau _ { k , N } } - x _ { \tau _ { k , N } } \right\| _ { 2 } ^ { 2 } } \\ & { \leq \int _ { \mathcal { E } _ { k } } \zeta _ { k , N } ( x _ { k } ) p _ { \tilde { X } _ { k } } ( x _ { k } ) \mathrm { d } x _ { k } + \frac { N \log ^ { 2 } T } { T ^ { 2 } } \displaystyle \sum _ { i = 0 } ^ { N - 1 } \hat { \tau } _ { k , i - 1 } \varepsilon _ { k , i } ^ { 2 } ( x _ { k } ) } \\ & { \lesssim \frac { d \log ^ { 5 } T } { T ^ { 3 } } \operatorname* { m i n } \Big \{ \frac { N d \hat { \tau } _ { k , - 1 } \log T } { T } + ( 1 - \hat { \tau } _ { k , N } ) \int _ { \tau _ { k , n } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } \left[ \mathsf { T r } ( \boldsymbol { \Sigma } _ { \tau } ^ { 2 } ( \boldsymbol { x } _ { \tau } ) ) \right] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , } \\ & { \quad \quad \quad \quad \frac { N L ^ { 2 } \hat { \tau } _ { k , - 1 } \log T } { T } \Big \} + \frac { \hat { \tau } _ { k , - 1 } N \log ^ { 2 } T } { T ^ { 2 } } \displaystyle \sum _ { i = 0 } ^ { N - 1 } \varepsilon _ { k , i } ^ { 2 } . } \end{array}
$$

because $\mathcal { E } _ { k } \subset \mathcal { E } _ { k , n }$ , $p _ { \widetilde { X } _ { k } } ( x _ { k } ) \le p _ { \widehat { X } _ { k } } ( x _ { k } )$ ( $x \neq \infty$ ), and

$$
\int _ { \mathcal { E } _ { k } } \zeta _ { k , n } ( x _ { k } ) p _ { \widetilde { X } _ { k } } ( x _ { k } ) \mathrm { d } x _ { k } \leq \int _ { \mathcal { E } _ { k , n } } \zeta _ { k , n } ( x _ { k } ) p _ { \widehat { X } _ { k } } ( x _ { k } ) \mathrm { d } x _ { k } .
$$

According to (33), we have that

$$
\begin{array} { l } { \displaystyle { \mathsf { K L } } \big ( p _ { \widehat { X } _ { k + 1 } | \widehat { X } _ { k } } \big ( \cdot | x _ { \tau _ { k , 0 } } \big ) \| p _ { Y _ { k + 1 } | Y _ { k } } \big ( \cdot | x _ { \tau _ { k , 0 } } \big ) \big ) = \frac { 1 - \tau _ { k + 1 , 0 } } { 2 \big ( \tau _ { k + 1 , 0 } - \tau _ { k , N } \big ) } \| y _ { \tau _ { k , N } } - x _ { \tau _ { k , N } } \| _ { 2 } ^ { 2 } } \\ { \displaystyle \lesssim \frac { T } { N \widehat { \tau } _ { k , N } \log T } \| y _ { \tau _ { k , N } } - x _ { \tau _ { k , N } } \| _ { 2 } ^ { 2 } . } \end{array}
$$

Inserting (77) into (78), we have

$$
\begin{array} { r l } & { \mathbb { E } _ { x _ { \tau _ { k , 0 } } \sim p _ { \tilde { X } _ { k } } } [ \mathrm { K L } ( p _ { \tilde { X } + 1 | \tilde { X } _ { k } } ( \cdot \vert x _ { \tau _ { k , 0 } } ) \| p _ { Y _ { k + 1 } | Y _ { k } } ( \cdot \vert x _ { \tau _ { k , 0 } } ) ) ] } \\ & { \lesssim \frac { T } { N \tilde { \tau } _ { k , \chi } \log ^ { 4 } T } \mathbb { E } _ { x _ { \tau _ { k , 0 } } \sim p _ { \tilde { X } _ { k } } } \| y _ { \tau _ { k , 0 } } - x _ { \tau _ { k , \chi } } \| _ { 2 } ^ { 2 } } \\ & { \stackrel { \mathrm { ( a ) } } { \lesssim } \frac { d K \log ^ { 4 } T } { T ^ { 3 } } \operatorname* { m i n } \{ \frac { N d \log T } { T } + \frac { 1 - \tilde { \tau } _ { k , N } } { \tilde { \tau } _ { k , N } } \int _ { \tau _ { k , N } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } [ \top ( \sum _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , \frac { N L ^ { 2 } \log T } { T } \} } \\ & { \quad + \frac { K N \log T } { T ^ { 2 } } \displaystyle \sum _ { i = 0 } ^ { N - 1 } \varepsilon _ { k , i } ^ { 2 } } \\ &  \stackrel { \mathrm { ( b ) } } { \lesssim } \frac { d \log ^ { 4 } T } { T ^ { 3 } } \operatorname* { m i n } \{ d \log T + \frac { K ( 1 - \widehat { \tau } _ { k , N } ) } { \widehat { \tau } _ { k , N } } \int _ { \tau _ { k , N } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } [ \top ( \sum _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d }  \end{array}
$$

where (a) uses (51), and (b) uses the fact that $K N \lesssim T$ . Notice that for any $0 < \tau _ { 1 } < \tau _ { 2 } < 1$ ,

$$
\int _ { \tau _ { 1 } } ^ { \tau _ { 2 } } \frac { \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau = \frac { \tau _ { 2 } } { 1 - \tau _ { 2 } } \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau _ { 2 } } ( x _ { \tau _ { 2 } } ) ) ] - \frac { \tau _ { 1 } } { 1 - \tau _ { 1 } } \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau _ { 1 } } ( x _ { \tau _ { 1 } } ) ) ] ,
$$

which has been proved in Li et al. (2024b) (cf. (90)). We have

$$
\begin{array} { r l } & { \quad \frac { 1 - \widehat \tau _ { k , N } } { \widehat \tau _ { k , N } } \displaystyle \int _ { \tau _ { k , N } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau } \\ & { = \frac { \tau _ { k , 0 } ( 1 - \widehat \tau _ { k , N } ) } { \widehat \tau _ { k , N } ( 1 - \tau _ { k , 0 } ) } \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau _ { k , 0 } } ( x _ { \tau _ { k , 0 } } ) ) ] - \frac { \tau _ { k , N } ( 1 - \widehat \tau _ { k , N } ) } { \widehat \tau _ { k , N } ( 1 - \tau _ { k , N } ) } \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau _ { k , N } } ( x _ { \tau _ { k , N } } ) ) ] . } \end{array}
$$

Furthermore, we have

$$
\begin{array} { r l } &  \begin{array} { r l } & { \displaystyle \sum _ { k = 0 } ^ { K - 1 } \frac { 1 - \widehat \gamma _ { k , N } } { \widehat \gamma _ { k , N } } \displaystyle \sum _ { f _ { \ell , N } , \ell = 0 , j = 1 \atop N } ^ { \ell - 1 } \frac { \widehat \gamma _ { k , N } ^ { \ell - 2 } } { ( 1 - \widehat \gamma _ { k , N } ^ { \ell } ) ^ { 2 } } \mathrm { d } f \tau } \\ & { \leq \displaystyle \sum _ { k = 0 } ^ { K - 1 } \frac { \widehat \gamma _ { k , N - 1 } } { \widehat \gamma _ { k , N } + 2 \gamma _ { k , N } } \displaystyle \sum _ { f _ { \ell , N } ^ { \ell } \geq 2 , \ell = 0 , \ell = 0 } ^ { K } \mathbb { E } \Big [ \tau ( \Sigma _ { \tau _ { k - 1 , \ell } , 0 , \ell = 0 } ^ { \ell - 1 } ( x _ { \tau _ { k + 1 , \ell } , 0 } ) ) \Big ] } \\ & { \leq \displaystyle \sum _ { k = 0 } ^ { K - 1 } \frac { \widehat \gamma _ { k , N - 1 } } { \widehat \gamma _ { k , N } + 2 \gamma _ { k , N } ( 1 - \widehat \gamma _ { k , N - 2 , 0 } ) } \mathbb { E } \Big [ \tau ( \Sigma _ { \tau _ { k - 1 , \ell } , 0 , \ell = 0 } ^ { \ell - 1 } ( x _ { \tau _ { k + 1 , \ell } , 0 } ) ) \Big ] } \\ &  \quad - \displaystyle \sum _ { k = 0 } ^ { K - 1 } \frac { \widehat \gamma _ { k , N - 1 } } { \widehat \gamma _ { k , N } ( 1 - \widehat \gamma _ { k , N } ) } \displaystyle \sum _ { | \tau | \in \mathcal { T } _ { \tau , N } } \Big [ \tau ( \Sigma _ { \tau _ { k - 1 , N } , \ell = 0 } ^ { \ell - 1 } ) \Big ] + \displaystyle \frac { \widehat \gamma _ { k , N } } { \widehat \gamma _ { k , N } ( 1 - \widehat \gamma _ { k , N } ) } \displaystyle \sum _ { | \tau | \in \mathcal { T } _ { \tau , N } } \Big [ \tau _ { \tau } ( \Sigma _  \tau _  \end{array} \end{array}
$$

where the last inequality uses the fact that $\tau _ { k + 2 , 0 } = \tau _ { k , N }$ . Notice that

$$
\mathbb { E } [ \Sigma _ { \tau } ( x _ { \tau } ) ] = \mathbb { E } [ \mathsf { C o v } ( Z \mid \sqrt { 1 - \tau } X _ { 0 } + \sqrt { \tau } Z = x _ { \tau } ) ] \preceq \mathsf { C o v } [ Z ] = I _ { d } ,
$$

and by using (51), we have

$$
\begin{array} { r } { \frac { \tau _ { 0 , 0 } ( 1 - \widehat \tau _ { 0 , N } ) } { \widehat \tau _ { 0 , N } ( 1 - \tau _ { 0 , 0 } ) } \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau _ { 0 , 0 } } ( x _ { \tau _ { 0 , 0 } } ) ) ] \lesssim d , \quad \frac { \tau _ { 1 , 0 } ( 1 - \widehat \tau _ { 1 , N } ) } { \widehat \tau _ { 1 , N } ( 1 - \tau _ { 1 , 0 } ) } \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau _ { 1 , 0 } } ( x _ { \tau _ { 1 , 0 } } ) ) ] \lesssim d . } \end{array}
$$

Moreover, we have

$$
\begin{array} { r l } & { \frac { \tau _ { k + 2 , 0 } \left( 1 - \widehat { \tau } _ { k + 2 , N } \right) } { \widehat { \tau } _ { k + 2 , N } \left( 1 - \tau _ { k + 2 , 0 } \right) } - \frac { \tau _ { k , N } \left( 1 - \widehat { \tau } _ { k , N } \right) } { \widehat { \tau } _ { k , N } \left( 1 - \tau _ { k , N } \right) } } \\ & { = \frac { \tau _ { k + 2 , 0 } \left( 1 - \widehat { \tau } _ { k + 2 , N } \right) } { \widehat { \tau } _ { k + 2 , N } \left( 1 - \tau _ { k + 2 , 0 } \right) } - \frac { \tau _ { k + 2 , 0 } \left( 1 - \widehat { \tau } _ { k + 2 , 0 } \right) } { \widehat { \tau } _ { k + 2 , 0 } \left( 1 - \tau _ { k + 2 , 0 } \right) } } \\ & { = \frac { \tau _ { k + 2 , 0 } \left( \widehat { \tau } _ { k + 2 , 0 } - \widehat { \tau } _ { k + 2 , N } \right) } { \widehat { \tau } _ { k + 2 , 0 } \widehat { \tau } _ { k + 2 , N } \left( 1 - \tau _ { k + 2 , 0 } \right) } \lesssim \frac { N \tau _ { k + 2 , 0 } \widehat { \tau } _ { k + 2 , 0 } \left( 1 - \widehat { \tau } _ { k + 2 , N } \right) \log { T } } { T \widehat { \tau } _ { k + 2 , 0 } \widehat { \tau } _ { k + 2 , N } \left( 1 - \tau _ { k + 2 , 0 } \right) } } \\ & { \lesssim \frac { \log { T } } { K } . } \end{array}
$$

Inserting into (80), we have

$$
\begin{array} { r l } & { \displaystyle \sum _ { k = 0 } ^ { K - 1 } \frac { 1 - \widehat \tau _ { k , N } } { \widehat \tau _ { k , N } } \int _ { \tau _ { k , N } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau \lesssim \displaystyle \sum _ { k = 0 } ^ { K - 3 } \frac { \log T } { K } \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau _ { k + 2 , 0 } } ( x _ { \tau _ { k + 2 , 0 } } ) ) ] + d } \\ & { \qquad \lesssim d \log T . } \end{array}
$$

Inserting (81) into (79), we have

$$
\sum _ { k = 0 } ^ { K - 1 } \mathbb { E } _ { \boldsymbol { x } _ { \tau _ { k , 0 } } \sim p _ { \widetilde { X } _ { k } } } \left[ { \mathsf { K L } } \left( p _ { \widehat { X } _ { k + 1 } | \widehat { X } _ { k } } \left( \cdot | \boldsymbol { x } _ { \tau _ { k , 0 } } \right) \parallel p _ { Y _ { k + 1 } | Y _ { k } } \left( \cdot | \boldsymbol { x } _ { \tau _ { k , 0 } } \right) \right) \right]
$$

$$
\begin{array} { r l } & { \lesssim \frac { K d \log ^ { 4 } T } { T ^ { 3 } } \operatorname* { m i n } \left\{ d \log T + d \log T , L ^ { 2 } \log T \right\} + \frac { \log T } { T } \underset { k = 0 } { \sum } \underset { i = 0 } { \sum } \varepsilon _ { k , i } ^ { - 1 } } \\ & { \lesssim \frac { K d \log ^ { 5 } T } { T ^ { 3 } } \operatorname* { m i n } \left\{ d , L ^ { 2 } \right\} + \varepsilon _ { \mathrm { s c o r e } } ^ { 2 } \log T , } \end{array}
$$

and we complete the proof of Lemma 4.

The remaining of this section shall use the following lemma to prove Lemma 11. Its proof is postponed to the end of this section.

Lemma 12. For any $x \in S _ { \tau } \cap \mathcal { L } _ { \tau }$ , $y \in S _ { \tau }$ , and any $0 < \tau < 1$ , we have

$$
\| s _ { \tau } ^ { \star } ( x ) - s _ { \tau } ^ { \star } ( y ) \| _ { 2 } \leq \frac { C \operatorname* { m i n } \{ d \log T , L \} } { \tau } \| x - y \| _ { 2 } ,
$$

where $C$ is a sufficiently large constant dependent on $\theta + c _ { 0 }$ .

Recalling the definition of $\xi _ { k , n }$ , we have

$$
\frac { \left\| y _ { \tau _ { k , n } } - x _ { \tau _ { k , n } } \right\| _ { 2 } } { \sqrt { 1 - \tau _ { k , n } } } \leq \frac { \left\| y _ { \tau _ { k , n } } ^ { \star } - z _ { \tau _ { k , n } } ^ { \star } \right\| _ { 2 } } { \sqrt { 1 - \tau _ { k , n } } } + \left\| \xi _ { k , n } \right\| _ { 2 } .
$$

According to the definitions of $y _ { \tau _ { k , n } } ^ { \star }$ and $z _ { \tau _ { k , n } } ^ { \star }$ , for $x _ { \tau _ { k , i } } , y _ { \tau _ { k , i } } \in S _ { \tau _ { k , i } }$ , we have

$$
\begin{array} { r l } { \frac { | | \hat { \mathbf { g } } _ { \star , - } ^ { \star } , - \hat { \mathbf { g } } _ { \star , + } ^ { \star } | } { \sqrt { 1 - \hbar } } | _ { \mathbf { g } } } & { \leq \frac { - 1 - \hbar } { 2 } | \frac { | \hat { \mathbf { g } } _ { \star , + } ^ { \star } , \hat { \mathbf { g } } _ { \star , - } ^ { \star } , - \hat { \mathbf { g } } _ { \star , - } ^ { \star } | } { 2 } \langle \hat { \mathbf { g } } _ { \star , + - } , \hat { \mathbf { g } } _ { \star , - } ^ { \star } \rangle | _ { \mathbf { g } } } \\ & { \leq \frac { 1 - \hbar } { 2 } | \frac { 1 } { \sqrt { 1 - \hbar } } \frac { | \hat { \mathbf { g } } _ { \star , - } ^ { \star } , \hat { \mathbf { g } } _ { \star , - } ^ { \star } | } { 2 } \frac { | \hat { \mathbf { g } } _ { \star , - } ^ { \star } , \hat { \mathbf { g } } _ { \star , - } ^ { \star } , - \hat { \mathbf { g } } _ { \star , - } ^ { \star } | } { 2 } \frac { | \hat { \mathbf { g } } _ { \star , - - } ^ { \star } , \hat { \mathbf { g } } _ { \star , - } ^ { \star } , - \hat { \mathbf { g } } _ { \star , - } ^ { \star } | } { 2 } } \\ &  \mathrm { \quad ~ \times ~ } | \hat { \mathbf { g } } _ { \star , - } ^ { \star } , \frac { | | \hat { \mathbf { g } } _ { \star , - } ^ { \star } , \hat { \mathbf { g } } _ { \star , - } ^ { \star } | } { 2 } \frac { \sqrt { 1 - \hbar } } { 2 } \mathrm { e } ^ { \mathrm { i } | \hat { \mathbf { g } } _ { \star , - } ^ { \star } | } \frac { \sqrt { 1 - \hbar } } { 2 } \mathrm { e } ^  \end{array}
$$

where (a) uses (50) and (51). Inserting into (83), we have

$$
\frac { \left\| y _ { \tau _ { k , n } } - x _ { \tau _ { k , n } } \right\| _ { 2 } } { \sqrt { 1 - \tau _ { k , n } } } \lesssim \frac { \operatorname* { m i n } \{ d \log T , L \} \log T } { T } \sum _ { i = 1 } ^ { n - 1 } \frac { \left\| y _ { \tau _ { k , i } } - x _ { \tau _ { k , i } } \right\| _ { 2 } } { \sqrt { 1 - \tau _ { k , i } } } + \left\| \xi _ { k , n } ( x _ { \tau _ { k , 0 } } ) \right\| _ { 2 } .
$$

By applying the above relation recursively, for $T \gtrsim \mathrm { m i n } \{ d \log T , L \} N \log T$ , we have

$$
\frac { \left\| y _ { \tau _ { k , n } } - x _ { \tau _ { k , n } } \right\| _ { 2 } } { \sqrt { 1 - \tau _ { k , n } } } \lesssim \frac { \operatorname* { m i n } \{ d \log T , L \} \log T } { T } \sum _ { i = 1 } ^ { n - 1 } \left\| \xi _ { k , i } ( x _ { \tau _ { k , 0 } } ) \right\| _ { 2 } + \left\| \xi _ { k , n } ( x _ { \tau _ { k , 0 } } ) \right\| _ { 2 } .
$$

Thus

$$
\frac { \left\| y _ { \tau _ { k , n } } - x _ { \tau _ { k , n } } \right\| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } \lesssim \frac { N \operatorname* { m i n } \{ d \log T , L \} ^ { 2 } \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 1 } ^ { n - 1 } \left\| \xi _ { k , i } ( x _ { \tau _ { k , 0 } } ) \right\| _ { 2 } ^ { 2 } + \left\| \xi _ { k , n } ( x _ { \tau _ { k , 0 } } ) \right\| _ { 2 } ^ { 2 } .
$$

Define

$$
\frac { \zeta _ { k , n } } { 1 - \tau _ { k , n } } = \frac { N \operatorname* { m i n } \{ d \log T , L \} ^ { 2 } \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 1 } ^ { n - 1 } \frac { \| x _ { \tau _ { k , i } } - z _ { \tau _ { k , i } } ^ { \star } \| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , i } } + \frac { \| x _ { \tau _ { k , n } } - z _ { \tau _ { k , n } } ^ { \star } \| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } .
$$

We have

$$
\begin{array} { r l } {  { \frac { \big \| y _ { \tau _ { k , n } } - x _ { \tau _ { k , n } } \big \| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } \lesssim \frac { \zeta _ { k , n } } { 1 - \tau _ { k , n } } + \frac { N \operatorname* { m i n } \{ d \log T , L \} ^ { 2 } \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 1 } ^ { n - 1 } \frac { \| y _ { \tau _ { k , i } } ^ { \star } - y _ { \tau _ { k , i } } \| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , i } } } \quad } & { } \\ & { \quad \quad + \frac { \| y _ { \tau _ { k , n } } ^ { \star } - y _ { \tau _ { k , n } } \| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } . } \end{array}
$$

According to Lemma 8, we have

$$
\frac { \lVert y _ { \tau _ { k , n } } ^ { \star } - y _ { \tau _ { k , n } } \rVert _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } \lesssim \frac { N \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 0 } ^ { n - 1 } \frac { \tau _ { k , i } \widetilde { \varepsilon } _ { k , i } ^ { 2 } } { 1 - \tau _ { k , i } } .
$$

Considering that $N ^ { 2 } \operatorname* { m i n } \{ d \log T , L \} ^ { 2 } \log ^ { 2 } T / T ^ { 2 } \lesssim 1$ , we have

$$
\frac { N \operatorname* { m i n } \{ d \log T , L \} ^ { 2 } \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 1 } ^ { n } \frac { \| y _ { \tau _ { k , i } } ^ { \star } - y _ { \tau _ { k , i } } \| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , i } } + \frac { \| y _ { \tau _ { k , n } } ^ { \star } - y _ { \tau _ { k , n } } \| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } \lesssim \frac { N \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 0 } ^ { n - 1 } \frac { \tau _ { k , i } \tilde { \varepsilon } _ { k , i } ^ { 2 } } { 1 - \tau _ { k , i } } .
$$

Thus we have

$$
\frac { \| y _ { \tau _ { k , n } } ( x _ { k } ) - x _ { \tau _ { k , n } } ( x _ { k } ) \| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } \leq \frac { \zeta _ { k , n } ( x _ { k } ) } { 1 - \tau _ { k , n } } + \frac { N \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 1 } ^ { n } \frac { \tau _ { k , i } \widetilde \varepsilon _ { k , i } ^ { 2 } ( x _ { k } ) } { 1 - \tau _ { k , i } } ,
$$

which establishes (75). Furthermore, we have

$$
\begin{array} { r l } & { \displaystyle \int _ { \mathcal { E } _ { k , n } } \frac { \zeta _ { k , n } ( x _ { k } ) } { 1 - \tau _ { k , n } } p _ { \widehat { X } _ { k } } ( x _ { k } ) \mathrm { d } x _ { k } } \\ & { \lesssim \displaystyle \int _ { \mathcal { E } _ { k , n } } \frac { N \operatorname* { m i n } \{ d \log T , L \} ^ { 2 } \log ^ { 2 } T } { T ^ { 2 } } \displaystyle \sum _ { i = 1 } ^ { n - 1 } \frac { \| x _ { \tau _ { k , i } } ( x _ { k } ) - z _ { \tau _ { k , i } } ^ { \star } ( x _ { k } ) \| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , i } } p _ { \widehat { X } _ { k } } ( x _ { k } ) \mathrm { d } x _ { k } } \\ & { \quad + \displaystyle \int _ { \mathcal { E } _ { k , n } } \frac { \| x _ { \tau _ { k , n } } ( x _ { k } ) - z _ { \tau _ { k , n } } ^ { \star } ( x _ { k } ) \| _ { 2 } ^ { 2 } } { 1 - \tau _ { k , n } } p _ { \widehat { X } _ { k } } ( x _ { k } ) \mathrm { d } x _ { k } . } \end{array}
$$

Recalling (53) in Lemma 8 and the fact that $N ^ { 2 } \operatorname* { m i n } \{ d \log T , L \} ^ { 2 } \log ^ { 2 } T / T ^ { 2 } \stackrel { < } { \sim } 1$ , we have

$$
\begin{array} { r l } & { \int _ { \varepsilon _ { k , n } } \frac { \zeta _ { k , n } \left( x _ { k } \right) } { 1 - \tau _ { k , n } } p _ { \widehat { X } _ { k } } ( x _ { k } ) \mathrm { d } x _ { k } } \\ & { \lesssim \frac { d \log ^ { 5 } T } { T ^ { 3 } } \operatorname* { m i n } \Big \{ \frac { N d \widehat { \tau } _ { k , - 1 } \log T } { T ( 1 - \widehat { \tau } _ { k , - 1 } ) } + \int _ { \tau _ { k , n } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } \left[ \mathsf { T } \mathsf { r } \left( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) \right) \right] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , \frac { N L ^ { 2 } \widehat { \tau } _ { k , - 1 } \log T } { T ( 1 - \widehat { \tau } _ { k , - 1 } ) } \Big \} , } \end{array}
$$

and thus

$$
\int _ { \mathscr { E } _ { k , n } } \zeta _ { k , n } ( x _ { k } ) p _ { \widehat { X } _ { k } } ( x _ { k } ) \mathrm { d } x _ { k }
$$

$$
\lesssim \frac { d \log ^ { 5 } T } { T ^ { 3 } } \operatorname* { m i n } \Big \{ \frac { N d \widehat { \tau } _ { k , - 1 } \log T } { T } + ( 1 - \widehat { \tau } _ { k , n } ) \int _ { \tau _ { k , n } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } \big [ \mathsf { T r } \big ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) \big ) \big ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , \frac { N L ^ { 2 } \widehat { \tau } _ { k , - 1 } \log T } { T } \Big \}
$$

and establish (76), where we use (51).

Proof of Lemma 12. Before diving into the proof details, we first present the following lemma. Its proof is postponed to Appendix F.3.

Lemma 13. For any $x , y$ satisfying

$$
\| x - y \| _ { 2 } \leq c \sqrt { \frac { \tau } { d \log T } }
$$

with $\begin{array} { r } { c \leq \sqrt { \frac { 1 } { \theta + c _ { 0 } } } / 1 2 . } \end{array}$ , if $x \in S _ { \tau }$ , we have

$$
p _ { X _ { \tau } } ( x ) - p _ { X _ { \tau } } ( y ) \leq 6 \| y - x \| _ { 2 } \sqrt { \frac { ( \theta + c _ { 0 } ) d \log T } { \tau } } p _ { X _ { \tau } } ( x ) .
$$

Moreover, if $x , y \in S _ { \tau }$ , then we have

$$
\frac { 1 } { 2 } \leq 1 - 6 \left\| x - y \right\| _ { 2 } \sqrt { \frac { ( \theta + c _ { 0 } ) d \log T } { \tau } } \leq \frac { p _ { X _ { \tau } } ( x ) } { p _ { X _ { \tau } } ( y ) } \leq \frac { 1 } { 1 - 6 \left\| x - y \right\| _ { 2 } \sqrt { \frac { ( \theta + c _ { 0 } ) d \log T } { \tau } } } \leq 2 .
$$

We first prove that for $x \in S _ { \tau }$ and $y \in S _ { \tau }$ , we have

$$
\| s _ { \tau } ^ { \star } ( x ) - s _ { \tau } ^ { \star } ( y ) \| _ { 2 } \lesssim \frac { d \log T } { \tau } \| x - y \| .
$$

To this end, without loss of generality, we assume that $p _ { X _ { \tau } } ( y ) \geq p _ { X _ { \tau } } ( x )$ . Then according to the definition of score function, we have

$$
\begin{array} { l } { { s _ { \tau } ^ { \star } ( y ) - s _ { \tau } ^ { \star } ( x ) = \displaystyle \frac { 1 } { \tau } ( x - y ) - \frac { 1 } { \tau } \int _ { x _ { 0 } } ( x - \sqrt { 1 - \tau } x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | y ) } } \\ { { \phantom { s _ { \tau } ^ { \star } ( y ) - s _ { \tau } ^ { \star } ( x ) = } + ( x - \sqrt { 1 - \tau } x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x ) \mathrm { d } x _ { 0 } . } } \end{array}
$$

Thus we have

$$
\begin{array} { l } { \displaystyle \left\| s _ { \tau } ^ { \star } ( y ) - s _ { \tau } ^ { \star } ( x ) \right\| _ { 2 } } \\ { \displaystyle \leq \frac { 1 } { \tau } \left\| x - y \right\| _ { 2 } + \frac { 1 } { \tau } \left\| \int _ { x _ { 0 } } ( x - \sqrt { 1 - \tau } x _ { 0 } ) \frac { p _ { X _ { \tau } | X _ { 0 } } ( y | x _ { 0 } ) - p _ { X _ { \tau } | X _ { 0 } } ( x | x _ { 0 } ) } { p _ { X _ { \tau } } ( y ) } p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } \right\| _ { 2 } } \\ { \displaystyle \phantom { \frac { 1 } { \tau } } + \frac { 1 } { \tau } \left\| \left( \frac { p _ { X _ { \tau } } ( x ) } { p _ { X _ { \tau } } ( y ) } - 1 \right) \int _ { x _ { 0 } } ( x - \sqrt { 1 - \tau } x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x ) \mathrm { d } x _ { 0 } \right\| _ { 2 } . } \end{array}
$$

For the second and the third term, we define $\mathcal { R } = \left. x _ { 0 } \in \mathbb { R } ^ { d } : \left\| x - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } \leq \left\| y - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } \right.$ , and have

$$
\begin{array} { r l } & { \quad \left\| \displaystyle \int _ { x _ { 0 } } ( x - \sqrt { 1 - \tau } x _ { 0 } ) \frac { p \boldsymbol { x } _ { \tau } | \boldsymbol { x } _ { 0 } } { p \boldsymbol { x } _ { \tau } ( y ) } \boldsymbol { - } p \boldsymbol { x } _ { \tau } | \boldsymbol { x } _ { 0 } ( x | x _ { 0 } ) \right\} p \boldsymbol { x } _ { 0 } ( x _ { 0 } ) \mathrm { d } x _ { 0 } | _ { 2 }  \\ & { \leq \left\| \displaystyle \int _ { \mathbb { R } ^ { c } } ( x - \sqrt { 1 - \tau } x _ { 0 } ) \frac { p \boldsymbol { x } _ { \tau } | \boldsymbol { x } _ { 0 } } { p \boldsymbol { x } _ { \tau } ( y ) } p _ { X _ { 0 } } ( x _ { 0 } ) \left( \exp \left( \frac { ( x - y ) ^ { \top } ( x + y - 2 \sqrt { 1 - \tau } x _ { 0 } } { 2 \tau } \right) - 1 \right) \mathrm { d } x _ { 0 } \right\| _ { 2 } } \\ & { \quad + \left\| \displaystyle \int _ { \mathbb { R } ^ { c } } ( x - \sqrt { 1 - \tau } x _ { 0 } ) \frac { p _ { X _ { \tau } | X _ { 0 } } ( y | x _ { 0 } ) } { p \boldsymbol { x } _ { \tau } ( y ) } p _ { X _ { 0 } } ( x _ { 0 } ) \left( 1 - \exp \left( \frac { ( y - x ) ^ { \top } ( x + y - 2 \sqrt { 1 - \tau } x _ { 0 } } { 2 \tau } \right) \right) \mathrm { d } x _ { 0 } \right\| _ { 2 } . } \end{array}
$$

The first term is smaller than

$$
\int _ { \mathcal { R } } | | x - \sqrt { 1 - \tau } x _ { 0 } | | _ { 2 } \frac { \| x - y \| \| x + y - 2 \sqrt { 1 - \tau } x _ { 0 } \| _ { 2 } } { 2 \tau } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x ) \mathrm { d } x _ { 0 } \frac { p _ { X _ { \tau } } ( x ) } { p _ { X _ { \tau } } ( y ) } ,
$$

and the second term is smaller than

$$
\begin{array} { l l } { \displaystyle \int _ { \mathbb { R } ^ { c } } \| y - \sqrt { 1 - \tau } x _ { 0 } \| _ { 2 } \frac { \| y - x \| _ { 2 } \| x + y - 2 \sqrt { 1 - \tau } x _ { 0 } \| _ { 2 } } { 2 \tau } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | y ) \mathrm { d } x _ { 0 } } \\ { \displaystyle + \left\| ( x - y ) \left( 1 - \frac { p _ { X _ { \tau } } ( x ) } { p _ { X _ { \tau } } ( y ) } \right) \right\| _ { 2 } . } \end{array}
$$

Combining these, we have

$$
\begin{array} { r l } & { \quad \left\| \int _ { \mathbf { x } _ { 0 } } ( x - \sqrt { 1 - \tau } x _ { 0 } ) \frac { y _ { X _ { \tau } } | x _ { 0 } ( \hat { y } | x _ { 0 } ) - p _ { X _ { \tau } } | x _ { 0 } ( x | x _ { 0 } ) } { p _ { X _ { \tau } } ( y ) } p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } \right\| _ { 2 } } \\ & { \leq \frac { \| y - x \| _ { 2 } } { \tau } \int _ { \mathbb { R } } \| x - \sqrt { 1 - \tau } x _ { 0 } \| _ { 2 } ^ { 2 } p _ { X _ { \tau } | X _ { \tau } } ( x _ { 0 } | x ) \mathrm { d } x _ { 0 } } \\ & { \quad + \frac { \| y - x \| _ { 2 } ^ { 2 } } { 2 \tau } \int _ { \mathbb { R } } \| x - \sqrt { 1 - \tau } x _ { 0 } \| _ { 2 } p _ { X _ { \tau } | X _ { \tau } } ( x _ { 0 } | x ) \mathrm { d } x _ { 0 } } \\ & { \quad + \frac { \| y - x \| _ { 2 } } { \tau } \int _ { \mathbb { R } } \left\| y - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } ^ { 2 } p _ { X _ { \tau } | X _ { \tau } } ( x _ { 0 } | y ) \mathrm { d } x _ { 0 } } \\ & { \quad + \frac { \| y - x \| _ { 2 } ^ { 2 } } { 2 \tau } \int _ { \mathbb { R } } \| y - \sqrt { 1 - \tau } x _ { 0 } \| _ { 2 } ^ { 2 } p _ { X _ { \tau } | X _ { \tau } } ( x _ { 0 } | y ) \mathrm { d } x _ { 0 } + \| x - y \| _ { 2 } } \\ &  \stackrel { \mathrm { G a } } { \lesssim } d \mathbf { g } T \left\| x - y \right\| _ { 2 } + \| x - y \| _ { 2 } ^ { 2 } \sqrt { d \mathbf { g } _ { \tau } } \| X _ { \tau } ( x _ { 0 } | y ) \mathrm { d } x _ { 0 } \end{array}
$$

for

$$
\| x - y \| _ { 2 } \leq c \sqrt { \frac { \tau } { d \log T } } ,
$$

where (a) uses Lemma 9. For the third term, by using Lemma 13, we have

$$
\begin{array} { r l } & { \quad \left\| \left( \frac { p _ { X _ { \tau } } ( x ) } { p _ { X _ { \tau } } ( y ) } - 1 \right) \int _ { x _ { 0 } } ( x - \sqrt { 1 - \tau } x _ { 0 } ) p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x ) \mathrm { d } x _ { 0 } \right\| _ { 2 } } \\ & { \leq 6 \left\| x - y \right\| _ { 2 } \sqrt { \frac { ( \theta + c _ { 0 } ) d \log T } { \tau } } \int _ { x _ { 0 } } \left\| x - \sqrt { 1 - \tau } x _ { 0 } \right\| _ { 2 } p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x ) \mathrm { d } x _ { 0 } } \\ & { \lesssim d \log T \left\| x - y \right\| _ { 2 } . } \end{array}
$$

Inserting (86) and (87) into (85), we have for $\begin{array} { r } { \| x - y \| _ { 2 } \leq c \sqrt { \frac { \tau } { d \log T } } } \end{array}$ ,

$$
\| s _ { \tau } ^ { \star } ( y ) - s _ { \tau } ^ { \star } ( x ) \| _ { 2 } \lesssim \frac { d \log T } { \tau } \| x - y \| _ { 2 } .
$$

For $\begin{array} { r } { \| x - y \| _ { 2 } \geq c \sqrt { \frac { \tau } { d \log T } } } \end{array}$ , we have

$$
\| s _ { \tau } ^ { \star } ( y ) - s _ { \tau } ^ { \star } ( x ) \| _ { 2 } \leq \| s _ { \tau } ^ { \star } ( y ) \| _ { 2 } + \| s _ { \tau } ^ { \star } ( x ) \| _ { 2 } \leq 1 0 \sqrt { \frac { ( \theta + c _ { 0 } ) d \log T } { \tau } } \lesssim \frac { d \log T } { \tau } \| x - y \| _ { 2 } .
$$

Thus we prove (84).

In addition, for $x \in { \mathcal { L } } _ { \tau }$ and $\begin{array} { r } { \| x - y \| \leq \frac { C \sqrt { d \tau } \log T } { L } } \end{array}$ , according to Definition 2, we have

$$
\| s _ { \tau } ^ { \star } ( y ) - s _ { \tau } ^ { \star } ( x ) \| _ { 2 } \leq \frac { L } { \tau } \| x - y \| _ { 2 } .
$$

Otherwise, for $x \in S _ { \tau }$ , $y \in S _ { \tau }$ , and $\begin{array} { r } { \| x - y \| > \frac { C \sqrt { d \tau } \log T } { L } } \end{array}$ , according to Lemma 9, we have

$$
\| s _ { \tau } ^ { \star } ( y ) - s _ { \tau } ^ { \star } ( x ) \| _ { 2 } \leq 2 \sqrt { \frac { 2 5 ( \theta + c _ { 0 } ) d \log T } { \tau } } \leq \frac { L } { \tau } \| x - y \| _ { 2 } ,
$$

for $C \geq 1 0 \sqrt { \theta + c _ { 0 } }$ . Thus we complete the proof.

# D.3 Proof of Lemma 5

According to Lemma 1, we know that $\widehat { X } _ { 0 } \stackrel { \mathrm { ~ d ~ } } { = } X _ { \tau _ { 0 , 0 } } = \sqrt { 1 - \tau _ { 0 , 0 } } X _ { 0 } + \sqrt { \tau _ { 0 , 0 } } Z$ for some standard Gaussian variable $Z$ . Then we have

$$
\begin{array} { r l } & { \mathsf { K L } \big ( p _ { \widehat { X } _ { 0 } } \| p _ { Y _ { 0 } } \big ) \le \mathsf { K L } \big ( p _ { X _ { \tau _ { 0 } , 0 } , X _ { 0 } } \| p _ { Y _ { 0 } } p _ { X _ { 0 } } \big ) = \mathbb { E } _ { X _ { 0 } \sim p _ { \mathsf { d a t a } } } [ \mathsf { K L } \big ( p _ { \widehat { X } _ { 0 } | X _ { 0 } } \| p _ { Y _ { 0 } } \big ) ] } \\ & { \qquad = \displaystyle \frac { 1 } { 2 } \left[ ( 1 - \tau _ { 0 , 0 } ) \mathbb { E } _ { X _ { 0 } \sim p _ { \mathsf { d a t a } } } \| X _ { 0 } \| ^ { 2 } - d \log ( \tau _ { 0 , 0 } ) + d \tau _ { 0 , 0 } - d \right] } \\ & { \qquad \le \displaystyle \frac { 1 } { 2 } T ^ { c _ { R } - c _ { 0 } } + d \frac { ( 1 - \tau _ { 0 , 0 } ) ^ { 2 } } { 2 \tau _ { 0 , 0 } } \le \displaystyle \frac { 1 } { 2 } T ^ { c _ { R } - c _ { 0 } } + \frac { d } { T ^ { 2 c _ { 0 } } } \le \frac { 1 } { T ^ { 1 0 } } , } \end{array}
$$

as long as $c _ { 0 } \geq \operatorname* { m a x } \{ c _ { R } + 1 0 , 1 0 \}$ , and $T \gtrsim d ^ { 1 / 1 0 }$ .

The remaining of this section focuses on the proof of (38). For $d \log T \geq L$ , we have $\mathcal { E } _ { k } = \emptyset$ and (38) holds trivially. Below we only consider the case of $d \log T \leq L$ . We decompose the set $\mathcal { E } _ { k } ^ { \mathrm { c } }$ as

$$
\begin{array} { r } { \mathcal E _ { k } ^ { \mathrm { c } } = \cup _ { n = 0 } ^ { N - 1 } \widehat { \mathcal E } _ { k , n } , } \end{array}
$$

where for $1 \leq n \leq N - 1$ ,

$$
\begin{array} { r l } & { \widehat { \mathcal { E } } _ { k , n } = \{ x _ { k } : x _ { \tau _ { k , i } } ( x _ { k } ) \in \widetilde { \mathcal { S } } _ { \tau _ { k , i } } \cap \mathcal { L } _ { \tau _ { k , i } } , \forall 0 \leq i \leq N - 1 , } \\ & { \qquad y _ { \tau _ { k , i } } ( x _ { k } ) \in \mathcal { S } _ { \tau _ { k , i } } , \forall 0 \leq i \leq n - 1 , y _ { \tau _ { k , n } } ( x _ { k } ) \notin \mathcal { S } _ { \tau _ { k , n } } \} , } \end{array}
$$

and

$$
\widehat { \mathcal { E } } _ { k , 0 } = \{ x _ { k } : \exists 0 \leq i \leq N - 1 , x _ { \tau _ { k , i } } ( x _ { k } ) \notin \widetilde { \mathcal { S } } _ { \tau _ { k , i } } \cap \mathcal { L } _ { \tau _ { k , i } } \} .
$$

Furthermore, we introduce another auxiliary set

$$
\mathcal { B } _ { k , n } = \left\{ x _ { k } : \frac { N \widehat { \tau } _ { k , - 1 } \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 0 } ^ { n - 1 } \| s _ { \tau _ { k , i } } ( y _ { \tau _ { k , i } } ( x _ { k } ) ) - s _ { \tau _ { k , i } } ^ { \star } ( y _ { \tau _ { k , i } } ( x _ { k } ) ) \| ^ { 2 } \leq \frac { C \tau _ { k , n } } { d \log T } \right\} ,
$$

where $C$ is a sufficiently small constant. Then we have

$$
\begin{array} { r } { \mathcal { E } _ { k } ^ { \mathrm { c } } \subset \widehat { \mathcal { E } } _ { k , 0 } \cup \left( \cup _ { n = 1 } ^ { N - 1 } \left( \widehat { \mathcal { E } } _ { k , n } \cap \mathcal { B } _ { k , n } \right) \right) \cup \left( \cup _ { n = 1 } ^ { N - 1 } \mathcal { B } _ { k , n } ^ { \mathrm { c } } \right) . } \end{array}
$$

Thus we have

$$
P ( \widehat { X } _ { k } \in \mathcal { E } _ { k } ^ { \mathrm { c } } ) \leq P ( \widehat { X } _ { k } \in \widehat { \mathcal { E } } _ { k , 0 } ) + \sum _ { n = 1 } ^ { N - 1 } P ( \widehat { X } _ { k } \in \widehat { \mathcal { E } } _ { k , n } \cap \mathcal { B } _ { k , n } ) + P ( \widehat { X } _ { k } \in \cup _ { n = 1 } ^ { N - 1 } \mathcal { B } _ { k , n } ^ { \mathrm { c } } ) .
$$

Below we shall calculate these terms separately. We start from considering $P ( \widehat { X } _ { k } \in \widehat { \mathcal { E } } _ { k , 0 } )$ . Noticing that $x _ { \tau _ { k , n } }$ has the identical distribution with $X _ { \tau _ { k , n } }$ , we have

$$
P ( \widehat { X } _ { k } \in \widehat { \mathcal { E } } _ { k , 0 } ) \leq \sum _ { n = 0 } ^ { N - 1 } P ( X _ { \tau _ { k , n } } \in \widetilde { \mathcal { S } } _ { \tau _ { k , n } } ^ { \mathrm { c } } ) + \sum _ { n = 0 } ^ { N } P ( X _ { \tau _ { k , n } } \in \mathcal { L } _ { \tau _ { k , n } } ^ { \mathrm { c } } ) \lesssim \frac { N } { T ^ { 4 } } ,
$$

where the last inequality comes from the fact that

$$
P ( X _ { \tau _ { k , n } } \in \widetilde { \mathcal { S } } _ { \tau _ { k , n } } ^ { \mathrm { c } } ) \lesssim \frac { 1 } { T ^ { 4 } } , \qquad P ( X _ { \tau _ { k , n } } \in \mathcal { L } _ { \tau _ { k , n } } ^ { \mathrm { c } } ) \lesssim \frac { 1 } { T ^ { 4 } } ,
$$

where the first inequality is proved in Li and Yan (2024a) (cf. (A.18)), and the second inequality comes from Definition 2.

Next, we analyze $P ( \widehat { X } _ { k } \in \widehat { \mathcal { E } } _ { k , n } \cap B _ { k , n } )$ . According to Lemma 13, we have

$$
p _ { X _ { \tau } } ( y ) \geq \left( 1 - 6 \| y - x \| \sqrt { \frac { ( \theta + c _ { 0 } ) d \log T } { \tau } } \right) p _ { X _ { \tau } } ( x ) , \quad \mathrm { i f } \ \| x - y \| ^ { 2 } \leq \frac { c \tau } { d \log T } ,
$$

where $c = 1 / ( 1 4 4 ( \theta + c _ { 0 } ) )$ . Recalling that $\log p _ { X _ { \tau } } ( x ) \geq \log 2 - \theta d \log T$ , we have

$$
\begin{array} { r l } & { \quad P \big ( \widehat { X } _ { k } \in \widehat { \mathcal { E } } _ { k , n } \cap \mathcal { B } _ { k , n } \big ) } \\ & { \leq P \Big ( \frac { N \widehat { \tau } _ { k , - 1 } \log ^ { 2 } T } { T ^ { 2 } } \displaystyle \sum _ { i = 0 } ^ { n - 1 } \widetilde { \varepsilon } _ { k , i } ^ { 2 } ( \widehat { X } _ { k } ) \leq \frac { C \tau _ { k , n } } { d \log T } , \| x _ { \tau _ { k , n } } ( \widehat { X } _ { k } ) - y _ { \tau _ { k , n } } ( \widehat { X } _ { k } ) \| _ { 2 } ^ { 2 } \geq \frac { c \tau _ { k , n } } { d \log T } , } \\ & { \quad x _ { \tau _ { k , i } } ( \widehat { X } _ { k } ) \in \widetilde { \mathcal { S } } _ { \tau _ { k , i } } \cap \mathcal { L } _ { \tau _ { k , i } } , y _ { \tau _ { k , i } } ( \widehat { X } _ { k } ) \in \mathcal { S } _ { \tau _ { k , i } } , \forall \ 0 \leq i < n \Big ) . } \end{array}
$$

According to (75), for

$$
\begin{array} { r l } & { x \in { \mathcal A } _ { k , n } : = \Bigl \{ x : \frac { N \widehat \tau _ { k , - 1 } \log ^ { 2 } T } { T ^ { 2 } } \displaystyle \sum _ { i = 0 } ^ { n - 1 } \widetilde \varepsilon _ { k , i } ^ { 2 } ( x ) \leq \frac { C \tau _ { k , n } } { d \log T } , } \\ & { \qquad x _ { \tau _ { k , i } } ( x ) \in \widetilde { \mathcal S } _ { \tau _ { k , i } } \cap { \mathcal L } _ { \tau _ { k , i } } , y _ { \tau _ { k , i } } ( x ) \in { \mathcal S } _ { \tau _ { k , i } } , \forall \ 0 \leq i < n \Bigr \} , } \end{array}
$$

which satisfies $\mathcal { A } _ { k , n } \subset \mathcal { E } _ { k , n }$ , we have

$$
\begin{array} { r l r } {  { \| x _ { \tau _ { k , n } } ( x ) - y _ { \tau _ { k , n } } ( x ) \| ^ { 2 } \lesssim \zeta _ { k , n } ( x ) + \frac { N \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 0 } ^ { n - 1 } \widehat { \tau } _ { k , i - 1 } \widehat { \varepsilon } _ { k , i } ^ { 2 } ( x ) } } \\ & { } & { \lesssim \zeta _ { k , n } ( x ) + \frac { \widehat { \tau } _ { k , - 1 } N \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 0 } ^ { n - 1 } \widehat { \varepsilon } _ { k , i } ^ { 2 } ( x ) \overset { ( \mathrm { a } ) } { \lesssim } \zeta _ { k , n } ( x ) + \frac { C \tau _ { k , n } } { d \log T } , } \end{array}
$$

where (a) comes from the condition that $\boldsymbol { x } \in \boldsymbol { A } _ { k , n }$ . Assuming $C \le c / 2$ , we have

$$
\begin{array} { r l } & { P ( \widehat { X } _ { k } \in \widehat { \mathcal { E } } _ { k , n } \cap B _ { k , n } ) } \\ & { \leq P \left( \widehat { X } _ { k } \in \mathcal { A } _ { k , n } , \zeta _ { k , n } ( \widehat { X } _ { k } ) \geq \frac { c \tau _ { k , n } } { 2 d \log T } \right) } \\ & { \overset { \mathrm { ( a ) } } { \leq } \frac { 2 d \log T } { c \tau _ { k , n } } \int _ { A _ { k , n } } \zeta _ { k , n } ( x ) p _ { \widehat { X } _ { k } } ( x ) \mathrm { d } x \leq \frac { 2 d \log T } { c \tau _ { k , n } } \int _ { \mathcal { E } _ { k , n } } \zeta _ { k , n } ( x ) p _ { \widehat { X } _ { k } } ( x ) \mathrm { d } x } \\ & { \overset { \mathrm { ( b ) } } { \lesssim } \frac { K d \log ^ { 4 } T } { T ^ { 3 } } \operatorname* { m i n } \Big \{ \frac { N d \log T } { T } + \frac { 1 - \widehat { \tau } _ { k , n } } { \widehat { \tau } _ { k , n } } \int _ { \tau _ { k , n } } ^ { \tau _ { k , n } } \frac { \| \nabla \| \tau ( \sum _ { \tau } ^ { 2 } ( x _ { \tau } ) ) \| } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , \frac { N L ^ { 2 } \log T } { T } \Big \} } \\ & { \lesssim \frac { d \log ^ { 5 } T } { T ^ { 3 } } \operatorname* { m i n } \Big \{ d + \frac { K ( 1 - \widehat { \tau } _ { k , N } ) } { \widehat { \tau } _ { k , N } \log T } \int _ { \tau _ { k , N } } ^ { \tau _ { k , n } } \frac { \| \nabla \| \tau ( \sum _ { \tau } ^ { 2 } ( x _ { \tau } ) ) \| } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , L ^ { 2 } \Big \} } \end{array}
$$

where (a) uses the Markov inequality, and (b) uses (76) and the fact that $K \stackrel { > } { \sim } \operatorname* { m i n } \{ d \log T , L \} \log T = d \log ^ { 2 } T$ for $d \log T \lesssim L$ . Summing from $n = 0$ to $N - 1$ , we have

$$
\sum _ { n = 1 } ^ { N - 1 } P ( \widehat { X } _ { k } \in \widehat { \mathcal { E } } _ { k , n } \cap \mathcal { B } _ { k , n } ) \lesssim \frac { \log ^ { 3 } T } { T ^ { 2 } } \operatorname* { m i n } \Big \{ d + \frac { K ( 1 - \widehat { \tau } _ { k , N } ) } { \widehat { \tau } _ { k , N } \log T } \int _ { \tau _ { k , N } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , L ^ { 2 } \Big \} .
$$

Finally, we analyze $P ( \widehat { X } _ { k } \in \cup _ { n = 1 } ^ { N - 1 } B _ { k , n } ^ { \mathrm { c } } )$ . Noticing that $B _ { k , n } \subset B _ { k , n - 1 }$ , we have

$$
\begin{array} { r l } & { \quad P ( \widehat { X } _ { k } \in \cup _ { n = 0 } ^ { N - 1 } \mathcal { B } _ { k , n } ^ { \mathrm { c } } ) \leq P ( \widehat { X } _ { k } \in \mathcal { B } _ { k , N - 1 } ^ { \mathrm { c } } ) } \\ & { \leq P \left( \frac { N \widehat { \tau } _ { k , 0 } \log ^ { 2 } T } { T ^ { 2 } } \sum _ { i = 1 } ^ { N - 1 } \| s _ { \tau _ { k , i } } ( y _ { \tau _ { k , i } } ( \widehat { X } _ { k } ) ) - s _ { \tau _ { k , i } } ^ { * } ( y _ { \tau _ { k , i } } ( \widehat { X } _ { k } ) ) \| ^ { 2 } > \frac { C \tau _ { k , N } } { d \log T } \right) } \\ & { \overset { \mathrm { ( a ) } } { \lesssim } \frac { N d \log ^ { 3 } T } { T ^ { 2 } } \overset { N - 1 } { \underset { i = 1 } { \overset { N } { \sum } } } \mathbb { E } _ { x _ { k } \sim \widehat { X } _ { k } } \| s _ { \tau _ { k , i } } ( y _ { \tau _ { k , i } } ( x _ { k } ) ) - s _ { \tau _ { k , i } } ^ { * } ( y _ { \tau _ { k , i } } ( x _ { k } ) ) \| ^ { 2 } } \\ & { \lesssim \frac { N d \log ^ { 3 } T } { T ^ { 2 } } \overset { N - 1 } { \underset { i = 0 } { \overset { N } { \sum } } } \varepsilon _ { k , i } ^ { 2 } , } \end{array}
$$

where (a) uses Markov inequality. Inserting (89), (90), and (91) into (88), we have

$$
\begin{array} { r l r } {  { P ( \widehat { X } _ { k } \in \mathcal { E } _ { k } ) \lesssim \frac { N } { T ^ { 4 } } + \frac { \log T } { T } \sum _ { i = 0 } ^ { N - 1 } \varepsilon _ { k , i } ^ { 2 } } } \\ & { } & { \quad + \frac { \log ^ { 3 } T } { T ^ { 2 } } \operatorname* { m i n } \Big \{ d + \frac { K ( 1 - \widehat { \tau } _ { k , N } ) } { \widehat { \tau } _ { k , N } \log T } \int _ { \tau _ { k , N } } ^ { \tau _ { k , 0 } } \frac { \mathbb { E } [ \mathsf { T r } ( \Sigma _ { \tau } ^ { 2 } ( x _ { \tau } ) ) ] } { ( 1 - \tau ) ^ { 2 } } \mathrm { d } \tau , L ^ { 2 } \Big \} . } \end{array}
$$

Moreover, we have

$$
\begin{array} { r l } & { \displaystyle \sum _ { k = 0 } ^ { K - 1 } P ( \widehat X _ { k } \in \mathcal { E } _ { k } ) \stackrel { \mathrm { ( a ) } } \lesssim \frac { 1 } { T ^ { 3 } } + \frac { \log T } { T } \displaystyle \sum _ { k = 0 } ^ { K - 1 } \sum _ { i = 0 } ^ { N - 1 } \varepsilon _ { k , i } ^ { 2 } + \frac { K \log ^ { 3 } T } { T ^ { 2 } } \operatorname* { m i n } \left\{ d + d , L ^ { 2 } \right\} } \\ & { \displaystyle \stackrel { \mathrm { ( b ) } } { \lesssim } \frac { d ^ { 2 } \log ^ { 5 } T } { T ^ { 2 } } + \frac { \log T } { T } \displaystyle \sum _ { k = 0 } ^ { K - 1 } \displaystyle \sum _ { i = 0 } ^ { N - 1 } \varepsilon _ { k , i } ^ { 2 } } \end{array}
$$

where (a) uses (81) and (b) uses the fact that $d \log T \lesssim L$ .

# D.4 Proof of (28)

We first show a basic inequality: for any two probability density functions $f$ , $g$ , and any set $\mathcal { R }$ , we have

$$
\int _ { \mathcal { R } } f ( x ) \log \frac { f ( x ) } { g ( x ) } \mathrm { d } x \geq \log \left( \frac { \int _ { \mathcal { R } } f ( x ) \mathrm { d } x } { \int _ { \mathcal { R } } g ( x ) \mathrm { d } x } \right) \int _ { \mathcal { R } } f ( x ) \mathrm { d } x
$$

which is proved in Lemma 6 in Li and Yan (2024a). Now we are ready to prove (28). To this end, let us prove a more general conclusion. For any two probability density functions $f$ and $g$ , and any set $\mathcal { R }$ , assume that $\ddot { f }$ and $\tilde { g }$ is defined as

$$
\tilde { f } ( x ) = f ( x ) \mathbb { 1 } \{ x \in \mathcal { R } \} + \int _ { x \notin \mathcal { R } } f ( x ) \mathrm { d } x \delta _ { \infty } , \quad \tilde { g } ( x ) = g ( x ) \mathbb { 1 } \{ x \in \mathcal { R } \} + \int _ { x \notin \mathcal { R } } g ( x ) \mathrm { d } x \delta _ { \infty } .
$$

Then

$$
{ \mathsf { K L } } ( { \tilde { f } } \| { \tilde { g } } ) \leq { \mathsf { K L } } ( f \| g ) .
$$

Towards this, according to definitions, we have

$$
\mathsf { K L } ( \tilde { f } \| \tilde { g } ) - \mathsf { K L } ( f \| g ) = \int _ { x \notin \mathcal R } f ( x ) \mathrm { d } x \log \left( \frac { \int _ { x \notin \mathcal R } f ( x ) \mathrm { d } x } { \int _ { x \notin \mathcal R } g ( x ) \mathrm { d } x } \right) - \int _ { x \notin \mathcal R } f ( x ) \log \left( \frac { f ( x ) } { g ( x ) } \right) \mathrm { d } x \leq 0
$$

and complete the proof.

# E Parallel sampling

# E.1 Parallel algorithm

The parallel sampling procedure follows the same structure as the original sampler described in (10): it consists of $K$ rounds, with each round comprising several iterations. The key difference is that, in each round, we use $N$ processors and perform $M \ll N$ iterations. In each iteration, each processor updates the sample $Y _ { m , k , n }$ using the outputs of other processors from the previous iteration $\{ Y _ { m - 1 , k , i } \} _ { i < n }$ . The implementation details for the $k$ -th round are as follows.

1. Initialization: for the $n$ -th parallel processor, the sample is initialized as

$$
\frac { Y _ { 0 , k , n } } { \sqrt { 1 - \tau _ { k , n } } } = \frac { Y _ { k } } { \sqrt { 1 - \tau _ { k , 0 } } } , \quad n = 1 , \cdots , N .
$$

2. Parallel updates: we use $N$ processors to update $Y _ { m , k , n }$ ( $n = 1 , \cdots , N )$ simultaneously for $M$ iterations. In the $m$ -th iteration, the update rule is:

$$
\begin{array} { l } { \displaystyle \frac { Y _ { m , k , n } } { \sqrt { 1 - \tau _ { k , n } } } = \frac { Y _ { k } } { \sqrt { 1 - \tau _ { k , 0 } } } + \frac { s _ { T - \frac { k N } { 2 } + 1 } ( Y _ { k } ) } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } ( \tau _ { k , 0 } - \hat { \tau } _ { k , 0 } ) } \\ { \displaystyle \quad + \sum _ { i = 1 } ^ { n - 1 } \frac { s _ { T - \frac { k N } { 2 } - i + 1 } ( Y _ { m - 1 , k , i } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) } \\ { \displaystyle \quad + \frac { s _ { T - \frac { k N } { 2 } - n + 2 } ( Y _ { m - 1 , k , n - 1 } ) } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , n - 1 } - \tau _ { k , n } ) . } \end{array}
$$

3. Noise injection: Once $Y _ { M , k , N }$ is obtained, we update $Y _ { k + 1 }$ as follows.

$$
Y _ { k + 1 } = \sqrt { \frac { 1 - \tau _ { k + 1 , 0 } } { 1 - \tau _ { k , N } } } Y _ { M , k , N } + \sqrt { \frac { \tau _ { k + 1 , 0 } - \tau _ { k , N } } { 1 - \tau _ { k , N } } } Z _ { k } ,
$$

where $Z _ { k } \sim { \mathcal { N } } ( 0 , I _ { d } )$ .

In this parallel framework, the total number of parallel rounds required to generate the final sample $Y _ { K }$ is $M K$ , and $N$ parallel processors are needed. The convergence rate of this procedure is established in Theorem 2. We remark that the implementation of this parallel algorithm assumes that the GPU memory is capable of supporting score estimations for a large batch of data simultaneously. Parallelizing across multiple GPUs introduces additional communication overhead, which may impact efficiency.

# E.2 Analysis for parallelization (Theorem 2)

By comparing the update rules for $Y _ { k , n }$ and $Y _ { m , k , n }$ , it is natural to control the difference of the following two sequences:

$$
\begin{array} { r l } & { \frac { y _ { \tau _ { k , n } } ( y _ { \tau _ { k , 0 } } ) } { \sqrt { 1 - \tau _ { k , n } } } = \frac { y _ { \tau _ { k , 0 } } } { \sqrt { 1 - \tau _ { k , 0 } } } + \frac { s _ { \tau _ { k , 0 } } ( y _ { \tau _ { k , 0 } } ) } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } ( \tau _ { k , 0 } - \widehat \tau _ { k , 0 } ) + \displaystyle \sum _ { i = 1 } ^ { n - 1 } \frac { s _ { \tau _ { k , i } } ( y _ { \tau _ { k , i } } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } ( \widehat \tau _ { k , i - 1 } - \widehat \tau _ { k , i } ) } \\ & { \qquad + \frac { s _ { \tau _ { k , n - 1 } } ( y _ { \tau _ { k , n - 1 } } ) } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } ( \widehat \tau _ { k , n - 1 } - \tau _ { k , n } ) , } \end{array}
$$

and

$$
\begin{array}{c} \frac { y _ { m , \tau _ { k , n } } ( y _ { \tau _ { k , 0 } } ) } { \sqrt { 1 - \tau _ { k , n } } } = \frac { y _ { \tau _ { k , 0 } } } { \sqrt { 1 - \tau _ { k , 0 } } } + \frac { s _ { \tau _ { k , 0 } } ( y _ { \tau _ { k , 0 } } ) } { 2 ( 1 - \tau _ { k , 0 } ) ^ { 3 / 2 } } ( \tau _ { k , 0 } - \widehat { \tau } _ { k , 0 } ) + \sum _ { i = 1 } ^ { n - 1 } \frac { s _ { \tau _ { k , i } } ( y _ { m - 1 , \tau _ { k , i } } ) } { 2 ( 1 - \tau _ { k , i } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } )  \\ { + \frac { s _ { \tau _ { k , n - 1 } } ( y _ { m - 1 , \tau _ { k , n - 1 } } ) } { 2 ( 1 - \tau _ { k , n - 1 } ) ^ { 3 / 2 } } ( \widehat { \tau } _ { k , n - 1 } - \tau _ { k , n } ) . \qquad } \end{array}
$$

We construct the typical set

$$
\begin{array} { r } { \mathcal { E } _ { k } : = \left\{ \begin{array} { l l } { \{ x _ { \tau _ { k , 0 } } : x _ { \tau _ { k , n } } ( x _ { \tau _ { k , 0 } } ) \in \widetilde { \mathcal { S } } _ { \tau _ { k , n } } \cap \mathcal { L } _ { \tau _ { k , n } } , y _ { m , \tau _ { k , n } } ( x _ { \tau _ { k , 0 } } ) \in \mathcal { S } _ { \tau _ { k , n } } , } \\ { \quad y _ { \tau _ { k , n } } ( x _ { \tau _ { k , 0 } } ) \in \mathcal { S } _ { \tau _ { k , n } } \cap \mathcal { L } _ { \tau _ { k , n } } , \forall 0 \leq n < N , \forall 0 \leq m < M \} , } & { \mathrm { i f ~ } L > d \log T , } \\ { \emptyset \quad \quad } & { \mathrm { i f ~ } \ L \leq d \log T , } \end{array} \right. } \end{array}
$$

and the auxiliary sequences $\smash { \widetilde { X } } _ { k }$ , $\widetilde { Y _ { k } }$ for $k = 0 \cdots , K$ as (22) and (24), with

$$
P _ { Y _ { k + 1 } | Y _ { k } } ( y | y _ { k } ) = \phi \left( y | y _ { M , \tau _ { k , N } } ( y _ { k } ) , \sigma _ { k } ^ { 2 } \right) .
$$

Similar with the proof of Theorem $^ { 1 }$ , we have

$$
\mathsf { T V } \big ( q _ { K } , p _ { Y _ { K } } \big ) \leq 2 \sum _ { k = 0 } ^ { K - 1 } P ( \widehat { X } _ { k } \in \mathcal { E } _ { k } ^ { \mathrm { c } } )
$$

$$
+ \sqrt { \frac { 1 } { 2 } \mathsf { K L } \big ( p _ { \widehat { X } _ { 0 } } \| p _ { Y _ { 0 } } \big ) + \frac { 1 } { 2 } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } _ { x _ { k } \sim p _ { \widetilde { X } _ { k } } } \left[ \mathsf { K L } \big ( p _ { \widehat { X } _ { k + 1 } | \widehat { X } _ { k } } \left( \cdot | x _ { k } \right) \| p _ { Y _ { k + 1 } | Y _ { k } } \left( \cdot | x _ { k } \right) \big ) \right] } .
$$

According to the Lipschitz condition of score estimates and Lemma 12, we have

$$
\frac { \left\| y _ { m , \tau _ { k , n } } - y _ { \tau _ { k , n } } \right\| _ { 2 } } { \sqrt { 1 - \tau _ { k , n } } } \lesssim \frac { \operatorname* { m i n } \{ d \log T , L \} \log T } { T } \sum _ { i = 1 } ^ { n - 1 } \frac { \left\| y _ { m - 1 , \tau _ { k , i } } - y _ { \tau _ { k , i } } \right\| _ { 2 } } { \sqrt { 1 - \tau _ { k , i } } } , \qquad y _ { \tau _ { k , 0 } } \in \mathcal { E } _ { k } .
$$

Applying the above relation recursively gives

$$
\operatorname* { m a x } _ { n } \frac { \big \| y _ { M , \tau _ { k , n } } - y _ { \tau _ { k , n } } \big \| _ { 2 } } { \sqrt { 1 - \tau _ { k , n } } } \leq \bigg ( \frac { N \operatorname* { m i n } \{ d \log T , L \} \log T } { T } \bigg ) ^ { M } \operatorname* { m a x } _ { n } \frac { \big \| y _ { 0 , \tau _ { k , n } } - y _ { \tau _ { k , n } } \big \| _ { 2 } } { \sqrt { 1 - \tau _ { k , n } } } \leq \frac { 1 } { \mathsf { p o l y } ( T ) } ,
$$

provided that $M \gtrsim \log T$ and $T \gtrsim N \operatorname* { m i n } \{ d \log T , L \} \log T$ . Thus as long as

$$
N \gtrsim \frac { ( \operatorname* { m i n } \{ d ^ { 2 / 3 } L ^ { - 2 / 3 } , d ^ { 1 / 3 } \} + 1 ) \log ^ { 5 / 3 } T } { \varepsilon ^ { 2 / 3 } } ,
$$

which guarantees that $\begin{array} { r } { T \gtrsim \frac { \operatorname* { m i n } \{ d , d ^ { 2 / 3 } L ^ { 1 / 3 } , d ^ { 1 / 3 } L \} \log ^ { 8 / 3 } T } { \varepsilon ^ { 2 / 3 } } } \end{array}$ , we can get the desired result immediately through just inserting the above error bound into Lemma 4 and Lemma 5. The probability of $P ( \widehat { X } _ { k } \in \mathcal { E } _ { k } ^ { \mathrm { c } } )$ is bounded by using Lemma 13 and the fact that $P ( x _ { \tau _ { k , n } } \in \mathcal { L } _ { \tau _ { k , n } } ^ { \mathrm { c } } ) \lesssim 1 / T ^ { 4 }$ . We omit the details here due to the similarity.

# F Proof of auxiliary lemmas in Theorem 1

Proof of (51). According to the definition of $\widehat { \tau } _ { k , i }$ , we have that

$$
\frac { 1 - \widehat \tau _ { k , N } } { 1 - \widehat \tau _ { k , - 1 } } = \frac { \widehat \alpha _ { T - \frac { k N } { 2 } - N } } { \widehat \alpha _ { T - \frac { k N } { 2 } + 1 } } \leq \left( 1 + \frac { c _ { 1 } \log T } { T } \right) ^ { N + 1 } \leq \exp \left( \frac { c _ { 1 } ( N + 1 ) \log T } { T } \right) \leq \mathrm { e } ,
$$

and

$$
\frac { \widehat { \tau } _ { k , - 1 } } { \widehat { \tau } _ { k , N } } = \frac { 1 - \widehat { \alpha } _ { T - \frac { k N } { 2 } + 1 } } { 1 - \widehat { \alpha } _ { T - \frac { k N } { 2 } - N } } \leq \left( 1 - \frac { c _ { 1 } \log T } { T } \right) ^ { - N - 1 } \leq \exp \left( \frac { 2 c _ { 1 } ( N + 1 ) \log T } { T } \right) \leq \mathrm { e } ,
$$

as long as $T \geq 4 c _ { 1 } N \log T$ . Thus we complete the proof.

# F.1 Proof of Lemma 1

Notice that the map $\Phi$ in (17) is just the integral form of (16), which is equivalent to

$$
\mathrm { d } x _ { \tau } = - \frac { 1 } { 2 ( 1 - \tau ) } \big ( x _ { \tau } + s _ { \tau } ^ { \star } ( x _ { \tau } ) \big ) \mathrm { d } \tau .
$$

This is the well-known probability ODE flow, which comes from Song et al. (2021) and is also used in Li et al. (2023).

The proof of (18) can be completed by using mathematical induction. Recalling that $\widehat { X } _ { 0 } \ \stackrel { d } { = } \ X _ { \tau _ { 0 , 0 } }$ , the (18) holds for $k + 1 = 0$ . Assume that (18) holds for $k + 1 = h$ . We have $\Phi _ { \tau _ { h - 1 , 0 }  \tau _ { h - 1 , N } } ( \widehat { X } _ { h - 1 } ) \stackrel { d } { = } X _ { \tau _ { h - 1 , N } }$ . According to (15), we could immediately get that (18) holds for $k = h$ .

# F.2 Proof of Lemma 2

We shall complete the remaining proof by mathematical induction. According to initializations, all inequalities in Lemma 2 hold for $k = 0$ . Assume (26) hold for $k = h$ . For $x , y \in \mathcal { E } _ { h + 1 } ^ { \mathrm { c } }$ , we have $p _ { \widetilde { X } _ { h + 1 } } ( x ) = 0 \leq$ $p _ { \widehat { X } _ { h + 1 } } ( x )$ , $p _ { \widetilde { Y } _ { h + 1 } } ( y ) = 0 \le p _ { Y _ { h + 1 } } ( y )$ . For $x , y \in \mathcal { E } _ { h + 1 }$ ,

$$
\begin{array} { r l } & { p _ { \widetilde { X } _ { h + 1 } } ( x ) = \displaystyle \int p _ { \widetilde { X } _ { h + 1 } | \widetilde { X } _ { h } } ( x \mid x _ { h } ) p _ { \widetilde { X } _ { h } } ( x _ { h } ) \mathrm { d } x _ { h } \le \displaystyle \int p _ { \widehat { X } _ { h + 1 } | \widehat { X } _ { h } } ( x \mid x _ { h } ) p _ { \widehat { X } _ { h } } ( x _ { h } ) \mathrm { d } x _ { h } = p _ { \widehat { X } _ { h } } ( x ) , } \\ & { p _ { \widetilde { Y } _ { h + 1 } } ( y ) = \displaystyle \int p _ { \widetilde { Y } _ { h + 1 } | \widetilde { Y } _ { h } } ( y \mid y _ { h } ) p _ { \widetilde { Y } _ { h } } ( y _ { h } ) \mathrm { d } y _ { h } \le \displaystyle \int p _ { Y _ { h + 1 } | Y _ { h } } ( y \mid y _ { h } ) p _ { Y _ { h } } ( y _ { h } ) \mathrm { d } y _ { h } = p _ { Y _ { h + 1 } } ( y ) . } \end{array}
$$

# F.3 Proof of Lemma 13

Notice that

$$
\begin{array} { l } { p _ { X _ { \tau } } ( y ) = \displaystyle \int p _ { X _ { \tau } | X _ { 0 } } ( y | x _ { 0 } ) p _ { X _ { 0 } } ( x _ { 0 } ) \mathrm { d } x _ { 0 } } \\ { \displaystyle = \int p _ { X _ { \tau } | X _ { 0 } } ( x \mid x _ { 0 } ) p _ { X _ { 0 } } ( x _ { 0 } ) \exp \left( \frac { ( x - y ) ^ { \top } ( x + y - 2 \sqrt { 1 - \tau } x _ { 0 } ) } { 2 \tau } \right) \mathrm { d } x _ { 0 } , } \\ { p _ { X _ { \tau } } ( x ) = \displaystyle \int p _ { X _ { \tau } | X _ { 0 } } ( y \mid x _ { 0 } ) p _ { X _ { 0 } } ( x _ { 0 } ) \exp \left( \frac { ( y - x ) ^ { \top } ( x + y - 2 \sqrt { 1 - \tau } x _ { 0 } ) } { 2 \tau } \right) \mathrm { d } x _ { 0 } . } \end{array}
$$

For $x \in S _ { \tau }$ , we have

$$
\begin{array} { r l } & { \begin{array} { r l } & { p _ { X _ { \tau } } ( x ) - p _ { X _ { \tau } } ( y ) } \\ & { = \displaystyle \int p _ { X _ { \tau } | X _ { 0 } } ( x | x _ { 0 } ) P _ { X _ { 0 } } ( x _ { 0 } ) \left( 1 - \exp \left( \frac { \left( x - y \right) ^ { \top } ( x + y - 2 \sqrt { 1 - \tau } x _ { 0 } ) } { 2 \tau } \right) \right) \mathrm { d } x _ { \theta } } \end{array} } \\ & { \leq \int p _ { X _ { \tau } | X _ { 0 } } ( x | x _ { 0 } ) p _ { X _ { 0 } } ( x _ { 0 } ) \frac { ( y - x ) ^ { \top } ( x + y - 2 \sqrt { 1 - \tau } x _ { 0 } ) } { 2 \tau } \mathrm { d } x _ { 0 } } \\ & { \leq \displaystyle \int p _ { X _ { \tau } | X _ { 0 } } ( x | x _ { 0 } ) p _ { X _ { 0 } } ( x _ { 0 } ) \frac { \| y - x \| ^ { 2 } } { 2 \tau } \mathrm { d } x _ { 0 } } \\ & { + \displaystyle \int p _ { X _ { \tau } | X _ { 0 } } ( x | x _ { 0 } ) P _ { X _ { 0 } } ( x _ { 0 } ) \frac { ( y - x ) ^ { \top } ( x - \sqrt { 1 - \tau } x _ { 0 } ) } { \tau } \mathrm { d } x _ { 0 } } \end{array}
$$

Furthermore, according to Lemma 9, and the fact that $\| x - y \| \leq 2 { \sqrt { ( \theta + c _ { 0 } ) d \tau \log T } }$ , we have

$$
\begin{array} { r l } & { \quad p _ { X _ { \tau } } ( x ) - p _ { X _ { \tau } } ( y ) } \\ & { \leq \left( \displaystyle \frac { \| y - x \| ^ { 2 } } { 2 \tau } + 5 \| y - x \| \sqrt { \displaystyle \frac { ( \theta + c _ { 0 } ) d \log T } { \tau } } \right) p _ { X _ { \tau } } ( x ) } \\ & { \leq 6 \| y - x \| \sqrt { \displaystyle \frac { ( \theta + c _ { 0 } ) d \log T } { \tau } } p _ { X _ { \tau } } ( x ) . } \end{array}
$$

Similarly, for $y \in S _ { \tau }$ , we have

$$
p _ { X _ { \tau } } ( y ) - p _ { X _ { \tau } } ( x ) \leq 6 \| y - x \| \sqrt { \frac { ( \theta + c _ { 0 } ) d \log T } { \tau } } p _ { X _ { \tau } } ( y ) .
$$

Thus we complete the proof.

# F.4 Proof of Lemma 7

Proof of (50). The first relation is immediately obtained by noticing that

$$
1 - \tau _ { 0 , 0 } = \overline { { \alpha } } _ { T + 1 } \leq \widehat { \alpha } _ { T } \leq 2 \widehat { \alpha } _ { T + 1 } ,
$$

provided that $\frac { c _ { 1 } \log T ^ { \prime } } { T } < 1$ .

Regarding the second inequality, for $\begin{array} { r } { T _ { 0 } = ( 1 - \frac { 3 c _ { 0 } } { c _ { 1 } } ) T + 1 } \end{array}$ and $c _ { 1 } > 3 c _ { 0 }$ , we claim that $\widehat { \alpha } _ { T _ { 0 } } \geq \frac { 1 } { 2 }$ ; otherwise, we have

$$
\widehat { \alpha } _ { t - 1 } = \widehat { \alpha } _ { t } \bigg ( 1 + \frac { c _ { 1 } \big ( 1 - \widehat { \alpha } _ { t } \big ) \log T } { T } \bigg ) > \widehat { \alpha } _ { t } \bigg ( 1 + \frac { c _ { 1 } \log T } { 2 T } \bigg ) ,
$$

and then

$$
\widehat { \alpha } _ { T _ { 0 } } > \widehat { \alpha } _ { T + 1 } \Bigl ( 1 + \frac { c _ { 1 } \log T } { 2 T } \Bigr ) ^ { T - T _ { 0 } + 1 } > \frac { 1 } { 2 } .
$$

Then if $c _ { 1 } > 5 c _ { 0 }$ , we have

$$
\tau _ { K , 0 } = 1 - \overline { { \alpha } } _ { 1 } \leq 1 - \widehat { \alpha } _ { 1 } \leq ( 1 - \widehat { \alpha } _ { T _ { 0 } } ) \left( 1 - \frac { c _ { 1 } \log T } { 2 T } \right) ^ { T _ { 0 } - 1 } \leq \frac { 1 } { T ^ { c _ { 0 } } } ,
$$

where we make use of the observation that for $t \leq T _ { 0 }$ ,

$$
1 - \widehat \alpha _ { t - 1 } = \big ( 1 - \widehat \alpha _ { t } \big ) \bigg ( 1 - \frac { c _ { 1 } \widehat \alpha _ { t } \log T } { T } \bigg ) \leq \big ( 1 - \widehat \alpha _ { t } \big ) \bigg ( 1 - \frac { c _ { 1 } \log T } { 2 T } \bigg ) .
$$

The third equation follows immediately from the definition of $\widehat { \tau } _ { k , n }$ that

$$
\frac { \widehat { \tau } _ { k , n - 1 } - \widehat { \tau } _ { k , n } } { \widehat { \tau } _ { k , n - 1 } ( 1 - \widehat { \tau } _ { k , n - 1 } ) } = \frac { \widehat { \alpha } _ { T - \frac { k N } { 2 } - n } - \widehat { \alpha } _ { T - \frac { k N } { 2 } - n + 1 } } { ( 1 - \widehat { \alpha } _ { T - \frac { k N } { 2 } - n + 1 } ) \widehat { \alpha } _ { T - \frac { k N } { 2 } - n + 1 } } = \frac { c _ { 1 } \log T } { T } .
$$

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