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:

$$X_0\cdots \cdots X_T\mathrm{.}X\; \_\; \{\; 0\; \}\; \{\; \backslash stackrel\; \{\; \backslash mathrm\; \{\; a\; d\; d\; \}\; \}\; \{\; \}\; \}\; \{\; \backslash stackrel\; \{\; \backslash mathrm\; \{\; n\; o\; i\; s\; e\; \}\; \}\; \{\; X\; \_\; \{\; 1\; \}\; \}\; \}\; \{\; \backslash stackrel\; \{\; \backslash mathrm\; \{\; a\; d\; d\; \}\; \}\; \{\; \}\; \}\; \backslash cdots\; \{\; \backslash stackrel\; \{\; \backslash mathrm\; \{\; a\; d\; d\; \}\; \}\; \{\; \}\; \}\; \backslash cdots\; X\; \_\; \{\; T\; \}\; .$$

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:

$$Y_0{Y}_1\cdot \cdot \cdot Y_T,\{\; Y\; \_\; \{\; 0\; \}\; \}\; \backslash stackrel\; \{\; \backslash mathrm\; \{\; d\; e\; n\; o\; i\; s\; e\; \}\; \}\; \{\; \}\; \{\; Y\; \_\; \{\; 1\; \}\; \}\; \backslash stackrel\; \{\; \backslash mathrm\; \{\; d\; e\; n\; o\; i\; s\; e\; \}\; \}\; \{\; \}\; \backslash cdot\; \backslash cdot\; \backslash cdot\; \backslash stackrel\; \{\; \backslash mathrm\; \{\; d\; e\; n\; o\; i\; s\; e\; \}\; \}\; \{\; \}\; \{\; Y\; \_\; \{\; T\; \}\; \}\; ,$$

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:

$$\min \{d,d^{2\mathrm{/}3}{L}^{1\mathrm{/}3},d^{1\mathrm{/}3}L\}{\epsilon }^{-2\mathrm{/}3},\backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; ,\; d\; ^\; \{\; 2\; /\; 3\; \}\; L\; ^\; \{\; 1\; /\; 3\; \}\; ,\; d\; ^\; \{\; 1\; /\; 3\; \}\; L\; \backslash \}\; \backslash varepsilon\; ^\; \{\; -\; 2\; /\; 3\; \}\; ,$$

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:

$$X_t=\sqrt{{\alpha }_t}{X}_{t-1}+\sqrt{1-{\alpha }_t}{W}_t,1\le t\le T,X\; \_\; \{\; t\; \}\; =\; \backslash sqrt\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; X\; \_\; \{\; t\; -\; 1\; \}\; +\; \backslash sqrt\; \{\; 1\; -\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; W\; \_\; \{\; t\; \}\; ,\; \backslash quad\; 1\; \backslash leq\; t\; \backslash leq\; T\; ,$$

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

$${\bar{\alpha }}_t:=\prod _{k=1}^t{\alpha }_k,1\le t\le T\mathrm{.}\backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; :\; =\; \backslash prod\; \_\; \{\; k\; =\; 1\; \}\; ^\; \{\; t\; \}\; \backslash alpha\; \_\; \{\; k\; \}\; ,\; \backslash quad\; 1\; \backslash leq\; t\; \backslash leq\; T\; .$$

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.

$$X_t=\sqrt{\overline{{\alpha }_t}}{X}_0+\sqrt{1-\overline{{\alpha }_t}}{\bar{W}}_t,{\bar{W}}_t\sim \mathcal{N}(0,I_d)\mathrm{.}X\; \_\; \{\; t\; \}\; =\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; X\; \_\; \{\; 0\; \}\; +\; \backslash sqrt\; \{\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \backslash overline\; \{\; \{\; W\; \}\; \}\; \_\; \{\; t\; \}\; ,\; \backslash quad\; \backslash overline\; \{\; \{\; W\; \}\; \}\; \_\; \{\; t\; \}\; \backslash sim\; \backslash mathcal\; \{\; N\; \}\; (\; 0\; ,\; I\; \_\; \{\; d\; \}\; )\; .$$

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:

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

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

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:

$$s_t^{\star }(x):=\mathrm{\nabla }\log p_{X_t}(x)=-\frac{1}{1-{\bar{\alpha }}_t}{\int }_{x_0}{p}_{X_0\mid X_t}(x_0\mid x)(x-\sqrt{{\bar{\alpha }}_t}{x}_0)\mathrm{d}{x}_0\mathrm{.}s\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; :\; =\; \backslash nabla\; \backslash log\; p\; \_\; \{\; X\; \_\; \{\; t\; \}\; \}\; (\; x\; )\; =\; -\; \backslash frac\; \{\; 1\; \}\; \{\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \}\; \backslash int\; \_\; \{\; x\; \_\; \{\; 0\; \}\; \}\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; \backslash mid\; X\; \_\; \{\; t\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; \backslash mid\; x\; )\; (\; x\; -\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \}\; x\; \_\; \{\; 0\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; .$$

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

$$s_{\tau }^{\star }(x):=\mathrm{\nabla }\log p_{X_{\tau }}(x)=-\frac{1}{\tau }{\int }_{x_0}{p}_{X_0\mathrm{\mid }{X}_{\tau }}(x_0\mathrm{\mid }x)(x-\sqrt{1-\tau }{x}_0)\mathrm{d}{x}_0\mathrm{.}s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; :\; =\; \backslash nabla\; \backslash log\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; )\; =\; -\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; \}\; \backslash int\; \_\; \{\; x\; \_\; \{\; 0\; \}\; \}\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; |\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; x\; )\; (\; x\; -\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; .$$

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

$${\widehat{\alpha }}_{T+1}=\frac{1}{T^{c_0}},{\widehat{\alpha }}_{t-1}={\widehat{\alpha }}_t+\frac{c_1{\widehat{\alpha }}_t(1-{\widehat{\alpha }}_t)\log T}{T},t=-\frac{N}{2}+1,\cdots ,T+1,\backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; +\; 1\; \}\; =\; \backslash frac\; \{\; 1\; \}\; \{\; T\; ^\; \{\; c\; \_\; \{\; 0\; \}\; \}\; \}\; ,\; \backslash quad\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; t\; -\; 1\; \}\; =\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; t\; \}\; +\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; t\; \}\; (\; 1\; -\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; t\; \}\; )\; \backslash log\; \{\; T\; \}\; \}\; \{\; T\; \}\; ,\; \backslash qquad\; t\; =\; -\; \backslash frac\; \{\; N\; \}\; \{\; 2\; \}\; +\; 1\; ,\; \backslash cdots\; ,\; T\; +\; 1\; ,$$

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

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

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

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

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:

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:

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

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

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

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

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

$$\begin{array}{l}{\displaystyle {\mathbb{P}}_{x\sim X_t}\{(1-\overline{{\alpha }_t})\mathrm{\parallel }{s}_t^{\star }(x^{\mathrm{\prime }})-s_t^{\star }(x){\mathrm{\parallel }}_2\le L\mathrm{\parallel }{x}^{\mathrm{\prime }}-x{\mathrm{\parallel }}_2,\mathrm{\forall }\mathrm{\parallel }{x}^{\mathrm{\prime }}-x{\mathrm{\parallel }}_2\le \frac{C\sqrt{d(1-{\bar{\alpha }}_t)\log T}}{L}\}}\\ {\displaystyle \ge 1-\frac{c}{(T+d{)}^4},}\end{array}\backslash begin\{array\}\; \{\; l\; l\; \}\; \{\; \backslash displaystyle\; \backslash mathbb\; \{\; P\; \}\; \_\; \{\; x\; \backslash sim\; X\; \_\; \{\; t\; \}\; \}\; \backslash left\backslash \{\; (\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; )\; \backslash |\; s\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; ^\; \{\; \backslash prime\; \}\; )\; -\; s\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; \backslash |\; \_\; \{\; 2\; \}\; \backslash le\; L\; \backslash |\; x\; ^\; \{\; \backslash prime\; \}\; -\; x\; \backslash |\; \_\; \{\; 2\; \}\; ,\; \backslash forall\; \backslash |\; x\; ^\; \{\; \backslash prime\; \}\; -\; x\; \backslash |\; \_\; \{\; 2\; \}\; \backslash le\; \backslash frac\; \{\; C\; \backslash sqrt\; \{\; d\; (\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; )\; \backslash log\; T\; \}\; \}\; \{\; L\; \}\; \backslash right\backslash \}\; \}\; \backslash \backslash \; \{\; \backslash displaystyle\; \backslash ge\; 1\; -\; \backslash frac\; \{\; c\; \}\; \{\; (\; T\; +\; d\; )\; ^\; \{\; 4\; \}\; \}\; ,\; \}\; \backslash end\{array\}$$

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

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 $\begin{array} { r } { X _ { 0 } \sim \sum _ { h = 1 } ^ { H } \gamma _ { h } \mathcal { N } ( \mu _ { h } , \sigma ^ { 2 } I _ { d } ) } \end{array}$ $\mu _ { h } \in \mathbb { R } ^ { d }$ $\sigma \geq 0$ $\gamma _ { h } \geq 0$ $\textstyle \sum _ { h = 1 } ^ { H } \gamma _ { h } = 1$ $t \geq 0$

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 $\overline { { \alpha } } _ { t } > 1 / 2$ ,

$$(1-{\bar{\alpha }}_t)\mathrm{\parallel }\mathrm{\nabla }{s}_t^{\star }(x){\mathrm{\parallel }}_{\mathsf{o}\mathsf{p}}\ge \frac{(1-{\bar{\alpha }}_t)\mathrm{\parallel }\mu {\mathrm{\parallel }}_2^2}{4(1-{\bar{\alpha }}_t+{\sigma }^2{)}^2}\mathrm{.}(\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; )\; \backslash |\; \backslash nabla\; s\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; \backslash |\; \_\; \{\; \backslash mathsf\; \{\; o\; p\; \}\; \}\; \backslash geq\; \backslash frac\; \{\; (\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; )\; \backslash |\; \backslash mu\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 4\; (\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; +\; \backslash sigma\; ^\; \{\; 2\; \}\; )\; ^\; \{\; 2\; \}\; \}\; .$$

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

$$\begin{array}{l}{\epsilon }_{\mathsf{s}\mathsf{c}\mathsf{o}\mathsf{r}\mathsf{e}}^2={\displaystyle \frac{1}{T}\sum _{k=0}^{K-1}\sum _{n=0}^{N-1}{\mathbb{E}}_{Y_k\sim q_k}[\mathrm{\parallel }{s}_{T-\frac{kN}{2}-n+1}(Y_{k,n})-s_{T-\frac{kN}{2}-n+1}^{\star }(Y_{k,n}){\mathrm{\parallel }}_2^2]}\\ =:{\displaystyle \frac{1}{T}\sum _{k=0}^{K-1}\sum _{n=0}^{N-1}{\epsilon }_{k,n}^2\mathrm{.}}\end{array}\backslash begin\{array\}\; \{\; l\; \}\; \{\; \backslash varepsilon\; \_\; \{\; \backslash mathsf\; \{\; s\; c\; o\; r\; e\; \}\; \}\; ^\; \{\; 2\; \}\; =\; \backslash displaystyle\; \backslash frac\; \{\; 1\; \}\; \{\; T\; \}\; \backslash sum\; \_\; \{\; k\; =\; 0\; \}\; ^\; \{\; K\; -\; 1\; \}\; \backslash sum\; \_\; \{\; n\; =\; 0\; \}\; ^\; \{\; N\; -\; 1\; \}\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Y\; \_\; \{\; k\; \}\; \backslash sim\; q\; \_\; \{\; k\; \}\; \}\; \backslash left[\; \backslash |\; s\; \_\; \{\; T\; -\; \backslash frac\; \{\; k\; N\; \}\; \{\; 2\; \}\; -\; n\; +\; 1\; \}\; (\; Y\; \_\; \{\; k\; ,\; n\; \}\; )\; -\; s\; \_\; \{\; T\; -\; \backslash frac\; \{\; k\; N\; \}\; \{\; 2\; \}\; -\; n\; +\; 1\; \}\; ^\; \{\; \backslash star\; \}\; (\; Y\; \_\; \{\; k\; ,\; n\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash right]\; \}\; \backslash \backslash \; \{\; =\; :\; \backslash displaystyle\; \backslash frac\; \{\; 1\; \}\; \{\; T\; \}\; \backslash sum\; \_\; \{\; k\; =\; 0\; \}\; ^\; \{\; K\; -\; 1\; \}\; \backslash sum\; \_\; \{\; n\; =\; 0\; \}\; ^\; \{\; N\; -\; 1\; \}\; \backslash varepsilon\; \_\; \{\; k\; ,\; n\; \}\; ^\; \{\; 2\; \}\; .\; \}\; \backslash end\{array\}$$

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

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

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

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

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 $p \geq 3$ .

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

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

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:

and

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

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

$$\begin{array}{r}{\tilde{\epsilon }}_{k,n}={\parallel s_{{\tau }_{k,n}}(Y_{{\tau }_{k,n}})-s_{{\tau }_{k,n}}^{\star }(Y_{{\tau }_{k,n}})\parallel }_2\mathrm{.}\end{array}\backslash begin\{array\}\; \{\; r\; \}\; \{\; \backslash widetilde\; \{\; \backslash varepsilon\; \}\; \_\; \{\; k\; ,\; n\; \}\; =\; \backslash left\backslash |\; s\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; (\; Y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; )\; -\; s\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash star\; \}\; (\; Y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; )\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; .\; \}\; \backslash end\{array\}$$

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

$$X_{\tau }\sqrt{1-\tau }{X}_0+\sqrt{\tau }Z,\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}Z\sim \mathcal{N}(0,I_d),\mathrm{f}\mathrm{o}\mathrm{r}0\le \tau \le 1.X\; \_\; \{\; \backslash tau\; \}\; \backslash stackrel\; \{\; \backslash mathrm\; \{\; d\; \}\; \}\; \{\; =\; \}\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; X\; \_\; \{\; 0\; \}\; +\; \backslash sqrt\; \{\; \backslash tau\; \}\; Z\; ,\; \backslash mathrm\; \{\; w\; i\; t\; h\; \}\; Z\; \backslash sim\; \{\; \backslash mathcal\; N\; \}\; (\; 0\; ,\; I\; \_\; \{\; d\; \}\; )\; ,\; \backslash mathrm\; \{\; f\; o\; r\; \}\; 0\; \backslash leq\; \backslash tau\; \backslash leq\; 1\; .$$

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

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

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

$${\mathrm{\Phi }}_{{\tau }_1{\tau }_2}(X_{{\tau }_1})X_{{\tau }_2}\mathrm{.}\backslash Phi\; \_\; \{\; \backslash tau\; \_\; \{\; 1\; \}\; \backslash tau\; \_\; \{\; 2\; \}\; \}\; (\; X\; \_\; \{\; \backslash tau\; \_\; \{\; 1\; \}\; \}\; )\; \backslash stackrel\; \{\; \backslash mathrm\; \{\; d\; \}\; \}\; \{\; =\; \}\; X\; \_\; \{\; \backslash tau\; \_\; \{\; 2\; \}\; \}\; .$$

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

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

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

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

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

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

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

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

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

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:

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

and (b) uses the fact that

$$P({\tilde{X}}_K=\mathrm{\infty })\le \mathsf{T}\mathsf{V}(p_{{\tilde{X}}_K},p_{{\widehat{X}}_K})\mathrm{.}P\; (\; \backslash tilde\; \{\; X\; \}\; \_\; \{\; K\; \}\; =\; \backslash infty\; )\; \backslash leq\; \backslash mathsf\; \{\; T\; V\; \}\; (\; p\; \_\; \{\; \backslash tilde\; \{\; X\; \}\; \_\; \{\; K\; \}\; \}\; ,\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; K\; \}\; \}\; )\; .$$

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

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

$$pX~k(x)=∫Ek−1pX~k∣X~k−1(x∣xk−1)pX~k−1(xk−1)dxk−1=∫Ek−1pX~k∣X~k−1(x∣xk−1)pX^k−1(xk−1)dxk−1−Δk(x)=∫Ek−1pX^k∣X^k−1(x∣xk−1)pX^k−1(xk−1)dxk−1−Δk(x)=pX^k(x)−∫Ek−1cpX^k∣X^k−1(x∣xk−1)pX^k−1(xk−1)dxk−1−Δk(x), \begin{array} { l } { p _ { \widetilde { X } _ { k } } ( x ) = \displaystyle \int _ { \mathcal { E } _ { k - 1 } } p _ { \widetilde { X } _ { k } | \widetilde { X } _ { k - 1 } } ( x | x _ { k - 1 } ) p _ { \widetilde { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } } \\ { \displaystyle = \int _ { \mathcal { E } _ { k - 1 } } p _ { \widetilde { X } _ { k } | \widetilde { X } _ { k - 1 } } ( x | x _ { k - 1 } ) p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } - \Delta _ { k } ( x ) } \\ { \displaystyle = \int _ { \mathcal { E } _ { k - 1 } } p _ { \widehat { X } _ { k } | \widehat { X } _ { k - 1 } } ( x | x _ { k - 1 } ) p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } - \Delta _ { k } ( x ) } \\ { \displaystyle = p _ { \widehat { X } _ { k } } ( x ) - \displaystyle \int _ { \mathcal { E } _ { k - 1 } ^ { c } } p _ { \widehat { X } _ { k } | \widehat { X } _ { k - 1 } } ( x | x _ { k - 1 } ) p _ { \widehat { X } _ { k - 1 } } ( x _ { k - 1 } ) \mathrm { d } x _ { k - 1 } - \Delta _ { k } ( x ) , } \end{array} $$

where

$${\mathrm{\Delta }}_k(x)={\int }_{{\mathcal{E}}_{k-1}}{p}_{{\tilde{X}}_k\mathrm{\mid }}{\tilde{x}}_{k-1}(x\mid x_{k-1})(p_{{\widehat{X}}_{k-1}}(x_{k-1})-p_{{\tilde{X}}_{k-1}}(x_{k-1}))\mathrm{d}{x}_{k-1}\mathrm{.}\backslash Delta\; \_\; \{\; k\; \}\; (\; x\; )\; =\; \backslash int\; \_\; \{\; \backslash mathscr\; \{\; E\; \}\; \_\; \{\; k\; -\; 1\; \}\; \}\; p\; \_\; \{\; \backslash widetilde\; \{\; X\; \}\; \_\; \{\; k\; \}\; |\; \}\; \backslash widetilde\; \{\; x\; \}\; \_\; \{\; k\; -\; 1\; \}\; (\; x\; \backslash mid\; x\; \_\; \{\; k\; -\; 1\; \}\; )\; \backslash left(\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; k\; -\; 1\; \}\; \}\; (\; x\; \_\; \{\; k\; -\; 1\; \}\; )\; -\; p\; \_\; \{\; \backslash widetilde\; \{\; X\; \}\; \_\; \{\; k\; -\; 1\; \}\; \}\; (\; x\; \_\; \{\; k\; -\; 1\; \}\; )\; \backslash right)\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; k\; -\; 1\; \}\; .$$

Thus we have

where (a) inserts (29). Notice that

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

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

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

considering that

$$\mathsf{T}\mathsf{V}(p_{{\tilde{X}}_0},p_{{\widehat{X}}_0})=P({\widehat{X}}_0\in {\mathcal{E}}_0),P({\widehat{X}}_K\in {\mathcal{E}}_K^{\mathrm{c}})=0.\backslash mathsf\; \{\; T\; V\; \}\; (\; p\; \_\; \{\; \backslash widetilde\; \{\; X\; \}\; \_\; \{\; 0\; \}\; \}\; ,\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; 0\; \}\; \}\; )\; =\; P\; (\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; 0\; \}\; \backslash in\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 0\; \}\; )\; ,\; \backslash quad\; P\; (\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; K\; \}\; \backslash in\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; K\; \}\; ^\; \{\; \backslash mathrm\; \{\; c\; \}\; \}\; )\; =\; 0\; .$$

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

$$\begin{array}{r}\mathsf{T}\mathsf{V}(q_K,p_{Y_K})\le \sqrt{\frac{1}{2}\mathsf{K}\mathsf{L}\left(p_{{\widehat{X}}_0}\mathrm{\parallel }{p}_{Y_0}\right)+\frac{1}{2}{\displaystyle \sum _{k=0}^{K-1}{\mathbb{E}}_{x_k\sim p_{{\tilde{X}}_k}}\left[\mathsf{K}\mathsf{L}\right(p_{{\widehat{X}}_{k+1}\mathrm{\mid }{\widehat{X}}_k}(\cdot \mathrm{\mid }{x}_k)\mathrm{\parallel }{p}_{Y_{k+1}\mathrm{\mid }{Y}_k}(\cdot \mathrm{\mid }{x}_k)\left)\right]}}\\ +2{\displaystyle \sum _{k=0}^{K-1}P({\widehat{X}}_k\in {\mathcal{E}}_k^{\circ })\mathrm{.}}\end{array}\backslash begin\{array\}\; \{\; r\; \}\; \{\; \backslash mathsf\; \{\; T\; V\; \}\; \backslash big\; (\; q\; \_\; \{\; K\; \}\; ,\; p\; \_\; \{\; Y\; \_\; \{\; K\; \}\; \}\; \backslash big\; )\; \backslash le\; \backslash sqrt\; \{\; \backslash frac\; \{\; 1\; \}\; \{\; 2\; \}\; \backslash mathsf\; \{\; K\; L\; \}\; \backslash big\; (\; p\; \_\; \{\; \backslash hat\; \{\; X\; \}\; \_\; \{\; 0\; \}\; \}\; \backslash |\; p\; \_\; \{\; Y\; \_\; \{\; 0\; \}\; \}\; \backslash big\; )\; +\; \backslash frac\; \{\; 1\; \}\; \{\; 2\; \}\; \backslash displaystyle\; \backslash sum\; \_\; \{\; k\; =\; 0\; \}\; ^\; \{\; K\; -\; 1\; \}\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; x\; \_\; \{\; k\; \}\; \backslash sim\; p\; \_\; \{\; \backslash widetilde\; X\; \_\; \{\; k\; \}\; \}\; \}\; \backslash left[\; \backslash mathsf\; \{\; K\; L\; \}\; \backslash big\; (\; p\; \_\; \{\; \backslash hat\; \{\; X\; \}\; \_\; \{\; k\; +\; 1\; \}\; |\; \backslash hat\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; \backslash left(\; \backslash cdot\; |\; x\; \_\; \{\; k\; \}\; \backslash right)\; \backslash |\; p\; \_\; \{\; Y\; \_\; \{\; k\; +\; 1\; \}\; |\; Y\; \_\; \{\; k\; \}\; \}\; \backslash left(\; \backslash cdot\; |\; x\; \_\; \{\; k\; \}\; \backslash right)\; \backslash big\; )\; \backslash right]\; \}\; \}\; \backslash \backslash \; \{\; +\; 2\; \backslash displaystyle\; \backslash sum\; \_\; \{\; k\; =\; 0\; \}\; ^\; \{\; K\; -\; 1\; \}\; P\; (\; \backslash widehat\; X\; \_\; \{\; k\; \}\; \backslash in\; \backslash mathcal\; E\; \_\; \{\; k\; \}\; ^\; \{\; \backslash circ\; \}\; )\; .\; \}\; \backslash 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

$$\mathsf{K}\mathsf{L}(p_{{\widehat{X}}_{k+1}\mid {\widehat{X}}_k}(\cdot \mid x_k)\parallel p_{Y_{k+1}\mid Y_k}(\cdot \mid x_k))=\frac{1-{\tau }_{k+1,0}}{2({\tau }_{k+1,0}-{\tau }_{k,N})}\parallel y_{{\tau }_{k,N}}\left(x_k\right)-x_{{\tau }_{k,N}}\left(x_k\right){\parallel }_2^2\mathrm{.}\backslash mathsf\; \{\; K\; L\; \}\; \backslash big\; (\; p\; \_\; \{\; \backslash hat\; \{\; X\; \}\; \_\; \{\; k\; +\; 1\; \}\; \backslash mid\; \backslash hat\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; \backslash left(\; \backslash cdot\; \backslash mid\; x\; \_\; \{\; k\; \}\; \backslash right)\; \backslash parallel\; p\; \_\; \{\; Y\; \_\; \{\; k\; +\; 1\; \}\; \backslash mid\; Y\; \_\; \{\; k\; \}\; \}\; \backslash left(\; \backslash cdot\; \backslash mid\; x\; \_\; \{\; k\; \}\; \backslash right)\; \backslash big\; )\; =\; \backslash frac\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; +\; 1\; ,\; 0\; \}\; \}\; \{\; 2\; (\; \backslash tau\; \_\; \{\; k\; +\; 1\; ,\; 0\; \}\; -\; \backslash tau\; \_\; \{\; k\; ,\; N\; \}\; )\; \}\; \backslash lVert\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; N\; \}\; \}\; \backslash left(\; x\; \_\; \{\; k\; \}\; \backslash right)\; -\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; N\; \}\; \}\; \backslash left(\; x\; \_\; \{\; k\; \}\; \backslash right)\; \backslash rVert\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; .$$

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 $x _ { \tau _ { k , n } }$ , $y _ { \tau _ { k , n } }$ , $y _ { \tau _ { k , n } } ^ { \star }$ ,

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

Moreover, we define the matrix

$${\mathrm{\Sigma }}_{\tau }(x)=\mathsf{C}\mathsf{o}\mathsf{v}[Z\mathrm{\mid }\sqrt{1-\tau }{X}_0+\sqrt{\tau }Z=x],\backslash Sigma\; \_\; \{\; \backslash tau\; \}\; (\; x\; )\; =\; \backslash mathsf\; \{\; C\; o\; v\; \}\; [\; Z\; |\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; X\; \_\; \{\; 0\; \}\; +\; \backslash sqrt\; \{\; \backslash tau\; \}\; Z\; =\; x\; ]\; ,$$

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 $1 - T ^ { - 1 0 0 }$ ,

$${\mathbb{E}}_{x_{{\tau }_{k,0}}\sim p_{\widehat{X_k}}}[\mathrm{\parallel }{\xi }_{k,n}(x_{{\tau }_{k,0}}){\mathrm{\parallel }}_2^2]\lesssim \frac{d{\log }^{4}T}{T^3}\min \{\frac{Nd{\widehat{\tau }}_{k,-1}\log T}{T(1-{\widehat{\tau }}_{k,-1})}+{\int }_{{\tau }_{k,n}}^{{\tau }_{k,0}}\frac{\mathbb{E}[\mathsf{T}\mathsf{r}({\mathrm{\Sigma }}_{\tau }^2(x_{\tau }))]}{(1-\tau {)}^2}\mathrm{d}\tau ,\backslash mathbb\; \{\; E\; \}\; \_\; \{\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; \backslash sim\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \_\; \{\; k\; \}\; \}\; \}\; \}\; \backslash left[\; \backslash |\; \backslash xi\; \_\; \{\; k\; ,\; n\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash right]\; \backslash lesssim\; \backslash frac\; \{\; d\; \backslash log\; ^\; \{\; 4\; \}\; T\; \}\; \{\; T\; ^\; \{\; 3\; \}\; \}\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash Big\; \backslash \{\; \backslash frac\; \{\; N\; d\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; -\; 1\; \}\; \backslash log\; T\; \}\; \{\; T\; (\; 1\; -\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; -\; 1\; \}\; )\; \}\; +\; \backslash int\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; \backslash frac\; \{\; \backslash mathbb\; \{\; E\; \}\; \backslash left[\; \backslash mathsf\; \{\; T\; r\; \}\; (\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; )\; \backslash right]\; \}\; \{\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 2\; \}\; \}\; \backslash mathrm\; \{\; d\; \}\; \backslash tau\; ,$$

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

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

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

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

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

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.

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 ${ \sqrt { d } } \lesssim L \lesssim d$ .

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

$$s_t^{\star }(x)=-{\mathrm{\Sigma }}_t^{-1}x,s\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; =\; -\; \backslash Sigma\; \_\; \{\; t\; \}\; ^\; \{\; -\; 1\; \}\; x\; ,$$

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

$$({\mathrm{\Sigma }}_t{)}_{i,i}={\bar{\alpha }}_t{\sigma }_i^2+1-{\bar{\alpha }}_t\mathrm{.}(\; \backslash Sigma\; \_\; \{\; t\; \}\; )\; \_\; \{\; i\; ,\; i\; \}\; =\; \backslash overline\; \{\; \{\; \{\; \backslash alpha\; \}\; \}\; \}\; \_\; \{\; t\; \}\; \backslash sigma\; \_\; \{\; i\; \}\; ^\; \{\; 2\; \}\; +\; 1\; -\; \backslash overline\; \{\; \{\; \{\; \backslash alpha\; \}\; \}\; \}\; \_\; \{\; t\; \}\; .$$

Thus we have

$$\mathrm{\parallel }{s}_t^{\star }(x)-s_t^{\star }(y){\mathrm{\parallel }}_2=\mathrm{\parallel }{\mathrm{\Sigma }}_t^{-1}(x-y){\mathrm{\parallel }}_2,\backslash |\; s\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; -\; s\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; \backslash |\; \_\; \{\; 2\; \}\; =\; \backslash |\; \backslash Sigma\; \_\; \{\; t\; \}\; ^\; \{\; -\; 1\; \}\; (\; x\; -\; y\; )\; \backslash |\; \_\; \{\; 2\; \}\; ,$$

and

$$\mathrm{\parallel }{\mathrm{\Sigma }}_t^{-1}{\mathrm{\parallel }}_2=\frac{1}{{\bar{\alpha }}_t\min {\sigma }_i^2+1-{\bar{\alpha }}_t}=\frac{1}{1-{\bar{\alpha }}_t}\mathrm{.}\backslash |\; \backslash Sigma\; \_\; \{\; t\; \}\; ^\; \{\; -\; 1\; \}\; \backslash |\; \_\; \{\; 2\; \}\; =\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash sigma\; \_\; \{\; i\; \}\; ^\; \{\; 2\; \}\; +\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \}\; =\; \backslash frac\; \{\; 1\; \}\; \{\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \}\; .$$

Then we complete the proof.

C.2 Computation of Example 2

It is easy to check that the score function is

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

where

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

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:

$$\mathbb{P}\{\mathrm{\exists }{x}^{\mathrm{\prime }}\in {\mathbb{R}}^d,(1-{\bar{\alpha }}_t)\mathrm{\parallel }{s}_t^{\star }(x)-s_t^{\star }(x^{\mathrm{\prime }}){\mathrm{\parallel }}_2>C\log (HT)\mathrm{\parallel }x-x^{\mathrm{\prime }}{\mathrm{\parallel }}_2\}\lesssim \frac{1}{T^4}\mathrm{.}\backslash mathbb\; \{\; P\; \}\; \backslash left\backslash \{\; \backslash exists\; x\; ^\; \{\; \backslash prime\; \}\; \backslash in\; \backslash mathbb\; \{\; R\; \}\; ^\; \{\; d\; \}\; ,\; (\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; )\; \backslash |\; s\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; -\; s\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; ^\; \{\; \backslash prime\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; >\; C\; \backslash log\; (\; H\; T\; )\; \backslash |\; x\; -\; x\; ^\; \{\; \backslash prime\; \}\; \backslash |\; \_\; \{\; 2\; \}\; \backslash right\backslash \}\; \backslash lesssim\; \backslash frac\; \{\; 1\; \}\; \{\; T\; ^\; \{\; 4\; \}\; \}\; .$$

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

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

$$(1−α‾t)∥st⋆(x)−st⋆(x′)∥2∥−1−α‾tσt2(x−x′)+α‾t(1−α‾t)σt2∑i=1H(πi(x)−πi(x′))(μi−μh)∥2≤1−α‾tσt2∥x−x′∥2+α‾t(1−α‾t)σt2∑i=1H∥(πi(x)−πi(x′))(μi−μh)∥2∥x−x′∥2+α‾t∑i=1H∥(πi(x)−πi(x′))(μi−μh)∥2, \begin{array} { l } { \displaystyle ( 1 - \overline { { \alpha } } _ { t } ) \| s _ { t } ^ { \star } ( x ) - s _ { t } ^ { \star } ( x ^ { \prime } ) \| _ { 2 } } \\ { \displaystyle \overset { \mathrm { ( a ) } } { = } \left\| - \frac { 1 - \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } ( x - x ^ { \prime } ) + \frac { \sqrt { \overline { { \alpha } } _ { t } } ( 1 - \overline { { \alpha } } _ { t } ) } { \sigma _ { t } ^ { 2 } } \displaystyle \sum _ { i = 1 } ^ { H } ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \right\| _ { 2 } } \\ { \displaystyle \leq \frac { 1 - \overline { { \alpha } } _ { t } } { \sigma _ { t } ^ { 2 } } \| x - x ^ { \prime } \| _ { 2 } + \frac { \sqrt { \overline { { \alpha } } _ { t } } ( 1 - \overline { { \alpha } } _ { t } ) } { \sigma _ { t } ^ { 2 } } \displaystyle \sum _ { i = 1 } ^ { H } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } } \\ { \displaystyle \overset { \mathrm { ( b ) } } { \leq } \| x - x ^ { \prime } \| _ { 2 } + \sqrt { \overline { { \alpha } } _ { t } } \displaystyle \sum _ { i = 1 } ^ { H } \| ( \pi _ { i } ( x ) - \pi _ { i } ( x ^ { \prime } ) ) ( \mu _ { i } - \mu _ { h } ) \| _ { 2 } , } \end{array} $$

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

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

$${\mathcal{T}}_t^h=\{x\in {\mathbb{R}}^d:\sqrt{\overline{{\alpha }_t}}\mid (x-\sqrt{\overline{{\alpha }_t}}{\mu }_h{)}^{\mathrm{\top }}({\mu }_i-{\mu }_h)\mid \le C_2{\sigma }_t\sqrt{\overline{{\alpha }_t}\log (HT)}\parallel {\mu }_i-{\mu }_h\parallel ,\mathrm{\forall }i\}\mathrm{.}\backslash mathcal\; \{\; T\; \}\; \_\; \{\; t\; \}\; ^\; \{\; h\; \}\; =\; \backslash left\backslash \{\; x\; \backslash in\; \backslash mathbb\; \{\; R\; \}\; ^\; \{\; d\; \}\; :\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \backslash lvert\; (\; x\; -\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \backslash mu\; \_\; \{\; h\; \}\; )\; ^\; \{\; \backslash top\; \}\; (\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; )\; \backslash rvert\; \backslash leq\; C\; \_\; \{\; 2\; \}\; \backslash sigma\; \_\; \{\; t\; \}\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \backslash log\; (\; H\; T\; )\; \}\; \backslash lVert\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash rVert\; ,\; \backslash forall\; i\; \backslash right\backslash \}\; .$$

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

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

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

$$T_t^h\subset \{x\in {\mathbb{R}}^d:\mathrm{\forall }{x}^{\mathrm{\prime }}\in {\mathbb{R}}^d,\sqrt{{\alpha }_t}\sum _{i=1}^H\mathrm{\parallel }({\pi }_i(x)-{\pi }_i(x^{\mathrm{\prime }}))({\mu }_i-{\mu }_h){\mathrm{\parallel }}_2\le \frac{C\log (HT)}{2}\mathrm{\parallel }x-x^{\mathrm{\prime }}{\mathrm{\parallel }}_2\}\mathrm{.}T\; \_\; \{\; t\; \}\; ^\; \{\; h\; \}\; \backslash subset\; \backslash left\backslash \{\; x\; \backslash in\; \backslash mathbb\; \{\; R\; \}\; ^\; \{\; d\; \}\; :\; \backslash forall\; x\; ^\; \{\; \backslash prime\; \}\; \backslash in\; \backslash mathbb\; \{\; R\; \}\; ^\; \{\; d\; \}\; ,\; \backslash sqrt\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; H\; \}\; \backslash |\; (\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; )\; -\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; ^\; \{\; \backslash prime\; \}\; )\; )\; (\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; \backslash frac\; \{\; C\; \backslash log\; (\; H\; T\; )\; \}\; \{\; 2\; \}\; \backslash |\; x\; -\; x\; ^\; \{\; \backslash prime\; \}\; \backslash |\; \_\; \{\; 2\; \}\; \backslash right\backslash \}\; .$$

Inserting it into (39), we have

$$\begin{array}{l}{\displaystyle \mathbb{P}\{\mathrm{\exists }{\mathit{x}}^{\mathrm{\prime }}\in {\mathbb{R}}^d,(1-{\bar{\alpha }}_t)\mathrm{\parallel }{\mathit{s}}_t^{\star }(\mathit{x})-{\mathit{s}}_t^{\star }({\mathit{x}}^{\mathrm{\prime }}){\mathrm{\parallel }}_2>C\log (HT)\mathrm{\parallel }\mathit{x}-{\mathit{x}}^{\mathrm{\prime }}{\mathrm{\parallel }}_2\}}\\ {\displaystyle \le \sum _{h=1}^H{\gamma }_h{\mathbb{P}}_h\{\mathit{x}\mathrm{\notin }{\mathcal{T}}_t^h\}\mathbb{1}({\gamma }_h\ge \frac{1}{H{T}^4})+\frac{1}{T^4}\lesssim \frac{1}{T^4},}\end{array}\backslash begin\{array\}\; \{\; l\; \}\; \{\; \backslash displaystyle\; \backslash mathbb\; \{\; P\; \}\; \backslash left\backslash \{\; \backslash exists\; \backslash boldsymbol\; \{\; x\; \}\; ^\; \{\; \backslash prime\; \}\; \backslash in\; \backslash mathbb\; \{\; R\; \}\; ^\; \{\; d\; \}\; ,\; (\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; )\; \backslash |\; \backslash boldsymbol\; \{\; s\; \}\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; \backslash boldsymbol\; \{\; x\; \}\; )\; -\; \backslash boldsymbol\; \{\; s\; \}\; \_\; \{\; t\; \}\; ^\; \{\; \backslash star\; \}\; (\; \backslash boldsymbol\; \{\; x\; \}\; ^\; \{\; \backslash prime\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; >\; C\; \backslash log\; (\; H\; T\; )\; \backslash |\; \backslash boldsymbol\; \{\; x\; \}\; -\; \backslash boldsymbol\; \{\; x\; \}\; ^\; \{\; \backslash prime\; \}\; \backslash |\; \_\; \{\; 2\; \}\; \backslash right\backslash \}\; \}\; \backslash \backslash \; \{\; \backslash displaystyle\; \backslash le\; \backslash sum\; \_\; \{\; h\; =\; 1\; \}\; ^\; \{\; H\; \}\; \backslash gamma\; \_\; \{\; h\; \}\; \backslash mathbb\; \{\; P\; \}\; \_\; \{\; h\; \}\; \backslash left\backslash \{\; \backslash boldsymbol\; \{\; x\; \}\; \backslash notin\; \backslash mathcal\; \{\; T\; \}\; \_\; \{\; t\; \}\; ^\; \{\; h\; \}\; \backslash right\backslash \}\; \backslash mathbb\; \{\; 1\; \}\; \backslash left(\; \backslash gamma\; \_\; \{\; h\; \}\; \backslash ge\; \backslash frac\; \{\; 1\; \}\; \{\; H\; T\; ^\; \{\; 4\; \}\; \}\; \backslash right)\; +\; \backslash frac\; \{\; 1\; \}\; \{\; T\; ^\; \{\; 4\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; 1\; \}\; \{\; T\; ^\; \{\; 4\; \}\; \}\; ,\; \}\; \backslash end\{array\}$$

and complete the proof.

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

$$\begin{array}{r}{\mathcal{F}}_h=i:\mathrm{\parallel }x-x^{\mathrm{\prime }}{\mathrm{\parallel }}_2\le c\sqrt{{\bar{\alpha }}_t}\mathrm{\parallel }{\mu }_i-{\mu }_h{\mathrm{\parallel }}_2,\end{array}\backslash begin\{array\}\; \{\; r\; \}\; \{\; \backslash mathcal\; \{\; F\; \}\; \_\; \{\; h\; \}\; =\; \backslash left.\; i\; :\; \backslash |\; x\; -\; x\; ^\; \{\; \backslash prime\; \}\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; c\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \}\; \backslash |\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash |\; \_\; \{\; 2\; \}\; \backslash right.\; ,\; \}\; \backslash end\{array\}$$

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

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 $\widetilde { x } _ { i } ^ { \prime } = x ^ { \prime } \mathbb { 1 } ( | x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { i } | _ { 2 } \leq | x ^ { \prime } - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { i } | _ { 2 } ) \ +$ $x \mathbf { 1 } [ \left| x ^ { \prime } - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { i } \right| _ { 2 } \leq \left| x - \sqrt { \overline { { \alpha } } _ { t } } \mu _ { i } \right| _ { 2 } )$

For the first term, we have

$${\pi }_i({\tilde{x}}_i)\mid 1-\exp (-\frac{\mathrm{\parallel }{\tilde{x}}_i^{\mathrm{\prime }}-\sqrt{\overline{{\alpha }_t}}{\mu }_i{\mathrm{\parallel }}_2^2}{2{\sigma }_t^2}+\frac{\mathrm{\parallel }{\tilde{x}}_i-\sqrt{\overline{{\alpha }_t}}{\mu }_i{\mathrm{\parallel }}_2^2}{2{\sigma }_t^2})\mid \backslash pi\; \_\; \{\; i\; \}\; (\; \backslash widetilde\; \{\; x\; \}\; \_\; \{\; i\; \}\; )\; \backslash left|\; 1\; -\; \backslash exp\; \backslash left(\; -\; \backslash frac\; \{\; \backslash |\; \backslash widetilde\; \{\; x\; \}\; \_\; \{\; i\; \}\; ^\; \{\; \backslash prime\; \}\; -\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \backslash mu\; \_\; \{\; i\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 2\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; +\; \backslash frac\; \{\; \backslash |\; \backslash widetilde\; \{\; x\; \}\; \_\; \{\; i\; \}\; -\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \backslash mu\; \_\; \{\; i\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 2\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash right)\; \backslash right|$$

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

$$s(\mathit{x}):=\sum _{\mathrm{\ell }=1}^H{\gamma }_{\mathrm{\ell }}\exp (-\frac{\mathrm{\parallel }\mathit{x}-\sqrt{\overline{{\alpha }_t}}{\mu }_{\mathrm{\ell }}{\mathrm{\parallel }}_2^2}{2{\sigma }_t^2}),s\; (\; \backslash boldsymbol\; \{\; x\; \}\; )\; :\; =\; \backslash sum\; \_\; \{\; \backslash ell\; =\; 1\; \}\; ^\; \{\; H\; \}\; \backslash gamma\; \_\; \{\; \backslash ell\; \}\; \backslash exp\; \backslash left(\; -\; \backslash frac\; \{\; \backslash |\; \backslash boldsymbol\; \{\; x\; \}\; -\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \backslash mu\; \_\; \{\; \backslash ell\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 2\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash right)\; ,$$

and (a) comes from the fact that

and the following inequality holds according to (43):

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

$$\sqrt{{\bar{\alpha }}_t}\mid {\pi }_i(x)-{\pi }_i(x^{\mathrm{\prime }})\mid \parallel {\mu }_i-{\mu }_h{\parallel }_2\le {ϵ}_{i,1}+\sum _{\mathrm{\ell }=1}^H{ϵ}_{i,2}^{(\mathrm{\ell })},\backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \}\; \backslash lvert\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; )\; -\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; ^\; \{\; \backslash prime\; \}\; )\; \backslash rvert\; \backslash lVert\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash rVert\; \_\; \{\; 2\; \}\; \backslash leq\; \backslash epsilon\; \_\; \{\; i\; ,\; 1\; \}\; +\; \backslash sum\; \_\; \{\; \backslash ell\; =\; 1\; \}\; ^\; \{\; H\; \}\; \backslash epsilon\; \_\; \{\; i\; ,\; 2\; \}\; ^\; \{\; (\; \backslash ell\; )\; \}\; ,$$

where

$${ϵ}_{i,1}:=\frac{\max \{{\pi }_i(x),{\pi }_i(x^{\mathrm{\prime }})\}}{{\sigma }_t^2}(\sqrt{\overline{{\alpha }_t}}\mathrm{\mid }(x-\sqrt{\overline{{\alpha }_t}}{\mu }_h{)}^{\mathrm{\top }}({\mu }_i-{\mu }_h)\mathrm{\mid }+2{\bar{\alpha }}_t\mathrm{\parallel }{\mu }_h-{\mu }_i{\mathrm{\parallel }}_2^2)\mathrm{\parallel }{x}^{\mathrm{\prime }}-x{\mathrm{\parallel }}_2,\backslash epsilon\; \_\; \{\; i\; ,\; 1\; \}\; :\; =\; \backslash frac\; \{\; \backslash operatorname*\; \{\; m\; a\; x\; \}\; \backslash \{\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; )\; ,\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; ^\; \{\; \backslash prime\; \}\; )\; \backslash \}\; \}\; \{\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash left(\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; |\; (\; x\; -\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \backslash mu\; \_\; \{\; h\; \}\; )\; ^\; \{\; \backslash top\; \}\; (\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; )\; |\; +\; 2\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \backslash |\; \backslash mu\; \_\; \{\; h\; \}\; -\; \backslash mu\; \_\; \{\; i\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash right)\; \backslash |\; x\; ^\; \{\; \backslash prime\; \}\; -\; x\; \backslash |\; \_\; \{\; 2\; \}\; ,$$

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

By using Lemma 6, we have

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

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

Thus we have

$${ϵ}_{i,2}^{(\mathrm{\ell })}\lesssim \max \{{\gamma }_{\mathrm{\ell }},{\pi }_{\mathrm{\ell }}(x),{\pi }_{\mathrm{\ell }}(x^{\mathrm{\prime }})\}\max \{{\gamma }_i,{\pi }_i(x),{\pi }_i(x^{\mathrm{\prime }})\}\log (HT)\mathrm{\parallel }x-x^{\mathrm{\prime }}{\mathrm{\parallel }}_2\mathrm{.}\backslash epsilon\; \_\; \{\; i\; ,\; 2\; \}\; ^\; \{\; (\; \backslash ell\; )\; \}\; \backslash lesssim\; \backslash operatorname*\; \{\; m\; a\; x\; \}\; \backslash \{\; \backslash gamma\; \_\; \{\; \backslash ell\; \}\; ,\; \backslash pi\; \_\; \{\; \backslash ell\; \}\; (\; x\; )\; ,\; \backslash pi\; \_\; \{\; \backslash ell\; \}\; (\; x\; ^\; \{\; \backslash prime\; \}\; )\; \backslash \}\; \backslash operatorname*\; \{\; m\; a\; x\; \}\; \backslash \{\; \backslash gamma\; \_\; \{\; i\; \}\; ,\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; )\; ,\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; ^\; \{\; \backslash prime\; \}\; )\; \backslash \}\; \backslash log\; (\; H\; T\; )\; \backslash |\; x\; -\; x\; ^\; \{\; \backslash prime\; \}\; \backslash |\; \_\; \{\; 2\; \}\; .$$

Inserting into (46), we have

Inserting it into (41) we have

$$\sqrt{{\bar{\alpha }}_t}\sum _{i=1}^H\mathrm{\parallel }({\pi }_i(x)-{\pi }_i(x^{\mathrm{\prime }}))({\mu }_i-{\mu }_h){\mathrm{\parallel }}_2\backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; H\; \}\; \backslash |\; (\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; )\; -\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; ^\; \{\; \backslash prime\; \}\; )\; )\; (\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}$$

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

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

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

$$\frac{\sqrt{\overline{{\alpha }_t}}}{{\sigma }_t^2}(x^{\mathrm{\prime }}-x{)}^{\mathrm{\top }}({\mu }_i-{\mu }_h)\le \frac{\sqrt{\overline{{\alpha }_t}}}{{\sigma }_t^2}\mathrm{\parallel }{x}^{\mathrm{\prime }}-x{\mathrm{\parallel }}_2\mathrm{\parallel }{\mu }_i-{\mu }_h{\mathrm{\parallel }}_2\le \frac{c\overline{{\alpha }_t}}{{\sigma }_t^2}\mathrm{\parallel }{\mu }_i-{\mu }_h{\mathrm{\parallel }}_2^2\le \frac{\overline{{\alpha }_t}}{24{\sigma }_t^2}\mathrm{\parallel }{\mu }_i-{\mu }_h{\mathrm{\parallel }}_2^2\mathrm{.}\backslash frac\; \{\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \}\; \{\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash big\; (\; x\; ^\; \{\; \backslash prime\; \}\; -\; x\; \backslash big\; )\; ^\; \{\; \backslash top\; \}\; \backslash big\; (\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash big\; )\; \backslash leq\; \backslash frac\; \{\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \}\; \{\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash |\; x\; ^\; \{\; \backslash prime\; \}\; -\; x\; \backslash |\; \_\; \{\; 2\; \}\; \backslash |\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; \backslash frac\; \{\; c\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \{\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash |\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash leq\; \backslash frac\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \{\; 2\; 4\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash |\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash |\; \_\; \{\; 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

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

$$4\log (HT)\le \frac{{\bar{\alpha }}_t\mathrm{\parallel }{\mu }_i-{\mu }_h{\mathrm{\parallel }}_2^2}{9C_2^2{\sigma }_t^2}\le \frac{{\bar{\alpha }}_t\mathrm{\parallel }{\mu }_i-{\mu }_h{\mathrm{\parallel }}_2^2}{24{\sigma }_t^2}\mathrm{.}4\; \backslash log\; (\; H\; T\; )\; \backslash leq\; \backslash frac\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \backslash |\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 9\; C\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash leq\; \backslash frac\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \backslash |\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 2\; 4\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; .$$

Furthermore, we have

and

$$\begin{array}{l}{\displaystyle \frac{\sqrt{\overline{{\alpha }_t}}}{{\sigma }_t^2}\max \{{\pi }_i(x),{\pi }_i(x^{\mathrm{\prime }})\}\mathrm{\mid }(x-\sqrt{\overline{{\alpha }_t}}{\mu }_h{)}^{\mathrm{\top }}({\mu }_i-{\mu }_h)\mathrm{\mid }}\\ {\displaystyle \le {\gamma }_i\exp (-\frac{5\overline{{\alpha }_t}\mathrm{\parallel }{\mu }_i-{\mu }_h{\mathrm{\parallel }}_2^2}{12{\sigma }_t^2}+\frac{2\sqrt{\overline{{\alpha }_t}}}{{\sigma }_t^2}\mathrm{\mid }(x-\sqrt{\overline{{\alpha }_t}}{\mu }_h{)}^{\mathrm{\top }}({\mu }_i-{\mu }_h)\mathrm{\mid })}\end{array}\backslash begin\{array\}\; \{\; l\; \}\; \{\; \backslash displaystyle\; \backslash frac\; \{\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \}\; \{\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash operatorname*\; \{\; m\; a\; x\; \}\; \backslash \{\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; )\; ,\; \backslash pi\; \_\; \{\; i\; \}\; (\; x\; ^\; \{\; \backslash prime\; \}\; )\; \backslash \}\; \backslash vert\; (\; x\; -\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \backslash mu\; \_\; \{\; h\; \}\; )\; ^\; \{\; \backslash top\; \}\; (\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; )\; \backslash vert\; \}\; \backslash \backslash \; \{\; \backslash displaystyle\; \backslash leq\; \backslash gamma\; \_\; \{\; i\; \}\; \backslash exp\; \backslash left(\; -\; \backslash frac\; \{\; 5\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \backslash |\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; 2\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; +\; \backslash frac\; \{\; 2\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \}\; \{\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash vert\; (\; x\; -\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \}\; \backslash mu\; \_\; \{\; h\; \}\; )\; ^\; \{\; \backslash top\; \}\; (\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; )\; \backslash vert\; \backslash right)\; \}\; \backslash end\{array\}$$

Thus to establish Lemma 6, it suffices to prove that

$$\exp (-\frac{3{\bar{\alpha }}_t\mathrm{\parallel }{\mu }_i-{\mu }_h{\mathrm{\parallel }}_2^2}{8{\sigma }_t^2}+\frac{2\sqrt{{\bar{\alpha }}_t}}{{\sigma }_t^2}\mathrm{\mid }(x-\sqrt{{\bar{\alpha }}_t}{\mu }_h{)}^{\mathrm{\top }}({\mu }_i-{\mu }_h)\mathrm{\mid })\le 1.\backslash exp\; \backslash left(\; -\; \backslash frac\; \{\; 3\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \backslash |\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 8\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; +\; \backslash frac\; \{\; 2\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \}\; \}\; \{\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; |\; (\; x\; -\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \}\; \backslash mu\; \_\; \{\; h\; \}\; )\; ^\; \{\; \backslash top\; \}\; (\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; )\; |\; \backslash right)\; \backslash leq\; 1\; .$$

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

$$\frac{2\sqrt{{\alpha }_t}}{{\sigma }_t^2}\mathrm{\mid }(x-\sqrt{{\alpha }_t}{\mu }_h{)}^{\mathrm{\top }}({\mu }_i-{\mu }_h)\mathrm{\mid }\le \frac{2C_2}{{\sigma }_t}\sqrt{{\bar{\alpha }}_t\log (HT)}\mathrm{\parallel }{\mu }_i-{\mu }_h{\mathrm{\parallel }}_2\le \frac{{\bar{\alpha }}_t}{3{\sigma }_t^2}\mathrm{\parallel }{\mu }_i-{\mu }_h{\mathrm{\parallel }}_2^2,\backslash frac\; \{\; 2\; \backslash sqrt\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \}\; \{\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; |\; (\; x\; -\; \backslash sqrt\; \{\; \backslash alpha\; \_\; \{\; t\; \}\; \}\; \backslash mu\; \_\; \{\; h\; \}\; )\; ^\; \{\; \backslash top\; \}\; (\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; )\; |\; \backslash leq\; \backslash frac\; \{\; 2\; C\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sigma\; \_\; \{\; t\; \}\; \}\; \backslash sqrt\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \backslash log\; (\; H\; T\; )\; \}\; \backslash |\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; \backslash frac\; \{\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; t\; \}\; \}\; \{\; 3\; \backslash sigma\; \_\; \{\; t\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash |\; \backslash mu\; \_\; \{\; i\; \}\; -\; \backslash mu\; \_\; \{\; h\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; ,$$

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

C.2.2 Proof of the lower bound

By calculation, we have

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

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

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

Thus we have

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

$$1-{\tau }_{0,0}\le {\widehat{\alpha }}_T\le \frac{2}{T^{c_0}},{\tau }_{K,0}\le 1-{\widehat{\alpha }}_1\le \frac{1}{T^{c_0}},and\frac{{\widehat{\tau }}_{k,n-1}-{\widehat{\tau }}_{k,n}}{{\widehat{\tau }}_{k,n-1}(1-{\widehat{\tau }}_{k,n-1})}=\frac{c_1\log T}{T}\mathrm{.}1\; -\; \backslash tau\; \_\; \{\; 0\; ,\; 0\; \}\; \backslash leq\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; \}\; \backslash leq\; \backslash frac\; \{\; 2\; \}\; \{\; T\; ^\; \{\; c\; \_\; \{\; 0\; \}\; \}\; \}\; ,\; \backslash qquad\; \backslash tau\; \_\; \{\; K\; ,\; 0\; \}\; \backslash leq\; 1\; -\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; 1\; \}\; \backslash leq\; \backslash frac\; \{\; 1\; \}\; \{\; T\; ^\; \{\; c\; \_\; \{\; 0\; \}\; \}\; \}\; ,\; \backslash quad\; a\; n\; d\; \backslash \; \backslash frac\; \{\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; -\; 1\; \}\; -\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; \}\; \}\; \{\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; -\; 1\; \}\; \backslash left(\; 1\; -\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; -\; 1\; \}\; \backslash right)\; \}\; =\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash log\; T\; \}\; \{\; T\; \}\; .$$

Moreover, we have

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

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 $1 - T ^ { - 1 0 0 }$ ,

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

Recalling Lemma 7, we have

Inserting into (54), we have that

$$\frac{\mathrm{\parallel }{y}_{{\tau }_{k,n}}(x_k)-y_{{\tau }_{k,n}}^{\star }(x_k){\mathrm{\parallel }}_2}{\sqrt{1-{\tau }_{k,n}}}\lesssim \sum _{i=0}^{n-1}\frac{{\tilde{\epsilon }}_{k,i}(x_k){\widehat{\tau }}_{k,i-1}\log T}{T(1-{\widehat{\tau }}_{k,i-1}{)}^{1\mathrm{/}2}}\mathrm{.}\backslash frac\; \{\; \backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; -\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \}\; \backslash lesssim\; \backslash sum\; \_\; \{\; i\; =\; 0\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash frac\; \{\; \backslash widetilde\; \{\; \backslash varepsilon\; \}\; \_\; \{\; k\; ,\; i\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; -\; 1\; \}\; \backslash log\; T\; \}\; \{\; T\; (\; 1\; -\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; -\; 1\; \}\; )\; ^\; \{\; 1\; /\; 2\; \}\; \}\; .$$

Thus we have

$$\frac{\mathrm{\parallel }{y}_{{\tau }_{k,n}}^{\star }(x_k)-y_{{\tau }_{k,n}}(x_k){\mathrm{\parallel }}_2^2}{1-{\tau }_{k,n}}\lesssim \frac{N{\log }^{2}T}{T^2}\sum _{i=0}^{n-1}\frac{{\tilde{\epsilon }}_{k,i}^2(x_k){\widehat{\tau }}_{k,i-1}}{1-{\widehat{\tau }}_{k,i-1}}\lesssim \frac{N{\log }^{2}T}{T^2}\sum _{i=0}^{n-1}\frac{{\tilde{\epsilon }}_{k,i}^2(x_k){\widehat{\tau }}_{k,i}}{1-{\widehat{\tau }}_{k,i}}\backslash frac\; \{\; \backslash Vert\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; -\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \backslash Vert\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; N\; \backslash log\; ^\; \{\; 2\; \}\; T\; \}\; \{\; T\; ^\; \{\; 2\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 0\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash frac\; \{\; \backslash tilde\; \{\; \backslash varepsilon\; \}\; \_\; \{\; k\; ,\; i\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \backslash hat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; -\; 1\; \}\; \}\; \{\; 1\; -\; \backslash hat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; -\; 1\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; N\; \backslash log\; ^\; \{\; 2\; \}\; T\; \}\; \{\; T\; ^\; \{\; 2\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 0\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash frac\; \{\; \backslash tilde\; \{\; \backslash varepsilon\; \}\; \_\; \{\; k\; ,\; i\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \backslash hat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; \}\; \}\; \{\; 1\; -\; \backslash hat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; \}\; \}$$

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

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

$$\frac{x_{{\tau }_{k,n}}(x_k)-z_{{\tau }_{k,n}}^{\star }(x_k)}{\sqrt{1-{\tau }_{k,n}}}={\mathcal{E}}_{1,1}(x_{{\tau }_{k,0}})+{\mathcal{E}}_{1,0}(x_{{\tau }_{k,0}})-{\mathcal{E}}_2(x_{{\tau }_{k,0}}),\backslash frac\; \{\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; -\; z\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \}\; =\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 1\; ,\; 1\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; +\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 1\; ,\; 0\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; -\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 2\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; ,$$

where

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

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

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

$$\parallel {\mathcal{E}}_{1,1}(x_{{\tau }_{k,0}}){\parallel }_2^2\le ({\widehat{\tau }}_{k,n-1}-{\tau }_{k,n}{)}^2{\left({\int }_{{\tau }_{k,n}}^{{\tau }_{k,n-1}}{\frac{\mathrm{\partial }}{\mathrm{\partial }\tau }\frac{s_{\tau }^{\star }(x_{\tau })}{2(1-\tau {)}^{3\mathrm{/}2}}}_2\mathrm{d}\tau \right)}^2\backslash lVert\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 1\; ,\; 1\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; \backslash rVert\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash leq\; (\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; -\; 1\; \}\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; )\; ^\; \{\; 2\; \}\; \backslash left(\; \backslash int\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; -\; 1\; \}\; \}\; \backslash left.\; \backslash frac\; \{\; \backslash partial\; \}\; \{\; \backslash partial\; \backslash tau\; \}\; \backslash frac\; \{\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \}\; \{\; 2\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 3\; /\; 2\; \}\; \}\; \backslash right.\; \_\; \{\; 2\; \}\; \backslash mathrm\; \{\; d\; \}\; \backslash tau\; \backslash right)\; ^\; \{\; 2\; \}$$

$$\le ({\widehat{\tau }}_{k,n-1}-{\widehat{\tau }}_{k,n}{)}^2({\widehat{\tau }}_{k,n-2}-{\widehat{\tau }}_{k,n}){\int }_{{\tau }_{k,n}}^{{\widehat{\tau }}_{k,n-2}}{\parallel \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }\frac{s_{\tau }^{\star }(x_{\tau })}{2(1-\tau {)}^{3\mathrm{/}2}}\parallel }_2^2\mathrm{d}\tau \mathrm{.}\backslash leq\; (\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; -\; 1\; \}\; -\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; \}\; )\; ^\; \{\; 2\; \}\; (\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; -\; 2\; \}\; -\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; \}\; )\; \backslash int\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; -\; 2\; \}\; \}\; \backslash left\backslash |\; \backslash frac\; \{\; \backslash partial\; \}\; \{\; \backslash partial\; \backslash tau\; \}\; \backslash frac\; \{\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \}\; \{\; 2\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 3\; /\; 2\; \}\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash mathrm\; \{\; d\; \}\; \backslash tau\; .$$

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,

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

We further have

$$\mathrm{\parallel }{\mathcal{E}}_{1,0}(x_{{\tau }_{k,0}}){\mathrm{\parallel }}_2^2\le ({\tau }_{k,0}-{\widehat{\tau }}_{k,0}{)}^3{\int }_{{\widehat{\tau }}_{k,0}}^{{\tau }_{k,0}}\parallel \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }\frac{s_{\tau }^{\star }(x_{\tau })}{2(1-\tau {)}^{3\mathrm{/}2}}{\parallel }_2^2\mathrm{d}\tau \mathrm{.}\backslash |\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 1\; ,\; 0\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash leq\; (\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; -\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; 0\; \}\; )\; ^\; \{\; 3\; \}\; \backslash int\; \_\; \{\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; 0\; \}\; \}\; ^\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; \backslash bigg\; \backslash |\; \backslash frac\; \{\; \backslash partial\; \}\; \{\; \backslash partial\; \backslash tau\; \}\; \backslash frac\; \{\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \}\; \{\; 2\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 3\; /\; 2\; \}\; \}\; \backslash bigg\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash mathrm\; \{\; d\; \}\; \backslash tau\; .$$

By taking expectation, we have

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

$$\mathbb{E}\left[\right({\mathbb{E}}_{{\mathit{x}}_{{\tau }_{k,0}}\sim p_{\widehat{X_k}}}[\mathrm{\parallel }{\mathcal{E}}_2({\mathit{x}}_{{\tau }_{k,0}}){\mathrm{\parallel }}_2^2\mid \tau ]{)}^r]=\mathbb{E}[{\mathbb{E}}_{{\mathit{x}}_k^{(j)}\sim p_{\widehat{X_k}}}[\prod _{1\le j\le r}\mathrm{\parallel }{\mathcal{E}}_2({\mathit{x}}_k^{(j)}){\mathrm{\parallel }}_2^2\mid \tau ]]\backslash mathbb\; \{\; E\; \}\; \backslash big\; [\; \backslash big\; (\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; \backslash sim\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \_\; \{\; k\; \}\; \}\; \}\; \}\; \backslash big\; [\; \backslash |\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 2\; \}\; (\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash mid\; \backslash tau\; \backslash big\; ]\; \backslash big\; )\; ^\; \{\; r\; \}\; \backslash big\; ]\; =\; \backslash mathbb\; \{\; E\; \}\; \backslash Big\; [\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; k\; \}\; ^\; \{\; (\; j\; )\; \}\; \backslash sim\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \_\; \{\; k\; \}\; \}\; \}\; \}\; \backslash Big\; [\; \backslash prod\; \_\; \{\; 1\; \backslash leq\; j\; \backslash leq\; r\; \}\; \backslash |\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 2\; \}\; (\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; k\; \}\; ^\; \{\; (\; j\; )\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash mid\; \backslash tau\; \backslash Big\; ]\; \backslash Big\; ]$$

$$={\mathbb{E}}_{{\mathit{x}}_k^{(j)}\sim p_{{\widehat{X}}_k}}\left[\mathbb{E}\right[\prod _{1\le j\le r}\mathrm{\parallel }{\mathcal{E}}_2({\mathit{x}}_k^{(j)}){\mathrm{\parallel }}_2^2\mathrm{\mid }{\mathit{x}}_k^{(j)}\left]\right]\mathrm{.}=\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; k\; \}\; ^\; \{\; (\; j\; )\; \}\; \backslash sim\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; \}\; \backslash left[\; \backslash mathbb\; \{\; E\; \}\; \backslash Big\; [\; \backslash prod\; \_\; \{\; 1\; \backslash leq\; j\; \backslash leq\; r\; \}\; \backslash |\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 2\; \}\; (\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; k\; \}\; ^\; \{\; (\; j\; )\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; |\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; k\; \}\; ^\; \{\; (\; j\; )\; \}\; \backslash Big\; ]\; \backslash right]\; .$$

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 $x _ { \tau _ { k , 0 } } = x _ { k } ^ { ( j ) }$ :

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

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

$$∥E2(xk(j))∥22≲log⁡rδ∑i=1n−1[(τ^k,i−1−τ^k,i)∫τ^k,iτ^k,i−1∥∂∂τsτ⋆(xτ)(1−τ)3/2∥2dτ]2log⁡rδ∑i=1n−1(τ^k,i−1−τ^k,i)3∫τ^k,iτ^k,i−1∥∂∂τsτ⋆(xτ)(1−τ)3/2∥22dτ, \begin{array} { r } { \lVert \mathcal { E } _ { 2 } ( x _ { k } ^ { ( j ) } ) \rVert _ { 2 } ^ { 2 } \lesssim \log { \frac { r } { \delta } } \displaystyle \sum _ { i = 1 } ^ { n - 1 } \left[ ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \Big \lVert \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big \rVert _ { 2 } \mathrm { d } \tau \right] ^ { 2 } } \\ { \stackrel { \mathrm { ( a ) } } { \lesssim } \log { \frac { r } { \delta } } \displaystyle \sum _ { i = 1 } ^ { n - 1 } ( \widehat { \tau } _ { k , i - 1 } - \widehat { \tau } _ { k , i } ) ^ { 3 } \int _ { \widehat { \tau } _ { k , i } } ^ { \widehat { \tau } _ { k , i - 1 } } \Big \lVert \frac { \partial } { \partial \tau } \frac { s _ { \tau } ^ { \star } ( x _ { \tau } ) } { ( 1 - \tau ) ^ { 3 / 2 } } \Big \rVert _ { 2 } ^ { 2 } \mathrm { d } \tau , } \end{array} $$

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.

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

Then the (56) implies that

$$\mathbb{E}[\prod _{1\le j\le r}\mathrm{\parallel }{\mathcal{E}}_2(x_k^{(j)}){\mathrm{\parallel }}_2^2\mathrm{\mid }{x}_k^{(j)}]\lesssim r!{(\sum _{i=1}^{n-1}({\widehat{\tau }}_{k,i-1}-{\widehat{\tau }}_{k,i}{)}^3{\int }_{{\widehat{\tau }}_{k,i}}^{{\widehat{\tau }}_{k,i-1}}\parallel \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }\frac{s_{\tau }^{\star }(x_{\tau })}{(1-\tau {)}^{3\mathrm{/}2}}{\parallel }_2^2\mathrm{d}\tau )}^r\mathrm{.}\backslash mathbb\; \{\; E\; \}\; \backslash Big\; [\; \backslash prod\; \_\; \{\; 1\; \backslash le\; j\; \backslash le\; r\; \}\; \backslash |\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 2\; \}\; (\; x\; \_\; \{\; k\; \}\; ^\; \{\; (\; j\; )\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; |\; x\; \_\; \{\; k\; \}\; ^\; \{\; (\; j\; )\; \}\; \backslash Big\; ]\; \backslash lesssim\; r\; !\; \backslash left(\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; n\; -\; 1\; \}\; (\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; -\; 1\; \}\; -\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; \}\; )\; ^\; \{\; 3\; \}\; \backslash int\; \_\; \{\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; \}\; \}\; ^\; \{\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; -\; 1\; \}\; \}\; \backslash Big\; \backslash |\; \backslash frac\; \{\; \backslash partial\; \}\; \{\; \backslash partial\; \backslash tau\; \}\; \backslash frac\; \{\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \}\; \{\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 3\; /\; 2\; \}\; \}\; \backslash Big\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash mathrm\; \{\; d\; \}\; \backslash tau\; \backslash right)\; ^\; \{\; r\; \}\; .$$

Inserting (57) into (55), we have

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

$${\mathbb{E}}_{{\mathit{x}}_k\sim p_{{\widehat{X}}_k}}[\mathrm{\parallel }{\mathcal{E}}_2({\mathit{x}}_k){\mathrm{\parallel }}_2^2]\lesssim \log T\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}}_{{\mathit{x}}_{\tau }\sim {\mathit{X}}_{\tau }}[\parallel \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }\frac{s_{\tau }^{\star }({\mathit{x}}_{\tau })}{(1-\tau {)}^{3\mathrm{/}2}}{\parallel }_2^2]\mathrm{d}\tau \mathrm{.}\backslash mathbb\; \{\; E\; \}\; \_\; \{\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; k\; \}\; \backslash sim\; p\; \_\; \{\; \backslash hat\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; \}\; \backslash big\; [\; \backslash |\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; 2\; \}\; (\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; k\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash big\; ]\; \backslash lesssim\; \backslash log\; T\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; n\; -\; 1\; \}\; (\; \backslash hat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; -\; 1\; \}\; -\; \backslash hat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; \}\; )\; ^\; \{\; 3\; \}\; \backslash int\; \_\; \{\; \backslash hat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; \}\; \}\; ^\; \{\; \backslash hat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; -\; 1\; \}\; \}\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; \backslash tau\; \}\; \backslash sim\; \backslash boldsymbol\; \{\; X\; \}\; \_\; \{\; \backslash tau\; \}\; \}\; \backslash bigg\; [\; \backslash Big\; \backslash |\; \backslash frac\; \{\; \backslash partial\; \}\; \{\; \backslash partial\; \backslash tau\; \}\; \backslash frac\; \{\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; \backslash tau\; \}\; )\; \}\; \{\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 3\; /\; 2\; \}\; \}\; \backslash Big\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash bigg\; ]\; \backslash mathrm\; \{\; d\; \}\; \backslash tau\; .$$

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

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

$${\mathbb{E}}_{x_{\tau }\sim X_{\tau }}[\parallel \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }\frac{s_{\tau }^{\star }(x_{\tau })}{(1-\tau {)}^{3\mathrm{/}2}}{\parallel }_2^2]\lesssim \frac{d\log T}{(1-\tau {)}^5{\tau }^3}\min \{d+\mathbb{E}[\mathsf{T}\mathsf{r}({\mathrm{\Sigma }}_{\tau }^2(x_{\tau }))],L^2\}\mathrm{.}\backslash mathbb\; \{\; E\; \}\; \_\; \{\; x\; \_\; \{\; \backslash tau\; \}\; \backslash sim\; X\; \_\; \{\; \backslash tau\; \}\; \}\; \backslash bigg\; [\; \backslash Big\; \backslash lVert\; \backslash frac\; \{\; \backslash partial\; \}\; \{\; \backslash partial\; \backslash tau\; \}\; \backslash frac\; \{\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \}\; \{\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 3\; /\; 2\; \}\; \}\; \backslash Big\; \backslash rVert\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash bigg\; ]\; \backslash lesssim\; \backslash frac\; \{\; d\; \backslash log\; T\; \}\; \{\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 5\; \}\; \backslash tau\; ^\; \{\; 3\; \}\; \}\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash left\backslash \{\; d\; +\; \backslash mathbb\; \{\; E\; \}\; [\; \backslash mathsf\; \{\; T\; r\; \}\; (\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; )\; ]\; ,\; L\; ^\; \{\; 2\; \}\; \backslash right\backslash \}\; .$$

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

$$\frac{{\mathbb{E}}_{{\mathit{x}}_k\sim p_{{\widehat{X}}_k}}[\mathrm{\parallel }{\mathit{x}}_{{\tau }_{k,n}}({\mathit{x}}_k)-{\mathit{z}}_{{\tau }_{k,n}}^{\star }({\mathit{x}}_k){\mathrm{\parallel }}_2^2]}{1-{\tau }_{k,n}}\backslash frac\; \{\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; k\; \}\; \backslash sim\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; \}\; \backslash left[\; \backslash |\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; (\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; k\; \}\; )\; -\; \backslash boldsymbol\; \{\; z\; \}\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash star\; \}\; (\; \backslash boldsymbol\; \{\; x\; \}\; \_\; \{\; k\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash right]\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}$$

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

and complete the proof.

Proof of (59). Notice that

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

For the first term, we have

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

where

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

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

$$\mathrm{\parallel }{s}_{\tau }^{\star }(x_{\tau }){\mathrm{\parallel }}_2^2\le \frac{1}{{\tau }^2}\int {\parallel x_{\tau }-\sqrt{1-\tau }{x}_0\parallel }_2^2{p}_{X_0\mathrm{\mid }{X}_{\tau }}(x_0\mathrm{\mid }{x}_{\tau })\mathrm{d}{x}_0\le \frac{25(\theta +c_0)d\log T}{\tau }\mathrm{.}\backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash leq\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; ^\; \{\; 2\; \}\; \}\; \backslash int\; \backslash left\backslash |\; x\; \_\; \{\; \backslash tau\; \}\; -\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; |\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \backslash leq\; \backslash frac\; \{\; 2\; 5\; (\; \backslash theta\; +\; c\; \_\; \{\; 0\; \}\; )\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; .$$

Moreover, we have

$${\int }_{x_{\tau }}\mathrm{\parallel }{J}_{\tau }(x_{\tau })s_{\tau }^{\star }(x_{\tau }){\mathrm{\parallel }}_2^2{p}_{\tau }(x_{\tau })\mathrm{d}{x}_{\tau }\lesssim \frac{d\log T}{{\tau }^3}\min \{d+\mathbb{E}[\mathsf{T}\mathsf{r}({\mathrm{\Sigma }}_{\tau }^2(x_{\tau }))],L^2\},\backslash int\; \_\; \{\; x\; \_\; \{\; \backslash tau\; \}\; \}\; \backslash Vert\; J\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash Vert\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; p\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; \backslash tau\; \}\; \backslash lesssim\; \backslash frac\; \{\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; ^\; \{\; 3\; \}\; \}\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash left\backslash \{\; d\; +\; \backslash mathbb\; \{\; E\; \}\; [\; \backslash mathsf\; \{\; T\; r\; \}\; (\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; )\; ]\; ,\; L\; ^\; \{\; 2\; \}\; \backslash right\backslash \}\; ,$$

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

By using Lemma 9, we have

For the second term, we have

Thus we have

where (i) comes from the following Lemma.

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

$$\mathbb{E}{\parallel \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }{\int }_{x_0}(y-x_0)p_{X_0\mathrm{\mid }{X}_{\tau }}(x_0\mathrm{\mid }\sqrt{1-\tau }y)\mathrm{d}{x}_0\parallel }_2^2\le \frac{d\min \{d\log T,L\}}{\tau (1-\tau {)}^3}\mathrm{.}\backslash mathbb\; \{\; E\; \}\; \backslash left\backslash |\; \backslash frac\; \{\; \backslash partial\; \}\; \{\; \backslash partial\; \backslash tau\; \}\; \backslash int\; \_\; \{\; x\; \_\; \{\; 0\; \}\; \}\; (\; y\; -\; x\; \_\; \{\; 0\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; |\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; y\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash leq\; \backslash frac\; \{\; d\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; \backslash log\; T\; ,\; L\; \backslash \}\; \}\; \{\; \backslash tau\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 3\; \}\; \}\; .$$

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

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

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

$R ^ { 2 } = ( 6 \theta + 3 c _ { 0 } ) d \tau \log T$ $\begin{array} { r } { p _ { X _ { 0 } | X _ { \tau } } ( x _ { 0 } | x _ { \tau } ) \leq p _ { X _ { 0 } } ( x _ { 0 } ) \exp ( - \frac { | x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } | _ { 2 } ^ { 2 } } { 3 \tau } ) } \end{array}$ $\left. x _ { \tau } - \sqrt { 1 - \tau } x _ { 0 } \right. _ { 2 } > R$ $\begin{array} { r } { z \exp ( - z ) \le \frac { R ^ { 2 } } { 3 \tau } \exp \bigl ( - \frac { R ^ { 2 } } { 3 \tau } \bigr ) } \end{array}$ $\begin{array} { r } { z \ge \frac { R ^ { 2 } } { 3 \tau } \ge 1 } \end{array}$ $( 2 \theta + c _ { 0 } ) d \log T \geq$ $^ { 1 }$ . 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

Noticing that

$$J_{\tau }(x_{\tau })=-\frac{1}{\tau }{I}_d+\frac{1}{\tau }{\mathrm{\Sigma }}_{\tau }(x_{\tau }),J\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; =\; -\; \backslash frac\; 1\; \backslash tau\; I\; \_\; \{\; d\; \}\; +\; \backslash frac\; 1\; \backslash tau\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; ,$$

we have

$$\mathrm{\parallel }{J}_{\tau }(x_{\tau }){\mathrm{\parallel }}_2^2\le \mathrm{\parallel }{J}_{\tau }(x_{\tau }){\mathrm{\parallel }}_F^2=\mathsf{T}\mathsf{r}(J_{\tau }^2(x_{\tau }))=\frac{1}{{\tau }^2}\mathsf{T}\mathsf{r}(I_d-2{\mathrm{\Sigma }}_{\tau }(x_{\tau })+{\mathrm{\Sigma }}_{\tau }^2(x_{\tau }))\backslash |\; J\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash leq\; \backslash |\; J\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash |\; \_\; \{\; F\; \}\; ^\; \{\; 2\; \}\; =\; \{\; \backslash mathsf\; \{\; T\; r\; \}\; \}\; (\; J\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; )\; =\; \{\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; ^\; \{\; 2\; \}\; \}\; \}\; \{\; \backslash mathsf\; \{\; T\; r\; \}\; \}\; (\; I\; \_\; \{\; d\; \}\; -\; 2\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; +\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; )$$

$$\le \frac{1}{{\tau }^2}\mathsf{T}\mathsf{r}(I_d+{\mathrm{\Sigma }}_{\tau }^2(x_{\tau }))\le \frac{d+\mathsf{T}\mathsf{r}({\mathrm{\Sigma }}_{\tau }^2(x_{\tau }))}{{\tau }^2}\mathrm{.}\backslash leq\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; ^\; \{\; 2\; \}\; \}\; \backslash mathsf\; \{\; T\; r\; \}\; (\; I\; \_\; \{\; d\; \}\; +\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; )\; \backslash leq\; \backslash frac\; \{\; d\; +\; \backslash mathsf\; \{\; T\; r\; \}\; (\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; )\; \}\; \{\; \backslash tau\; ^\; \{\; 2\; \}\; \}\; .$$

Thus we have

$$\int \parallel J_{\tau }(x_{\tau }){\parallel }_2^2{p}_{\tau }(x_{\tau })\mathrm{d}{x}_{\tau }\le \frac{d+\mathbb{E}[\mathsf{T}\mathsf{r}({\mathrm{\Sigma }}_{\tau }^2(x_{\tau }))]}{{\tau }^2}\mathrm{.}\backslash int\; \backslash Big\; \backslash |\; J\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash Big\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; p\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; \backslash tau\; \}\; \backslash leq\; \backslash frac\; \{\; d\; +\; \backslash mathbb\; \{\; E\; \}\; [\; \backslash mathsf\; \{\; T\; r\; \}\; (\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; )\; ]\; \}\; \{\; \backslash tau\; ^\; \{\; 2\; \}\; \}\; .$$

To bound the second term, we notice that

Then according to Jensen’s inequality, we have

$$\begin{array}{l}{\displaystyle {\int }_{S_{\tau }^c}\parallel J_{\tau }(x_{\tau })s_{\tau }^{\star }(x_{\tau }){\parallel }_2^2{p}_{\tau }(x_{\tau })\mathrm{d}{x}_{\tau }\le \frac{1}{{\tau }^3}{\int }_{S_{\tau }^c}\parallel \sqrt{\tau }{s}_{\tau }^{\star }(x_{\tau }){\parallel }_2^2(1+\mathrm{\parallel }{\mathrm{\Sigma }}_{\tau }(x_{\tau }){\mathrm{\parallel }}_2{)}^2{p}_{\tau }(x_{\tau })\mathrm{d}{x}_{\tau }}\\ {\displaystyle \lesssim \frac{1}{{\tau }^3}{(\int \mathrm{\parallel }\sqrt{\tau }{s}_{\tau }^{\star }(x_{\tau }){\mathrm{\parallel }}_2^4(1+\mathrm{\parallel }{\mathrm{\Sigma }}_{\tau }(x_{\tau }){\mathrm{\parallel }}_2{)}^4{p}_{\tau }(x_{\tau })\mathrm{d}{x}_{\tau })}^{1\mathrm{/}2}{({\int }_{S_{\tau }^c}{p}_{\tau }(x_{\tau })\mathrm{d}{x}_{\tau })}^{1\mathrm{/}2}\mathrm{.}}\end{array}\backslash begin\{array\}\; \{\; l\; \}\; \{\; \backslash displaystyle\; \backslash int\; \_\; \{\; S\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; c\; \}\; \}\; \backslash Big\; \backslash |\; J\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash Big\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; p\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; \backslash tau\; \}\; \backslash leq\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; ^\; \{\; 3\; \}\; \}\; \backslash int\; \_\; \{\; S\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; c\; \}\; \}\; \backslash Big\; \backslash |\; \backslash sqrt\; \{\; \backslash tau\; \}\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash Big\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash big\; (\; 1\; +\; \backslash |\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; \backslash big\; )\; ^\; \{\; 2\; \}\; p\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; \backslash tau\; \}\; \}\; \backslash \backslash \; \{\; \backslash displaystyle\; \backslash lesssim\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; ^\; \{\; 3\; \}\; \}\; \backslash left(\; \backslash int\; \backslash |\; \backslash sqrt\; \{\; \backslash tau\; \}\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 4\; \}\; (\; 1\; +\; \backslash |\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; )\; ^\; \{\; 4\; \}\; p\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; \backslash tau\; \}\; \backslash right)\; ^\; \{\; 1\; /\; 2\; \}\; \backslash left(\; \backslash int\; \_\; \{\; S\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; c\; \}\; \}\; p\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; \backslash tau\; \}\; \backslash right)\; ^\; \{\; 1\; /\; 2\; \}\; .\; \}\; \backslash end\{array\}$$

Noticing that

we have

Moreover, we have

$$\le \frac{1}{T^4}+\frac{\mathbb{E}\mathrm{\parallel }{X}_0\mathrm{\parallel }}{T^{c_R+4}}+\frac{1}{T^4}\lesssim \frac{1}{T^4},\backslash leq\; \backslash frac\; \{\; 1\; \}\; \{\; T\; ^\; \{\; 4\; \}\; \}\; +\; \backslash frac\; \{\; \backslash mathbb\; \{\; E\; \}\; \backslash |\; X\; \_\; \{\; 0\; \}\; \backslash |\; \}\; \{\; T\; ^\; \{\; c\; \_\; \{\; R\; \}\; +\; 4\; \}\; \}\; +\; \backslash frac\; \{\; 1\; \}\; \{\; T\; ^\; \{\; 4\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; 1\; \}\; \{\; T\; ^\; \{\; 4\; \}\; \}\; ,$$

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

$${\int }_{S_{\tau }^{\mathrm{c}}}\parallel J_{\tau }(x_{\tau })s_{\tau }^{\star }(x_{\tau }){\parallel }^2{p}_{\tau }(x_{\tau })\mathrm{d}{x}_{\tau }\le \frac{d^3}{{\tau }^3{T}^2}\lesssim \frac{d^2}{{\tau }^3}\backslash int\; \_\; \{\; S\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash mathrm\; \{\; c\; \}\; \}\; \}\; \backslash Big\; \backslash |\; J\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash Big\; \backslash |\; ^\; \{\; 2\; \}\; p\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; \backslash tau\; \}\; \backslash leq\; \backslash frac\; \{\; d\; ^\; \{\; 3\; \}\; \}\; \{\; \backslash tau\; ^\; \{\; 3\; \}\; T\; ^\; \{\; 2\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; d\; ^\; \{\; 2\; \}\; \}\; \{\; \backslash tau\; ^\; \{\; 3\; \}\; \}$$

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

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

For the first term, we have

$${\int }_{{\mathcal{L}}_{\tau }}{\parallel s_{\tau }^{\star }(x_{\tau })\parallel }_2^2{p}_{\tau }(x_{\tau })\mathrm{d}{x}_{\tau }\le \frac{1}{{\tau }^2}\int {\parallel x_{\tau }-\sqrt{1-\tau }{x}_0\parallel }_2^2{p}_{X_0}{\mathrm{\mid }}_{X_{\tau }}(x_0\mathrm{\mid }{x}_{\tau })p_{\tau }(x_{\tau })\mathrm{d}{x}_0\mathrm{d}{x}_{\tau }\le \frac{d}{\tau }\mathrm{.}\backslash int\; \_\; \{\; \backslash mathcal\; \{\; L\; \}\; \_\; \{\; \backslash tau\; \}\; \}\; \backslash left\backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; p\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; \backslash tau\; \}\; \backslash leq\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; ^\; \{\; 2\; \}\; \}\; \backslash int\; \backslash left\backslash |\; x\; \_\; \{\; \backslash tau\; \}\; -\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; \}\; |\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; x\; \_\; \{\; \backslash tau\; \}\; )\; p\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; \backslash tau\; \}\; \backslash leq\; \backslash frac\; \{\; d\; \}\; \{\; \backslash tau\; \}\; .$$

For the second term, we have

Thus we have

$${\mathbb{E}}_{x_{\tau }\sim X_{\tau }}\parallel J_{\tau }(x_{\tau })s_{\tau }^{\star }(x_{\tau }){\parallel }_2^2\lesssim \frac{d{L}^2}{{\tau }^3}+\frac{d}{{\tau }^3}\lesssim \frac{d{L}^2}{{\tau }^3}\mathrm{.}\backslash mathbb\; \{\; E\; \}\; \_\; \{\; x\; \_\; \{\; \backslash tau\; \}\; \backslash sim\; X\; \_\; \{\; \backslash tau\; \}\; \}\; \backslash Big\; \backslash |\; J\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; \backslash Big\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash lesssim\; \backslash frac\; \{\; d\; L\; ^\; \{\; 2\; \}\; \}\; \{\; \backslash tau\; ^\; \{\; 3\; \}\; \}\; +\; \backslash frac\; \{\; d\; \}\; \{\; \backslash tau\; ^\; \{\; 3\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; d\; L\; ^\; \{\; 2\; \}\; \}\; \{\; \backslash tau\; ^\; \{\; 3\; \}\; \}\; .$$

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

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

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

$$\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }{\int }_{x_0}(y-x_0)p_{X_0\mathrm{\mid }{X}_{\tau }}(x_0\mathrm{\mid }\sqrt{1-\tau }y)\mathrm{d}{x}_0}\\ {\displaystyle =\frac{\mathrm{\partial }}{\mathrm{\partial }t}\frac{\int (y-x_0)\varphi (y\mathrm{\mid }{x}_0,t{I}_d)p_{X_0}(x_0)\mathrm{d}{x}_0}{\int \varphi (y\mathrm{\mid }{x}_0,t{I}_d)p_{X_0}(x_0)\mathrm{d}{x}_0}\frac{\mathrm{\partial }\frac{\tau }{1-\tau }}{\mathrm{\partial }\tau }}\end{array}\backslash begin\{array\}\; \{\; l\; \}\; \{\; \{\; \backslash displaystyle\; \{\; \backslash frac\; \{\; \backslash partial\; \}\; \{\; \backslash partial\; \backslash tau\; \}\; \backslash int\; \_\; \{\; x\; \_\; \{\; 0\; \}\; \}\; (\; y\; -\; x\; \_\; \{\; 0\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; |\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; y\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \}\; \}\; \}\; \backslash \backslash \; \{\; \{\; \backslash displaystyle\; \{\; =\; \backslash frac\; \{\; \backslash partial\; \}\; \{\; \backslash partial\; t\; \}\; \backslash frac\; \{\; \backslash int\; (\; y\; -\; x\; \_\; \{\; 0\; \}\; )\; \backslash phi\; (\; y\; |\; x\; \_\; \{\; 0\; \}\; ,\; t\; I\; \_\; \{\; d\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \}\; \{\; \backslash int\; \backslash phi\; (\; y\; |\; x\; \_\; \{\; 0\; \}\; ,\; t\; I\; \_\; \{\; d\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \}\; \backslash frac\; \{\; \backslash partial\; \backslash frac\; \{\; \backslash tau\; \}\; \{\; 1\; -\; \backslash tau\; \}\; \}\; \{\; \backslash partial\; \backslash tau\; \}\; \}\; \}\; \}\; \backslash end\{array\}$$

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

$$\begin{array}{r}{\mathbb{E}}_{Z\mathrm{\mid }y}[\mathrm{\parallel }Z{\mathrm{\parallel }}_2^{2}Z-{\mathbb{E}}_{Z\mathrm{\mid }y}Z{\mathbb{E}}_{Z\mathrm{\mid }y}\mathrm{\parallel }Z{\mathrm{\parallel }}_2^2]={\mathbb{E}}_{Z\mathrm{\mid }y}\left[(Z-{\mathbb{E}}_{Z\mathrm{\mid }y}Z)(\mathrm{\parallel }Z{\mathrm{\parallel }}_2^2-{\mathbb{E}}_{Z\mathrm{\mid }y}\mathrm{\parallel }Z{\mathrm{\parallel }}_2^2)\right]\mathrm{.}\end{array}\backslash begin\{array\}\; \{\; r\; \}\; \{\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; \backslash left[\; \backslash |\; Z\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; Z\; -\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; Z\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; \{\; \backslash |\; Z\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash right]\; =\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; \backslash left[\; \backslash left(\; Z\; -\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; Z\; \backslash right)\; \backslash left(\; \backslash |\; Z\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; -\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; \{\; \backslash |\; Z\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash right)\; \backslash right]\; .\; \}\; \backslash end\{array\}$$

According to Cauchy-Switch inequality, we have

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

$$\begin{array}{r}{\mathbb{E}}_{Z\mathrm{\mid }y}{\mid u^{\mathrm{\top }}(Z-{\mathbb{E}}_{Z\mathrm{\mid }y}Z)\mid }^2=u^{\mathrm{\top }}{\mathbb{E}}_{Z\mathrm{\mid }y}(Z-{\mathbb{E}}_{Z\mathrm{\mid }y}Z){(Z-{\mathbb{E}}_{Z\mathrm{\mid }y}Z)}^{\mathrm{\top }}u\\ =u^{\mathrm{\top }}(I_d+\tau J_{\tau }(X_{\tau }))u\le \mathrm{\parallel }{I}_d+\tau J_{\tau }(X_{\tau }){\mathrm{\parallel }}_2\mathrm{.}\end{array}\backslash begin\{array\}\; \{\; r\; \}\; \{\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; \backslash left|\; u\; ^\; \{\; \backslash top\; \}\; \backslash left(\; Z\; -\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; Z\; \backslash right)\; \backslash right|\; ^\; \{\; 2\; \}\; =\; u\; ^\; \{\; \backslash top\; \}\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; \backslash left(\; Z\; -\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; Z\; \backslash right)\; \backslash left(\; Z\; -\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; Z\; |\; y\; \}\; Z\; \backslash right)\; ^\; \{\; \backslash top\; \}\; u\; \}\; \backslash \backslash \; \{\; =\; u\; ^\; \{\; \backslash top\; \}\; \backslash left(\; I\; \_\; \{\; d\; \}\; +\; \backslash tau\; J\; \_\; \{\; \backslash tau\; \}\; (\; X\; \_\; \{\; \backslash tau\; \}\; )\; \backslash right)\; u\; \backslash le\; \backslash |\; I\; \_\; \{\; d\; \}\; +\; \backslash tau\; J\; \_\; \{\; \backslash tau\; \}\; (\; X\; \_\; \{\; \backslash tau\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; .\; \}\; \backslash end\{array\}$$

For the second factor, we have

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

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

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

$$\mathbb{E}{\parallel \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }{\int }_{x_0}(y-x_0)p_{X_0\mathrm{\mid }{X}_{\tau }}(x_0\mathrm{\mid }\sqrt{1-\tau }y)\mathrm{d}{x}_0\parallel }_2^2\le \frac{d\min \{d\log T,L\}}{\tau (1-\tau {)}^3}\mathrm{.}\backslash mathbb\; \{\; E\; \}\; \backslash left\backslash |\; \backslash frac\; \{\; \backslash partial\; \}\; \{\; \backslash partial\; \backslash tau\; \}\; \backslash int\; \_\; \{\; x\; \_\; \{\; 0\; \}\; \}\; (\; y\; -\; x\; \_\; \{\; 0\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; |\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; y\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash leq\; \backslash frac\; \{\; d\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; \backslash log\; T\; ,\; L\; \backslash \}\; \}\; \{\; \backslash tau\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 3\; \}\; \}\; .$$

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 $0 \leq n \leq N$ ,

$$\mathrm{\parallel }{y}_{{\tau }_{k,n}}(x_k)-x_{{\tau }_{k,n}}(x_k){\mathrm{\parallel }}_2^2\le {\zeta }_{k,n}(x_k)+\frac{N{\log }^{2}T}{T^2}\sum _{i=0}^{n-1}{\widehat{\tau }}_{k,i-1}{\widehat{\epsilon }}_{k,i}^2(x_k),x_k\in {\mathcal{E}}_k,\backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; -\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \backslash leq\; \backslash zeta\; \_\; \{\; k\; ,\; n\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; +\; \backslash frac\; \{\; N\; \backslash log\; ^\; \{\; 2\; \}\; T\; \}\; \{\; T\; ^\; \{\; 2\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 0\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; i\; -\; 1\; \}\; \backslash widehat\; \{\; \backslash varepsilon\; \}\; \_\; \{\; k\; ,\; i\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; ,\; x\; \_\; \{\; k\; \}\; \backslash in\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; k\; \}\; ,$$

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

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

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

$${\int }_{{\mathcal{E}}_k}{\zeta }_{k,n}(x_k)p_{{\tilde{X}}_k}(x_k)\mathrm{d}{x}_k\le {\int }_{{\mathcal{E}}_{k,n}}{\zeta }_{k,n}(x_k)p_{{\widehat{X}}_k}(x_k)\mathrm{d}{x}_k\mathrm{.}\backslash int\; \_\; \{\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; k\; \}\; \}\; \backslash zeta\; \_\; \{\; k\; ,\; n\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; p\; \_\; \{\; \backslash widetilde\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; k\; \}\; \backslash leq\; \backslash int\; \_\; \{\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash zeta\; \_\; \{\; k\; ,\; n\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; k\; \}\; .$$

According to (33), we have that

$$\begin{array}{l}{\displaystyle \mathsf{K}\mathsf{L}\left(p_{{\widehat{X}}_{k+1}\mathrm{\mid }{\widehat{X}}_k}\right(\cdot \mathrm{\mid }{x}_{{\tau }_{k,0}}\left)\mathrm{\parallel }{p}_{Y_{k+1}\mathrm{\mid }{Y}_k}\right(\cdot \mathrm{\mid }{x}_{{\tau }_{k,0}}\left)\right)=\frac{1-{\tau }_{k+1,0}}{2({\tau }_{k+1,0}-{\tau }_{k,N})}\mathrm{\parallel }{y}_{{\tau }_{k,N}}-x_{{\tau }_{k,N}}{\mathrm{\parallel }}_2^2}\\ {\displaystyle \lesssim \frac{T}{N{\widehat{\tau }}_{k,N}\log T}\mathrm{\parallel }{y}_{{\tau }_{k,N}}-x_{{\tau }_{k,N}}{\mathrm{\parallel }}_2^2\mathrm{.}}\end{array}\backslash begin\{array\}\; \{\; l\; \}\; \{\; \backslash displaystyle\; \{\; \backslash mathsf\; \{\; K\; L\; \}\; \}\; \backslash big\; (\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; k\; +\; 1\; \}\; |\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; \backslash big\; (\; \backslash cdot\; |\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; \backslash big\; )\; \backslash |\; p\; \_\; \{\; Y\; \_\; \{\; k\; +\; 1\; \}\; |\; Y\; \_\; \{\; k\; \}\; \}\; \backslash big\; (\; \backslash cdot\; |\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; \backslash big\; )\; \backslash big\; )\; =\; \backslash frac\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; +\; 1\; ,\; 0\; \}\; \}\; \{\; 2\; \backslash big\; (\; \backslash tau\; \_\; \{\; k\; +\; 1\; ,\; 0\; \}\; -\; \backslash tau\; \_\; \{\; k\; ,\; N\; \}\; \backslash big\; )\; \}\; \backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; N\; \}\; \}\; -\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; N\; \}\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \backslash \backslash \; \{\; \backslash displaystyle\; \backslash lesssim\; \backslash frac\; \{\; T\; \}\; \{\; N\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; N\; \}\; \backslash log\; T\; \}\; \backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; N\; \}\; \}\; -\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; N\; \}\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; .\; \}\; \backslash end\{array\}$$

Inserting (77) into (78), we have

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

$${\int }_{{\tau }_1}^{{\tau }_2}\frac{\mathbb{E}[\mathsf{T}\mathsf{r}({\mathrm{\Sigma }}_{\tau }^2(x_{\tau }))]}{(1-\tau {)}^2}\mathrm{d}\tau =\frac{{\tau }_2}{1-{\tau }_2}\mathbb{E}[\mathsf{T}\mathsf{r}({\mathrm{\Sigma }}_{{\tau }_2}(x_{{\tau }_2}))]-\frac{{\tau }_1}{1-{\tau }_1}\mathbb{E}[\mathsf{T}\mathsf{r}({\mathrm{\Sigma }}_{{\tau }_1}(x_{{\tau }_1}))],\backslash int\; \_\; \{\; \backslash tau\; \_\; \{\; 1\; \}\; \}\; ^\; \{\; \backslash tau\; \_\; \{\; 2\; \}\; \}\; \backslash frac\; \{\; \backslash mathbb\; \{\; E\; \}\; [\; \backslash mathsf\; \{\; T\; r\; \}\; (\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; )\; ]\; \}\; \{\; (\; 1\; -\; \backslash tau\; )\; ^\; \{\; 2\; \}\; \}\; \backslash mathrm\; \{\; d\; \}\; \backslash tau\; =\; \backslash frac\; \{\; \backslash tau\; \_\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; 2\; \}\; \}\; \backslash mathbb\; \{\; E\; \}\; [\; \backslash mathsf\; \{\; T\; r\; \}\; (\; \backslash Sigma\; \_\; \{\; \backslash tau\; \_\; \{\; 2\; \}\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; 2\; \}\; \}\; )\; )\; ]\; -\; \backslash frac\; \{\; \backslash tau\; \_\; \{\; 1\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; 1\; \}\; \}\; \backslash mathbb\; \{\; E\; \}\; [\; \backslash mathsf\; \{\; T\; r\; \}\; (\; \backslash Sigma\; \_\; \{\; \backslash tau\; \_\; \{\; 1\; \}\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; 1\; \}\; \}\; )\; )\; ]\; ,$$

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

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

$$\mathbb{E}[{\mathrm{\Sigma }}_{\tau }(x_{\tau })]=\mathbb{E}[\mathsf{C}\mathsf{o}\mathsf{v}(Z\mid \sqrt{1-\tau }{X}_0+\sqrt{\tau }Z=x_{\tau })]⪯\mathsf{C}\mathsf{o}\mathsf{v}[Z]=I_d,\backslash mathbb\; \{\; E\; \}\; [\; \backslash Sigma\; \_\; \{\; \backslash tau\; \}\; (\; x\; \_\; \{\; \backslash tau\; \}\; )\; ]\; =\; \backslash mathbb\; \{\; E\; \}\; [\; \backslash mathsf\; \{\; C\; o\; v\; \}\; (\; Z\; \backslash mid\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; X\; \_\; \{\; 0\; \}\; +\; \backslash sqrt\; \{\; \backslash tau\; \}\; Z\; =\; x\; \_\; \{\; \backslash tau\; \}\; )\; ]\; \backslash preceq\; \backslash mathsf\; \{\; C\; o\; v\; \}\; [\; Z\; ]\; =\; I\; \_\; \{\; d\; \}\; ,$$

and by using (51), we have

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

Moreover, we have

Inserting into (80), we have

Inserting (81) into (79), we have

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

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

$$\mathrm{\parallel }{s}_{\tau }^{\star }(x)-s_{\tau }^{\star }(y){\mathrm{\parallel }}_2\le \frac{C\min \{d\log T,L\}}{\tau }\mathrm{\parallel }x-y{\mathrm{\parallel }}_2,\backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; -\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; \backslash frac\; \{\; C\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; \backslash log\; T\; ,\; L\; \backslash \}\; \}\; \{\; \backslash tau\; \}\; \backslash |\; x\; -\; y\; \backslash |\; \_\; \{\; 2\; \}\; ,$$

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

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

$$\frac{{\parallel y_{{\tau }_{k,n}}-x_{{\tau }_{k,n}}\parallel }_2}{\sqrt{1-{\tau }_{k,n}}}\le \frac{{\parallel y_{{\tau }_{k,n}}^{\star }-z_{{\tau }_{k,n}}^{\star }\parallel }_2}{\sqrt{1-{\tau }_{k,n}}}+{\parallel {\xi }_{k,n}\parallel }_2\mathrm{.}\backslash frac\; \{\; \backslash left\backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; -\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \}\; \backslash leq\; \backslash frac\; \{\; \backslash left\backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash star\; \}\; -\; z\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash star\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \}\; +\; \backslash left\backslash |\; \backslash xi\; \_\; \{\; k\; ,\; n\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; .$$

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

$$\frac{{\parallel y_{{\tau }_{k,n}}-x_{{\tau }_{k,n}}\parallel }_2}{\sqrt{1-{\tau }_{k,n}}}\lesssim \frac{\min \{d\log T,L\}\log T}{T}\sum _{i=1}^{n-1}\frac{{\parallel y_{{\tau }_{k,i}}-x_{{\tau }_{k,i}}\parallel }_2}{\sqrt{1-{\tau }_{k,i}}}+{\parallel {\xi }_{k,n}(x_{{\tau }_{k,0}})\parallel }_2\mathrm{.}\backslash frac\; \{\; \backslash left\backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; -\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; \backslash log\; T\; ,\; L\; \backslash \}\; \backslash log\; T\; \}\; \{\; T\; \}\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash frac\; \{\; \backslash left\backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; -\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; \}\; +\; \backslash left\backslash |\; \backslash xi\; \_\; \{\; k\; ,\; n\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; .$$

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

$$\frac{{\parallel y_{{\tau }_{k,n}}-x_{{\tau }_{k,n}}\parallel }_2}{\sqrt{1-{\tau }_{k,n}}}\lesssim \frac{\min \{d\log T,L\}\log T}{T}\sum _{i=1}^{n-1}{\parallel {\xi }_{k,i}(x_{{\tau }_{k,0}})\parallel }_2+{\parallel {\xi }_{k,n}(x_{{\tau }_{k,0}})\parallel }_2\mathrm{.}\backslash frac\; \{\; \backslash left\backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; -\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; \backslash log\; T\; ,\; L\; \backslash \}\; \backslash log\; T\; \}\; \{\; T\; \}\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash left\backslash |\; \backslash xi\; \_\; \{\; k\; ,\; i\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; +\; \backslash left\backslash |\; \backslash xi\; \_\; \{\; k\; ,\; n\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; .$$

Thus

$$\frac{{\parallel y_{{\tau }_{k,n}}-x_{{\tau }_{k,n}}\parallel }_2^2}{1-{\tau }_{k,n}}\lesssim \frac{N\min \{d\log T,L{\}}^2{\log }^{2}T}{T^2}\sum _{i=1}^{n-1}{\parallel {\xi }_{k,i}(x_{{\tau }_{k,0}})\parallel }_2^2+{\parallel {\xi }_{k,n}(x_{{\tau }_{k,0}})\parallel }_2^2\mathrm{.}\backslash frac\; \{\; \backslash left\backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; -\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; N\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; \backslash log\; T\; ,\; L\; \backslash \}\; ^\; \{\; 2\; \}\; \backslash log\; ^\; \{\; 2\; \}\; T\; \}\; \{\; T\; ^\; \{\; 2\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash left\backslash |\; \backslash xi\; \_\; \{\; k\; ,\; i\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; +\; \backslash left\backslash |\; \backslash xi\; \_\; \{\; k\; ,\; n\; \}\; (\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; )\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; .$$

Define

$$\frac{{\zeta }_{k,n}}{1-{\tau }_{k,n}}=\frac{N\min \{d\log T,L{\}}^2{\log }^{2}T}{T^2}\sum _{i=1}^{n-1}\frac{\mathrm{\parallel }{x}_{{\tau }_{k,i}}-z_{{\tau }_{k,i}}^{\star }{\mathrm{\parallel }}_2^2}{1-{\tau }_{k,i}}+\frac{\mathrm{\parallel }{x}_{{\tau }_{k,n}}-z_{{\tau }_{k,n}}^{\star }{\mathrm{\parallel }}_2^2}{1-{\tau }_{k,n}}\mathrm{.}\backslash frac\; \{\; \backslash zeta\; \_\; \{\; k\; ,\; n\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; =\; \backslash frac\; \{\; N\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; \backslash log\; T\; ,\; L\; \backslash \}\; ^\; \{\; 2\; \}\; \backslash log\; ^\; \{\; 2\; \}\; T\; \}\; \{\; T\; ^\; \{\; 2\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash frac\; \{\; \backslash |\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; -\; z\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; ^\; \{\; \backslash star\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; +\; \backslash frac\; \{\; \backslash |\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; -\; z\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash star\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; .$$

We have

According to Lemma 8, we have

$$\frac{\parallel y_{{\tau }_{k,n}}^{\star }-y_{{\tau }_{k,n}}{\parallel }_2^2}{1-{\tau }_{k,n}}\lesssim \frac{N{\log }^{2}T}{T^2}\sum _{i=0}^{n-1}\frac{{\tau }_{k,i}{\tilde{\epsilon }}_{k,i}^2}{1-{\tau }_{k,i}}\mathrm{.}\backslash frac\; \{\; \backslash lVert\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash star\; \}\; -\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash rVert\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; N\; \backslash log\; ^\; \{\; 2\; \}\; T\; \}\; \{\; T\; ^\; \{\; 2\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 0\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash frac\; \{\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \backslash widetilde\; \{\; \backslash varepsilon\; \}\; \_\; \{\; k\; ,\; i\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; .$$

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

$$\frac{N\min \{d\log T,L{\}}^2{\log }^{2}T}{T^2}\sum _{i=1}^n\frac{\mathrm{\parallel }{y}_{{\tau }_{k,i}}^{\star }-y_{{\tau }_{k,i}}{\mathrm{\parallel }}_2^2}{1-{\tau }_{k,i}}+\frac{\mathrm{\parallel }{y}_{{\tau }_{k,n}}^{\star }-y_{{\tau }_{k,n}}{\mathrm{\parallel }}_2^2}{1-{\tau }_{k,n}}\lesssim \frac{N{\log }^{2}T}{T^2}\sum _{i=0}^{n-1}\frac{{\tau }_{k,i}{\tilde{\epsilon }}_{k,i}^2}{1-{\tau }_{k,i}}\mathrm{.}\backslash frac\; \{\; N\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; \backslash log\; T\; ,\; L\; \backslash \}\; ^\; \{\; 2\; \}\; \backslash log\; ^\; \{\; 2\; \}\; T\; \}\; \{\; T\; ^\; \{\; 2\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; n\; \}\; \backslash frac\; \{\; \backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; ^\; \{\; \backslash star\; \}\; -\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; +\; \backslash frac\; \{\; \backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; ^\; \{\; \backslash star\; \}\; -\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; N\; \backslash log\; ^\; \{\; 2\; \}\; T\; \}\; \{\; T\; ^\; \{\; 2\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 0\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash frac\; \{\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \backslash tilde\; \{\; \backslash varepsilon\; \}\; \_\; \{\; k\; ,\; i\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; .$$

Thus we have

$$\frac{\mathrm{\parallel }{y}_{{\tau }_{k,n}}(x_k)-x_{{\tau }_{k,n}}(x_k){\mathrm{\parallel }}_2^2}{1-{\tau }_{k,n}}\le \frac{{\zeta }_{k,n}(x_k)}{1-{\tau }_{k,n}}+\frac{N{\log }^{2}T}{T^2}\sum _{i=1}^n\frac{{\tau }_{k,i}{\tilde{\epsilon }}_{k,i}^2(x_k)}{1-{\tau }_{k,i}},\backslash frac\; \{\; \backslash |\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; -\; x\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \backslash |\; \_\; \{\; 2\; \}\; ^\; \{\; 2\; \}\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash leq\; \backslash frac\; \{\; \backslash zeta\; \_\; \{\; k\; ,\; n\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; +\; \backslash frac\; \{\; N\; \backslash log\; ^\; \{\; 2\; \}\; T\; \}\; \{\; T\; ^\; \{\; 2\; \}\; \}\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; n\; \}\; \backslash frac\; \{\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \backslash widetilde\; \backslash varepsilon\; \_\; \{\; k\; ,\; i\; \}\; ^\; \{\; 2\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \}\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; ,$$

which establishes (75). Furthermore, we have

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

and thus

$${\int }_{{\mathcal{E}}_{k,n}}{\zeta }_{k,n}(x_k)p_{{\widehat{X}}_k}(x_k)\mathrm{d}{x}_k\backslash int\; \_\; \{\; \backslash mathscr\; \{\; E\; \}\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash zeta\; \_\; \{\; k\; ,\; n\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; (\; x\; \_\; \{\; k\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; k\; \}$$

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

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

$$\mathrm{\parallel }x-y{\mathrm{\parallel }}_2\le c\sqrt{\frac{\tau }{d\log T}}\backslash |\; x\; -\; y\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; c\; \backslash sqrt\; \{\; \backslash frac\; \{\; \backslash tau\; \}\; \{\; d\; \backslash log\; T\; \}\; \}$$

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

$$p_{X_{\tau }}(x)-p_{X_{\tau }}(y)\le 6\mathrm{\parallel }y-x{\mathrm{\parallel }}_2\sqrt{\frac{(\theta +c_0)d\log T}{\tau }}{p}_{X_{\tau }}(x)\mathrm{.}p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; )\; -\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; y\; )\; \backslash leq\; 6\; \backslash |\; y\; -\; x\; \backslash |\; \_\; \{\; 2\; \}\; \backslash sqrt\; \{\; \backslash frac\; \{\; (\; \backslash theta\; +\; c\; \_\; \{\; 0\; \}\; )\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; \}\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; )\; .$$

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

$$\frac{1}{2}\le 1-6{\parallel x-y\parallel }_2\sqrt{\frac{(\theta +c_0)d\log T}{\tau }}\le \frac{p_{X_{\tau }}(x)}{p_{X_{\tau }}(y)}\le \frac{1}{1-6{\parallel x-y\parallel }_2\sqrt{\frac{(\theta +c_0)d\log T}{\tau }}}\le 2.\backslash frac\; \{\; 1\; \}\; \{\; 2\; \}\; \backslash leq\; 1\; -\; 6\; \backslash left\backslash |\; x\; -\; y\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \backslash sqrt\; \{\; \backslash frac\; \{\; (\; \backslash theta\; +\; c\; \_\; \{\; 0\; \}\; )\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; \}\; \backslash leq\; \backslash frac\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; )\; \}\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; y\; )\; \}\; \backslash leq\; \backslash frac\; \{\; 1\; \}\; \{\; 1\; -\; 6\; \backslash left\backslash |\; x\; -\; y\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \backslash sqrt\; \{\; \backslash frac\; \{\; (\; \backslash theta\; +\; c\; \_\; \{\; 0\; \}\; )\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; \}\; \}\; \backslash leq\; 2\; .$$

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

$$\mathrm{\parallel }{s}_{\tau }^{\star }(x)-s_{\tau }^{\star }(y){\mathrm{\parallel }}_2\lesssim \frac{d\log T}{\tau }\mathrm{\parallel }x-y\mathrm{\parallel }\mathrm{.}\backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; -\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; \backslash |\; \_\; \{\; 2\; \}\; \backslash lesssim\; \backslash frac\; \{\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; \backslash |\; x\; -\; y\; \backslash |\; .$$

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

$$\begin{array}{l}{s}_{\tau }^{\star }(y)-s_{\tau }^{\star }(x)={\displaystyle \frac{1}{\tau }(x-y)-\frac{1}{\tau }{\int }_{x_0}(x-\sqrt{1-\tau }{x}_0)p_{X_0\mathrm{\mid }{X}_{\tau }}(x_0\mathrm{\mid }y)}\\ \phantom{s_{\tau }^{\star }(y)-s_{\tau }^{\star }(x)=}+(x-\sqrt{1-\tau }{x}_0)p_{X_0\mathrm{\mid }{X}_{\tau }}(x_0\mathrm{\mid }x)\mathrm{d}{x}_0\mathrm{.}\end{array}\backslash begin\{array\}\; \{\; l\; \}\; \{\; \{\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; -\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; =\; \backslash displaystyle\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; \}\; (\; x\; -\; y\; )\; -\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; \}\; \backslash int\; \_\; \{\; x\; \_\; \{\; 0\; \}\; \}\; (\; x\; -\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; |\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; y\; )\; \}\; \}\; \backslash \backslash \; \{\; \{\; \backslash phantom\; \{\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; -\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; =\; \}\; +\; (\; x\; -\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; |\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; x\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; .\; \}\; \}\; \backslash end\{array\}$$

Thus we have

$$\begin{array}{l}{\displaystyle {\parallel s_{\tau }^{\star }(y)-s_{\tau }^{\star }(x)\parallel }_2}\\ {\displaystyle \le \frac{1}{\tau }{\parallel x-y\parallel }_2+\frac{1}{\tau }{\parallel {\int }_{x_0}(x-\sqrt{1-\tau }{x}_0)\frac{p_{X_{\tau }\mathrm{\mid }{X}_0}(y\mathrm{\mid }{x}_0)-p_{X_{\tau }\mathrm{\mid }{X}_0}(x\mathrm{\mid }{x}_0)}{p_{X_{\tau }}(y)}{p}_{X_0}(x_0)\mathrm{d}{x}_0\parallel }_2}\\ {\displaystyle \phantom{\frac{1}{\tau }}+\frac{1}{\tau }{\parallel (\frac{p_{X_{\tau }}(x)}{p_{X_{\tau }}(y)}-1){\int }_{x_0}(x-\sqrt{1-\tau }{x}_0)p_{X_0\mathrm{\mid }{X}_{\tau }}(x_0\mathrm{\mid }x)\mathrm{d}{x}_0\parallel }_2\mathrm{.}}\end{array}\backslash begin\{array\}\; \{\; l\; \}\; \{\; \backslash displaystyle\; \backslash left\backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; -\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \}\; \backslash \backslash \; \{\; \backslash displaystyle\; \backslash leq\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; \}\; \backslash left\backslash |\; x\; -\; y\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; +\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; \}\; \backslash left\backslash |\; \backslash int\; \_\; \{\; x\; \_\; \{\; 0\; \}\; \}\; (\; x\; -\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; )\; \backslash frac\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; |\; X\; \_\; \{\; 0\; \}\; \}\; (\; y\; |\; x\; \_\; \{\; 0\; \}\; )\; -\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; |\; X\; \_\; \{\; 0\; \}\; \}\; (\; x\; |\; x\; \_\; \{\; 0\; \}\; )\; \}\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; y\; )\; \}\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \}\; \backslash \backslash \; \{\; \backslash displaystyle\; \backslash phantom\; \{\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; \}\; \}\; +\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash tau\; \}\; \backslash left\backslash |\; \backslash left(\; \backslash frac\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; )\; \}\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; y\; )\; \}\; -\; 1\; \backslash right)\; \backslash int\; \_\; \{\; x\; \_\; \{\; 0\; \}\; \}\; (\; x\; -\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; |\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; x\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; .\; \}\; \backslash 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

$${\int }_{\mathcal{R}}\mathrm{\mid }\mathrm{\mid }x-\sqrt{1-\tau }{x}_0\mathrm{\mid }{\mathrm{\mid }}_2\frac{\mathrm{\parallel }x-y\mathrm{\parallel }\mathrm{\parallel }x+y-2\sqrt{1-\tau }{x}_0{\mathrm{\parallel }}_2}{2\tau }{p}_{X_0\mathrm{\mid }{X}_{\tau }}(x_0\mathrm{\mid }x)\mathrm{d}{x}_0\frac{p_{X_{\tau }}(x)}{p_{X_{\tau }}(y)},\backslash int\; \_\; \{\; \backslash mathcal\; \{\; R\; \}\; \}\; |\; |\; x\; -\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; |\; |\; \_\; \{\; 2\; \}\; \backslash frac\; \{\; \backslash |\; x\; -\; y\; \backslash |\; \backslash |\; x\; +\; y\; -\; 2\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; \backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; 2\; \backslash tau\; \}\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; |\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; x\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \backslash frac\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; )\; \}\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; y\; )\; \}\; ,$$

and the second term is smaller than

$$\begin{array}{l}{\displaystyle {\int }_{{\mathbb{R}}^c}\mathrm{\parallel }y-\sqrt{1-\tau }{x}_0{\mathrm{\parallel }}_2\frac{\mathrm{\parallel }y-x{\mathrm{\parallel }}_2\mathrm{\parallel }x+y-2\sqrt{1-\tau }{x}_0{\mathrm{\parallel }}_2}{2\tau }{p}_{X_0\mathrm{\mid }{X}_{\tau }}(x_0\mathrm{\mid }y)\mathrm{d}{x}_0}\\ {\displaystyle +{\parallel (x-y)(1-\frac{p_{X_{\tau }}(x)}{p_{X_{\tau }}(y)})\parallel }_2\mathrm{.}}\end{array}\backslash begin\{array\}\; \{\; l\; l\; \}\; \{\; \backslash displaystyle\; \backslash int\; \_\; \{\; \backslash mathbb\; \{\; R\; \}\; ^\; \{\; c\; \}\; \}\; \backslash |\; y\; -\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; \backslash |\; \_\; \{\; 2\; \}\; \backslash frac\; \{\; \backslash |\; y\; -\; x\; \backslash |\; \_\; \{\; 2\; \}\; \backslash |\; x\; +\; y\; -\; 2\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; \backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; 2\; \backslash tau\; \}\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; |\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; |\; y\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \}\; \backslash \backslash \; \{\; \backslash displaystyle\; +\; \backslash left\backslash |\; (\; x\; -\; y\; )\; \backslash left(\; 1\; -\; \backslash frac\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; )\; \}\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; y\; )\; \}\; \backslash right)\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; .\; \}\; \backslash end\{array\}$$

Combining these, we have

for

$$\mathrm{\parallel }x-y{\mathrm{\parallel }}_2\le c\sqrt{\frac{\tau }{d\log T}},\backslash |\; x\; -\; y\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; c\; \backslash sqrt\; \{\; \backslash frac\; \{\; \backslash tau\; \}\; \{\; d\; \backslash log\; T\; \}\; \}\; ,$$

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

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

$$\mathrm{\parallel }{s}_{\tau }^{\star }(y)-s_{\tau }^{\star }(x){\mathrm{\parallel }}_2\lesssim \frac{d\log T}{\tau }\mathrm{\parallel }x-y{\mathrm{\parallel }}_2\mathrm{.}\backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; -\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; \backslash |\; \_\; \{\; 2\; \}\; \backslash lesssim\; \backslash frac\; \{\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; \backslash |\; x\; -\; y\; \backslash |\; \_\; \{\; 2\; \}\; .$$

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

$$\mathrm{\parallel }{s}_{\tau }^{\star }(y)-s_{\tau }^{\star }(x){\mathrm{\parallel }}_2\le \mathrm{\parallel }{s}_{\tau }^{\star }(y){\mathrm{\parallel }}_2+\mathrm{\parallel }{s}_{\tau }^{\star }(x){\mathrm{\parallel }}_2\le 10\sqrt{\frac{(\theta +c_0)d\log T}{\tau }}\lesssim \frac{d\log T}{\tau }\mathrm{\parallel }x-y{\mathrm{\parallel }}_2\mathrm{.}\backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; -\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; \backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; \backslash |\; \_\; \{\; 2\; \}\; +\; \backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; 1\; 0\; \backslash sqrt\; \{\; \backslash frac\; \{\; (\; \backslash theta\; +\; c\; \_\; \{\; 0\; \}\; )\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; \backslash |\; x\; -\; y\; \backslash |\; \_\; \{\; 2\; \}\; .$$

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

$$\mathrm{\parallel }{s}_{\tau }^{\star }(y)-s_{\tau }^{\star }(x){\mathrm{\parallel }}_2\le \frac{L}{\tau }\mathrm{\parallel }x-y{\mathrm{\parallel }}_2\mathrm{.}\backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; -\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; \backslash frac\; \{\; L\; \}\; \{\; \backslash tau\; \}\; \backslash |\; x\; -\; y\; \backslash |\; \_\; \{\; 2\; \}\; .$$

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

$$\mathrm{\parallel }{s}_{\tau }^{\star }(y)-s_{\tau }^{\star }(x){\mathrm{\parallel }}_2\le 2\sqrt{\frac{25(\theta +c_0)d\log T}{\tau }}\le \frac{L}{\tau }\mathrm{\parallel }x-y{\mathrm{\parallel }}_2,\backslash |\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; y\; )\; -\; s\; \_\; \{\; \backslash tau\; \}\; ^\; \{\; \backslash star\; \}\; (\; x\; )\; \backslash |\; \_\; \{\; 2\; \}\; \backslash leq\; 2\; \backslash sqrt\; \{\; \backslash frac\; \{\; 2\; 5\; (\; \backslash theta\; +\; c\; \_\; \{\; 0\; \}\; )\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; \}\; \backslash leq\; \backslash frac\; \{\; L\; \}\; \{\; \backslash tau\; \}\; \backslash |\; x\; -\; y\; \backslash |\; \_\; \{\; 2\; \}\; ,$$

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

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

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

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

and

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

Furthermore, we introduce another auxiliary set

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

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

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

Thus we have

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

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

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

where the last inequality comes from the fact that

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

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

$$p_{X_{\tau }}(y)\ge (1-6\mathrm{\parallel }y-x\mathrm{\parallel }\sqrt{\frac{(\theta +c_0)d\log T}{\tau }})p_{X_{\tau }}(x),\mathrm{i}\mathrm{f}\mathrm{\parallel }x-y{\mathrm{\parallel }}^2\le \frac{c\tau }{d\log T},p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; y\; )\; \backslash geq\; \backslash left(\; 1\; -\; 6\; \backslash |\; y\; -\; x\; \backslash |\; \backslash sqrt\; \{\; \backslash frac\; \{\; (\; \backslash theta\; +\; c\; \_\; \{\; 0\; \}\; )\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; \}\; \backslash right)\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; )\; ,\; \backslash quad\; \backslash mathrm\; \{\; i\; f\; \}\; \backslash \; \backslash |\; x\; -\; y\; \backslash |\; ^\; \{\; 2\; \}\; \backslash leq\; \backslash frac\; \{\; c\; \backslash tau\; \}\; \{\; d\; \backslash log\; T\; \}\; ,$$

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

According to (75), for

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

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

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

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

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

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

Moreover, we have

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

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

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

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

Then

$$\mathsf{K}\mathsf{L}(\tilde{f}\mathrm{\parallel }\tilde{g})\le \mathsf{K}\mathsf{L}(f\mathrm{\parallel }g)\mathrm{.}\{\; \backslash mathsf\; \{\; K\; L\; \}\; \}\; (\; \{\; \backslash tilde\; \{\; f\; \}\; \}\; \backslash |\; \{\; \backslash tilde\; \{\; g\; \}\; \}\; )\; \backslash leq\; \{\; \backslash mathsf\; \{\; K\; L\; \}\; \}\; (\; f\; \backslash |\; g\; )\; .$$

Towards this, according to definitions, we have

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

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

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

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

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

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

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:

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

We construct the typical set

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

$$P_{Y_{k+1}\mathrm{\mid }{Y}_k}(y\mathrm{\mid }{y}_k)=\varphi (y\mathrm{\mid }{y}_{M,{\tau }_{k,N}}(y_k),{\sigma }_k^2)\mathrm{.}P\; \_\; \{\; Y\; \_\; \{\; k\; +\; 1\; \}\; |\; Y\; \_\; \{\; k\; \}\; \}\; (\; y\; |\; y\; \_\; \{\; k\; \}\; )\; =\; \backslash phi\; \backslash left(\; y\; |\; y\; \_\; \{\; M\; ,\; \backslash tau\; \_\; \{\; k\; ,\; N\; \}\; \}\; (\; y\; \_\; \{\; k\; \}\; )\; ,\; \backslash sigma\; \_\; \{\; k\; \}\; ^\; \{\; 2\; \}\; \backslash right)\; .$$

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

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

$$+\sqrt{\frac{1}{2}\mathsf{K}\mathsf{L}\left(p_{{\widehat{X}}_0}\mathrm{\parallel }{p}_{Y_0}\right)+\frac{1}{2}\sum _{k=0}^{K-1}{\mathbb{E}}_{x_k\sim p_{{\tilde{X}}_k}}\left[\mathsf{K}\mathsf{L}\right(p_{{\widehat{X}}_{k+1}\mathrm{\mid }{\widehat{X}}_k}(\cdot \mathrm{\mid }{x}_k)\mathrm{\parallel }{p}_{Y_{k+1}\mathrm{\mid }{Y}_k}(\cdot \mathrm{\mid }{x}_k)\left)\right]}\mathrm{.}+\; \backslash sqrt\; \{\; \backslash frac\; \{\; 1\; \}\; \{\; 2\; \}\; \backslash mathsf\; \{\; K\; L\; \}\; \backslash big\; (\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; 0\; \}\; \}\; \backslash |\; p\; \_\; \{\; Y\; \_\; \{\; 0\; \}\; \}\; \backslash big\; )\; +\; \backslash frac\; \{\; 1\; \}\; \{\; 2\; \}\; \backslash sum\; \_\; \{\; k\; =\; 0\; \}\; ^\; \{\; K\; -\; 1\; \}\; \backslash mathbb\; \{\; E\; \}\; \_\; \{\; x\; \_\; \{\; k\; \}\; \backslash sim\; p\; \_\; \{\; \backslash widetilde\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; \}\; \backslash left[\; \backslash mathsf\; \{\; K\; L\; \}\; \backslash big\; (\; p\; \_\; \{\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; k\; +\; 1\; \}\; |\; \backslash widehat\; \{\; X\; \}\; \_\; \{\; k\; \}\; \}\; \backslash left(\; \backslash cdot\; |\; x\; \_\; \{\; k\; \}\; \backslash right)\; \backslash |\; p\; \_\; \{\; Y\; \_\; \{\; k\; +\; 1\; \}\; |\; Y\; \_\; \{\; k\; \}\; \}\; \backslash left(\; \backslash cdot\; |\; x\; \_\; \{\; k\; \}\; \backslash right)\; \backslash big\; )\; \backslash right]\; \}\; .$$

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

$$\frac{{\parallel y_{m,{\tau }_{k,n}}-y_{{\tau }_{k,n}}\parallel }_2}{\sqrt{1-{\tau }_{k,n}}}\lesssim \frac{\min \{d\log T,L\}\log T}{T}\sum _{i=1}^{n-1}\frac{{\parallel y_{m-1,{\tau }_{k,i}}-y_{{\tau }_{k,i}}\parallel }_2}{\sqrt{1-{\tau }_{k,i}}},y_{{\tau }_{k,0}}\in {\mathcal{E}}_k\mathrm{.}\backslash frac\; \{\; \backslash left\backslash |\; y\; \_\; \{\; m\; ,\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; -\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \}\; \backslash lesssim\; \backslash frac\; \{\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; \backslash log\; T\; ,\; L\; \backslash \}\; \backslash log\; T\; \}\; \{\; T\; \}\; \backslash sum\; \_\; \{\; i\; =\; 1\; \}\; ^\; \{\; n\; -\; 1\; \}\; \backslash frac\; \{\; \backslash left\backslash |\; y\; \_\; \{\; m\; -\; 1\; ,\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; -\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; \backslash right\backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; i\; \}\; \}\; \}\; ,\; \backslash qquad\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; 0\; \}\; \}\; \backslash in\; \backslash mathcal\; \{\; E\; \}\; \_\; \{\; k\; \}\; .$$

Applying the above relation recursively gives

$$\underset{n}{\max }\frac{\parallel y_{M,{\tau }_{k,n}}-y_{{\tau }_{k,n}}{\parallel }_2}{\sqrt{1-{\tau }_{k,n}}}\le (\frac{N\min \{d\log T,L\}\log T}{T}{)}^M\underset{n}{\max }\frac{\parallel y_{0,{\tau }_{k,n}}-y_{{\tau }_{k,n}}{\parallel }_2}{\sqrt{1-{\tau }_{k,n}}}\le \frac{1}{\mathsf{p}\mathsf{o}\mathsf{l}\mathsf{y}(T)},\backslash operatorname*\; \{\; m\; a\; x\; \}\; \_\; \{\; n\; \}\; \backslash frac\; \{\; \backslash big\; \backslash |\; y\; \_\; \{\; M\; ,\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; -\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash big\; \backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \}\; \backslash leq\; \backslash bigg\; (\; \backslash frac\; \{\; N\; \backslash operatorname*\; \{\; m\; i\; n\; \}\; \backslash \{\; d\; \backslash log\; T\; ,\; L\; \backslash \}\; \backslash log\; T\; \}\; \{\; T\; \}\; \backslash bigg\; )\; ^\; \{\; M\; \}\; \backslash operatorname*\; \{\; m\; a\; x\; \}\; \_\; \{\; n\; \}\; \backslash frac\; \{\; \backslash big\; \backslash |\; y\; \_\; \{\; 0\; ,\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; -\; y\; \_\; \{\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \backslash big\; \backslash |\; \_\; \{\; 2\; \}\; \}\; \{\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \_\; \{\; k\; ,\; n\; \}\; \}\; \}\; \backslash leq\; \backslash frac\; \{\; 1\; \}\; \{\; \backslash mathsf\; \{\; p\; o\; l\; y\; \}\; (\; T\; )\; \}\; ,$$

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

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

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

$$\frac{1-{\widehat{\tau }}_{k,N}}{1-{\widehat{\tau }}_{k,-1}}=\frac{{\widehat{\alpha }}_{T-\frac{kN}{2}-N}}{{\widehat{\alpha }}_{T-\frac{kN}{2}+1}}\le {(1+\frac{c_1\log T}{T})}^{N+1}\le \exp \left(\frac{c_1(N+1)\log T}{T}\right)\le \mathrm{e},\backslash frac\; \{\; 1\; -\; \backslash widehat\; \backslash tau\; \_\; \{\; k\; ,\; N\; \}\; \}\; \{\; 1\; -\; \backslash widehat\; \backslash tau\; \_\; \{\; k\; ,\; -\; 1\; \}\; \}\; =\; \backslash frac\; \{\; \backslash widehat\; \backslash alpha\; \_\; \{\; T\; -\; \backslash frac\; \{\; k\; N\; \}\; \{\; 2\; \}\; -\; N\; \}\; \}\; \{\; \backslash widehat\; \backslash alpha\; \_\; \{\; T\; -\; \backslash frac\; \{\; k\; N\; \}\; \{\; 2\; \}\; +\; 1\; \}\; \}\; \backslash leq\; \backslash left(\; 1\; +\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash log\; T\; \}\; \{\; T\; \}\; \backslash right)\; ^\; \{\; N\; +\; 1\; \}\; \backslash leq\; \backslash exp\; \backslash left(\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; (\; N\; +\; 1\; )\; \backslash log\; T\; \}\; \{\; T\; \}\; \backslash right)\; \backslash leq\; \backslash mathrm\; \{\; e\; \}\; ,$$

and

$$\frac{{\widehat{\tau }}_{k,-1}}{{\widehat{\tau }}_{k,N}}=\frac{1-{\widehat{\alpha }}_{T-\frac{kN}{2}+1}}{1-{\widehat{\alpha }}_{T-\frac{kN}{2}-N}}\le {(1-\frac{c_1\log T}{T})}^{-N-1}\le \exp \left(\frac{2c_1(N+1)\log T}{T}\right)\le \mathrm{e},\backslash frac\; \{\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; -\; 1\; \}\; \}\; \{\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; N\; \}\; \}\; =\; \backslash frac\; \{\; 1\; -\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; -\; \backslash frac\; \{\; k\; N\; \}\; \{\; 2\; \}\; +\; 1\; \}\; \}\; \{\; 1\; -\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; -\; \backslash frac\; \{\; k\; N\; \}\; \{\; 2\; \}\; -\; N\; \}\; \}\; \backslash leq\; \backslash left(\; 1\; -\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash log\; T\; \}\; \{\; T\; \}\; \backslash right)\; ^\; \{\; -\; N\; -\; 1\; \}\; \backslash leq\; \backslash exp\; \backslash left(\; \backslash frac\; \{\; 2\; c\; \_\; \{\; 1\; \}\; (\; N\; +\; 1\; )\; \backslash log\; T\; \}\; \{\; T\; \}\; \backslash right)\; \backslash leq\; \backslash mathrm\; \{\; e\; \}\; ,$$

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

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

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 $x , y \in \mathcal { E } _ { h + 1 }$ ,

F.3 Proof of Lemma 13

Notice that

$$\begin{array}{l}{p}_{X_{\tau }}(y)={\displaystyle \int p_{X_{\tau }\mathrm{\mid }{X}_0}(y\mathrm{\mid }{x}_0)p_{X_0}(x_0)\mathrm{d}{x}_0}\\ {\displaystyle =\int p_{X_{\tau }\mathrm{\mid }{X}_0}(x\mid x_0)p_{X_0}(x_0)\exp \left(\frac{(x-y{)}^{\mathrm{\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 }\mathrm{\mid }{X}_0}(y\mid x_0)p_{X_0}(x_0)\exp \left(\frac{(y-x{)}^{\mathrm{\top }}(x+y-2\sqrt{1-\tau }{x}_0)}{2\tau }\right)\mathrm{d}{x}_0\mathrm{.}}\end{array}\backslash begin\{array\}\; \{\; l\; \}\; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; y\; )\; =\; \backslash displaystyle\; \backslash int\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; |\; X\; \_\; \{\; 0\; \}\; \}\; (\; y\; |\; x\; \_\; \{\; 0\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; )\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; \}\; \backslash \backslash \; \{\; \backslash displaystyle\; =\; \backslash int\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; |\; X\; \_\; \{\; 0\; \}\; \}\; (\; x\; \backslash mid\; x\; \_\; \{\; 0\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; )\; \backslash exp\; \backslash left(\; \backslash frac\; \{\; (\; x\; -\; y\; )\; ^\; \{\; \backslash top\; \}\; (\; x\; +\; y\; -\; 2\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; )\; \}\; \{\; 2\; \backslash tau\; \}\; \backslash right)\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; ,\; \}\; \backslash \backslash \; \{\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; )\; =\; \backslash displaystyle\; \backslash int\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; |\; X\; \_\; \{\; 0\; \}\; \}\; (\; y\; \backslash mid\; x\; \_\; \{\; 0\; \}\; )\; p\; \_\; \{\; X\; \_\; \{\; 0\; \}\; \}\; (\; x\; \_\; \{\; 0\; \}\; )\; \backslash exp\; \backslash left(\; \backslash frac\; \{\; (\; y\; -\; x\; )\; ^\; \{\; \backslash top\; \}\; (\; x\; +\; y\; -\; 2\; \backslash sqrt\; \{\; 1\; -\; \backslash tau\; \}\; x\; \_\; \{\; 0\; \}\; )\; \}\; \{\; 2\; \backslash tau\; \}\; \backslash right)\; \backslash mathrm\; \{\; d\; \}\; x\; \_\; \{\; 0\; \}\; .\; \}\; \backslash end\{array\}$$

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

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

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

$$p_{X_{\tau }}(y)-p_{X_{\tau }}(x)\le 6\mathrm{\parallel }y-x\mathrm{\parallel }\sqrt{\frac{(\theta +c_0)d\log T}{\tau }}{p}_{X_{\tau }}(y)\mathrm{.}p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; y\; )\; -\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; x\; )\; \backslash leq\; 6\; \backslash |\; y\; -\; x\; \backslash |\; \backslash sqrt\; \{\; \backslash frac\; \{\; (\; \backslash theta\; +\; c\; \_\; \{\; 0\; \}\; )\; d\; \backslash log\; T\; \}\; \{\; \backslash tau\; \}\; \}\; p\; \_\; \{\; X\; \_\; \{\; \backslash tau\; \}\; \}\; (\; y\; )\; .$$

Thus we complete the proof.

F.4 Proof of Lemma 7

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

$$1-{\tau }_{0,0}={\bar{\alpha }}_{T+1}\le {\widehat{\alpha }}_T\le 2{\widehat{\alpha }}_{T+1},1\; -\; \backslash tau\; \_\; \{\; 0\; ,\; 0\; \}\; =\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; T\; +\; 1\; \}\; \backslash leq\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; \}\; \backslash leq\; 2\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; +\; 1\; \}\; ,$$

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

$${\widehat{\alpha }}_{t-1}={\widehat{\alpha }}_t(1+\frac{c_1(1-{\widehat{\alpha }}_t)\log T}{T})>{\widehat{\alpha }}_t(1+\frac{c_1\log T}{2T}),\backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; t\; -\; 1\; \}\; =\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; t\; \}\; \backslash bigg\; (\; 1\; +\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash big\; (\; 1\; -\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; t\; \}\; \backslash big\; )\; \backslash log\; T\; \}\; \{\; T\; \}\; \backslash bigg\; )\; >\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; t\; \}\; \backslash bigg\; (\; 1\; +\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash log\; T\; \}\; \{\; 2\; T\; \}\; \backslash bigg\; )\; ,$$

and then

$${\widehat{\alpha }}_{T_0}>{\widehat{\alpha }}_{T+1}(1+\frac{c_1\log T}{2T}{)}^{T-T_0+1}>\frac{1}{2}\mathrm{.}\backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; \_\; \{\; 0\; \}\; \}\; >\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; +\; 1\; \}\; \backslash Bigl\; (\; 1\; +\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash log\; T\; \}\; \{\; 2\; T\; \}\; \backslash Bigr\; )\; ^\; \{\; T\; -\; T\; \_\; \{\; 0\; \}\; +\; 1\; \}\; >\; \backslash frac\; \{\; 1\; \}\; \{\; 2\; \}\; .$$

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

$${\tau }_{K,0}=1-{\bar{\alpha }}_1\le 1-{\widehat{\alpha }}_1\le (1-{\widehat{\alpha }}_{T_0}){(1-\frac{c_1\log T}{2T})}^{T_0-1}\le \frac{1}{T^{c_0}},\backslash tau\; \_\; \{\; K\; ,\; 0\; \}\; =\; 1\; -\; \backslash overline\; \{\; \{\; \backslash alpha\; \}\; \}\; \_\; \{\; 1\; \}\; \backslash leq\; 1\; -\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; 1\; \}\; \backslash leq\; (\; 1\; -\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; \_\; \{\; 0\; \}\; \}\; )\; \backslash left(\; 1\; -\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash log\; T\; \}\; \{\; 2\; T\; \}\; \backslash right)\; ^\; \{\; T\; \_\; \{\; 0\; \}\; -\; 1\; \}\; \backslash leq\; \backslash frac\; \{\; 1\; \}\; \{\; T\; ^\; \{\; c\; \_\; \{\; 0\; \}\; \}\; \}\; ,$$

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

$$1-{\widehat{\alpha }}_{t-1}=(1-{\widehat{\alpha }}_t)(1-\frac{c_1{\widehat{\alpha }}_t\log T}{T})\le (1-{\widehat{\alpha }}_t)(1-\frac{c_1\log T}{2T})\mathrm{.}1\; -\; \backslash widehat\; \backslash alpha\; \_\; \{\; t\; -\; 1\; \}\; =\; \backslash big\; (\; 1\; -\; \backslash widehat\; \backslash alpha\; \_\; \{\; t\; \}\; \backslash big\; )\; \backslash bigg\; (\; 1\; -\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash widehat\; \backslash alpha\; \_\; \{\; t\; \}\; \backslash log\; T\; \}\; \{\; T\; \}\; \backslash bigg\; )\; \backslash leq\; \backslash big\; (\; 1\; -\; \backslash widehat\; \backslash alpha\; \_\; \{\; t\; \}\; \backslash big\; )\; \backslash bigg\; (\; 1\; -\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash log\; T\; \}\; \{\; 2\; T\; \}\; \backslash bigg\; )\; .$$

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

$$\frac{{\widehat{\tau }}_{k,n-1}-{\widehat{\tau }}_{k,n}}{{\widehat{\tau }}_{k,n-1}(1-{\widehat{\tau }}_{k,n-1})}=\frac{{\widehat{\alpha }}_{T-\frac{kN}{2}-n}-{\widehat{\alpha }}_{T-\frac{kN}{2}-n+1}}{(1-{\widehat{\alpha }}_{T-\frac{kN}{2}-n+1}){\widehat{\alpha }}_{T-\frac{kN}{2}-n+1}}=\frac{c_1\log T}{T}\mathrm{.}\backslash frac\; \{\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; -\; 1\; \}\; -\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; \}\; \}\; \{\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; -\; 1\; \}\; (\; 1\; -\; \backslash widehat\; \{\; \backslash tau\; \}\; \_\; \{\; k\; ,\; n\; -\; 1\; \}\; )\; \}\; =\; \backslash frac\; \{\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; -\; \backslash frac\; \{\; k\; N\; \}\; \{\; 2\; \}\; -\; n\; \}\; -\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; -\; \backslash frac\; \{\; k\; N\; \}\; \{\; 2\; \}\; -\; n\; +\; 1\; \}\; \}\; \{\; (\; 1\; -\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; -\; \backslash frac\; \{\; k\; N\; \}\; \{\; 2\; \}\; -\; n\; +\; 1\; \}\; )\; \backslash widehat\; \{\; \backslash alpha\; \}\; \_\; \{\; T\; -\; \backslash frac\; \{\; k\; N\; \}\; \{\; 2\; \}\; -\; n\; +\; 1\; \}\; \}\; =\; \backslash frac\; \{\; c\; \_\; \{\; 1\; \}\; \backslash log\; T\; \}\; \{\; T\; \}\; .$$

References

Benton, J., De Bortoli, V., Doucet, A., and Deligiannidis, G. (2023). Nearly $d$ -linear convergence bounds for diffusion models via stochastic localization. In The Twelfth International Conference on Learning Representations.

Chen, H., Lee, H., and Lu, J. (2023). Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions. In International Conference on Machine Learning, pages 4735–4763. PMLR.

Chen, H., Ren, Y., Ying, L., and Rotskoff, G. M. (2024a). Accelerating diffusion models with parallel sampling: Inference at sub-linear time complexity. arXiv preprint arXiv:2405.15986.

Chen, S., Chewi, S., Lee, H., Li, Y., Lu, J., and Salim, A. (2024b). The probability flow ode is provably fast. Advances in Neural Information Processing Systems, 36.

Chen, S., Chewi, S., Li, J., Li, Y., Salim, A., and Zhang, A. R. (2022). Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions. arXiv preprint arXiv:2209.11215.

Croitoru, F.-A., Hondru, V., Ionescu, R. T., and Shah, M. (2023). Diffusion models in vision: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(9):10850–10869.

De Bortoli, V. (2022). Convergence of denoising diffusion models under the manifold hypothesis. Transactions on Machine Learning Research.

Dhariwal, P. and Nichol, A. (2021). Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780–8794.

Gao, X., Nguyen, H. M., and Zhu, L. (2023). Wasserstein convergence guarantees for a general class of score-based generative models. arXiv preprint arXiv:2311.11003.
Gupta, S., Cai, L., and Chen, S. (2024). Faster diffusion-based sampling with randomized midpoints: Sequential and parallel. arXiv preprint arXiv:2406.00924.
Ho, J., Jain, A., and Abbeel, P. (2020). Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840–6851.
Huang, D. Z., Huang, J., and Lin, Z. (2024a). Convergence analysis of probability flow ode for score-based generative models. arXiv preprint arXiv:2404.09730.
Huang, X., Zou, D., Dong, H., Zhang, Y., Ma, Y.-A., and Zhang, T. (2024b). Reverse transition kernel: A flexible framework to accelerate diffusion inference. arXiv preprint arXiv:2405.16387.
Huang, Z., Wei, Y., and Chen, Y. (2024c). Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality. arXiv preprint arXiv:2410.18784.
Hyvärinen, A. (2007). Some extensions of score matching. Computational statistics & data analysis, 51(5):2499–2512.
Hyvärinen, A. and Dayan, P. (2005). Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6(4).
Lee, H., Lu, J., and Tan, Y. (2022). Convergence for score-based generative modeling with polynomial complexity. Advances in Neural Information Processing Systems, 35:22870–22882.
Lee, H., Lu, J., and Tan, Y. (2023). Convergence of score-based generative modeling for general data distributions. In International Conference on Algorithmic Learning Theory, pages 946–985. PMLR.
Li, G. and Cai, C. (2024). Provable acceleration for diffusion models under minimal assumptions. arXiv preprint arXiv:2410.23285.
Li, G., Huang, Y., Efimov, T., Wei, Y., Chi, Y., and Chen, Y. (2024a). Accelerating convergence of scorebased diffusion models, provably. In Forty-first International Conference on Machine Learning.
Li, G. and Jiao, Y. (2024). Improved convergence rate for diffusion probabilistic models. arXiv preprint arXiv:2410.13738.
Li, G., Wei, Y., Chen, Y., and Chi, Y. (2023). Towards faster non-asymptotic convergence for diffusion-based generative models. arXiv preprint arXiv:2306.09251.
Li, G., Wei, Y., Chi, Y., and Chen, Y. (2024b). A sharp convergence theory for the probability flow odes of diffusion models. arXiv preprint arXiv:2408.02320.
Li, G. and Yan, Y. (2024a). $o ( d / t )$ convergence theory for diffusion probabilistic models under minimal assumptions. arXiv preprint arXiv:2409.18959.
Li, G. and Yan, Y. (2024b). Adapting to unknown low-dimensional structures in score-based diffusion models. arXiv preprint arXiv:2405.14861.
Liang, J., Huang, Z., and Chen, Y. (2025). Low-dimensional adaptation of diffusion models: Convergence in total variation. arXiv preprint arXiv:2501.12982.
Pang, T., Xu, K., Li, C., Song, Y., Ermon, S., and Zhu, J. (2020). Efficient learning of generative models via finite-difference score matching. Advances in Neural Information Processing Systems, 33:19175–19188.
Ramesh, A., Dhariwal, P., Nichol, A., Chu, C., and Chen, M. (2022). Hierarchical text-conditional image generation with clip latents. arXiv preprint arXiv:2204.06125, 1(2):3.
Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. (2022). High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10684–10695.
Saharia, C., Chan, W., Saxena, S., Li, L., Whang, J., Denton, E. L., Ghasemipour, K., Gontijo Lopes, R., Karagol Ayan, B., Salimans, T., et al. (2022). Photorealistic text-to-image diffusion models with deep language understanding. Advances in neural information processing systems, 35:36479–36494.
Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N., and Ganguli, S. (2015). Deep unsupervised learning using nonequilibrium thermodynamics. In International conference on machine learning, pages 2256–2265. PMLR.
Song, J., Meng, C., and Ermon, S. (2021). Denoising diffusion implicit models. In International Conference on Learning Representations.
Song, Y. and Ermon, S. (2019). Generative modeling by estimating gradients of the data distribution. Advances in neural information processing systems, 32.
Tang, W. and Zhao, H. (2024). Score-based diffusion models via stochastic differential equations–a technical tutorial. arXiv preprint arXiv:2402.07487.
Villegas, R., Babaeizadeh, M., Kindermans, P.-J., Moraldo, H., Zhang, H., Saffar, M. T., Castro, S., Kunze, J., and Erhan, D. (2022). Phenaki: Variable length video generation from open domain textual descriptions. In International Conference on Learning Representations.
Vincent, P. (2011). A connection between score matching and denoising autoencoders. Neural computation, 23(7):1661–1674.
Yang, L., Zhang, Z., Song, Y., Hong, S., Xu, R., Zhao, Y., Zhang, W., Cui, B., and Yang, M.-H. (2023). Diffusion models: A comprehensive survey of methods and applications. ACM Computing Surveys, 56(4):1– 39.
Zhang, J., Liu, Z., Yan, L., Li, G., and Gu, Y. (2025). Improving diffusion-based inverse algorithms under few-step constraint via learnable linear extrapolation. arXiv preprint arXiv:2503.10103.