Stochastic Recursive Variance-Reduced Cubic Regularization Methods

Introduction

Many machine learning problems can be formulated as empirical risk minimization, which is in the form of finite-sum optimization as follows:

where $\epsilon_{g},\epsilon_{H}>0$ are predefined precision parameters. The most classic algorithm to find the approximate local minimum is cubic-regularized (CR) Newton method, which was originally proposed in the seminal paper by Nesterov and Polyak (2006). Generally speaking, in the $k$-th iteration, cubic regularization method solves a subproblem, which minimizes a cubic-regularized second-order Taylor expansion at the current iterate $\mathbf{x}_{k}$. The update rule can be written as follows:

where $M>0$ is a penalty parameter. Nesterov and Polyak (2006) proved that to find an $(\epsilon,\sqrt{\epsilon})$-approximate local minimum of a nonconvex function $F$, cubic regularization requires at most $O(\epsilon^{-3/2})$ iterations. However, when applying cubic regularization to nonconvex finite-sum optimization in (1.1), a major bottleneck of cubic regularization is that it needs to compute $n$ individual gradients $\nabla f_{i}(\mathbf{x}_{k})$ and Hessian matrices $\nabla^{2}f_{i}(\mathbf{x}_{k})$ at each iteration, which leads to a total $O(n\epsilon^{-3/2})$ gradient complexity (i.e., number of queries to the stochastic gradient oracle $\nabla f_{i}(\mathbf{x})$ for some $i$ and $\mathbf{x}$) and $O(n\epsilon^{-3/2})$ Hessian complexity (i.e., number of queries to the stochastic Hessian oracle $\nabla^{2}f_{i}(\mathbf{x})$ for some $i$ and $\mathbf{x}$). Such computational overhead will be extremely expensive when $n$ is large as is in many large-scale machine learning applications.

To overcome the aforementioned computational burden of cubic regularization, Kohler and Lucchi (2017); Xu et al. (2017) used subsampled gradient and subsampled Hessian, which achieve $\widetilde{O}(n\epsilon^{-3/2}\land\epsilon^{-7/2})$ gradient complexity and $\widetilde{O}(n\epsilon^{-3/2}\land\epsilon^{-5/2})$ Hessian complexity. Zhou et al. (2018d) proposed a stochastic variance reduced cubic regularization method (SVRC), which uses novel semi-stochastic gradient and semi-stochastic Hessian estimators inspired by variance reduction for first-order finite-sum optimization (Johnson and Zhang, 2013; Reddi et al., 2016; Allen-Zhu and Hazan, 2016), which attains $O(n^{4/5}\epsilon^{-3/2})$ Second-order Oracle (SO) complexitySecond-order Oracle (SO) returns triple $[f_{i}(\mathbf{x}),\nabla f_{i}(\mathbf{x}),\nabla^{2}f_{i}(\mathbf{x})]$ for some $i$ and $\mathbf{x}$, hence the SO complexity can be seen as the maximum of gradient and Hessian complexities.. Zhou et al. (2018b); Wang et al. (2018b); Zhang et al. (2018a) used a simpler semi-stochastic gradient compared with (Zhou et al., 2018d), and semi-stochastic Hessian, which achieves a better Hessian complexity, i.e., $O(n^{2/3}\epsilon^{-3/2})$. However, it is unclear whether the gradient and Hessian complexities of the aforementioned SVRC algorithms can be further improved. Furthermore, all these algorithms need to use the semi-stochastic Hessian estimator, which is not compatible with Hessian-vector product-based cubic subproblem solvers (Agarwal et al., 2017; Carmon and Duchi, 2016, 2018). Therefore, the cubic subproblem (1.4) in each iteration of existing SVRC algorithms has to be solved by computing the inverse of the Hessian matrix, whose computational complexity is at least $O(d^{w})$$w$ is the matrix multiplication constant, where $w=2.37...$ (Golub and Van Loan, 1996). . This makes existing SVRC algorithms not very practical for high-dimensional problems.

In this paper, we first show that the gradient and Hessian complexities of SVRC-type algorithms can be further improved. The core idea is to use novel recursively updated semi-stochastic gradient and Hessian estimators, which are inspired by the stochastic path-integrated differential estimator (SPIDER) (Fang et al., 2018) and the StochAstic Recursive grAdient algoritHm (SARAH) (Nguyen et al., 2017) for first-order optimization. We show that such kind of estimators can be extended to second-order optimization to reduce the Hessian complexity. Nevertheless, our analysis is very different from that in Fang et al. (2018); Nguyen et al. (2017), because we study a fundamentally different optimization problem (i.e., finding local minima against finding first-order stationary points) and a completely different optimization algorithm (i.e., cubic regularization versus gradient method). In addition, in order to reduce the runtime complexity of existing SVRC algorithms, we further propose a Hessian-free SVRC method that can not only use the novel semi-stochastic gradient estimator, but also leverage the Hessian-vector product-based fast cubic subproblem solvers. Experiments on benchmark nonconvex finite-sum optimization problems illustrate the superiority of our newly proposed SVRC algorithms over the state-of-the-art (Due to space limit, we include the experiments in Appendix 6).

In detail, our contributions are summarized as follows:

We propose a new SVRC algorithm, namely SRVRC, which can find an $(\epsilon,\sqrt{\epsilon})$-approximate local minimum with $\widetilde{O}(n\epsilon^{-3/2}\land\epsilon^{-3})$ gradient complexity and $\widetilde{O}(n\land\epsilon^{-1}+n^{1/2}\epsilon^{-3/2}\land\epsilon^{-2})$ Hessian complexity. Compared with previous work in cubic regularization, the gradient and Hessian complexity of SRVRC is strictly better than the algorithms in Zhou et al. (2018b); Wang et al. (2018b); Zhang et al. (2018a), and better than that in Zhou et al. (2018d); Shen et al. (2019) in a wide regime.

We further propose a new algorithm $\text{SRVRC}_{\text{free}}$, which requires $\widetilde{O}(\epsilon^{-3}\land n\epsilon^{-2})$ stochastic gradient and Hessian-vector product computations to find an $(\epsilon,\sqrt{\epsilon})$-approximate local minimum. $\text{SRVRC}_{\text{free}}$ is strictly better than the algorithms in (Agarwal et al., 2017; Carmon and Duchi, 2016; Tripuraneni et al., 2018) when $n\gg 1$. The runtime complexity of $\text{SRVRC}_{\text{free}}$ is also better than that of SRVRC when the problem dimension $d$ is large.

In an independent and concurrent work (Shen et al., 2019), two stochastic trust region methods namely STR1 and STR2 were proposed, which are based on the same idea of variance reduction using SPIDER, and are related to our first algorithm SRVRC. Our SRVRC is better than STR1 because it enjoys the same Hessian complexity but a better gradient complexity than STR1. Compared with STR2, our SRVRC has a consistently lower Hessian complexity and lower gradient complexity in a wide regime (i.e., $\epsilon\gg n^{-1/2}$). Since Hessian complexity is the dominating term in cubic regularization method (Zhou et al., 2018b; Wang et al., 2018b), our SRVRC is arguably better than STR2, as verified by our experiments.

For the ease of comparison, we summarize the comparison of methods which need to compute the Hessian explicitly in Table 1, the Hessian-free or Hessian-vector product based methods in Table 2.

Additional Related Work

In this section, we review additional related work that is not discussed in the introduction section.

Cubic Regularization and Trust-Region Methods Since cubic regularization was first proposed by Nesterov and Polyak (2006), there has been a line of followup research. It was extended to adaptive regularized cubic methods (ARC) by Cartis et al. (2011a, b), which enjoy the same iteration complexity as standard cubic regularization while having better empirical performance. The first attempt to make cubic regularization a Hessian-free method was done by Carmon and Duchi (2016), which solves the cubic sub-problem by gradient descent, requiring in total $\widetilde{O}(n\epsilon^{-2})$ stochastic gradient and Hessian-vector product computations. Agarwal et al. (2017) solved cubic sub-problem by fast matrix inversion based on accelerated gradient descent, which requires $\widetilde{O}(n\epsilon^{-3/2}+n^{3/4}\epsilon^{-7/4})$ stochastic gradient and Hessian-vector product computations. In the pure stochastic optimization setting, Tripuraneni et al. (2018) proposed stochastic cubic regularization method, which uses subsampled gradient and Hessian-vector product-based cubic subproblem solver, and requires $\widetilde{O}(\epsilon^{-3.5})$ stochastic gradient and Hessian-vector product computations. A closely related second-order method to cubic regularization methods are trust-region methods (Conn et al., 2000; Cartis et al., 2009, 2012, 2013). Recent studies (Blanchet et al., 2016; Curtis et al., 2017; Martínez and Raydan, 2017) proved that the trust-region method can achieve the same iteration complexity as the cubic regularization method. Xu et al. (2017) also extended trust-region method to subsampled trust-region method for nonconvex finite-sum optimization.

Local Minima Finding Besides cubic regularization and trust-region type methods, there is another line of research for finding approximate local minima, which is based on first-order optimization. Ge et al. (2015); Jin et al. (2017a) proved that (stochastic) gradient methods with additive noise are able to escape from nondegenerate saddle points and find approximate local minima. Carmon et al. (2018); Royer and Wright (2017); Allen-Zhu (2017); Xu et al. (2018); Allen-Zhu and Li (2018); Jin et al. (2017b); Yu et al. (2017, 2018); Zhou et al. (2018a); Fang et al. (2018) showed that by alternating first-order optimization and Hessian-vector product based negative curvature descent, one can find approximate local minima even more efficiently. Very recently, Fang et al. (2019); Jin et al. (2019) showed that stochastic gradient descent itself can escape from saddle points.

Variance Reduction Variance reduction techniques play an important role in our proposed algorithms. Variance reduction techniques were first proposed for convex finite-sum optimization, which use semi-stochastic gradient to reduce the variance of the stochastic gradient and improve the gradient complexity. Representative algorithms include Stochastic Average Gradient (SAG) (Roux et al., 2012), Stochastic Variance Reduced Gradient (SVRG) (Johnson and Zhang, 2013; Xiao and Zhang, 2014), SAGA (Defazio et al., 2014) and SARAH (Nguyen et al., 2017), to mention a few. For nonconvex finite-sum optimization problems, Garber and Hazan (2015); Shalev-Shwartz (2016) studied the case where each individual function is nonconvex, but their sum is still (strongly) convex. Reddi et al. (2016); Allen-Zhu and Hazan (2016) extended SVRG to noncovnex finite-sum optimization, which is able to converge to first-order stationary point with better gradient complexity than vanilla gradient descent. Fang et al. (2018); Zhou et al. (2018c); Wang et al. (2018a); Nguyen et al. (2019) further improved the gradient complexity for nonconvex finite-sum optimization to be (near) optimal.

Notation and Preliminaries

We begin with a few assumptions that are needed for later theoretical analyses of our algorithms.

The following assumption says that the gap between the function value at the initial point $\mathbf{x}_{0}$ and the minimal function value is bounded.

We also need the following $L$-gradient Lipschitz and $\rho$-Hessian Lipschitz assumption.

Note that $L$-gradient Lipschitz is not required in the original cubic regularization algorithm (Nesterov and Polyak, 2006) and the SVRC algorithm (Zhou et al., 2018d). However, for most other SVRC algorithms (Zhou et al., 2018b; Wang et al., 2018b; Zhang et al., 2018a), they need the $L$-gradient Lipschitz assumption.

In addition, we need the difference between the stochastic gradient and the full gradient to be bounded.

It is worth noting that Assumption 3.3 is weaker than the assumption that each $f_{i}$ is Lipschitz continuous, which has been made in Kohler and Lucchi (2017); Zhou et al. (2018b); Wang et al. (2018b); Zhang et al. (2018a). We would also like to point out that we can make additional assumptions on the variances of the stochastic gradient and Hessian, such as the ones made in Tripuraneni et al. (2018). Nevertheless, making these additional assumptions does not improve the dependency of the gradient and Hessian complexities or the stochastic gradient and Hessian-vector product computations on $\epsilon$ and $n$. Therefore we chose not making these additional assumptions on the variances.

The Proposed SRVRC Algorithm

In this section, we present SRVRC, a novel algorithm which utilizes new semi-stochastic gradient and Hessian estimators compared with previous SVRC algorithms. We also provide a convergence analysis of the proposed algorithm.

In order to reduce the computational complexity for calculating full gradient and full Hessian in (1.3), several ideas such as subsampled/stochastic gradient and Hessian (Kohler and Lucchi, 2017; Xu et al., 2017; Tripuraneni et al., 2018) and variance-reduced semi-stochastic gradient and Hessian (Zhou et al., 2018d; Wang et al., 2018b; Zhang et al., 2018a) have been used in previous work. SRVRC follows this line of work. The key idea is to use a new construction of semi-stochastic gradient and Hessian estimators, which are recursively updated in each iteration, and reset periodically after certain number of iterations (i.e., an epoch). This is inspired by the first-order variance reduction algorithms SPIDER (Fang et al., 2018) and SARAH (Nguyen et al., 2017). SRVRC constructs semi-stochastic gradient and Hessian as in (3.1) and (3.2) respectively. To be more specific, in the $t$-th iteration when $\text{mod}(t,S^{(g)})=0$ or $\text{mod}(t,S^{(h)})=0$, where $S^{(g)},S^{(h)}$ are the epoch lengths of gradient and Hessian, SRVRC will set the semi-stochastic gradient $\mathbf{v}_{t}$ and Hessian $\mathbf{U}_{t}$ to be a subsampled gradient $\nabla f_{\mathcal{J}_{t}}(\mathbf{x}_{t})$ and Hessian $\nabla^{2}f_{\mathcal{J}_{t}}(\mathbf{x}_{t})$ at point $\mathbf{x}_{t}$, respectively. In the $t$-th iteration when $\text{mod}(t,S)\neq 0$ or $\text{mod}(t,S^{(h)})\neq 0$, SRVRC constructs semi-stochastic gradient and Hessian $\mathbf{v}_{t}$ and $\mathbf{U}_{t}$ based on previous estimators $\mathbf{v}_{t-1}$ and $\mathbf{U}_{t-1}$ recursively. With semi-stochastic gradient $\mathbf{v}_{t}$, semi-stochastic Hessian $\mathbf{U}_{t}$ and $t$-th Cubic penalty parameter $M_{t}$, SRVRC constructs the $t$-th Cubic subproblem $m_{t}$ and solves for the solution to $m_{t}$ as $t$-th update direction as (3.3). If $\|\mathbf{h}_{t}\|_{2}$ is less than a given threshold which we set it as $\sqrt{\epsilon/\rho}$, SRVRC returns $\mathbf{x}_{t+1}=\mathbf{x}_{t}+\mathbf{h}_{t}$ as its output. Otherwise, SRVRC updates $\mathbf{x}_{t+1}=\mathbf{x}_{t}+\mathbf{h}_{t}$ and continues the loop.

The main difference between SRVRC and previous stochastic cubic regularization algorithms (Kohler and Lucchi, 2017; Xu et al., 2017; Zhou et al., 2018d, b; Wang et al., 2018b; Zhang et al., 2018a) is that SRVRC adapts new semi-stochastic gradient and semi-stochastic Hessian estimators, which are defined recursively and have smaller asymptotic variance. The use of such semi-stochastic gradient has been proved to help reduce the gradient complexity in first-order nonconvex finite-sum optimization for finding stationary points (Fang et al., 2018; Wang et al., 2018a; Nguyen et al., 2019). Our work takes one step further to apply it to Hessian, and we will later show that it helps reduce the gradient and Hessian complexities in second-order nonconvex finite-sum optimization for finding local minima.

2 Convergence Analysis

In this subsection, we present our theoretical results about SRVRC. While the idea of using variance reduction technique for cubic regularization is hardly new, the new semi-stochastic gradient and Hessian estimators in (3.1) and (3.2) bring new technical challenges in the convergence analysis.

To describe whether a point $\mathbf{x}$ is a local minimum, we follow the original cubic regularization work (Nesterov and Polyak, 2006) to use the following criterion $\mu(\mathbf{x})$:

For any $\mathbf{x}$, define $\mu(\mathbf{x})$ as \mu(\mathbf{x})=\max\{\|\nabla F(\mathbf{x})\|_{2}^{3/2},-\lambda_{\min}^{3}\big{(}\nabla^{2}F(\mathbf{x})\big{)}/{\rho^{3/2}}\}.

It is easy to note that $\mu(\mathbf{x})\leq\epsilon^{3/2}$ if and only if $\mathbf{x}$ is an $(\epsilon,\sqrt{\rho\epsilon})$-approximate local minimum. Thus, in order to find an $(\epsilon,\sqrt{\rho\epsilon})$-approximate local minimum, it suffices to find a point $\mathbf{x}$ which satisfies $\mu(\mathbf{x})\leq\epsilon^{3/2}$.

The following theorem provides the convergence guarantee of SRVRC for finding an $(\epsilon,\sqrt{\rho\epsilon})$-approximate local minimum.

Under Assumptions 3.1, 3.2 and 3.3, set the cubic penalty parameter $M_{t}=4\rho$ for any $t$ and the total iteration number $T\geq 40\Delta_{F}\rho^{1/2}\epsilon^{-3/2}$. For $t$ such that $\text{mod}(t,S^{(g)})\neq 0$ or $\text{mod}(t,S^{(h)})\neq 0$, set the gradient sample size $B_{t}^{(g)}$ and Hessian sample size $B_{t}^{(h)}$ as

For $t$ such that $\text{mod}(t,S^{(g)})=0$ or $\text{mod}(t,S^{(h)})=0$, set the gradient sample size $B_{t}^{(g)}$ and Hessian sample size $B_{t}^{(h)}$ as

Then with probability at least $1-\xi$, SRVRC outputs $\mathbf{x}_{\text{out}}$ satisfying $\mu(\mathbf{x}_{\text{out}})\leq 600\epsilon^{3/2}$, i.e., an $(\epsilon,\sqrt{\rho\epsilon})$-approximate local minimum.

Next corollary spells out the exact gradient complexity and Hessian complexity of SRVRC to find an $(\epsilon,\sqrt{\rho\epsilon})$-approximate local minimum.

Under the same conditions as Theorem 4.2, if we set $S^{(g)},S^{(h)}$ as $S^{(g)}=\sqrt{\rho\epsilon}/L\cdot\sqrt{n\land M^{2}/\epsilon^{2}}$ and $S^{(h)}=\sqrt{n\land L/(\rho\epsilon)}$, and set $T,\{B^{(g)}_{t}\},\{B^{(h)}_{t}\}$ as their lower bounds in (4.1)- (4.4), then with probability at least $1-\xi$, SRVRC will output an $(\epsilon,\sqrt{\rho\epsilon})$-approximate local minimum within

For SRVRC, if we assume $M,L,\rho,\Delta_{F}$ to be constants, then its gradient complexity is $\widetilde{O}(n/\epsilon^{3/2}\land\sqrt{n}/\epsilon^{2}\land\epsilon^{-3})$, and its Hessian complexity is $\widetilde{O}(n\land\epsilon^{-1}+n^{1/2}\epsilon^{-3/2}\land\epsilon^{-2})$. Regarding Hessian complexity, suppose that $\epsilon\ll 1$, then the Hessian complexity of SRVRC can be simplified as $\widetilde{O}(n^{1/2}\epsilon^{-3/2}\land\epsilon^{-2})$. Compared with existing SVRC algorithms (Zhou et al., 2018b; Zhang et al., 2018a; Wang et al., 2018b), SRVRC outperforms the best-known Hessian sample complexity by a factor of $n^{1/6}\land n^{2/3}\epsilon^{1/2}$. In terms of gradient complexity, SRVRC outperforms STR2 (Shen et al., 2019) by a factor of $n^{3/4}\epsilon^{3/2}$ when $\epsilon\gg n^{-1/2}$.

Note that both Theorem 4.2 and Corollary 4.3 still hold when Assumption 3.3 does not hold. In that case, $M=\infty$ and SRVRC’s Hessian complexity remains the same, while its gradient complexity can be potentially worse, i.e., $\widetilde{O}(n/\epsilon^{3/2}\land\sqrt{n}/\epsilon^{2})$, which degenerates to that of STR1 (Shen et al., 2019).

Hessian-Free SRVRC

While SRVRC adapts novel semi-stochastic gradient and Hessian estimators to reduce both the gradient and Hessian complexities, it has three limitations for high-dimensional problems with $d\gg 1$: (1) it needs to compute and store the Hessian matrix, which needs $O(d^{2})$ computational time and storage space; (2) it needs to solve cubic subproblem $m_{t}$ exactly, which requires $O(d^{w})$ computational time because it needs to compute the inverse of a Hessian matrix (Nesterov and Polyak, 2006); and (3) it cannot leverage the Hessian-vector product-based cubic subproblem solvers (Agarwal et al., 2017; Carmon and Duchi, 2016, 2018) because of the use of the semi-stochastic Hessian estimator. It is interesting to ask whether we can modify SRVRC to overcome these shortcomings.

Note that although the subproblem depends on $\mathbf{U}_{t}$, $\text{SRVRC}_{\text{free}}$ never explicitly computes this matrix. Instead, it only provides the subproblem solver access to $\mathbf{U}_{t}$ through stochastic Hessian-vector product function $\mathbf{U}_{t}[\cdot]$. The subproblem solver performs gradient-based optimization to solve the subproblem $m_{t}(\mathbf{h})$ as $\nabla m_{t}(\mathbf{h})$ depends on $\mathbf{U}_{t}$ only via $\mathbf{U}_{t}[\mathbf{h}]$. In detail, following Tripuraneni et al. (2018), $\text{SRVRC}_{\text{free}}$ uses Cubic-Subsolver (See Algorithms 3 and 4 in Appendix G) and Cubic-Finalsolver from (Carmon and Duchi, 2016), to find an approximate solution $\mathbf{h}_{t}$ to the cubic subproblem in (3.3). Both Cubic-Subsolver and Cubic-Finalsolver only need to access gradient $\mathbf{v}_{t}$ and Hessian-vector product function $\mathbf{U}_{t}[\cdot]$ along with other problem-dependent parameters. With the output $\mathbf{h}_{t}$ from Cubic-Subsolver, $\text{SRVRC}_{\text{free}}$ decides either to update $\mathbf{x}_{t}$ as $\mathbf{x}_{t+1}\leftarrow\mathbf{x}_{t}+\mathbf{h}_{t}$ or to exit the loop. For the later case, $\text{SRVRC}_{\text{free}}$ will call Cubic-Finalsolver to output $\mathbf{h}_{t}$, and takes $\mathbf{x}_{t+1}=\mathbf{x}_{t}+\mathbf{h}_{t}$ as its final output.

The main differences between SRVRC and $\text{SRVRC}_{\text{free}}$ are two-fold. First, $\text{SRVRC}_{\text{free}}$ only needs to compute stochastic gradient and Hessian-vector product. Since both of these two computations only take $O(d)$ time in many applications in machine learning, $\text{SRVRC}_{\text{free}}$ is suitable for high-dimensional problems. In the sequel, following Agarwal et al. (2017); Carmon et al. (2018); Tripuraneni et al. (2018), we do not distinguish stochastic gradient and Hessian-vector product computations and consider them to have the same runtime complexity. Second, instead of solving cubic subproblem $m_{t}$ exactly, $\text{SRVRC}_{\text{free}}$ adopts approximate subproblem solver Cubic-Subsolver and Cubic-Finalsolver, both of which only need to access gradient and Hessian-vector product function, and again only take $O(d)$ time. Thus, $\text{SRVRC}_{\text{free}}$ is computational more efficient than SRVRC when $d\gg 1$.

2 Convergence Analysis

We now provide the convergence guarantee of $\text{SRVRC}_{\text{free}}$, which ensures that $\text{SRVRC}_{\text{free}}$ will output an $(\epsilon,\sqrt{\rho\epsilon})$-approximate local minimum.

Under Assumptions 3.1, 3.2, 3.3, suppose $\epsilon<L/(4\rho)$. Set the cubic penalty parameter $M_{t}=4\rho$ for any $t$ and the total iteration number $T\geq 25\Delta_{F}\rho^{1/2}\epsilon^{-3/2}$. Set the Hessian-vector product sample size $B_{t}^{(h)}$ as

For $t$ such that $\text{mod}(t,S^{(g)})\neq 0$, set the gradient sample size $B^{(g)}_{t}$ as

For $t$ such that $\text{mod}(t,S^{(g)})=0$, set the gradient sample size $B^{(g)}_{t}$ as

Then with probability at least $1-\xi$, $\text{SRVRC}_{\text{free}}$ outputs $\mathbf{x}_{\text{out}}$ satisfying $\mu(\mathbf{x}_{\text{out}})\leq 1300\epsilon^{3/2}$, i.e., an $(\epsilon,\sqrt{\rho\epsilon})$-approximate local minimum.

The following corollary calculates the total amount of stochastic gradient and Hessian-vector product computations of $\text{SRVRC}_{\text{free}}$ to find an $(\epsilon,\sqrt{\rho\epsilon})$-approximate local minimum.

Under the same conditions as Theorem 5.1, if set $S^{(g)}=\sqrt{\rho\epsilon}/L\cdot\sqrt{n\land M^{2}/\epsilon^{2}}$ and set $T,\{B^{(g)}_{t}\},\{B^{(h)}_{t}\}$ as their lower bounds in (5.1)-(5.3), then with probability at least $1-\xi$, $\text{SRVRC}_{\text{free}}$ will output an $(\epsilon,\sqrt{\rho\epsilon})$-approximate local minimum within

stochastic gradient and Hessian-vector product computations.

For $\text{SRVRC}_{\text{free}}$, if we assume $\rho,L,M,\Delta_{F}$ are constants, then (5.4) is $\widetilde{O}(n\epsilon^{-2}\land\epsilon^{-3})$. For stochastic algorithms, the regime $n\rightarrow\infty$ is of most interest. In this regime, (5.4) becomes $\widetilde{O}(\epsilon^{-3})$. Compared with other local minimum finding algorithms based on stochastic gradient and Hessian-vector product, $\text{SRVRC}_{\text{free}}$ outperforms the results achieved by Tripuraneni et al. (2018) and Allen-Zhu (2018) by a factor of $\epsilon^{-1/2}$. $\text{SRVRC}_{\text{free}}$ also matches the best-known result achieved by a recent first-order algorithm proposed in (Fang et al., 2018). Note that the algorithm proposed by Fang et al. (2018) needs to alternate the first-order finite-sum optimization algorithm SPIDER and negative curvature descent. In sharp contrast, $\text{SRVRC}_{\text{free}}$ is a pure cubic regularization type algorithm and does not need to calculate the negative curvature direction.

It is worth noting that both Theorem 5.1 and Corollary 5.2 still hold when Assumption 3.3 does not hold, and $\text{SRVRC}_{\text{free}}$’s runtime complexity remains the same. The only difference is: without Assumption 3.3, we need to use full gradient (i.e., $B_{t}^{(g)}=n$) instead of subsampled gradient at each iteration $t$.

3 Discussions on runtime complexity

We would like to further compare the runtime complexity between SRVRC and $\text{SRVRC}_{\text{free}}$. In specific, SRVRC needs $O(d)$ time to construct semi-stochastic gradient and $O(d^{2})$ time to construct semi-stochastic Hessian. SRVRC also needs $O(d^{w})$ time to solve cubic subproblem $m_{t}$ for each iteration. Thus, with the fact that the total number of iterations is $T=O(\epsilon^{-3/2})$ by Corollary 4.3, SRVRC needs

runtime to find an $(\epsilon,\sqrt{\epsilon})$-approximate local minimum if we regard $M,L,\rho,\Delta_{F}$ as constants. As we mentioned before, for many machine learning problems, both stochastic gradient and Hessian-vector product computations only need $O(d)$ time, therefore the runtime of $\text{SRVRC}_{\text{free}}$ is $\widetilde{O}(dn\epsilon^{-2}\land d\epsilon^{-3})$. We conclude that $\text{SRVRC}_{\text{free}}$ outperforms SRVRC when $d$ is large, which is in accordance with the fact that Hessian-free methods are superior for high dimension machine learning tasks. On the other hand, a careful calculation can show that the runtime of SRVRC can be less than that of $\text{SRVRC}_{\text{free}}$ when $d$ is moderately small. This is also reflected in our experiments in Section 6.

Experiments

In this section, we present numerical experiments on different nonconvex empirical risk minimization (ERM) problems and on different datasets to validate the advantage of our proposed SRVRC and $\text{SRVRC}_{\text{free}}$ algorithms for finding approximate local minima.

Baselines: We compare our algorithms with the following algorithms: SPIDER+ (Fang et al., 2018), which is the local minimum finding version of SPIDER, stochastic trust region (STR1, STR2) (Shen et al., 2019), subsampled cubic regularization (SCR) (Kohler and Lucchi, 2017), stochastic cubic regularization (STC) (Tripuraneni et al., 2018), stochastic variance-reduced cubic regularization (SVRC) (Zhou et al., 2018d), sample efficient SVRC (Lite-SVRC) (Zhou et al., 2018b; Wang et al., 2018b; Zhang et al., 2018a).

Parameter Settings and Subproblem Solver For each algorithm, we set the cubic penalty parameter $M_{t}$ adaptively based on how well the model approximates the real objective as suggested in (Cartis et al., 2011a, b; Kohler and Lucchi, 2017). For SRVRC, we set $S^{(g)}=S^{(h)}=S$ for the simplicity and set gradient and Hessian batch sizes $B_{t}^{(g)}$ and $B_{t}^{(h)}$ as follows:

For $\text{SRVRC}_{\text{free}}$, we set gradient batch sizes $B_{t}^{(g)}$ the same as SRVRC and Hessian batch sizes $B_{t}^{(h)}=B^{(h)}$. We tune $S$ over the grid $\{5,10,20,50\}$, $B^{(g)}$ over the grid $\{n,n/10,n/20,n/100\}$, and $B^{(h)}$ over the grid $\{50,100,500,1000\}$ for the best performance. For SCR, SVRC, Lite-SVRC, and SRVRC, we solve the cubic subproblem using the cubic subproblem solver discussed in (Nesterov and Polyak, 2006). For STR1 and STR2, we solve the trust-region subproblem using the exact trust-region subproblem solver discussed in (Conn et al., 2000). For STC and $\text{SRVRC}_{\text{free}}$, we use Cubic-Subsolver (Algorithm 3 in Appendix G) to approximately solve the cubic subproblem. All algorithms are carefully tuned for a fair comparison.

where $\text{softmax}(\mathbf{a})=\exp(\mathbf{a})/\sum_{i=1}^{d}\exp(a_{i})$ is the softmax function and $\lambda=10^{-3}$.

We plot the logarithmic function value gap with respect to CPU time in Figure 1. From Figure 1(a) to 1(f), we can see that for the low dimension optimization task on a9a, covtype and ijcnn1, our SRVRC outperforms all the other algorithms with respect to CPU time. We can also observe that the stochastic trust region method STR1 is better than STR2, which is well-aligned with our discussion before. The SPIDER+ does not perform as well as other second-order methods, even though its stochastic gradient and Hessian complexity is comparable to second-order methods in theory. Meanwhile, we also notice that $\text{SRVRC}_{\text{free}}$ always outperforms STC, which suggests that the variance reduction technique is useful. For high dimension optimization task mnist, cifar10 and SVHN, only SPIDER+, STC and $\text{SRVRC}_{\text{free}}$ are able to make notable progress and $\text{SRVRC}_{\text{free}}$ outperforms the other two. This is again consistent with our theory and discussions in Section 5. Overall, our experiments clearly validate the advantage of SRVRC and $\text{SRVRC}_{\text{free}}$, and corroborate the theory of both algorithms.

Conclusions and Future Work

In this work we present two faster SVRC algorithms namely SRVRC and $\text{SRVRC}_{\text{free}}$ to find approximate local minima for nonconvex finite-sum optimization problems. SRVRC outperforms existing SVRC algorithms in terms of gradient and Hessian complexities, while $\text{SRVRC}_{\text{free}}$ further outperforms the best-known runtime complexity for existing CR based algorithms. Whether our algorithms have achieved the optimal complexity under the current assumptions is still an open problem, and we leave it as a future work.

Appendix A Proofs in Section 4

To prove Theorem 4.2, we need the following lemma adapted from Zhou et al. (2018d), which characterizes that $\mu(\mathbf{x}_{t}+\mathbf{h})$ can be bounded by $\|\mathbf{h}\|_{2}$ and the norm of difference between semi-stochastic gradient and Hessian.

Next lemma gives bounds on the inner products $\langle\nabla F(\mathbf{x}_{t})-\mathbf{v}_{t},\mathbf{h}\rangle$ and \langle\big{(}\nabla^{2}F(\mathbf{x}_{t})-\mathbf{U}_{t}\big{)}\mathbf{h},\mathbf{h}\rangle.

We also need the following two lemmas, which show that semi-stochastic gradient and Hessian $\mathbf{v}_{t}$ and $\mathbf{U}_{t}$ estimators are good approximations to true gradient and Hessian.

Suppose that $\{B_{k}^{(g)}\}$ satisfies (4.1) and (4.3), then conditioned on $\mathcal{F}_{\lfloor t/S^{(g)}\rfloor\cdot S^{(g)}}$, with probability at least $1-\delta\cdot(t-\lfloor t/S^{(g)}\rfloor\cdot S^{(g)})$, we have that for all $\lfloor t/S^{(g)}\rfloor\cdot S^{(g)}\leq k\leq t$,

Suppose that $\{B_{k}^{(h)}\}$ satisfies (4.2) and (4.4), then conditioned on $\mathcal{F}_{\lfloor t/S^{(h)}\rfloor\cdot S^{(h)}}$, with probability at least $1-\delta\cdot(t-\lfloor t/S^{(h)}\rfloor\cdot S^{(h)})$, we have that for all $\lfloor t/S^{(h)}\rfloor\cdot S^{(h)}\leq k\leq t$,

Given all the above lemmas, we are ready to prove Theorem 4.2.

Suppose that SRVRC terminates at iteration $T^{*}-1$, then $\|\mathbf{h}_{t}\|_{2}>\sqrt{\epsilon/\rho}$ for all $0\leq t\leq T^{*}-1$. We have

where the second inequality holds due to the fact that $m_{t}(\mathbf{h}_{t})\leq m_{t}({\bm{0}})=0$, $M_{t}=4\rho$ and Lemma A.2. By Lemmas A.3 and A.4, with probability at least $1-2T\delta$, for all $0\leq t\leq T-1$, we have that

for all $0\leq t\leq T-1$. Substituting (A.4) into (A.3), we have

Telescoping (A.5) from $t=0,\dots,T^{*}-1$, we have

Recall that we have $T\geq 40\Delta_{F}\sqrt{\rho}/\epsilon^{3/2}$ from the condition of Theorem 4.2, then by (A.6), we have $T^{*}\leq T$. Thus, we have $\|\mathbf{h}_{T^{*}-1}\|_{2}\leq\sqrt{\epsilon/\rho}$. Denote $\widetilde{T}=T^{*}-1$, then we have

where the first inequality holds due to Lemma A.1 with $\nabla m_{\widetilde{T}}(\mathbf{h}_{\widetilde{T}})=0$ and $\|\mathbf{h}_{\widetilde{T}}\|_{2}=\|\mathbf{h}_{\widetilde{T}}^{*}\|_{2}$. This completes our proof. ∎

A.2 Proof of Corollary 4.3

Suppose that SRVRC terminates at $T^{*}-1\leq T-1$ iteration. Telescoping (A.5) from $t=0$ to $T^{*}-1$, we have

where the last inequality holds since $T$ is set to be $40\Delta_{F}\sqrt{\rho}/\epsilon^{3/2}$ as the conditions of Corollary 4.3 suggests. (A.7) implies that $\sum_{t=0}^{T^{*}-1}\|\mathbf{h}_{t}\|_{2}^{3}\leq 40\Delta_{F}/\rho$. Thus, we have

where the first inequality holds due to Hölder’s inequality inequality, and the second inequality is due to $T^{*}\leq T=40\Delta_{F}\sqrt{\rho}/\epsilon^{3/2}$. We first consider the total gradient sample complexity $\sum_{t=0}^{T^{*}-1}B_{t}^{(g)}$, which can be bounded as

where $C_{1}=1440\log^{2}(d/\delta)$, the second inequality holds due to (A.8), and the last equality holds due to the choice of $S^{(g)}=\sqrt{\rho\epsilon}/L\cdot\sqrt{n\land M^{2}/\epsilon^{2}}$. We then consider the total Hessian sample complexity $\sum_{t=0}^{T^{*}-1}B_{t}^{(h)}$, which can be bounded as

where $C_{2}=800\log^{2}(d/\delta)$, the second inequality holds due to (A.8), and the last equality holds due to the choice of $S^{(h)}=\sqrt{n\land L/(\rho\epsilon)}$. ∎

Appendix B Proofs in Section 5

In this section, we denote $\delta=\xi/(3T)$ for simplicity.

We need the following two lemmas, which bound the variance of semi-stochastic gradient and Hessian estimators.

Suppose that $\{B_{k}^{(g)}\}$ satisfies (5.2) and (5.3), then conditioned on $\mathcal{F}_{\lfloor t/S\rfloor\cdot S}$, with probability at least $1-\delta\cdot(t-\lfloor t/S\rfloor\cdot S)$, we have that for all $\lfloor t/S\rfloor\cdot S\leq k\leq t$,

The proof is very similar to that of Lemma A.3, hence we omit it. ∎

Suppose that $\{B_{k}^{(h)}\}$ satisfies (5.1), then conditioned on $\mathcal{F}_{k}$, with probability at least $1-\delta$, we have that

The proof is very similar to that of Lemma A.4, hence we omit it. ∎

We have the following lemma to guarantee that by Algorithm 3 Cubic-Subsolver, the output $\mathbf{h}_{t}$ satisfies that sufficient decrease of function value will be made and the total number of iterations is bounded by $T^{\prime}$.

For any $t\geq 0$, suppose that $\|\mathbf{h}_{t}^{*}\|_{2}\geq\sqrt{\epsilon/\rho}$ or $\|\mathbf{v}_{t}\|_{2}\geq\max\{M_{t}\epsilon/(2\rho),\sqrt{LM_{t}/2}(\epsilon/\rho)^{3/4}\}$. We set $\eta=1/(16L)$. Then for $\epsilon<16L^{2}\rho/M_{t}^{2}$, with probability at least $1-\delta$, $\text{Cubic-Subsolver}(\mathbf{U}_{t},\mathbf{v}_{t},M_{t},\eta,\sqrt{\epsilon/\rho},0.5,\delta)$ will return $\mathbf{h}_{t}$ satisfying $m_{t}(\mathbf{h}_{t})\leq-M_{t}\rho^{-3/2}\epsilon^{3/2}/24$. within

iterations, where $C_{S}>0$ is a constant.

We have the following lemma which provides the guarantee for the dynamic of gradient steps in Cubic-Finalsolver.

Then for Cubic-Finalsolver, suppose that $\eta<(4(L+\tau R))^{-1}$, then each iterate $\Delta$ in Cubic-Finalsolver satisfies that $\|\Delta\|_{2}\leq\|\mathbf{s}\|_{2}$, and $g(\mathbf{h})$ is $(L+2\tau R)$-smooth.

With these lemmas, we begin our proof of Theorem 5.1.

Suppose that $\text{SRVRC}_{\text{free}}$ terminates at iteration $T^{*}-1$. Then $T^{*}\leq T$. We first claim that $T^{*}<T$. Otherwise, suppose $T^{*}=T$, then we have that for all $0\leq t<T^{*}$,

where the second inequality holds due to $M_{t}=4\rho$ and Lemma A.2. By Lemma B.3 and union bound, we know that with probability at least $1-T\delta$, we have

where we use the fact that $M_{t}=4\rho$. By Lemmas B.1 and B.2, we know that with probability at least $1-2T\delta$, for all $0\leq t\leq T^{*}-1$, we have

Substituting (B.2) and (B.3) into (B.1), we have

Telescoping (B.4) from $t=0$ to $T^{*}-1$, we have

where the last inequality holds since we assume $T^{*}=T\geq 25\Delta_{F}\rho^{1/2}\epsilon^{-3/2}$ from the condition of Theorem 5.1. (B.5) leads to a contradiction, thus we have $T^{*}<T$. Therefore, by union bound, with probability at least $1-3T\delta$, Cubic-Finalsolver is executed by $\text{SRVRC}_{\text{free}}$ at $T^{*}-1$ iteration. We have that $\|\mathbf{v}_{T^{*}-1}\|_{2}<\max\{M_{T^{*}-1}\epsilon/(2\rho),\sqrt{LM_{T^{*}-1}/2}(\epsilon/\rho)^{3/4}\}$ and $\|\mathbf{h}_{T^{*}-1}^{*}\|_{2}<\sqrt{\epsilon/\rho}$ by Lemma B.3.

The only thing left is to check that we indeed find a second-order stationary point, $\mathbf{x}_{T^{*}}$, by Cubic-Finalsolver. We first need to check that the choice of $\eta=1/(16L)$ satisfies that $1/\eta>4(L+M_{t}R)$ by Lemma B.4, where

We can check that with the assumption that $\|\mathbf{v}_{T^{*}-1}\|_{2}<\max\{M_{T^{*}-1}\epsilon/(2\rho),\sqrt{LM_{T^{*}-1}/2}(\epsilon/\rho)^{3/4}\}$, if $\epsilon<4L^{2}\rho/M_{T^{*}-1}^{2}$, then $1/\eta>4(L+M_{T^{*}-1}R)$ holds.

For simplicity, we denote $\widetilde{T}=T^{*}-1$. Then we have

where the first inequality holds due to Lemma A.1, the second inequality holds due to the fact that $\|\mathbf{h}_{\widetilde{T}}\|_{2}\leq\|\mathbf{h}_{\widetilde{T}}^{*}\|_{2}$ from Lemma B.4, the last inequality holds due to the facts that $\|\nabla m_{\widetilde{T}}(\mathbf{h}_{\widetilde{T}})\|_{2}\leq\epsilon$ from Cubic-Finalsolver and $\|\mathbf{h}_{\widetilde{T}}^{*}\|_{2}\leq\sqrt{\epsilon/\rho}$ by Lemma B.3. ∎

B.2 Proof of Corollary 5.2

We have the following lemma to bound the total number of iterations $T^{\prime\prime}$ of Algorithm 4 Cubic-Finalsolver.

If $\epsilon<4L^{2}\rho/M_{t}^{2}$, then Cubic-Finalsolver will terminate within $T^{\prime\prime}=C_{F}L/\sqrt{\rho\epsilon}$ iterations, where $C_{F}>0$ is a constant.

where the first inequality holds due to Hölder’s inequality, the second inequality holds due to the facts that $T^{*}\leq T=25\Delta_{F}\rho^{1/2}/\epsilon^{3/2}$ and $\Delta_{F}\geq\rho\sum_{t=0}^{T^{*}-1}\|\mathbf{h}_{t}\|_{2}^{3}/4$ by (B.5). We first consider the total stochastic gradient computations, $\sum_{t=0}^{T^{*}-1}B_{t}^{(g)}$, which can be bounded as

where $C_{1}=2640\log^{2}(d/\delta)$, the second inequality holds due to (B.6), the last equality holds due to the fact $S^{(g)}=\sqrt{\rho\epsilon}/L\cdot\sqrt{n\land M^{2}/\epsilon^{2}}$. We now consider the total amount of Hessian-vector product computations ${\mathcal{T}}$, which includes ${\mathcal{T}}_{1}$ from Cubic-Subsolver and ${\mathcal{T}}_{2}$ from Cubic-Finalsolver. By Lemma B.3, we know that at $k$-th iteration of $\text{SRVRC}_{\text{free}}$, Cubic-Subsolver has $T^{\prime}$ iterations, which needs $B_{k}^{(h)}$ Hessian-vector product computations each time. Thus, we have

where $C_{2}=1200\log^{2}(d/\delta)$, the first inequality holds due to the fact that $B_{k}^{(h)}=C_{2}n\land(L^{2}/\rho\epsilon)$, the second inequality holds due to the fact that $T=25\Delta_{F}\rho^{1/2}/\epsilon^{3/2}$, the last inequality holds due to the fact that $T^{\prime}=C_{S}L/M_{t}\cdot\sqrt{\rho/\epsilon}=C_{S}L/(4\sqrt{\rho\epsilon})$. For ${\mathcal{T}}_{2}$, we have

where the first inequality holds due to the fact that $B_{T^{*}-1}^{(h)}=C_{2}n\land(L^{2}/\rho\epsilon)$, the second inequality holds due to the fact that $T^{\prime\prime}=C_{F}L/\sqrt{\rho\epsilon}$ by Lemma B.5. Since at each iteration we need $B_{T^{*}-1}^{(h)}$ Hessian-vector computations.

Combining (B.7), (B.8) and (B.9), we know that the total stochastic gradient and Hessian-vector product computations are bounded as

Appendix C Proofs of Technical Lemmas in Appendix A

We have the following lemmas from Zhou et al. (2018d)

(Zhou et al., 2018d) If $M_{t}\geq 2\rho$, then we have

(Zhou et al., 2018d) If $M_{t}\geq 2\rho$, then we have

where the second inequality holds due to the fact that for any $a,b,c\geq 0$, we have $(a+b+c)^{3/2}\leq 2(a^{3/2}+b^{3/2}+c^{3/2})$. By Lemma C.2, we have

where the second inequality holds due to the fact that for any $a,b,c\geq 0$, we have $(a+b+c)^{3}\leq 9(a^{3}+b^{3}+c^{3})$. Thus we have

where the inequality holds due to (C.1), (C.2) and the fact that $M_{t}\geq 4\rho$. ∎

C.2 Proof of Lemma A.2

where the first inequality holds due to Cauchy–Schwarz inequality, the second inequality holds due to Young’s inequality. We also have

where the first inequality holds due to Cauchy–Schwarz inequality, the second inequality holds due to Young’s inequality. ∎

C.3 Proof of Lemma A.3

Conditioned on $\mathcal{F}_{k}$, with probability at least $1-\delta$ , we have

First, we have $\mathbf{v}_{t}-\nabla F(\mathbf{x}_{t})=\sum_{k=\lfloor t/S^{(g)}\rfloor\cdot S^{(g)}}^{t}\mathbf{u}_{k}$, where

where the second inequality holds due to (4.1). For $\text{mod}(k,S^{(g)})=0$, with probability at least $1-\delta$, we have

where the second inequality holds due to (4.3). Conditioned on $\mathcal{F}_{\lfloor t/S^{(g)}\rfloor\cdot S^{(g)}}$, by union bound, with probability at least $1-\delta\cdot(t-\lfloor t/S^{(g)}\rfloor\cdot S^{(g)})$ (C.5) or (C.6) holds for all $\lfloor t/S^{(g)}\rfloor\cdot S^{(g)}\leq k\leq t$. Then for given $k$, by vector Azuma-Hoeffding inequality in Lemma F.1, conditioned on$\mathcal{F}_{k}$, with probability at least $1-\delta$ we have

Finally, by union bound, we have that with probability at least $1-2\delta\cdot(t-\lfloor t/S^{(g)}\rfloor\cdot S^{(g)})$, for all $\lfloor t/S^{(g)}\rfloor\cdot S^{(g)}\leq k\leq t$, we have (C.7) holds. ∎

C.4 Proof of Lemma A.4

Conditioned on $\mathcal{F}_{k}$, with probability at least $1-\delta$ , we have the following concentration inequality

First, we have $\mathbf{U}_{t}-\nabla^{2}F(\mathbf{x}_{t})=\sum_{k=\lfloor t/S^{(h)}\rfloor\cdot S^{(h)}}^{t}\mathbf{V}_{k}$, where

where the second inequality holds due to (4.1). For $\text{mod}(k,S^{(h)})=0$, with probability at least $1-\delta$, we have

where the second inequality holds due to (4.3). Conditioned on $\mathcal{F}_{\lfloor t/S^{(h)}\rfloor\cdot S^{(h)}}$, by union bound, with probability at least $1-\delta\cdot(t-\lfloor t/S^{(h)}\rfloor\cdot S^{(h)})$ (C.10) or (C.11) holds for all $\lfloor t/S^{(h)}\rfloor\cdot S^{(h)}\leq k\leq t$. Then for given $k$, by Matrix Azuma inequality Lemma F.2, conditioned on$\mathcal{F}_{k}$, with probability at least $1-\delta$ we have

Finally, by union bound, we have that with probability at least $1-2\delta\cdot(t-\lfloor t/S^{(h)}\rfloor\cdot S^{(h)})$, for all $\lfloor t/S^{(h)}\rfloor\cdot S^{(h)}\leq k\leq t$, we have (C.12) holds.

Appendix D Proofs of Technical Lemmas in Appendix B

We have the following lemma which guarantees the effectiveness of Cubic-Subsolver in Algorithm 3.

then $\mathbf{x}=\text{Cubic-Subsolver}(\mathbf{A},\mathbf{b},\tau,\eta,\zeta,\epsilon^{\prime},\delta^{\prime})$ satisfies that $g(\mathbf{x})\leq-(1-\epsilon^{\prime})\tau\zeta^{3}/12$.

We simply set $\mathbf{A}=\mathbf{U}_{t}$, $\mathbf{b}=\mathbf{v}_{t}$, $\tau=M_{t}$, $\eta=(16L)^{-1}$, $\zeta=\sqrt{\epsilon/\rho}$, $\epsilon^{\prime}=0.5$ and $\delta^{\prime}=\delta$. We have $\|\mathbf{U}_{t}\|_{2}\leq L$, then we set $\beta=L$. With the choice of $M_{t}$ where $M_{t}=4\rho$ and the assumption that $\epsilon<4L^{2}\rho/M_{t}^{2}$, we can check that $\eta<1/(8\beta+2\tau\zeta)$. We also have that $\mathbf{s}=\mathbf{h}_{t}^{*}$ and (D.1) holds. Thus, by Lemma D.1, we have

By the choice of $T^{\prime}$ in Cubic-Subsolver, we have

D.2 Proof of Lemma B.5

We have the following lemma which provides the guarantee for the function value in Cubic-Finalsolver.

In Cubic-Finalsolver we are focusing on minimizing $m_{T^{*}-1}(\mathbf{h})$. We have that $\|\mathbf{v}_{t}\|_{2}<\max\{M_{t}\epsilon/(2\rho),\sqrt{LM_{t}/2}(\epsilon/\rho)^{3/4}\}$ and $\|\mathbf{h}_{T^{*}-1}^{*}\|_{2}\leq\sqrt{\epsilon/\rho}$ by Lemma B.3. We can check that $\eta=(16L)^{-1}$ satisfies that $\eta<(4(L+\tau R))^{-1}$, where $R$ is defined in Lemma B.4, when $\epsilon<4L^{2}\rho/M_{t}^{2}$. From Lemma B.4 we also know that $m_{T^{*}-1}$ is $(L+2M_{T^{*}-1}R)$-smooth, which satisfies that $1/\eta>2(L+2M_{T^{*}-1}R)$. Thus, by standard gradient descent analysis, to get a point $\Delta$ where $\|\nabla m_{T^{*}-1}(\Delta)\|_{2}\leq\epsilon$, Cubic-Finalsolver needs to run

iterations, where we denote by $\Delta_{0}$ the starting point of Cubic-Finalsolver. By directly computing, we have $m_{T^{*}-1}(\Delta_{0})\leq 0$. By Lemma D.2, we have

Thus, (D.2) can be further bounded as $T^{\prime\prime}=O(L/\sqrt{\rho\epsilon})$. ∎

Appendix E Proofs of Additional Lemmas in Appendix C

We only need to consider the case where $B^{(g)}_{k}=|\mathcal{J}_{k}|<n$. For each $i\in\mathcal{J}_{k}$, let

where the second inequality holds due to the $L$-smoothness of $f_{i}$ and $F$. Thus by vector Azuma-Hoeffding inequality in Lemma F.1, we have that with probability at least $1-\delta$,

E.2 Proof of Lemma C.4

We only need to consider the case where $B^{(h)}_{k}=|\mathcal{I}_{k}|<n$. For each $i\in\mathcal{I}_{k}$, let

where the second inequality holds due to $\rho$-Hessian Lipschitz continuous of $f_{i}$ and $F$. Then by Matrix Azuma inequality Lemma F.2, we have that with probability at least $1-\delta$,

Appendix F Auxiliary Lemmas

We have the following vector Azuma-Hoeffding inequality:

We have the following Matrix Azuma inequality :

(Tropp, 2012) Consider a finite adapted sequence $\{\mathbf{X}_{k}\}$ of self-adjoint matrices in dimension $d$, and a fixed sequence $\{\mathbf{A}_{k}\}$ of self-adjoint matrices that satisfy

Then we have that with probability at least $1-\delta$,

Appendix G Additional Algorithms and Functions

Due to space limit, we include the approximate solvers (Carmon and Duchi, 2016) for the cubic subproblem in this section for the purpose of self-containedness.

References