On the Adaptivity of Stochastic Gradient-Based Optimization

TL;DR

This paper introduces the SCSG method, an adaptive stochastic gradient algorithm for convex optimization that adjusts to different problem regimes, achieving better theoretical complexity with minimal tuning.

Contribution

The paper proposes SCSG, a novel adaptive algorithm that automatically adjusts to strong convexity and accuracy levels using batch variance reduction and geometrization techniques.

Findings

SCSG outperforms existing adaptive algorithms in theoretical complexity.

SCSG requires tuning parameters only based on smoothness, simplifying practical use.

The method effectively adapts to various regimes without prior regime knowledge.

Abstract

Stochastic-gradient-based optimization has been a core enabling methodology in applications to large-scale problems in machine learning and related areas. Despite the progress, the gap between theory and practice remains significant, with theoreticians pursuing mathematical optimality at a cost of obtaining specialized procedures in different regimes (e.g., modulus of strong convexity, magnitude of target accuracy, signal-to-noise ratio), and with practitioners not readily able to know which regime is appropriate to their problem, and seeking broadly applicable algorithms that are reasonably close to optimality. To bridge these perspectives it is necessary to study algorithms that are adaptive to different regimes. We present the stochastically controlled stochastic gradient (SCSG) method for composite convex finite-sum optimization problems and show that SCSG is adaptive to both strong convexity and target accuracy. The adaptivity is achieved by batch variance reduction with adaptive batch sizes and a novel technique, which we referred to as geometrization, which sets the length of each epoch as a geometric random variable. The algorithm achieves strictly better theoretical complexity than other existing adaptive algorithms, while the tuning parameters of the algorithm only depend on the smoothness parameter of the objective.

Equations

$$\underset{x\in \mathcal{X}}{\min }F(x)=f(x)+\psi (x),\text{where }f(x)=\frac{1}{n}\sum _{i=1}^n{f}_i(x),$$

$$\nabla f_{\mathcal{I}}(x)=\frac{1}{|\mathcal{I}|}\sum _{i\in \mathcal{I}}\nabla f_i(x).$$

$$f_i(x)-f_i(y)-⟨\nabla f_i(y),x-y⟩\le \frac{L}{2}\Vert x-y{\Vert }_2^2,\forall i=1,\dots ,n,$$

$$f(x)-f(x^{\ast })\ge \frac{\mu }{2}\Vert x-x^{\ast }{\Vert }_2^2,$$

$$\mathcal{H}(f)=\frac{1}{n}\sum _{i=1}^n\Vert \nabla f_i(x^{\ast }){\Vert }_2^2,$$

$$D_x=L\mathbb{E}\Vert {\tilde{x}}_0-x^{\ast }{\Vert }_2^2,D_H=\mathcal{H}/L,D=\max \{D_x,D_H\}.$$

$$\mathbb{E}N=\frac{\gamma }{1-\gamma }.$$

$$T(ϵ)=\min \{T^{\prime }:\mathbb{E}(f({\tilde{x}}_T)-f(x^{\ast }))\le ϵ,\forall T\ge T^{\prime }\}.$$

$$C_{\mathrm{comp}}(ϵ)=\sum _{j=1}^{T(ϵ)}(b_j{N}_j+B_j).$$

$$\mathbb{E}{C}_{\mathrm{comp}}(ϵ)=\sum _{j=1}^{T(ϵ)}\left(b_j\frac{m_j}{b_j}+B_j\right)=\sum _{j=1}^{T(ϵ)}(m_j+B_j).$$

$$B_j=m_j\equiv n,b_j=1,{\eta }_j\equiv \eta =O\left(\frac{1}{L}\right).$$

$$\frac{1}{\mu \eta (1-2\eta L)m}+\frac{2\eta L}{1-2\eta L}<1.$$

$$2\eta \mathbb{E}⟨\nabla f(x_k^{(j)}),x_k^{(j)}-x^{\ast }⟩-4{\eta }^{2}L\mathbb{E}(f(x_k^{(j)})-f(x^{\ast }))$$

$$\le 4{\eta }^{2}L\mathbb{E}(f(x_0^{(j)})-f(x^{\ast }))+\mathbb{E}\Vert x_k^{(j)}-x^{\ast }{\Vert }_2^2-\mathbb{E}\Vert x_{k+1}^{(j)}-x^{\ast }{\Vert }_2^2.$$

$$2\eta (1-2\eta L)\mathbb{E}(f(x_k^{(j)})-f(x^{\ast }))+\eta \mu \mathbb{E}\Vert x_k^{(j)}-x^{\ast }{\Vert }_2^2$$

$$\le 4{\eta }^{2}L\mathbb{E}(f(x_0^{(j)})-f(x^{\ast }))+\mathbb{E}\Vert x_k^{(j)}-x^{\ast }{\Vert }_2^2-\mathbb{E}\Vert x_{k+1}^{(j)}-x^{\ast }{\Vert }_2^2.$$

$$2\eta (1-2\eta L)\mathbb{E}(f({\tilde{x}}_j)-f(x^{\ast }))+\eta \mu \mathbb{E}\Vert {\tilde{x}}_j-x^{\ast }{\Vert }_2^2$$

$$\le 4{\eta }^{2}L\mathbb{E}(f({\tilde{x}}_{j-1})-f(x^{\ast }))+\mathbb{E}\Vert x_{N_j}^{(j)}-x^{\ast }{\Vert }_2^2-\mathbb{E}\Vert x_{N_j+1}^{(j)}-x^{\ast }{\Vert }_2^2.$$

$$2\eta (1-2\eta L)\mathbb{E}(f({\tilde{x}}_j)-f(x^{\ast }))+\eta \mu \mathbb{E}\Vert {\tilde{x}}_j-x^{\ast }{\Vert }_2^2$$

$$\le 4{\eta }^{2}L\mathbb{E}(f({\tilde{x}}_{j-1})-f(x^{\ast }))+\frac{1}{m}\mathbb{E}\Vert x_0^{(j)}-x^{\ast }{\Vert }_2^2-\frac{1}{m}\mathbb{E}\Vert x_m^{(j)}-x^{\ast }{\Vert }_2^2$$

$$=4{\eta }^{2}L\mathbb{E}(f({\tilde{x}}_{j-1})-f(x^{\ast }))+\frac{1}{m}\mathbb{E}\Vert {\tilde{x}}_{j-1}-x^{\ast }{\Vert }_2^2-\frac{1}{m}\mathbb{E}\Vert x_m^{(j)}-x^{\ast }{\Vert }_2^2.$$

$$2\eta (1-2\eta L)\mathbb{E}(f({\tilde{x}}_j)-f(x^{\ast }))+\eta \mu \mathbb{E}\Vert {\tilde{x}}_j-x^{\ast }{\Vert }_2^2$$

$$\le 4{\eta }^{2}L\mathbb{E}(f({\tilde{x}}_{j-1})-f(x^{\ast }))+\frac{1}{m}\mathbb{E}\Vert {\tilde{x}}_{j-1}-x^{\ast }{\Vert }_2^2.$$

$$2\eta (1-2\eta L)\mathbb{E}(f({\tilde{x}}_j)-f(x^{\ast }))\le \left(4{\eta }^{2}L+\frac{2}{m\mu }\right)\mathbb{E}(f({\tilde{x}}_{j-1})-f(x^{\ast })).$$

$$\mathbb{E}(D_N-D_{N+1})=\left(\frac{1}{\gamma }-1\right)\left(D_0-\mathbb{E}{D}_N\right).$$

$$\mathbb{E}(D_N-D_{N+1})=\sum _{n\ge 0}(D_n-D_{n+1}){\gamma }^n(1-\gamma )$$

$$=(1-\gamma )\left(D_0-\sum _{n\ge 1}{D}_n({\gamma }^{n-1}-{\gamma }^n)\right)=(1-\gamma )\left(\frac{1}{\gamma }{D}_0-\sum _{n\ge 0}{D}_n({\gamma }^{n-1}-{\gamma }^n)\right)$$

$$=(1-\gamma )\left(\frac{1}{\gamma }{D}_0-\frac{1}{\gamma }\sum _{n\ge 0}{D}_n{\gamma }^n(1-\gamma )\right)=\left(\frac{1}{\gamma }-1\right)(D_0-\mathbb{E}{D}_N),$$

$$2\eta (1-2\eta L)\mathbb{E}(f({\tilde{x}}_j)-f(x^{\ast }))+\eta \mu \mathbb{E}\Vert {\tilde{x}}_j-x^{\ast }{\Vert }_2^2$$

$$\le 4{\eta }^{2}L\mathbb{E}(f({\tilde{x}}_{j-1})-f(x^{\ast }))+\frac{1}{n}\mathbb{E}\Vert x_0^{(j)}-x^{\ast }{\Vert }_2^2-\frac{1}{n}\mathbb{E}\Vert x_{N_j}^{(j)}-x^{\ast }{\Vert }_2^2$$

$$=4{\eta }^{2}L\mathbb{E}(f({\tilde{x}}_{j-1})-f(x^{\ast }))+\frac{1}{n}\mathbb{E}\Vert {\tilde{x}}_{j-1}-x^{\ast }{\Vert }_2^2-\frac{1}{n}\mathbb{E}\Vert {\tilde{x}}_j-x^{\ast }{\Vert }_2^2.$$

$$2\eta (1-2\eta L)\mathbb{E}(f({\tilde{x}}_j)-f(x^{\ast }))+\left(\frac{1}{n}+\eta \mu \right)\mathbb{E}\Vert {\tilde{x}}_j-x^{\ast }{\Vert }_2^2$$

$$\le 4{\eta }^{2}L\mathbb{E}(f({\tilde{x}}_{j-1})-f(x^{\ast }))+\frac{1}{n}\mathbb{E}\Vert {\tilde{x}}_{j-1}-x^{\ast }{\Vert }_2^2.$$

$${\mathcal{L}}_j=2\eta (1-2\eta L)\mathbb{E}(f({\tilde{x}}_j)-f(x^{\ast }))+\left(\frac{1}{n}+\eta \mu \right)\mathbb{E}\Vert {\tilde{x}}_j-x^{\ast }{\Vert }_2^2.$$

$${\mathcal{L}}_j\le \max \left\{\frac{2\eta L}{1-2\eta L},\frac{1}{1+n\eta \mu }\right\}{\mathcal{L}}_{j-1}\triangleq {\lambda }^{-1}{\mathcal{L}}_{j-1}.$$

$${\mathcal{L}}_T\le ϵ,\forall T\ge \frac{\log \frac{{\mathcal{L}}_0}{ϵ}}{\log \lambda }⟹T(ϵ)\le \frac{\log \frac{{\mathcal{L}}_0}{ϵ}}{\log \lambda },$$

Full text

{AMS}

90C15, 90C25, 90C06

1 Introduction

The application of gradient-based optimization methodology to statistical machine learning has been a major success story, in practice and in theory. Indeed, there is an increasingly detailed theory available for gradient-based algorithms that helps to explain their practical success. There remains, however, a significant gap between theory and practice, in that the designer of machine learning algorithms is required to make numerous choices that depend on parameters that are unlikely to be known in a real-world machine-learning setting. For example, existing theory asserts that different algorithms are preferred if a problem is strongly convex or merely convex, if the target accuracy is high or low, if the signal-to-noise is high or low and if data are independent or correlated. This poses a serious challenge to builders of machine-learning software, and to users of that software. Indeed, a distinctive aspect of machine-learning problems, especially large-scale problems, is that the user of an algorithm can be expected to know little or nothing about quantitative structural properties of the functions being optimized. It is hoped that the data and the data analysis will inform such properties, not the other way around.

To take a classical example, the stochastic gradient descent (SGD) algorithm takes different forms for strongly convex objectives and non-strongly convex objectives. In the former case, letting $\mu$ denote the strong-convexity parameter, if the stepsize is set as $O(1/\mu t)$ then SGD exhibits a convergence rate of $O(1/\mu\epsilon)$, where $\epsilon$ is the target accuracy (nesterov04, ). In the latter case setting the stepsize to $O(1/\sqrt{t})$ yields a rate of $O(1/\epsilon^{2})$ (Nemirovsky09, ). Using the former scheme for non-strongly convex objectives can significantly deteriorate the convergence (Nemirovsky09, ). It is sometimes suggested that one can insure strong convexity by simply adding a quadratic regularizer to the objective, using the coefficient of the regularizer as a conservative estimate of the strong-convexity parameter. But this produces a significantly faster rate only if $\mu\gg\epsilon$, a regime that is unrealistic in many machine-learning applications, where $\epsilon$ is relatively large. Setting $\mu$ to such a large value would have a major effect on the statistical properties of the optimizer.

Similar comments apply to presumptions of knowledge of Lipschitz parameters, mini-batch sizes, variance-reduction tuning parameters, etc. Current practice often involves heuristics in setting these tuning parameters, but the use of these heuristics can change the algorithm and the optimality guarantees may disappear.

Our goal, therefore, should be that our algorithms are adaptive, in the sense that they perform as well as an algorithm that is assumed to know the “correct” choice of tuning parameters, even if they do not know those parameters. In particular, in the convex setting, we wish to derive an algorithm that does not involve $\mu$ in its implementation but whose convergence rate would be better for larger $\mu$ while still reasonable for smaller $\mu$, including the non-strongly convex case where $\mu=0$.

Such adaptivity has been studied implicitly in the classic literature. ruppert1988efficient and polyak1990new and polyak1992acceleration showed that the average iterate of SGD with stepsize $O(t^{-\alpha})$ for $\alpha\in(1/2,1)$ satisfies a central limit theorem with information-theoretically optimal asymptotic variance. This implies adaptivity because the performance adapts to the underlying parameters of the problem, including the modulus of strong convexity, even though the algorithm does not require knowledge of them. The analysis by polyak1992acceleration is, however, asymptotic and relies on the smoothness of Hessian. Under similar assumptions on the Hessian, moulines2011non provided a non-asymptotic analysis establishing adaptivity of SGD with Polyak-Ruppert averaging. Further contributions to this line of work include bach13 ; flammarion15 ; dieuleveut17 , who prove the adaptivity of certain versions of SGD with refined rates for self-concordant objectives, including least-square regression and logistic regression.

This line of work relies on conditions on higher-order derivatives which are not required in the modern literature on stochastic gradient methods. In fact, under fairly standard assumptions for first-order methods, moulines2011non provided a non-asymptotic analysis for SGD with stepsize $O(t^{-\alpha})$ without averaging and showed that this algorithm exhibits adaptivity to strong convexity while having reasonable guarantee for non-strongly convex objectives. Specifically, if $\alpha=2/3$, their results show that the rate to achieve an $\epsilon$-accurate solution for the expected function value is \tilde{O}\big{(}\min\big{\{}1/\mu^{3}+1/\mu\epsilon^{2}, 1/\epsilon^{3}\big{\}}\big{)}, where $\tilde{O}$ hides logarithmic factors. This result was taken further by studying alternative stepsize schemes; in particular, xu2019accelerate proposed a variant of projected SGD with stagewise diminishing stepsizes and diameters. Unlike the aforementioned work, the adaptivity in this case is weaker because it requires knowledge of the strong convexity parameter, as well as the initial suboptimality, in the complexity analysis (in particular, they require a sufficiently large initial diameter). A weaker form of adaptivity, to smoothness but not to strong convexity, was established by levy2018online when the polynomial decaying stepsize is replaced by an AdaGrad-type stepsize (Adagrad, ), assuming a bounded domain and bounded stochastic gradients. Finally, chen2018sadagrad presented a restarting variant of AdaGrad with provable adaptivity to strong convexity given an initial overestimate of the strong convexity parameter.

Further progress has been made by focusing on a setting that is particularly relevant to machine learning—that of finite-sum optimization. The objective function in this setting takes the following form:

[TABLE]

where $\mathcal{X}$ is the parameter space, $n$ is the number of data points, the functions $f_{i}(x)$ are data-point-specific loss functions and $\psi(x)$ is the regularization term. We assume that each $f_{i}(x)$ is differentiably convex and $\psi(x)$ is convex but can be non-differentiable. The introduction of the parameter $n$ into the optimization problem has two important implications. First, it implies that the number of operations to obtain a full gradient is $O(n)$, which is generally impractical in modern machine-learning applications, where the value of $n$ can be in the tens to hundreds of millions. This fact motivates us to make use of stochastic estimates of gradients. Such randomness introduces additional variance that interacts with the variability of the data, and tuning parameters are often introduced to control this variance.

Second, the finite-sum formulation highlights the need for adaptivity to the target accuracy $\epsilon$, where that accuracy is related to the number of data points $n$ for statistical reasons. Unfortunately, different algorithms perform better in high-accuracy versus low-accuracy regimes, and the choice of regime is generally not clear to a user of machine-learning algorithms, given that target accuracy varies not only as a function of $n$, but also as a function of other parameters, such as the signal-to-noise ratio, that the user is not likely to know. Ideally, therefore, optimization algorithms should be adaptive to target accuracy, performing well in either regime.

Deterministic gradient-descent-based methods can be made adaptive to strong convexity and smoothness simultaneously by exploiting the Polyak stepsize (hazan2019revisiting, ). However, computation of a full gradient is expensive, rendering the method undesirable for finite-sum optimization. A recent line of research has shown that algorithms with lower complexity can be designed in the finite-sum setting with some adaptivity, generally via careful control of the variance. The stochastic average gradient (SAG) method opened this line of research, establishing a complexity of $\tilde{O}\left(\min\{n/\epsilon,n+L/\mu\}\right)$ (SAG, ). Importantly, this result shows that SAG is adaptive to strong convexity. To achieve such adaptivity, however, SAG requires two sequences of iterates, the average iterate and the last iterate. SAGA propose a single-sequence variant of SAG that is also adaptive to strong convexity, yet under stronger assumption that each $f_{i}$ is strongly convex. Both methods suffer, however, from a prohibitive storage cost of $O(nd)$, where $d$ is the dimension of $\mathcal{X}$. Further developments in this vein include the stochastic variance reduced gradient (SVRG) method (SVRG, ) and the stochastic dual coordinate ascent (SDCA) method (SVRG, ); they achieve the same computational complexity as SAG while reducing the storage cost to $O(d)$. They are not, however, adaptive to strong convexity.

SCSG presented a randomized variant of SVRG that achieves the same convergence rate and adaptivity as SAG but with the same storage cost as SVRG. However, as is the case with SAG, the complexity of $O(n/\epsilon)$ for the non-strongly convex case is much larger than the oracle lower bound of $O(n+\sqrt{n/\epsilon})$ (woodworth16, ). nguyen2019finite propose another variance-reduction method that is provably adaptive to strong convexity, though the result is proved for the expected gradient norm and is thus weaker than SCSG . xu17 develop another variant of SVRG which adapts to a more general condition, called a “Hölderian error bound,” with strong convexity being a special case. In contrast to SCSG , they required an initial conservative estimate of the strong convexity parameter. Under an extra strong assumption of gradient interpolation—that all individual loss functions have vanishing gradients at the optimum—vaswani2019painless developed an algorithm that achieves adaptivity to the smoothness parameter and to the modulus of strong convexity simultaneously. On the other hand, recent work of lan2019unified that came after our work presents an algorithm that achieves certain adaptivity to the target accuracy. However, they need to know the modulus of strong convexity and the adaptivity is only obtained in the high-accuracy regime because it requires full gradient computations periodically. Finally, while our focus is convex optimization, we note that adaptivity has also been studied for nonconvex finite-sum optimization (e.g., lei17 ; paquette2017catalyst ).

In this article we present an algorithm, the stochastically controlled stochastic gradient (SCSG) algorithm, that exhibits adaptivity both to strong convexity and to target accuracy. SCSG is a nested procedure that is similar to the SVRG algorithm. Crucially, it does not require the computation of a full gradient in the outer loop as performed by SVRG, but makes use of stochastic estimates of gradients in both the outer loop and the inner loop. Moreover, it makes essential use of a randomization technique (“geometrization”) that allows terms to telescope across the outer loop and the inner loops; such telescoping does not happen in SVRG, a fact which leads to the loss of adaptivity for SVRG.

The rest of the article is organized as follows: Section 2 introduces notation, assumptions and definitions. In Section 3 and Section 4, we focus on the relatively simple setting of unregularized problems and Euclidean geometry, introducing the key ideas of geometrization and adaptive batching. We provide key proofs in Section 4 and leave non-essential proofs into Appendix A in the Supplementary Material. We extend these results to regularized problems and to non-Euclidean geometry in Section 5. The extension relaxes standard assumptions for analyzing mirror descent methods and may be of independent interest. All proofs for the general case are relegated into Appendix B and some miscellaneous results are presented in Appendix C in Supplementary Material. Finally, the desirable empirical performance of SCSG is demonstrated in Appendix D.

2 Notation, Assumptions and Definitions

We write $a\wedge b$ (resp. $a\vee b$) for $\min\{a,b\}$ (resp. $\max\{a,b\}$), and $(a)_{*}^{\xi}$ (or $[a]_{*}^{\xi}$) for $\max\{a,1\}^{\xi}$ throughout the paper. The symbol $\mathbb{E}$ denotes the expectation of a random element and $\mathbb{E}_{X}$ denotes an expectation over the randomness of $X$ while conditioning on all other random elements. We adopt Landau’s notation ($O(\cdot),o(\cdot)$), and we occasionally use $\tilde{O}(\cdot)$ to hide logarithmic factors. We define computational cost by making use of the IFO framework of Agarwal14 ; reddi16svrg , where we assume that sampling an index $i$ and computing the pair $(f_{i}(x),\nabla f_{i}(x))$ incurs a unit of cost. For notational convenience, given a subset $\mathcal{I}\subset\{1,\ldots,n\}$, we denote by $\nabla f_{\mathcal{I}}(x)$ the batch gradient:

[TABLE]

By definition, computing $\nabla f_{\mathcal{I}}(x)$ incurs $|\mathcal{I}|$ units of cost.

In this section and the following two sections we focus on unregularized problems and Euclidean geometry, turning to regularized problems and non-Euclidean geometry in Section 5. Specifically, we consider the case $\mathcal{X}=\mathbb{R}^{d}$, $\psi\equiv 0$ and make the following assumptions that target the finite-sum optimization problem:

1. A1

$f_{i}$ is convex with $L$-Lipschitz gradient

[TABLE]

for some $L<\infty$; 2. A2

$F=f$ is strongly convex at $x^{*}$ with

[TABLE]

for some $\mu\geq 0$.

Note that assumption A2 always holds with $\mu=0$, corresponding to the non-strongly convex case. Note also that with the exception of SAG , this assumption is weaker than most of the the literature on smooth finite-sum optimization, where strong convexity of $f$ is required at every point.

Our analysis will make use of the following key quantity (SCSG, ):

[TABLE]

where $x^{*}$ denotes the optimum of $f$. If multiple optima exist we take one that minimizes $\mathcal{H}(f)$. We use $\mathcal{H}(f)$, an average squared norm at the optimum, in place of the uniform upper bound on the gradient that is often assumed in other work. The latter is not realistic for many practical problems in machine learning, including least squares, where the gradient is unbounded. On the other hand, it is noteworthy that $\mathcal{H}(f)$ only depends on the optimum. For instance, $\mathcal{H}(f)=0$ if all gradients vanish at the optimum, as studied for over-parametrized models (e.g., (ma2017power, ; vaswani2018fast, ; vaswani2019painless, )). As will be shown later, the complexity of our algorithm only depends on $\mathcal{H}(f)$ so it can be applied to study the case with data interpolation. We will write $\mathcal{H}(f)$ as $\mathcal{H}$ when no confusion can arise.

We let $\tilde{x}_{0}$ denote the initial value (possibly random) and define the following measures of complexity:

[TABLE]

Recall that a geometric random variable, $N\sim\mathrm{Geom}(\gamma)$, is a discrete random variable with probability mass function $P(N=k)=(1-\gamma)\gamma^{k},\mbox{for}\ k=0,1,\ldots$, and expectation:

[TABLE]

Geometric random variables will play a key role in the design and analysis of our algorithm.

Finally, we introduce two fundamental definitions that serve to clarify desirable properties of optimization algorithms. We refer to the first property as $\epsilon$-independence. {definition*} An algorithm is $\mathbf{\epsilon}$-independent if it guarantees convergence at all target accuracies $\epsilon$.

$\epsilon$-independence is a crucial property in practice because a target accuracy is usually not exactly known apriori. An $\epsilon$-independent algorithm satisfies the “one-pass-for-all” property where the theoretical complexity analysis applies to the whole path of the iterates. In contrast, an $\epsilon$-dependent algorithm only has a theoretical guarantee for a particular $\epsilon$, whose value is often unknown in practice. To illustrate we consider SGD, where the iterate is updated by $x_{k+1}=x_{k}-\eta_{k}\nabla f_{i_{k}}(x_{k})$ and where $i_{k}$ is a uniform index from $\{1,\ldots,n\}$. There are two popular schemes for theoretical analysis: (1) $\eta_{k}=O(1/\sqrt{k})$ or (2) $\eta_{k}\equiv 1/\sqrt{T}$ and the iterates are updated for $O(T)$ steps where $T=O(1/\epsilon^{2})$. Although both versions have theoretical complexity $\tilde{O}\left(\epsilon^{-2}\right)$, only the former is $\epsilon$-independent.

The second important property is referred to as almost universality. {definition*} An algorithm is almost universal if it only requires the knowledge of the smoothness parameters $L$.

The term almost universality is motivated by the notion of universality introduced by nesterov14 which does not require the knowledge of $L$ or other parameters such as the variance of the stochastic gradients. Returning to the previous example, both versions of SGD are universal. It is noteworthy that universal gradient methods are usually either $\epsilon$-dependent (e.g. (nesterov14, )) or require imposing other assumptions such as uniformly bounded $\nabla f_{i}$ (e.g. (Nemirovsky09, )). The SCSG algorithm developed in this paper is both $\epsilon$-independent and almost universal. This category also includes SGD for general convex functions (Nemirovsky09, ), SAG (SAG, ), SAGA (SAGA, ), SVRG++ (Zhu15, ), Katyusha for non-strongly convex functions (Katyusha, ), and AMSVRG (nitanda16, ). In contrast, algorithms such as SGD for strongly convex functions (Nemirovsky09, ), SVRG (SVRG, ), SDCA (SDCA, ), APCG (APCG, ), Katyusha for strongly convex functions (Katyusha, ) and adaptive SVRG (xu17, ) are $\epsilon$-independent but not almost universal because they need full or partial knowledge of $\mu$. Furthermore, algorithms such as Catalyst (Catalyst, ) and AdaptReg (blackbox, ) even depend on unknown quantities such as $F(x_{0})-F(x^{*})$ or the variance of the $\nabla f_{i}$. In comparing algorithms we believe that clarity on these distinctions is critical, in addition to comparison of convergence rates.

3 Stochastically Controlled Stochastic Gradient (SCSG)

In this section we present SCSG, a computationally efficient framework for variance reduction in stochastic gradient descent algorithms. SCSG builds on the SVRG algorithm of SVRG , incorporating several essential modifications that yield not only computational efficiency but also adaptivity. Recall that SVRG is a nested procedure that computes a full gradient in each outer loop and uses that gradient as a baseline to reduce the variance of the stochastic gradients that are computed in an inner loop. The need to compute a full gradient, at a cost of $n$ operations, unfortunately makes the SVRG procedure impractical for large-scale applications. SCSG seeks to remove this bottleneck by replacing the full gradient with an approximate, stochastic gradient, one that is based on a batch size that is significantly smaller than $n$ but larger than the size used for the stochastic gradients in the inner loop. By carefully weighing the contributions to the bias and variance of these sampling-based estimates, SCSG achieves a small iteration complexity while also keeping the per-iteration complexity feasibly small.

Further support for the SCSG framework comes from the comparison with SVRG in the setting of strongly convex objectives. In this setting, SVRG relies heavily on a presumption of knowledge of the strong convexity parameter $\mu$. In particular, to achieve a complexity of $O((n+\kappa)\log(1/\epsilon))$, the number of stochastic gradients queried in the inner loop of SVRG needs to scale as $\kappa$. By contrast, the SCSG framework achieves the same complexity without knowledge of $\mu$. This is achieved by setting the number of inner-loop stochastic gradients to be a geometric random variable. As we discuss below, the usage of a geometric random variable—a technique that we refer to as “geometrization”—is crucial in the design and analysis of SCSG. We believe that it is a key theoretical tool for achieving adaptivity to strong convexity.

The original version of SCSG was $\epsilon$-dependent and not almost universal, because it required knowledge of the parameter $\mathcal{H}$ (SCSG, ). Moreover the algorithm had a sub-optimal rate in the high-accuracy regime. In further development of the SCSG framework, in the context of nonconvex optimization (lei17, ), we found that $\epsilon$-independence and almost universality could be achieved by employing an increasing sequence of batch sizes.

In the remainder of this section, we bring these ideas together and present the general form of the SCSG algorithm, incorporating adaptive batching, geometrization and mini-batches in the inner loop. The resulting algorithm is adaptive, $\epsilon$-independent and almost universal. Roughly speaking, the adaptive batching enables the adaptivity to target accuracy and the geometrization enables the adaptivity to strong convexity. The pseudocode for SCSG is shown in Algorithm 1. Guidelines for practical choice for the parameters is provided in Remark 4.3 in Section 4.2. As can be seen, the algorithm is superficially complex, but, as in the case of line-search and trust-region methods that augment simple gradient-based methods in deterministic optimization, the relative lack of dependence on hyperparameters makes the algorithm robust and relatively easy to deploy.

Note that in Algorithm 1, and throughout the paper, we use $\tilde{x}_{j}$ to denote the iterate in the $j$th outer loop and $x_{k}^{(j)}$ to denote the iterate in the $k$th step of the $j$th inner loop.

To measure the computational complexity of SCSG, let $T(\epsilon)$ denote the first time step at which $\tilde{x}_{T}$ is an $\epsilon$-approximate solution, as well as all following iterates $\tilde{x}_{T+1},\tilde{x}_{T+2},\ldots$:

[TABLE]

This criterion is more stringent than those considered in some other work which neglects the performance of $\tilde{x}_{T}$ for $T>T(\epsilon)$. The computational cost incurred for computing $\tilde{x}_{T}$ is

[TABLE]

Noting that $C_{\mathrm{comp}}(\epsilon)$ is random, we consider the average complexity obtained by taking the expectation of $C_{\mathrm{comp}}(\epsilon)$. Since $N_{j}\sim\mathrm{Geom}(\frac{m_{j}}{m_{j}+b_{j}})$, we have:

[TABLE]

3.1 Two key ideas: adaptive batching and geometrization

The adaptivity of SCSG is achieved via two techniques: adaptive batching and geometrization. We provide intuitive motivation for these two ideas in this section.

The motivation for adaptive batching is straightforward. Heuristically, at the early stages of the optimization process, the iterate is far from the optimum and a small subset of data is sufficient to reduce the variance. On the other hand, at later stages, finer variance reduction is required to prevent the iterate from moving in the wrong direction. By allowing the batch sizes to increase, SCSG behaves like SGD for the purposes of low-accuracy computation while it behaves like SVRG for high-accuracy computation.

The motivation for geometrization is more subtle. To isolate its effect, let us consider a special case of SCSG in which the parameters are set as follows:

[TABLE]

Note that the above setting is only used to illustrate the effect of geometrization and the setting that leads to adaptivity to both strong convexity and target accuracy is more involved and given in Section 4. In this simplified setting, SCSG reduces to SVRG if we replace line 5 by $N_{j}\sim\mathrm{Unif}\left(\{0,\ldots,m_{j}-1\}\right)$, with $m_{j}\equiv m$ for some positive integer $m$. (Although SVRG is usually implemented in practice by setting $N_{j}$ to be a fixed $m$, a uniform random $N_{j}$ is crucial for the analysis of SVRG (SVRG, ).) SVRG achieves a rate of $O\left((n+\kappa)\log(1/\epsilon)\right)$ rate only if

[TABLE]

This requires $m>\frac{1}{\mu\eta}$; hence, SVRG requires knowledge of $\mu$ to achieve the theoretical rate. We briefly sketch the step in the proof of the convergence of SVRG where this limitation arises, and we show how geometrization circumvents the need to know $\mu$. To simplify our arguments we follow SVRG and make the assumption that strong convexity holds everywhere for $f$; note that this is stronger than our assumption A2.

In Theorem 1 of SVRG , the following argument appears:

[TABLE]

Strong convexity implies that

[TABLE]

Note that this conclusion is independent of the choice of $N_{j}$ and hence holds for both SVRG and SCSG. To assess the overall effect of the $j$th inner loop on the left-hand side, we let $k=N_{j}$, thereby focusing on the last step of the inner loop, and we substitute $\tilde{x}_{j}$ for $x^{(j)}_{N_{j}}$ and $\tilde{x}_{j-1}$ for $x^{(j)}_{0}$. We have:

[TABLE]

For SVRG, $N_{j}\sim\mathrm{Unif}\{0,\ldots,m-1\}$, and thus (8) reduces to

[TABLE]

Unfortunately, given that $x^{(j)}_{m}\not=\tilde{x}_{j}$, the last two terms do not telescope, and one has to drop the final term, leading to the following conservative bound:

[TABLE]

Without strong convexity (i.e., when $\mu=0$), $\mathbb{E}\|\tilde{x}_{j-1}-x^{*}\|_{2}^{2}$ can be arbitrarily larger than $\mathbb{E}(f(\tilde{x}_{j-1})-f(x^{*}))$ and hence (9) is not helpful. Thus SVRG exploit strong convexity at this point, using $\mathbb{E}\|\tilde{x}_{j-1}-x^{*}\|_{2}^{2}\leq\frac{2}{\mu}\mathbb{E}(f(\tilde{x}_{j-1})-f(x^{*}))$. Then (9) implies that

[TABLE]

This requires the coefficient on the left-hand side to be larger than that on the right-hand side, leading to the condition (6).

Summarizing, the reason that SVRG rely on the knowledge of $\mu$ is that it permits the removal of the last term in (8). By contrast, if $N_{j}$ is a geometric random variable instead of a uniform random variable, the problem is completely circumvented, by making use of the following elementary lemma.

Lemma 3.1**.**

Let $N\sim\mathrm{Geom}(\gamma)$ for $\gamma>0$. Then for any sequence $D_{0},D_{1},\ldots$ with $\mathbb{E}|D_{N}|<\infty$,

[TABLE]

Remark 3.2**.**

*The requirement $\mathbb{E}|D_{N}|<\infty$ is essential. A useful sufficient condition if $|D_{k}|=O(\mathrm{poly}(k))$ because a geometric random variable has finite moments of any order. *

Proof 3.3**.**

By definition,

[TABLE]

*where the last equality is followed by the condition that $\mathbb{E}|D_{N}|<\infty$. *

Returning to (8) for SCSG with Lemma 3.1 in hand, where $N_{j}\sim\mathrm{Geom}(\frac{n}{n+1})$ and $D_{k}=\mathbb{E}\|x^{(j)}_{k}-x^{*}\|_{2}^{2}$, and assuming that $\mathbb{E}D_{N_{j}}<\infty$, we obtain

[TABLE]

The assumption that $\mathbb{E}D_{N_{j}}<\infty$ will be justified in our general theory and is taken for granted here to avoid distraction. (10) can be rearranged to yield a function that provides a better assessment of progress than the function in (9):

[TABLE]

We accordingly view the left-hand side of (3.1) as a Lyapunov function and define:

[TABLE]

We then have:

[TABLE]

As a result,

[TABLE]

and, by (4),

[TABLE]

Suppose $\eta L<\frac{1}{6}$,

[TABLE]

Therefore the complexity of SCSG is

[TABLE]

In summary, the better control provided by geometrization enables SCSG to achieve the fast rate of SVRG without knowledge of $\mu$.

4 Convergence Analysis of SCSG for Unregularized Smooth Problems

4.1 One-epoch analysis

We start with the analysis for a single epoch. The key difficulty lies in controlling the bias of $\nu^{(j)}_{k}$, conditional on $\mathcal{I}_{j}$ drawn at the beginning of the $j$th epoch. We have:

[TABLE]

We deal with this extra bias by exploiting Lemma 3.1 and obtaining the following theorem which connects the iterates produced in consecutive epochs. The proof of the theorem is relegated to Section 4.5.

Theorem 4.1**.**

Fix any $\Gamma\leq 1/4$. Assume that

[TABLE]

Then under assumptions A1 and A2,

[TABLE]

4.2 Multi-epoch analysis

We now turn to the multi-epoch analysis, focusing on using the one-epoch analysis to determine the setting of the hyperparameters. Interestingly, we require that the batch size $B_{j}$ scales as the square of the number of inner-loop iterations $m_{j}$. The proof is relegated to Section 4.5.

Theorem 4.2**.**

Fix any constant $\alpha>1$, $m_{0}>0$ and $\xi\in(0,1)$. Let

[TABLE]

Take $\Gamma=1/4\alpha^{1/\xi}$ and assume that

[TABLE]

Then

[TABLE]

where $\Lambda_{T}=\lambda_{T}\lambda_{T-1}\cdots\lambda_{1}$, $\tilde{\Lambda}_{T}=\tilde{\lambda}_{T}\tilde{\lambda}_{T-1}\cdots\tilde{\lambda}_{1}$,

[TABLE]

and $T_{\kappa}^{*},T_{n}^{*}$ be positive numbers such that

[TABLE]

Remark 4.3**.**

In practice, we recommend the following setting as a default:

[TABLE]

*This setting works well as demonstrated in Appendix D in the Supplementary Material. For those examples, $b$ was chosen as $10^{-4}n$ for fair comparison. Here we choose $B_{0}$ smaller than $m_{0}$ because the batch size $B_{t}$ grows faster than the inner loop size $m_{t}$ and the variance reduction is more effective in later stages. Under this setting, SCSG takes $\lceil\log(n/B_{0})/2\log\alpha\rceil=16$ passes for $B_{t}$ to reach $n$. This creates a reasonably long transition from little to full variance reduction. *

4.3 Complexity analysis

Under the specification of Theorem 4.2 and recalling the definition of $T(\epsilon)$ in (3), we have

[TABLE]

On the other hand,

[TABLE]

By (5), we conclude that

[TABLE]

The following theorem gives the size of $T(\epsilon)$ and thus provides the theoretical complexity of SCSG. The proof is relegated to Section 4.5.

Theorem 4.4**.**

Under the specification of Theorem 4.2, we have

[TABLE]

where

[TABLE]

In particular,

[TABLE]

Remark 4.5**.**

The version of SCSG considered in Theorem 4.4 that achieves the complexity (15) is $\epsilon$-independent and almost universal.

4.4 Discussion

Our complexity result involves the unusual terms $\left(\frac{D_{x}}{\epsilon\kappa}\right)^{2\xi}_{*}$ and $\left(\frac{D}{\epsilon\kappa}\right)^{2\xi}_{*}$. However, they are relatively insignificant as the exponent $\xi$ can be made arbitrarily small and $\frac{1}{\epsilon\kappa}$ is small in practice unless the target accuracy is unusually high. For instance, under the setting given in Remark 4.3, $\xi$ can be as small as $\log\alpha/\log(m_{0}/4b)\approx 0.088$ to guarantee the condition $m_{0}\geq b/\Gamma$ as long as the stepsize is sufficiently small. Thus, the term $\left(\frac{D_{x}}{\epsilon\kappa}\right)^{2\xi}_{*}$ is generally negligible. If we use $\tilde{\tilde{O}}$ to denote a bound that hides these terms and the logarithmic terms, we have

[TABLE]

We discuss some of the consequences of (16).

4.4.1 Adaptivity to target accuracy

For non-strongly convex objectives, (16) implies that

[TABLE]

whereas, for strongly convex objectives, (16) implies that

[TABLE]

Both (17) and (18) exhibit the adaptivity of SCSG to the target accuracy: for low-accuracy computation (i.e., large $\epsilon$), SCSG achieves the same complexity as SGD for non-strongly convex objectives, which can be much more efficient than SVRG-type algorithms in the setting of large datasets (i.e., large $n$). On the other hand, for high-accuracy computation (i.e., small $\epsilon$), SCSG avoids the high variance of SGD and achieves the same complexity as SVRG++ (Zhu15, ) for non-strongly convex objectives and as SVRG for strongly convex objectives (SVRG, ).

4.4.2 Adaptivity to strong convexity

The first two terms of (16) are independent of $n$:

[TABLE]

The last two terms of (16) depend on $n$ but have better dependence on $\epsilon$:

[TABLE]

Both (19) and (20) show the adaptivity of SCSG to strong convexity. In both cases, if $\kappa<\!\!<\frac{1}{\epsilon}$, SCSG benefits from the strong convexity: for the former, (19) yields

[TABLE]

which can be much smaller than $O\left(\frac{D^{2}}{\epsilon^{2}}\right)$, if $D_{x}>\!\!>D_{H}$. For the latter, (20) yields

[TABLE]

which is the same as SVRG, but without the knowledge of $\mu$. On the other hand, in ill-conditioned problems where $\kappa>\!\!>\frac{1}{\epsilon}$, SCSG still achieves the best of SGD and SVRG++ (Zhu15, ) for non-strongly convex objectives. This is not achieved by Adaptive SVRG (xu17, ) and only partially achieved by R-SVRG (SCSG, ), which requires two sequences of iterates. Finally, although SAG and SAGA provide guarantees in ill-conditioned problems, they have an inferior complexity of $\tilde{O}\left(\frac{n}{\epsilon}\wedge(n+\kappa)\right)$.

4.4.3 Weaker requirement on gradients

For algorithms without access to full gradients, it is necessary to impose some conditions on $\nabla f_{i}(x)$. The strongest condition imposes a uniform bound (see, e.g., (Nemirovsky09, )):

[TABLE]

while a slightly milder condition imposes the following bound (see, e.g., (minibatchSGD, ))

[TABLE]

These two types of conditions are typical in analyses of SGD when $f_{i}$ is not assumed to be convex. This is satisfied by many practical problems; e.g., generalized linear models. In our situation, the extra assumption on the convexity of each component allows us to relax assumptions such as (21) or (22) into

[TABLE]

First it is easy to show that

[TABLE]

More importantly, $\mathcal{H}$ can be much smaller than the other two measures, and there are common situations where $A^{2}=\sigma^{2}=\infty$ while $\mathcal{H}<\infty$. For instance, in least-squares problems where $f_{i}(x)=\frac{1}{2}(a_{i}^{T}x-b_{i})^{2}$, $A^{2}=\sigma^{2}=\infty$ unless the domain is bounded. Although assuming a bounded domain is a common assumption in the literature, there is generally no guarantee, at least for algorithms involving stochasticity, that the iterate will stay in the domain unless a projection step is performed. However, the projection step is never performed in practice and thus the bounded domain assumption is artificial. By contrast, SCSG show that

[TABLE]

for least-squares problems, without a bounded domain. This implies that $\mathcal{H}=O(1)$ provided that $\frac{1}{n}\sum_{i=1}^{n}b_{i}^{2}=O(1)$. Similar bounds can be derived for generalized linear models (SCSG, ). It turns out that $\mathcal{H}=O(1)$ for various applications where there is no guarantee for $\sigma^{2}$ or $A^{2}$. We refer the readers to SCSG for an extensive discussion.

4.4.4 Optimality of the complexity bound

To the best of our knowledge, SCSG is the first stochastic algorithm that achieves adaptivity to both target accuracy and strong convexity. However, it is still illuminating to compare each component of (16) separately with the best achievable rate in the literature.

• •

The first component $\tilde{\tilde{O}}\left(\frac{D^{2}}{\epsilon^{2}}\right)$ is optimal in terms of $\epsilon$-dependence for non-strongly convex objectives (Agarwal10, ; woodworth16, ). Under slightly stronger assumptions on the gradient bounds (but without the convexity of each $f_{i}$), mini-batched SGD achieves the $O\left(\frac{1}{\epsilon^{2}}\right)$ rate (Nemirovsky09, ; minibatchSGD, ). However, the dependence on $D$ is suboptimal. Without knowing $D_{x}$ and $\sigma^{2}$ or $A^{2}$, defined in (21) and (22) , the resulting complexity of (mini-batched) SGD can be no better than $O\left(\frac{(D_{x}\vee A)^{2}}{\epsilon^{2}}\right)\geq O\left(\frac{D^{2}}{\epsilon^{2}}\right)$. When they are known, Nemirovsky09 and minibatchSGD are able to improve it to $O\left(\frac{D_{x}\sigma}{\epsilon^{2}}\right)$ and $O\left(\frac{D_{x}A}{\epsilon^{2}}\right)$. With averaging, juditsky11 improve it to

[TABLE]

With momentum acceleration, lan12 further improve the rate to

[TABLE]

• •

The second component $\tilde{\tilde{O}}\left(\kappa^{2}+\frac{D_{H}^{2}}{\epsilon^{2}}\right)$ is new to the best of our knowledge. When $\mu$ is known and $\mathbb{E}\|\nabla f_{i}(x)\|_{2}^{2}$ is uniformly bounded for all $i$ and $x$, a requirement that is more stringent than our setting, it is known that the optimal complexity in terms of $\epsilon$-dependence and $\mu$-dependence is $O\left(\frac{\kappa}{\epsilon}\right)$; see, e.g., hazan10 ; Rakhlin12 for the upper bound and woodworth16 for the lower bound. However, the lower bound is established under the condition that $\mu$ is known. It remains an interesting direction to derive a tight lower bound when $\mu$ is unknown.

• •

The third component $\tilde{\tilde{O}}\left(n+\frac{D}{\epsilon}\right)$ should be sub-optimal in terms of both $\epsilon$ and $D$. SVRG++ (Zhu15, ) achieves the $\tilde{O}\left(n+\frac{D_{x}}{\epsilon}\right)$ rate. By adding momentum terms, Adaptive SVRG (nitanda16, ) slightly improves the rate to $\tilde{O}\left(\left\{n+\frac{D_{x}}{\epsilon}\right\}\wedge n\sqrt{\frac{D_{x}}{\epsilon}}\right)$. On the other hand, woodworth16 prove a lower bound $\tilde{O}\left(n+\sqrt{\frac{nD_{x}}{\epsilon}}\right)$. This can be achieved by Accelerated SDCA (APSDCA, ) or Katyusha (Katyusha, ). However, the former has only been established for particular problems such as generalized linear models, and the latter involves black-box acceleration (blackbox, ), which requires setting the parameters based on unknown quantities such as $D_{x}$. The Varag algorithm lan2019unified , which came after our work, is the first algorithm that achieves the lower bound for generic finite-sum optimization problems.

• •

The last component $\tilde{\tilde{O}}\left(n+\kappa\right)$ has been proved by arjevani17 to be optimal, up to small factors $\left(\frac{D_{H}}{\epsilon\kappa}\right)^{2\xi}_{*}$, for a large class of algorithms when $\mu$ is unknown. The story is different when $\mu$ is known. The optimal complexity can be improved to $\tilde{O}(n+\sqrt{n\kappa})$ and can be achieved by, for instance, by Katyusha (Katyusha, ).

In summary, a major remaining challenges is to derive oracle lower bounds involving all of $\epsilon,n,D_{x},D_{H}$, for $\epsilon$-independent and almost universal algorithms.

4.5 Proofs

4.5.1 Lemmas

We start by four lemmas with proofs presented in Section A in Supplementary Material. The first lemma gives an upper bound of the expected squared norm of $\nu^{(j)}_{k}$, which is standard in the analyses of most first-order methods.

Lemma 4.6**.**

Under assumption A1,

[TABLE]

*where $e_{j}$ is defined in (12). *

The second lemma gives an upper bound for $\mathbb{E}\|e_{j}\|_{2}^{2}$.

Lemma 4.7**.**

Under assumption A1,

[TABLE]

The third lemma connects the iterates $\tilde{x}_{j}$ and $\tilde{x}_{j-1}$ in adjacent epochs. The proof exploits the elegant property of geometrization.

Lemma 4.8**.**

Let $u\in\mathbb{R}^{d}$ be any variable that is independent of $\mathcal{I}_{j}$ and subsequent random subsets within the $j$th epoch, $\tilde{\mathcal{I}}^{(j)}_{0},\tilde{\mathcal{I}}^{(j)}_{1},\ldots$, with $\mathbb{E}\|u-x^{*}\|_{2}^{2}<\infty$. Then under assumption A1,

[TABLE]

where

[TABLE]

The term $\mathbb{E}\left\langle e_{j},\tilde{x}_{j-1}-\tilde{x}_{j}\right\rangle$ is non-standard. We derive an upper bound in the following lemma. Surprisingly, this lemma is a direct consequence of Lemma 4.8.

Lemma 4.9**.**

Fix any $\gamma_{j}>0$. Under assumption A1,

[TABLE]

4.5.2 Proof of Theorem 4.1

We start from a more general version of Theorem 4.1.

Theorem 4.10**.**

Fix any $\Gamma_{j}\in(0,1)$. Assume that

[TABLE]

Then under assumption A1,

[TABLE]

Proof 4.11**.**

Letting $u=x^{*}$ in Lemma 4.8 and applying Lemma 4.9 with $\gamma_{j}=\Gamma_{j}/(1-\Gamma_{j})$ (i.e. $\Gamma_{j}=\gamma_{j}/(1+\gamma_{j})$), we obtain:

[TABLE]

First, by Lemma C.1 with $x=x^{*},y=\tilde{x}_{j}$ and the fact that $\nabla f(x^{*})=0$,

[TABLE]

Second, since $\gamma_{j}\geq\frac{(1+\gamma_{j})\eta_{j}L}{b_{j}}$ by (24), by convexity of $f$ we obtain that

[TABLE]

Combining (25)-(27) yields

[TABLE]

Since $1\geq 2(1+\gamma_{j})\eta_{j}L$ by (24), by convexity of $f$,

[TABLE]

By (28) and Lemma 4.7,

[TABLE]

Rearranging the terms, we have

[TABLE]

*Dividing both sides by $2\eta_{j}(1+\gamma_{j})$ and recalling the definition that $\Gamma_{j}=\gamma_{j}/(1+\gamma_{j})$, we complete the proof. *

Proof 4.12** (Proof of Theorem 4.1).**

Let $\Gamma_{j}\equiv\Gamma$ in Theorem 4.10. Under condition (13), (24) is satisfied because $\Gamma\leq 1/4$. Moreover,

[TABLE]

and

[TABLE]

By assumption A2,

[TABLE]

By Theorem 4.10 we have

[TABLE]

*The proof is completed by multiplying by $4/3$ in both sides and that $D_{H}=\mathcal{H}/L$. *

4.5.3 Other Proofs

Proof 4.13** (Proof of Theorem 4.2).**

By Theorem 4.1 with $\Gamma=1/4\alpha^{1/\xi}$,

[TABLE]

Let

[TABLE]

Then

[TABLE]

where the last line uses the condition that

[TABLE]

For any $j\geq 0$,

[TABLE]

When $j\geq T_{\kappa}^{*}$, we have $\alpha^{j}\geq\kappa/\eta L=1/\eta\mu$, and thus

[TABLE]

In summary,

[TABLE]

Plugging this into (29), we conclude that

[TABLE]

Finally we prove the following statement by induction.

[TABLE]

It is obvious that (31) holds for $T=0$. Suppose it holds for $T-1$, then by (30),

[TABLE]

where the last line uses the fact that $\tilde{\lambda}_{T}\leq\lambda_{T}$ for all $T>0$. If $T<T_{n}^{*}$, then $\tilde{\Lambda}_{T}=\alpha^{-T}$ and thus

[TABLE]

If $T>T_{n}^{*}$,

[TABLE]

Therefore, (31) is proved. The proof is then completed by noting that

[TABLE]

and

[TABLE]

*where (i) uses assumption A1 and the definition of $D_{x}$, (ii) uses the fact that $\mu\leq L$ and the condition $b/m_{0}\leq\Gamma\leq 1/4$, (iii) uses the fact that $\Gamma\leq 1/4$ and thus $\eta L\leq(1-\Gamma)/2\leq 3/8$. *

Proof 4.14** (Proof of Theorem 4.4).**

Let

[TABLE]

Then for any $T\geq\max\{T^{(1)}(\epsilon),T^{(2)}(\epsilon)\}$,

[TABLE]

This entails

[TABLE]

By definition, when $T\leq T_{\kappa}^{*}$,

[TABLE]

and when $T>T_{\kappa}^{*}$, since $\xi<1$,

[TABLE]

As a result,

[TABLE]

and

[TABLE]

Similarly, when $T\leq T_{n}^{*}\vee T_{\kappa}^{*}$

[TABLE]

and when $T>T_{n}^{*}\vee T_{\kappa}^{*}$,

[TABLE]

Thus,

[TABLE]

As a result,

[TABLE]

and

[TABLE]

In summary, by (34)

[TABLE]

where the last line uses the monotonicity of the mapping $x\mapsto x^{1-\xi}$. By definition, $T_{n}^{*}=O(\log n)=\tilde{O}(1)$. As a result, $D_{x}+D_{H}T_{n}^{*}=\tilde{O}(\max\{D_{x},D_{H}\})=\tilde{O}(D)$ and thus

[TABLE]

*The proof is then completed by replacing $\alpha^{T(\epsilon)}$ by $A(\epsilon)$ and equation 14. *

5 Mirror-Proximal SCSG for Composite Problems

In this section we extend SCSG to composite problems in non-Euclidean spaces. Throughout this section we deal with problem (1) with $\mathcal{X}$ assumed to be a subset of a Hilbert space $\mathcal{X}_{0}$, equipped with an inner product $\left\langle\cdot,\cdot\right\rangle$. Let $\|\cdot\|_{2}$ denote the norm induced by the inner product; i.e. $\|x\|_{2}=\sqrt{\left\langle x,x\right\rangle}$. For any convex function $g$, let $g^{*}$ denote the convex conjugate of $g$:

[TABLE]

For any differential convex function $w$, let $B_{w}(\cdot,\cdot)$ denote the Bregman divergence:

[TABLE]

We denote by $\mathbb{R}^{+}$ the set of nonnegative reals.

We define Mirror-Proximal SCSG as a variant of Algorithm 1 designed for composite problems (with $\psi\not=0$). The algorithm is detailed below. The only difference lies in line 9 where the gradient step is replaced by a mirror-proximal step. This is the standard extension to composite problems in general Hilbert spaces (see, e.g., (duchi10, ; lan15, ; Katyusha, )). Note that when $\psi(x)\equiv 0$ and $w(x)=\|x\|_{2}^{2}/2$, Algorithm 2 reduces to Algorithm 1. Whenever $w(x)=\|x\|_{2}^{2}/2$, line 9 reduces to the proximal gradient step.

Unlike most of the literature on composite mirror descent algorithms, our analysis requires a weaker condition on the distance-generating function $w(x)$. To state the condition, we define a class of functions which we refer to as Convex sup-Homogeneous Envelope Functions (CHEF).

Definition 5.1**.**

Given any increasing function $g:\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$ such that

[TABLE]

a function $G:\mathcal{X}_{0}\mapsto\mathbb{R}^{+}$ is a CHEF with parameters $(g(\cdot),C_{G})$ if

1. C1

$G$* is non-negative with $G(0)=0$, convex and symmetric in the sense that $G(w)=G(-w)$ for any $w\in\mathcal{X}_{0}$;* 2. C2

for any $w\in\mathcal{X}_{0}$ and $\lambda>0$,

[TABLE] 3. C3

$G^{*}$, the convex conjugate of $G$, satisfies a generalized Nemirovsky inequality in the sense that for any set of independent mean-zero $\mathcal{X}_{0}$-valued random vectors $Z_{1},\ldots,Z_{m}$,

[TABLE]

Note that $G^{*}$ is non-negative, since for any $w\in\mathcal{X}_{0}$,

[TABLE]

Our first condition is imposed on the Bregman divergence induced by $w(x)$.

1. B1

There exists a CHEF such that for any $x,y\in\mathcal{X}$,

[TABLE]

In the literature (see, e.g., (beck03, ; duchi10, ; Katyusha, )), it is common to consider a special case where

[TABLE]

where $\|\cdot\|$ can be any norm, not necessarily $\|\cdot\|_{2}$, on $\mathcal{X}_{0}$. srebro11 considered a more general class of $G$’s in the form of

[TABLE]

It is clear that $G_{q}$ satisfies C1 and C2 for any $q>1$ with $g(\lambda)=\lambda^{q-1}$.

To see that $G_{q}(w;\|\cdot\|)$ satisfies C3, we first consider the case where $q=2$, where $\mathcal{X}_{0}=\mathbb{R}^{d}$ is the Euclidean space and where $\|\cdot\|=\|\cdot\|_{r}$ for some $r\geq 1$, with

[TABLE]

Then $\|\cdot\|_{*}=\|\cdot\|_{r^{\prime}}$ where $r^{\prime}=r/(r-1)$. By Lemma C.3,

[TABLE]

By Nemirovsky’s inequality (Theorem 2.2 of dumbgen10 ), for any independent mean-zero random vectors $Z_{1},\ldots,Z_{m}\in\mathbb{R}^{d}$,

[TABLE]

where

[TABLE]

Thus, whenever $r^{\prime}=O(1)$, $K_{\mathrm{Nem}}(d,r^{\prime})=O(1)$. Even when $r^{\prime}=\infty$, in which case $r=1$, $K_{\mathrm{Nem}}(d,r^{\prime})$ scales as $\log d$.

Generally, given $G(x)=\|x\|^{q}/q$ for $q\in(1,2)$ and a norm $\|\cdot\|$ on a general Hilbert space $\mathcal{X}$, Lemma C.3 implies that $G^{*}(x)=\|x\|_{*}^{p}/p$ where $p=q/(q-1)$. Then the property C3 is equivalent to the condition that $\mathcal{X}_{0}$ has Martingale Type $p$ (see, e.g., (pisier75, )). In particular, when $\mathcal{X}_{0}=\mathbb{R}^{d}$ and $\|\cdot\|=\|\cdot\|_{r}$ with $r\leq q$, we prove in Proposition C.6 that $G^{*}(x)$ satisfies the property C3, using Hanner’s inequality (hanner56, ). In summary, the property C3 is satisfied in almost all cases that have been commonly studied in the literature on mirror descent methods.

Besides assumption B1 we need the analogous assumptions of A1 and A2 for the smoothness and strong convexity of the objectives.

1. B2

$\displaystyle 0\leq B_{f_{i}}(x,y)\leq LG(x-y)$ for all $i$ and $x,y\in\mathcal{X}$; 2. B3

$\displaystyle F(x)-F(x^{*})\geq\mu B_{w}(x^{*},x)$ for some $\mu\geq 0$.

It is easy to see that assumptions B2 and B3 reduce to A1 and A2 when $\mathcal{X}=\mathbb{R}^{d},G(x)=w(x)=\|x\|_{2}^{2}/2$. Note that B3 only requires strong convexity at $x^{*}$; it does not requires global strong convexity.

Finally, we modify the definitions of $D_{x},D_{H}$ as

[TABLE]

where $x^{*}$ is the optimum of $F$. It is straightforward to show that $D_{x}$ and $D_{H}$ coincides with (2) up to a constant two when $\mathcal{X}_{0}=\mathcal{X}=\mathbb{R}^{d}$ and $w(x)=\|x\|_{2}^{2}/2$. We also define an extra quantity $D_{F}$ as

[TABLE]

In the unregularized case, assumption B1 and B2 imply that $D_{F}\leq D_{x}$. However, this comparison may not hold in the regularized case. Finally we re-define $D$ as the maximum of $D_{x}$, $D_{H}$ and $D_{F}$.

5.1 Main results

Similar to the unregularized case, we present results on the one-epoch analysis, the multiple-epoch analysis and the complexity analysis. The results are almost the same as those in Section 4 though the proofs are much more involved. All proofs are relegated to Appendix B.

Theorem 5.2**.**

Fix any $\xi>0$ and $\Gamma=1/4\alpha^{1/\xi}$. Assume that

[TABLE]

and

[TABLE]

Then under assumptions B1 - B3,

[TABLE]

Theorem 5.2 and Theorem 4.1 give almost the same result, up to constants, except that Theorem 4.1 has an additional term $\frac{\eta L}{b}$ in the coefficient of $D_{H}$. In the cases where $\eta$ is small and $b$ is large, Theorem 4.1 gives a better guarantee. However, in our settings for SCSG, $\eta L$ and $b$ are both taken as $O(1)$ and thus the theorems yield the same results up to the constant.

To set the parameters for SCSG in this general case, we still take a constant stepsize, a constant mini-batch size and a geometrically increasing sequence for $m_{j}$. In contrast, $B_{j}$ should scale as $m_{j}/g(1/m_{j})$. This coincides with Theorem 4.2 since $g(x)=x$ in the unregularized case with the usual strong convexity condition (assumption A2).

Theorem 5.3**.**

Fix and constant $\alpha>1$, $m_{0}>0$ and $\xi\in(0,1)$. Let

[TABLE]

Assume that

[TABLE]

Then under assumptions B1 - B3,

[TABLE]

*where $\Lambda_{T}$ and $\tilde{\Lambda}_{T}$ are defined as in Theorem 4.2. *

Theorem 5.3 gives almost the same result as Theorem 4.2, except that the second term is loose up to a term $\frac{\eta L}{b}$. As mentioned before, this is a constant in our setting and thus the gap is negligible in terms of the theoretical complexity.

Applying the same argument as in Theorem 4.4, we can derive the theoretical complexity. Again it coincides with Theorem 4.4 in the unregularized case with the usual strong convexity condition (i.e., assumption A2).

Theorem 5.4**.**

Under the specification of Theorem 5.3, we have

[TABLE]

where $\tilde{\kappa}=\alpha^{T_{n}^{*}}+\kappa$ and

[TABLE]

Interestingly, in the uniformly convex case (juditsky2014deterministic, ), where

[TABLE]

we can set $g(\lambda)=\lambda^{q-1}$. Then Theorem 5.4 implies that

[TABLE]

Recalling that $\tilde{\tilde{O}}$ hides the negligible terms $\left(\frac{D}{\epsilon\kappa}\right)^{2\xi}$ and $\tilde{\kappa}=\alpha^{T_{n}^{*}}+\kappa=O(n+\kappa)$, we can rewrite (35) as

[TABLE]

The first term matches the bound in srebro11 . However, the other terms have not been investigated in the literature to the best of our knowledge.

6 Conclusions

We have presented SCSG, a gradient-based algorithm for the convex finite-sum optimization problem, which is $\epsilon$-independent and almost-universal. These properties arise from two ideas: geometrization and batching variance reduction. SCSG achieves strong adaptivity to both the target accuracy and to strong convexity with complexity $\tilde{\tilde{O}}\left(\frac{D^{2}}{\epsilon^{2}}\wedge\left(\kappa^{2}+\frac{D_{H}^{2}}{\epsilon^{2}}\right)\wedge\left(n+\frac{D}{\epsilon}\right)\wedge\left(n+\kappa\right)\right)$ up to negligible terms. This is strictly better than other existing adaptive algorithms. We also present a mirror-proximal version of SCSG for problems involving non-Euclidean geometry. Our analysis requires the Bregman divergence to be lower bounded by a CHEF, a construct which unifies and generalizes existing work on mirror-descent methods. We derive a set of technical tools to deal with CHEFs which may be of interest in other problems.

A major direction for further research is to delineate optimal rates for algorithms that exhibit adaptivity. Our conjecture is that the optimal complexity for a reasonably large class of algorithms that do not require knowledge of $\mu$ is

[TABLE]

We believe that momentum terms are required to achieve such a rate.

Appendix A Technical Proofs in Section 4

Proof A.1** (Proof of Lemma 4.6).**

Using the fact that $\mathbb{E}\|Z\|_{2}^{2}=\mathbb{E}\|Z-\mathbb{E}Z\|_{2}^{2}+\|\mathbb{E}Z\|_{2}^{2}$ (for any random vector $Z$), we have

[TABLE]

where the last inequality is obtained from Lemma C.2. By Lemma C.1 with $g=f_{i},x=x^{(j)}_{0},y=x^{(j)}_{k}$,

[TABLE]

*The proof is completed by the fact that $\|a+b\|_{2}^{2}=2\|a\|_{2}^{2}+2\|b\|_{2}^{2}-\|a-b\|_{2}^{2}\leq 2\|a\|_{2}^{2}+2\|b\|_{2}^{2}$. *

Proof A.2** (Proof of Lemma 4.7).**

Using the fact that $\|a+b\|_{2}^{2}\leq 2\|a\|_{2}^{2}+2\|b\|_{2}^{2}$ and $\nabla f(x^{*})=0$, we have

[TABLE]

Then

[TABLE]

On the other hand, by Lemma C.2 again, we obtain that

[TABLE]

*Putting the pieces together we prove the result. *

To prove the remaining two lemmas, we need an extra lemma that justifies the condition $\mathbb{E}|D_{N}|<\infty$ when applying geometrization (Lemma 3.1) in our proofs for different choices of $\{D_{k}\}$. The proof is distracting and relegated to the end of this section.

Lemma A.3**.**

Assume that $\eta_{j}L\leq 1/2$. Then for any $j$,

[TABLE]

Proof A.4** (Proof of Lemma 4.8).**

By definition,

[TABLE]

Let $\mathbb{E}_{j}$ denote the expectation with respect to $\mathcal{I}_{j}$ and $\tilde{\mathcal{I}}^{(j)}_{0},\tilde{\mathcal{I}}^{(j)}_{1},\ldots$. Then

[TABLE]

Since $u$ is independent of $\mathcal{I}_{j}$ and $\mathbb{E}_{\mathcal{I}_{j}}e_{j}=0$, we have

[TABLE]

Similarly, since $x^{(j)}_{0}$ is also independent of $\mathcal{I}_{j}$,

[TABLE]

Therefore,

[TABLE]

Let $D_{k}=\mathbb{E}\|x^{(j)}_{k}-u\|_{2}^{2}$. By Lemma A.3,

[TABLE]

Let $k=N_{j}$ in (36) and by Lemma 3.1 and Lemma A.3,

[TABLE]

By definition, $\tilde{x}_{j}=x^{(j)}_{N_{j}},\tilde{x}_{j-1}=x^{(j)}_{0}$ and thus

[TABLE]

*The proof is completed by Lemma 4.6 which yields $\eta_{j}^{2}\mathbb{E}\|\nu^{(j)}_{N_{j}}\|_{2}^{2}\leq\mathbb{E}W_{j}$. *

Proof A.5** (Proof of Lemma 4.9).**

Let $u=\tilde{x}_{j-1}$, then it is independent of $\mathcal{I}_{j}$ and $\tilde{\mathcal{I}}^{(j)}_{0},\tilde{\mathcal{I}}^{(j)}_{1},\ldots$. By Lemma 4.8,

[TABLE]

Using the fact that $2\left\langle a,b\right\rangle\leq\frac{1}{\beta}\|a\|_{2}^{2}+\beta\|b\|_{2}^{2}$ for any $\beta>0$, we have

[TABLE]

This implies that

[TABLE]

By (37), we obtain that

[TABLE]

*The proof is then completed by multiplying both sides by $\gamma_{j}$. *

Proof A.6** (Proof of Lemma A.3).**

We prove the first claim by induction. When $j=0$, the claim is obvious. Suppose we prove the claim for $j-1$, i.e.

[TABLE]

Let $y^{(j)}_{k}$ be another sequence constructed as follows:

[TABLE]

In other words, $y^{(j)}_{k}$ is a hypothetical sequence of iterates produced by SVRG initialized at $x^{(j)}_{0}$ and updated using the same sequence of random subsets. Let $\mathrm{Id}$ denote the identity mapping. Then

[TABLE]

where we use the fact that $\mu_{j}=\nabla f(x^{(j)}_{0})+e_{j}=\nabla f(y^{(j)}_{0})+e_{j}$. Since $f_{\tilde{\mathcal{I}}^{(j)}_{k-1}}$ is $L$-smooth and convex and $\eta_{j}\leq 1/2L$, it is well known that $\mathrm{Id}-\eta_{j}\nabla f_{\tilde{\mathcal{I}}^{(j)}_{k-1}}$ is a non-expansive operator. Thus,

[TABLE]

As a result,

[TABLE]

On the other hand, By (7) (Theorem 1 of SVRG ) and the convexity of $f$,

[TABLE]

Since $f(y^{(j)}_{k})-f(x^{*})\geq 0$ and $1-2\eta_{j}L\geq 0$, we have

[TABLE]

As a result,

[TABLE]

Putting (38) and (39) together, and using the fact that $\|a+b\|_{2}^{2}\leq 2\|a\|_{2}^{2}+2\|b\|_{2}^{2}$, we obtain that

[TABLE]

By Lemma 4.7,

[TABLE]

By (40),

[TABLE]

By the induction hypothesis,

[TABLE]

and

[TABLE]

By Lemma 4.6,

[TABLE]

Then the first claim yields

[TABLE]

Finally,

[TABLE]

Appendix B Technical Proofs in Section 5

B.1 Technical tools to handle CHEFs

In this section we establish several technical tools to tackle CHEFs. These results can be of independent interest because they unify and generalize various fundamental results which are widely used in the analysis of first-order methods.

The first lemma shows the sub-homogeneity of $G^{*}$.

Lemma B.1**.**

[A key property of CHEF] For any $w\in\mathcal{X}_{0}$ and $\lambda\geq 0$,

[TABLE]

Proof B.2**.**

When $\lambda=0$, since $G$ is nonnegative,

[TABLE]

We assume that $\lambda>0$ throughout the rest of the proof. Note that $\mathcal{X}_{0}$ is a Hilbert space and hence a cone; i.e., $x\in\mathcal{X}_{0}\Longrightarrow\lambda x\in\mathcal{X}_{0}$ for any $\lambda>0$. By definition,

[TABLE]

By property C2 of CHEF,

[TABLE]

Thus,

[TABLE]

The second lemma gives the Fenchel-Young inequality which involves the Bregman divergence and the corresponding dual Bregman divergence. In the special case where $w(x)=\|x\|_{2}^{2}/2$, it reduces to the basic inequality that $2\left\langle u,z\right\rangle\leq\alpha\|z\|_{2}^{2}+\alpha^{-1}\|u\|_{2}^{2}$ for any $\alpha>0$.

Lemma B.3**.**

For any $u,y,z\in\mathcal{X}$ and $\alpha>0$,

[TABLE]

Proof B.4**.**

Let $F_{y}(z)=\alpha B_{w}(y+z,y)$. By the Fenchel-Young inequality,

[TABLE]

By definition,

[TABLE]

where (i) uses the property that

[TABLE]

It is left to prove that

[TABLE]

By the Fenchel-Young inequality,

[TABLE]

On the other hand, for any $y^{\prime}\in\mathcal{X}_{0}$, by convexity of $w$,

[TABLE]

Taking the supremum over $y^{\prime}$ we obtain that

[TABLE]

*Putting two pieces together, we prove (41). The proof is then completed. *

The third lemma generalizes the co-coercive property of smooth convex functions.

Lemma B.5**.**

[Generalized co-coercive property] Let $h$ and $H$ be arbitrary convex functions on $\mathcal{X}_{0}$ such that

[TABLE]

for any $x,y\in\mathcal{X}_{0}$. Then

[TABLE]

Proof B.6**.**

Let $h_{y}(x)=B_{h}(x,y)$. Fix any $x^{\prime}\in\mathcal{X}_{0}$. Then

[TABLE]

Since $h$ is convex,

[TABLE]

The above two inequalities imply that

[TABLE]

Thus,

[TABLE]

Taking a supremum over $x^{\prime}\in\mathcal{X}_{0}$, we obtain that

[TABLE]

The last lemma gives the relationship between the Bregman divergence and its dual divergence.

Lemma B.7**.**

Let $h$ and $H$ be arbitrary convex functions on $\mathcal{X}$ and $\mathcal{X}_{0}$ such that

[TABLE]

Assume further that $H$ is symmetric in the sense that $H(x)=H(-x)$. Then

[TABLE]

Proof B.8**.**

Interchanging $x$ and $y$, we obtain that

[TABLE]

By symmetry of $H$,

[TABLE]

By the Fenchel-Young inequality, for any $\alpha>0$,

[TABLE]

Thus,

[TABLE]

which implies that

[TABLE]

Replacing $x$ and $y$ by $\nabla h^{*}(x)$ and $\nabla h^{*}(y)$, respectively, and noting that $\nabla h(\nabla h^{*})(x)=x$, we obtain that

[TABLE]

By Taylor’s expansion,

[TABLE]

*where (i) uses the Fenchel-Young inequality and (ii) uses (42). *

B.2 Lemmas

We prove six lemmas for the one-epoch analysis. The first lemma connects the consecutive iterates within an epoch.

Lemma B.9**.**

For any $u\in\mathcal{X}$,

[TABLE]

where

[TABLE]

Proof B.10**.**

By definition (line 9 of Algorithm 2), there exists $\xi\in\partial\psi(x^{(j)}_{k+1})$ such that

[TABLE]

By convexity of $\psi$,

[TABLE]

It is easy to verify that

[TABLE]

*The proof is completed by rearranging the terms. *

The second lemma provides a bound for $M^{(j)}_{k}$ as an analogue of Lemma 4.6 for the unregularized case, though the proof is much more involved.

Lemma B.11**.**

Assume that $\eta_{j}L\leq\frac{1}{3}$. Under assumptions B1 and B2,

[TABLE]

Proof B.12**.**

We decompose $M^{(j)}_{k}$ into three components:

[TABLE]

where

[TABLE]

First we bound $\mathbb{E}_{\tilde{\mathcal{I}}^{(j)}_{k}}M^{(j)}_{k1}$. By Lemma B.3 with $u=-(\nu^{(j)}_{k}-\nabla f(x^{(j)}_{k})-e_{j}),z=x^{(j)}_{k+1}-x^{(j)}_{k},y=x^{(j)}_{k},\alpha=\frac{2}{3\eta_{j}}$,

[TABLE]

where (i) uses Lemma B.7 and assumption B1. By definition of $\nu^{(j)}_{k}$,

[TABLE]

Let $Z_{1},\ldots,Z_{b_{j}}$ be i.i.d. random elements on $\mathcal{X}_{0}$ such that

[TABLE]

where $\mathbf{\tilde{i}}$ is a uniform random variable in $\{1,\ldots,n\}$ that is independent of $x^{(j)}_{k}$. Since $G^{*}$ is convex and $\mathbb{E}Z_{j}=0$, by Hoeffding’s lemma (Proposition C.5),

[TABLE]

Then by the property C3 of a CHEF,

[TABLE]

Then we have

[TABLE]

where (i) uses the convexity and the symmetry of $G^{*}$, (ii) is implied by Lemma B.1 and (iii) is implied by Lemma B.5. Taking an expectation over $\mathbf{\tilde{i}}$, we obtain that

[TABLE]

Putting (44)- (47) together, we conclude that

[TABLE]

To bound $\mathbb{E}_{\tilde{\mathcal{I}}^{(j)}_{k}}M^{(j)}_{k2}$, by assumption B2 and B1,

[TABLE]

Thus,

[TABLE]

*The proof is completed by combining (48) and (49). *

The third lemma controls the magnitude of certain transformations of $e_{j}$ that will be useful in the following analysis. Note that in the unregularized case where $G(x)=\|x\|_{2}^{2}/2$, $G^{*}(\beta e_{j})=\beta^{2}\|e_{j}\|_{2}^{2}/2$. Thus Lemma B.13 is essentially an analogue of Lemma 4.7.

Lemma B.13**.**

Under assumptions B1 and B2, for any $\beta>0$,

[TABLE]

Proof B.14**.**

Note that

[TABLE]

Clearly $e_{j}=0$ when $B_{j}=n$, and then

[TABLE]

Throughout the rest of the proof we assume that $B_{j}<n$. Similar to the proof of Lemma B.11, let $Z_{1},\ldots,Z_{B_{j}}$ be i.i.d. random elements in $\mathcal{X}_{0}$ such that

[TABLE]

where $\mathbf{\tilde{i}}$ is a uniform random variable from $\{1,\ldots,n\}$. Then $\mathbb{E}Z_{j}=0$ and hence by Hoeffding’s lemma (Proposition C.5),

[TABLE]

By the property C3 of a CHEF,

[TABLE]

*By definition, *

[TABLE]

where (i) uses the convexity and the symmetry of $G^{*}$, (ii) uses Lemma B.1 and (iii) uses Lemma B.5. Finally, note that there exists $\xi\in\partial\psi(x^{*})$ such that

[TABLE]

By convexity of $\psi$, $\left\langle\xi,\tilde{x}_{j-1}-x^{*}\right\rangle\leq\psi(\tilde{x}_{j-1})-\psi(x^{*})$. Thus,

[TABLE]

*The proof is then completed by combining (50)-(51). *

The fourth lemma is an analogue of Lemma A.3 which enables the geometrization in the following proofs. Again, the proof is relegated to the end of this subsection to avoid distraction.

Lemma B.15**.**

Assume that

[TABLE]

Then for any $j$,

[TABLE]

The fifth lemma connects the iterates $\tilde{x}_{j}$ and $\tilde{x}_{j-1}$ in adjacent epochs. As with Lemma 4.8, the proof exploits the elegant property of geometrization.

Lemma B.16**.**

Assume that $\eta_{j}$ satisfies 52. Let $u\in\mathbb{R}^{d}$ be any variable that is independent of $\mathcal{I}_{j}$ and subsequent random subsets $(\tilde{\mathcal{I}}^{(j)}_{k})_{k\geq 1}$ within the $j$-th epoch. Then under assumptions B1 and B2,

[TABLE]

Proof B.17**.**

By Lemma B.9 and Lemma B.11,

[TABLE]

Note that

[TABLE]

Since $x^{(j)}_{k}$ and $u$ are independent of $\tilde{\mathcal{I}}^{(j)}_{k}$, the first term on the left-hand side is

[TABLE]

Letting $\mathbb{E}_{j}$ denote the expectation over $\mathcal{I}_{j}$ and $(\tilde{\mathcal{I}}^{(j)}_{k})_{k\geq 0}$. Then

[TABLE]

Note that $\mathcal{I}_{j}$ is independent of $x^{(j)}_{0}$,

[TABLE]

Thus,

[TABLE]

Letting $k=N_{j}$, we obtain

[TABLE]

On the other hand, letting $k=N_{j}$, by Lemma 3.1 with $D_{k}=B_{w}(u,x^{(j)}_{k})$ and Lemma B.15,

[TABLE]

Note that $x^{(j)}_{0}=\tilde{x}_{j-1}$, $x^{(j)}_{N_{j}}=\tilde{x}_{j}$, we have

[TABLE]

Similarly, by Lemma 3.1 and Lemma B.15,

[TABLE]

and

[TABLE]

*The proof is completed by putting (53)-(54) together. *

The last lemma bounds the non-standard term $\mathbb{E}\left\langle e_{j},\tilde{x}_{j-1}-\tilde{x}_{j}\right\rangle$ as in Lemma 4.9. Similarly, this lemma is a direct consequence of Lemma B.16.

Lemma B.18**.**

Assume that $\eta_{j}$ satisfies 52. Under assumptions B1 and B2, for any $\gamma_{j}>0$,

[TABLE]

Proof B.19**.**

By Lemma B.16 with $u=\tilde{x}_{j-1}$,

[TABLE]

This simplifies into

[TABLE]

Fix $\gamma_{j}>0$. By Lemma B.3 with $u=e_{j},z=\tilde{x}_{j-1}-\tilde{x}_{j},y=\tilde{x}_{j},\alpha=\frac{b_{j}}{\eta_{j}(m_{j}+b_{j})}\frac{\gamma_{j}}{1+\gamma_{j}}$,

[TABLE]

By Lemma B.7 with $h=w$ and $H=G$,

[TABLE]

*The proof is completed by combining (55) and (56). *

Proof B.20** (Proof of Lemma B.15).**

We prove the first two claims by induction. When $j=0$, the claim is obvious. Suppose we prove the claim for the case of $j-1$, i.e.

[TABLE]

Let $M^{(j)}_{k}$ be defined in (43). Similar to the proof of Lemma B.11, we decompose it into three terms:

[TABLE]

where

[TABLE]

By (44), we have

[TABLE]

Using the convexity and the symmetry of $G^{*}$, we have

[TABLE]

Using the same arguments as (45) and (46),

[TABLE]

where (i) is implied by Lemma B.1 and (ii) is implied by Lemma B.5. Analogously,

[TABLE]

Thus,

[TABLE]

Using the same argument as (49) and the assumption that $\eta_{j}L\leq 1/6$, we have

[TABLE]

By Fenchel-Young inequality, for any convex function $H$,

[TABLE]

Let $H(z)=G(z)/6\eta_{j}$, then $H^{*}(z)=G^{*}(6\eta_{j}z)/6\eta_{j}$ and thus,

[TABLE]

By Lemma B.13,

[TABLE]

Putting (57) - (59) together and using the monotonicity of $g^{-1}$, we conclude that

[TABLE]

where the last inequality uses the condition on $\eta_{j}L$ that

[TABLE]

Similar to (51), we have

[TABLE]

Thus,

[TABLE]

where

[TABLE]

Next, by Lemma B.9 with $u=x^{*}$ and by (60)

[TABLE]

In addition,

[TABLE]

The above two inequalities imply that

[TABLE]

By Fenchel-Young inequality, for any constant $c_{j}>0$,

[TABLE]

By (61) and Lemma B.13,

[TABLE]

where

[TABLE]

Let $c_{j}=1/2m_{j}$. Then it is easy to see that $\tilde{M}_{0}^{\prime}\leq\tilde{M}_{0}$. Further let

[TABLE]

Then (62) implies that

[TABLE]

Therefore,

[TABLE]

Recalling that

[TABLE]

Thus,

[TABLE]

As a result,

[TABLE]

By the induction hypothesis, we know that

[TABLE]

and thus,

[TABLE]

To prove the last claim, by Fenchel-Young inequality, we have

[TABLE]

*The finiteness of the expectation is then followed by the first two claims and Lemma B.13. *

B.3 One-epoch analysis

First we prove a more general version of Theorem 5.2.

Theorem B.21**.**

Fix any $\Gamma_{j}\in(0,1)$. Assume that

[TABLE]

Then under assumptions B1 and B2,

[TABLE]

where

[TABLE]

Proof B.22**.**

Let $\gamma_{j}=\Gamma_{j}/(1-\Gamma_{j})$ (and hence $\Gamma_{j}=\gamma_{j}/(1+\gamma_{j})$). For convenience, let

[TABLE]

By Lemma B.13,

[TABLE]

By Lemma B.18,

[TABLE]

By Lemma B.16 with $u=x^{*}$,

[TABLE]

where (i) uses (63) and (ii) uses the condition that

[TABLE]

By convexity of $f$, we have

[TABLE]

Therefore, we have

[TABLE]

Rearanging the terms yields

[TABLE]

*The proof is completed by dividing both sides by $1+\gamma_{j}=\frac{1}{1-\Gamma_{j}}$. *

Proof B.23** (Proof of Theorem 5.2).**

Let

[TABLE]

Since $6\eta_{j}L\leq b_{j}g\left(\frac{\Gamma_{j}}{3C_{G}}\right)$, $m_{j}\geq 4b$ and $g^{-1}$ is increasing,

[TABLE]

By definition,

[TABLE]

Thus,

[TABLE]

By assumption B3,

[TABLE]

By Theorem B.21,

[TABLE]

Since $m_{j}\geq 4\alpha^{1/\xi}\max\{b,4C_{G}\}$,

[TABLE]

and

[TABLE]

Therefore, (64) implies that

[TABLE]

*where the last line uses the fact that $\Gamma\leq\frac{1}{4}$. The proof is completed by multiplying both sides by $\frac{4}{3}$. *

B.4 Multi-epoch analysis

Proof B.24** (Proof of Theorem 5.3).**

Let

[TABLE]

Then Theorem (5.2) implies that

[TABLE]

where (i) uses the fact that

[TABLE]

For any $j\geq 0$,

[TABLE]

When $j\geq T_{\kappa}^{*}$, $\alpha^{j}\geq\kappa/\eta L=1/\eta\mu$ and thus

[TABLE]

In summary,

[TABLE]

Plugging this into (65), we conclude that

[TABLE]

Using the same argument as in the proof of Theorem 4.2, we can prove that

[TABLE]

The proof is then completed by noting that

[TABLE]

and

[TABLE]

*where (i) uses the fact that $\mu\leq L$ and the condition $b/m_{0}\leq\Gamma$ and (ii) uses the fact that $\eta L\leq 1/6$ and $\Gamma\leq 1/4$. *

B.5 Complexity analysis

Proof B.25** (Proof of Theorem 5.4).**

Let

[TABLE]

Then for any $T\geq\max\{T^{(1)}(\epsilon),T^{(2)}(\epsilon)\}$,

[TABLE]

where we use the fact that $\eta L\leq 1/6\leq 1/3$. This entails that

[TABLE]

The same arguments as in the proof of Theorem 4.4 imply that

[TABLE]

On the other hand, since $g$ is increasing, whenever $j\leq T_{n}^{*}$,

[TABLE]

Thus, if $T(\epsilon)\leq T_{n}^{*}$,

[TABLE]

If $T(\epsilon)>T_{n}^{*}$, since $B_{j}\leq n$,

[TABLE]

Putting the two pieces together, we obtain that

[TABLE]

Finally, by (5), we conclude that

[TABLE]

*The proof is then completed by replacing $\alpha^{T(\epsilon)}$ ith $A(\epsilon)$. *

Appendix C Miscellaneous

Lemma C.1** (Theorem 2.1.5 of nesterov04 .).**

Let $g$ be a convex function that satisfies the assumption $\textbf{A}1$,

[TABLE]

Lemma C.2** (Lemma A.1 of lei17 ).**

Let $z_{1},\ldots,z_{M}\in\mathbb{R}^{d}$ be an arbitrary population and $\mathcal{J}$ be a uniform random subset of $[M]$ with size $m$. Then

[TABLE]

Lemma C.3**.**

Let $\|\cdot\|$ be an arbitrary norm on a Hilbert space and let $\|\cdot\|_{*}$ be its dual norm. Letting $G_{q}(x)=\|x\|^{q}/q$, we have

[TABLE]

*where $p=q/(q-1)$. *

Proof C.4**.**

By definition of $\|\cdot\|_{*}$,

[TABLE]

Thus,

[TABLE]

By the Fenchel-Young inequality, for any nonnegative numbers $a,b$

[TABLE]

Let $a=\|x\|_{*},b=\|y\|$,

[TABLE]

By definition,

[TABLE]

Thus, we have proved that

[TABLE]

To show the equality, it remains to find a $y(x)$ for any $x$ that achieves the equality. Let

[TABLE]

Then $\|y(x)\|=\|x\|_{*}^{p-1}$ and

[TABLE]

Therefore,

[TABLE]

*The proof is completed by putting the two pieces together. *

Proposition C.5** (Theorem 4, hoeffding63 ).**

Let $C=(c_{1},\ldots,c_{n})$ be a finite population of random elements. Let $Z_{1},\ldots,Z_{m}$ be a random sample with replacement from $C$ and $W_{1},\ldots,W_{m}$ be a random sample without replacement from $C$. If the function $f(x)$ is continuous and convex, then

[TABLE]

Proposition C.6**.**

*Let $\mathcal{X}=\mathbb{R}^{d}$ and $G(x)=\|x\|_{r}^{q}/q$ with $r\leq q$. Then property C3 holds with a constant that only depends on $(d,r,q)$. *

Proof C.7**.**

By Lemma C.3,

[TABLE]

where $r^{\prime}=r/(r-1)$ and $p=q/(q-1)$. Let $\rho_{r}(t)$ denote the modulus of smoothness:

[TABLE]

Let $\mathcal{F}_{n}$ denote the sigma field generated by $\{Z_{1},\ldots,Z_{n}\}$ for $n\leq m$ and $\mathcal{F}_{n}=\mathcal{F}_{m}$ for all $n>m$. Further let

[TABLE]

Clearly $\{X_{n}\}_{n\geq 0}$ is a martingale with respect to the filtration $(\mathcal{F}_{n})_{n\geq 0}$. By Proposition 2.4 (b) of pisier75 , it is left to show that

[TABLE]

for some constant $A>0$. By Hanner’s inequality (hanner56, ),

[TABLE]

This implies that

[TABLE]

Note that $r\leq q$ implies that $r^{\prime}\geq p$. First we notice that

[TABLE]

Thus there exists a constant $c$ such that for any $t\leq c$,

[TABLE]

When $t>c$,

[TABLE]

where the last inequality uses the fact that $p\geq 1$. Putting the pieces together, we prove (66) with

[TABLE]

Appendix D Empirical Demonstration

Although our paper is focusing on the provable adaptivity of SCSG which is not easy to be illustrated numerically, it is worth demonstrating the empirical performance of SCSG as a sanity check. We consider multi-class classification problems on five datasets from UCI machine learning repository (https://archive.ics.uci.edu/ml/index.php, UCI ) – adult111Data source: https://archive.ics.uci.edu/ml/datasets/Adult ; contributed by adult ., connect222Data source: https://archive.ics.uci.edu/ml/datasets/Connect-4; contributed by John Tromp., credit333Data source: https://archive.ics.uci.edu/ml/datasets/default+of+credit+card+clients ; contributed by credit ., crowdsource444Data source: https://archive.ics.uci.edu/ml/datasets/Crowdsourced+Mapping ; contributed by crowdsource . and sensor555Data source: https://archive.ics.uci.edu/ml/datasets/Gas+Sensor+Array+Drift+Dataset+at

+Different+Concentrations ; contributed by sensor1 and sensor2 .. We also include the well-known MNIST dataset666Data source: http://yann.lecun.com/exdb/mnist/ ; contributed by lecun1998gradient . For each dataset, we fit a multi-class logistic regression with individual loss

[TABLE]

where $K$ denotes the number of classes and $x\in\mathbb{R}^{p\times(K-1)}$ denotes the concatenation of $x_{1},x_{2},\ldots,x_{K-1}$. By convention, we add an $L_{2}$ regularizer $(1/n)\|x\|_{2}^{2}$ on the loss function to guarantee the existence of (finite) optimizers, i.e.

[TABLE]

Although $F$ is guaranteed to be $1/n$ strongly convex, the parameter is too small to be useful for algorithms that takes it as an input.

For each dataset, we optimize the objective function by the following six methods:

1. Method 1.

SCSG with $\alpha=1.25,B_{0}=0.001n,m_{0}=0.005n$, $b=0.0001n$, as suggested in Remark 4.3, and a constant step size $\eta$ (to be discussed below). 2. Method 2.

SVRG with inner loop size $m=2n$ as suggested in SVRG , mini-batch size $b=0.0001n$ and a constant step size $\eta$; 3. Method 3.

SARAH (nguyen2019finite, ) with inner loop size $m=2n$, mini-batch size $b=0.0001n$ and a constant step size $\eta$; 4. Method 4.

Katyushans, which works for non-strongly convex functions, with inner loop size $m=2n$ (Algorithm 2 of Katyusha ). We choose option II to update $y_{k+1}$ (line 12 of Algorithm 2) because this yields better results than optionI. In addition, we find that the default option $\alpha_{s}=1/3\tau_{1,s}L$ may be too conservative. To explore the best performance of Katyusha, we set it to be $\eta/\eta_{1,s}$ and find the best-tuned $\eta$. Indeed, the best-tuned $\eta$ is at least $1/L$ on all six datasets and is as large as $8/L$ for mnist and sensor. 5. Method 5.

SGD with a constant stepsize $\eta$ and mini-batch size $b=0.0001n$; 6. Method 6.

SGD with a linearly decayed step size $\eta_{t}=\eta/(1+t)$ and mini-batch size $b=0.0001n$; 7. Method 7.

GD with a constant step size $\eta$.

We run each algorithm and each parameter $\eta$ for $50$ effective passes of data and choose the step size that yields the smallest function value. In particular, we set

[TABLE]

where $L$ is estimated by

[TABLE]

where $L_{i}$ is obtained from Proposition E.1 of SCSG . Note that we compute the average smoothness as opposed to the worst-case smoothness in our theory because the former yields a better scaling for $c$ empirically. In principle we could set $L=\max_{i}L_{i}$ and change the scale of $c$ correspondingly. In addition, we remove 5% outliers with largest $L_{i}$’s from each dataset to avoid unstable performance. We emphasize that outlier removal is helpful to stabilize all algorithms considered here.

Finally, we estimate the true optimum $f(x^{*})$ by running SCSG with the best-tuned $\eta$ with 5000 effective passes of data. As a sanity check, we also estimate $f(x^{*})$ via SVRG with the best-tuned $\eta$ with 5000 effective passes of data and find that SCSG consistently yields better solution than SVRG for all datasets.

The programs to replicate all results are available at https://github.com/lihualei71/ScsgAdaptivity. The best-tuned results for each algorithm are displayed in Figure 1. We find that the best-tuned step size does not lie in the boundary of the tuning set, namely $2^{-10}/L$ or $2^{10}/L$, for all methods and all datasets. This indicates that we tune each method sufficiently. For each plot, the y-axis gives the ratio $(f(x)-f(x^{*}))/(f(x_{0})-f(x^{*}))$ in log-scales so that the initial value is always $10^{0}$. It is clear that SCSG performs well on all these datasets.