Stop Wasting My Gradients: Practical SVRG

Introduction

We consider the problem of optimizing the average of a finite but large sum of smooth functions,

A huge proportion of the model-fitting procedures in machine learning can be mapped to this problem. This includes classic models like least squares and logistic regression but also includes more advanced methods like conditional random fields and deep neural network models. In the high-dimensional setting (large $d$), the traditional approaches for solving (1) are: full gradient (FG) methods which have linear convergence rates but need to evaluate the gradient $f_{i}$ for all $n$ examples on every iteration, and stochastic gradient (SG) methods which make rapid initial progress as they only use a single gradient on each iteration but ultimately have slower sublinear convergence rates.

Le Roux et al. proposed the first general method, stochastic average gradient (SAG), that only considers one training example on each iteration but still achieves a linear convergence rate. Other methods have subsequently been shown to have this property , but these all require storing a previous evaluation of the gradient $f_{i}^{\prime}$ or the dual variables for each $i$. For many objectives this only requires $O(n)$ space, but for general problems this requires $O(np)$ space making them impractical.

Recently, several methods have been proposed with similar convergence rates to SAG but without the memory requirements . They are known as mixed gradient, stochastic variance-reduced gradient (SVRG), and semi-stochastic gradient methods (we will use SVRG). We give a canonical SVRG algorithm in the next section, but the salient features of these methods are that they evaluate two gradients on each iteration and occasionally must compute the gradient on all examples. SVRG methods often dramatically outperform classic FG and SG methods, but these extra evaluations mean that SVRG is slower than SG methods in the important early iterations. They also mean that SVRG methods are typically slower than memory-based methods like SAG.

In this work we first show that SVRG is robust to inexact calculation of the full gradients it requires (§3), provided the accuracy increases over time. We use this to explore growing-batch strategies that require fewer gradient evaluations when far from the solution, and we propose a mixed SG/SVRG method that may improve performance in the early iterations (§4). We next explore using support vectors to reduce the number of gradients required when close to the solution (§5), give a justification for the regularized SVRG update that is commonly used in practice (§6), consider alternative mini-batch strategies (§7), and finally consider the generalization error of the method (§8).

Notation and SVRG Algorithm

SVRG assumes $f$ is $\mu$-strongly convex, each $f_{i}$ is convex, and each gradient $f_{i}^{\prime}$ is Lipschitz-continuous with constant $L$. The method begins with an initial estimate $x^{0}$, sets $x_{0}=x^{0}$ and then generates a sequence of iterates $x_{t}$ using

where $\eta$ is the positive step size, we set $\mu^{s}=f^{\prime}(x^{s})$, and $i_{t}$ is chosen uniformly from $\{1,2,\dots,n\}$. After every $m$ steps, we set $x^{s+1}=x_{t}$ for a random $t\in\{1,\dots,m\}$, and we reset $t=0$ with $x_{0}=x^{s+1}$.

To analyze the convergence rate of SVRG, we will find it convenient to define the function

as it appears repeatedly in our results. We will use $\rho(a)$ to indicate the value of $\rho(a,b)$ when $a=b$, and we will simply use $\rho$ for the special case when $a=b=L$. Johnson & Zhang show that if $\eta$ and $m$ are chosen such that $0<\rho<1$, the algorithm achieves a linear convergence rate of the form

where $x^{*}$ is the optimal solution. This convergence rate is very fast for appropriate $\eta$ and $m$. While this result relies on constants we may not know in general, practical choices with good empirical performance include setting $m=n$, $\eta=1/L$, and using $x^{s+1}=x_{m}$ rather than a random iterate.

Unfortunately, the SVRG algorithm requires $2m+n$ gradient evaluations for every $m$ iterations of (2), since updating $x_{t}$ requires two gradient evaluations and computing $\mu^{s}$ require $n$ gradient evaluations. We can reduce this to $m+n$ if we store the gradients $f_{i}^{\prime}(x^{s})$, but this is not practical in most applications. Thus, SVRG requires many more gradient evaluations than classic SG iterations of memory-based methods like SAG.

SVRG with Error

We first give a result for the SVRG method where we assume that $\mu^{s}$ is equal to $f^{\prime}(x^{s})$ up to some error $e^{s}$. This is in the spirit of the analysis of , who analyze FG methods under similar assumptions. We assume that $\|x_{t}-x^{*}\|\leq Z$ for all $t$, which has been used in related work and is reasonable because of the coercity implied by strong-convexity.

If $\mu^{s}=f^{\prime}(x^{s})+e^{s}$ and we set $\eta$ and $m$ so that $\rho<1$, then the SVRG algorithm (2) with $x^{s+1}$ chosen randomly from $\{x_{1},x_{2},\dots,x_{m}\}$ satisfies

Xiao & Zhang show that non-uniform sampling (NUS) improves the performance of SVRG. They assume each $f_{i}^{\prime}$ is $L_{i}$-Lipschitz continuous, and sample $i_{t}=i$ with probability $L_{i}/n\bar{L}$ where $\bar{L}=\frac{1}{n}\sum_{i=1}^{n}L_{i}$. The iteration is then changed to

which maintains that the search direction is unbiased. In Appendix A, we show that if $\mu^{s}$ is computed with error for this algorithm and if we set $\eta$ and $m$ so that $0<\rho(\bar{L})<1$, then we have a convergence rate of

which can be faster since the average $\bar{L}$ may be much smaller than the maximum value $L$.

2 SVRG with Batching

There are many ways we could allow an error in the calculation of $\mu^{s}$ to speed up the algorithm. For example, if evaluating each $f_{i}^{\prime}$ involves solving an optimization problem, then we could solve this optimization problem inexactly. For example, if we are fitting a graphical model with an iterative approximate inference method, we can terminate the iterations early to save time.

When the $f_{i}$ are simple but $n$ is large, a natural way to approximate $\mu^{s}$ is with a subset (or ‘batch’) of training examples $\mathcal{B}^{s}$ (chosen without replacement),

The batch size $|\mathcal{B}^{s}|$ controls the error in the approximation, and we can drive the error to zero by increasing it to $n$. Existing SVRG methods correspond to the special case where $|\mathcal{B}^{s}|=n$ for all $s$.

Algorithm 1 gives pseudo-code for an SVRG implementation that uses this sub-sampling strategy. If we assume that the sample variance of the norms of the gradients is bounded by $S^{2}$ for all $x^{s}$,

If the batch size satisfies the above condition then

and the convergence rate of SVRG is unchanged compared to using the full batch on all iterations.

Mixed SG and SVRG Method

An approximate $\mu^{s}$ can drastically reduce the computational cost of the SVRG algorithm, but does not affect the $2$ in the $2m+n$ gradients required for $m$ SVRG iterations. This factor of $2$ is significant in the early iterations, since this is when stochastic methods make the most progress and when we typically see the largest reduction in the test error.

To reduce this factor, we can consider a mixed strategy: if $i_{t}$ is in the batch $\mathcal{B}^{s}$ then perform an SVRG iteration, but if $i_{t}$ is not in the current batch then use a classic SG iteration. We illustrate this modification in Algorithm 2. This modification allows the algorithm to take advantage of the rapid initial progress of SG, since it predominantly uses SG iterations when far from the solution. Below we give a convergence rate for this mixed strategy.

We give the proof in Appendix B. The extra term depending on the variance $\sigma^{2}$ is typically the bottleneck for SG methods. Classic SG methods require the step-size $\eta$ to converge to zero because of this term. However, the mixed SG/SVRG method can keep the fast progress from using a constant $\eta$ since the term depending on $\sigma^{2}$ converges to zero as $\alpha$ converges to one. Since $\alpha<1$ implies that $\rho(L,\alpha L)<\rho$, this result implies that when $[f(x^{s})-f(x^{*})]$ is large compared to $e^{s}$ and $\sigma^{2}$ that the mixed SG/SVRG method actually converges faster.

Sharing a single step size $\eta$ between the SG and SVRG iterations in Proposition 2 is sub-optimal. For example, if $x$ is close to $x^{*}$ and $|\mathcal{B}^{s}|\approx n$, then the SG iteration might actually take us far away from the minimizer. Thus, we may want to use a decreasing sequence of step sizes for the SG iterations. In Appendix B, we show that using $\eta=O^{*}(\sqrt{(n-|\mathcal{B}|)/n|\mathcal{B}|})$ for the SG iterations can improve the dependence on the error $e^{s}$ and variance $\sigma^{2}$.

Using Support Vectors

Using a batch $\mathcal{B}^{s}$ decreases the number of gradient evaluations required when SVRG is far from the solution, but its benefit diminishes over time. However, for certain objectives we can further reduce the number of gradient evaluations by identifying support vectors. For example, consider minimizing the Huberized hinge loss (HSVM) with threshold $\epsilon$ ,

In terms of (1), we have $f_{i}(x)=f(b_{i}a_{i}^{T}x)$. The performance of this loss function is similar to logistic regression and the hinge loss, but it has the appealing properties of both: it is differentiable like logistic regression meaning we can apply methods like SVRG, but it has support vectors like the hinge loss meaning that many examples will have $f_{i}(x^{*})=0$ and $f_{i}^{\prime}(x^{*})=0$. We can also construct Huberized variants of many non-smooth losses for regression and multi-class classification.

If we knew the support vectors where $f_{i}(x^{*})>0$, we could solve the problem faster by ignoring the non-support vectors. For example, if there are $100000$ training examples but only $100$ support vectors in the optimal solution, we could solve the problem $1000$ times faster. While we typically don’t know the support vectors, in this section we outline a heuristic that gives large practical improvements by trying to identify them as the algorithm runs.

Our heuristic has two components. The first component is maintaining the list of non-support vectors at $x^{s}$. Specifically, we maintain a list of examples $i$ where $f_{i}^{\prime}(x^{s})=0$. When SVRG picks an example $i_{t}$ that is part of this list, we know that $f_{i_{t}}^{\prime}(x^{s})=0$ and thus the iteration only needs one gradient evaluation. This modification is not a heuristic, in that it still applies the exact SVRG algorithm. However, at best it can only cut the runtime in half.

The heuristic part of our strategy is to skip $f_{i}^{\prime}(x^{s})$ or $f_{i}^{\prime}(x_{t})$ if our evaluation of $f_{i}^{\prime}$ has been zero more than two consecutive times (and skipping it an exponentially larger number of times each time it remains zero). Specifically, for each example $i$ we maintain two variables, $sk_{i}$ (for ‘skip’) and $ps_{i}$ (for ‘pass’). Whenever we need to evaluate $f_{i}^{\prime}$ for some $x^{s}$ or $x_{t}$, we run Algorithm 3 which may skip the evaluation. This strategy can lead to huge computational savings in later iterations if there are few support vectors, since many iterations will require no gradient evaluations.

Identifying support vectors to speed up computation has long been an important part of SVM solvers, and is related to the classic shrinking heuristic . While it has previously been explored in the context of dual coordinate ascent methods , this is the first work exploring it for linearly-convergent stochastic gradient methods.

Regularized SVRG

We are often interested in the special case where problem (1) has the decomposition

A common choice of $h$ is a scaled $1$-norm of the parameter vector, $h(x)=\lambda\|x\|_{1}$. This non-smooth regularizer encourages sparsity in the parameter vector, and can be addressed with the proximal-SVRG method of Xiao & Zhang . Alternately, if we want an explicit $Z$ we could set $h$ to the indicator function for a $2$-norm ball containing $x^{*}$. In Appendix C, we give a variant of Proposition 1 that allows errors in the proximal-SVRG method for non-smooth/constrained settings like this.

where $\mu^{s}=\frac{1}{n}\sum_{i=1}^{n}g_{i}(x^{s})$. That is, they take an exact gradient step with respect to the regularizer and an SVRG step with respect to the $g_{i}$ functions. When the $g_{i}^{\prime}$ are sparse, this form of the update allows us to implement the iteration without needing full-vector operations. A related update is used by Le Roux et al. to avoid full-vector operations in the SAG algorithm [1, §4]. In Appendix C, we prove the below convergence rate for this update.

Consider instances of problem (1) that can be written in the form (4) where $h^{\prime}$ is $L_{h}$-Lipschitz continuous and each $g_{i}^{\prime}$ is $L_{g}$-Lipschitz continuous, and assume that we set $\eta$ and $m$ so that $0<\rho(L_{m})<1$ with $L_{m}=\max\{L_{g},L_{h}\}$. Then the regularized SVRG iteration (5) has

Mini-Batching Strategies

Konečný et al. have also recently considered using batches of data within SVRG. They consider using ‘mini-batches’ in the inner iteration (the update of $x_{t}$) to decrease the variance of the method, but still use full passes through the data to compute $\mu^{s}$. This prior work is thus complimentary to the current work (in practice, both strategies can be used to improve performance). In Appendix D we show that sampling the inner mini-batch proportional to $L_{i}$ achieves a convergence rate of

where $M$ is the size of the mini-batch while

and we assume $0<\rho_{M}<1$. This generalizes the standard rate of SVRG and improves on the result of Konečný et al. in the smooth case. This rate can be faster than the rate of the standard SVRG method at the cost of a more expensive iteration, and may be clearly advantageous in settings where parallel computation allows us to compute several gradients simultaneously.

The regularized SVRG form (5) suggests an alternate mini-batch strategy for problem (1): consider a mini-batch that contains a ‘fixed’ set $\mathcal{B}_{f}$ and a ‘random’ set $\mathcal{B}_{t}$. Without loss of generality, assume that we sort the $f_{i}$ based on their $L_{i}$ values so that $L_{1}\geq L_{2}\geq\dots\geq L_{n}$. For the fixed $\mathcal{B}_{f}$ we will always choose the $M_{f}$ values with the largest $L_{i}$, $\mathcal{B}_{f}=\{f_{1},f_{2},\dots,f_{M_{f}}\}$. In contrast, we choose the members of the random set $\mathcal{B}_{t}$ by sampling from $B_{r}=\{f_{M_{f}+1},\dots,f_{n}\}$ proportional to their Lipschitz constants, $p_{i}=\frac{L_{i}}{(M_{r})\bar{L}_{r}}$ with $\bar{L}_{r}=(1/M_{r})\sum_{i=M_{f}+1}^{n}L_{i}$. In Appendix D, we show the following convergence rate for this strategy:

Let $g(x)=(1/n)\sum_{i\notin[\mathcal{B}_{f}]}f_{i}(x)$ and $h(x)=(1/n)\sum_{i\in[\mathcal{B}_{f}]}f_{i}(x)$. If we replace the SVRG update with

where $\zeta=\frac{(n-M_{f})\bar{L}_{r}}{(M-M_{f})n}$ and $\kappa=\max\{\frac{L_{1}}{n},\zeta\}$.

If $L_{1}\leq n\bar{L}/M$ and $M_{f}<\frac{(\alpha-1)nM}{\alpha n-M}$ with $\alpha=\frac{\bar{L}}{\bar{L}_{r}}$, then we get a faster convergence rate than SVRG with a mini-batch of size $M$. The scenario where this rate is slower than the existing mini-batch SVRG strategy is when $L_{1}\leq n\bar{L}/M$. But we could relax this assumption by dividing each element of the fixed set $\mathcal{B}_{f}$ into two functions: $\beta f_{i}$ and $(1-\beta)f_{i}$, where $\beta=1/M$, then replacing each function $f_{i}$ in $\mathcal{B}_{f}$ with $\beta f_{i}$ and adding $(1-\beta)f_{i}$ to the random set $B_{r}$. This result may be relevant if we have access to a field-programmable gate array (FPGA) or graphical processing unit (GPU) that can compute the gradient for a fixed subset of the examples very efficiently. However, our experiments (Appendix F) indicate this strategy only gives marginal gains.

In Appendix F, we also consider constructing mini-batches by sampling proportional to $f_{i}(x^{s})$ or $\|f_{i}^{\prime}(x^{s})\|$. These seemed to work as well as Lipschitz sampling on all but one of the datasets in our experiments, and this strategy is appealing because we have access to these values while we may not know the $L_{i}$ values. However, these strategies diverged on one of the datasets.

Learning efficiency

In this section we compare the performance of SVRG as a large-scale learning algorithm compared to FG and SG methods. Following Bottou & Bousquet , we can formulate the generalization error $\mathcal{E}$ of a learning algorithm as the sum of three terms

where the approximation error $\mathcal{E}_{\text{app}}$ measures the effect of using a limited class of models, the estimation error $\mathcal{E}_{\text{est}}$ measures the effect of using a finite training set, and the optimization error $\mathcal{E}_{\text{opt}}$ measures the effect of inexactly solving problem (1). Bottou & Bousquet study asymptotic performance of various algorithms for a fixed approximation error and under certain conditions on the distribution of the data depending on parameters $\alpha$ or $\nu$. In Appendix E, we discuss how SVRG can be analyzed in their framework. The table below includes SVRG among their results.

In this table, the condition number is $\kappa=L/\mu$. In this setting, linearly-convergent stochastic gradient methods can obtain better bounds for ill-conditioned problems, with a better dependence on the dimension and without depending on the noise variance $\nu$.

Experimental Results

We plot the experimental results for one run of the algorithms on one dataset in Figure 1, while Appendix F reports results on the other $8$ datasets over $10$ different runs. In our results, the growing batch strategy (Grow) always had better test error performance than using the full batch, while for large datasets it also performed substantially better in terms of the training objective. In contrast, the Mixed strategy sometimes helped performance and sometimes hurt performance. Utilizing support vectors often improved the training objective, often by large margins, but its effect on the test objective was smaller.

Discussion

As SVRG is the only memory-free method among the new stochastic linearly-convergent methods, it represents the natural method to use for a huge variety of machine learning problems. In this work we show that the convergence rate of the SVRG algorithm can be preserved even under an inexact approximation to the full gradient. We also showed that using mini-batches to approximate $\mu^{s}$ gives a natural way to do this, explored the use of support vectors to further reduce the number of gradient evaluations, gave an analysis of the regularized SVRG update, and considered several new mini-batch strategies. Our theoretical and experimental results indicate that many of these simple modifications should be considered in any practical implementation of SVRG.

Acknowledgements

We would like to thank the reviewers for their helpful comments. This research was supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN 312176-2010, RGPIN 311661-08, RGPIN-06068-2015). Jakub Konečný is supported by a Google European Doctoral Fellowship.

Appendix A Convergence Rate of SVRG with Error

We first give the proof of Proposition 1, which gives a convergence rate for SVRG with an error and uniform sampling. We then turn to the case of non-uniform sampling.

We follow a similar argument to Johnson & Zhang , but propagating the error $e^{s}$ through the analysis. We begin by deriving a simple bound on the variance of the sub-optimality of the gradients.

Because each $f_{i}^{\prime}$ is $L$-Lipschitz continuous, we have [22, Theorem 2.1.5]

Setting $y=x^{*}$ and summing this inequality times $(1/n)$ over all $i$ we obtain the result. ∎

By using the inequality $\|x+y\|^{2}\leq 2\|x\|^{2}+2\|y\|^{2}$ and the property

The following Lemma gives a bound on the distance to the optimal solution.

In every iteration $t$ of the inner loop,

The inequality above follows from convexity of $f$. The result follows from applying Cauchy-Schwartz to the linear term in $e$ and that $\|x_{t-1}-x^{*}\|\leq Z$. ∎

To prove Proposition 1 from the main paper, we first sum the inequality in Lemma 3 for all $t=1,...,m$ and take the expectation with respect to the choice of $x^{s}$ to get

where the last inequality uses strong-convexity and that $f^{\prime}(x^{*})=0$. By dividing both sides by $2\eta(1-2L\eta)m$ (which is positive due to the constraint $\eta\leq 1/2L$ implied by $0<\rho<1$ and $\eta>0$), we get

A.2 Non-Uniform Sampling

If we sample $i_{t}$ proportional to the individual Lipschitz constants $L_{i}$, then we have the following analogue of Lemma 1.

Because each $f_{i}^{\prime}$ is $L_{i}$-Lipschitz continuous, we have [22, Theorem 2.1.5]

Setting $y=x^{*}$ and summing this inequality times $(1/n)$ over all $i$ we have

With this modified lemma, we can derive the convergence rate under this non-uniform sampling scheme by following an identical sequence of steps but where each instance of $L$ is replaced by $\bar{L}$.

Appendix B Mixed SVRG and SG Method

We first give the proof of Proposition 2 in the paper, which analyzes a method that mixes SG and SVRG updates using a constant step size. We then consider a variant where the SG and SVRG updates use different step sizes.

Using this in Lemma 3 and following a similar argument we have

where the second inequality uses convexity of $f$ and we have defined $\beta=(1-\alpha)$. We now sum up both sides and take the expectation with respect to the history,

and by dividing both sides by $2\eta(1-2L\eta)m$ we get the result.

B.2 Mixed SVRG and SG with Different Step Sizes

Consider a variant where we use a step size of $\eta$ in the SVRG update and a step-size $\eta_{s}$ in the SG update (which will decrease as the iterations proceed). Analyzing the mixed algorithm in this setting gives

As before, we take the expectation for all $t$ and sum up these values, then rearranage and use strong-convexity of $f$ to get

If we now divide both side by $2m(\alpha\eta(1-2\eta L)+\beta\eta_{s})$, we get

To improve the dependence on the error $e^{s}$ and variance $\sigma^{2}$ compared to the basic SVRG algorithm with error $e^{s}$ (Proposition 1), we require that the terms depending on these values are smaller,

Thus, it is sufficient that $\eta_{s}$ satisfies

Using the relationship between expected error and $S^{2}$, while noting that $S^{2}\leq\sigma^{2}$ and $\frac{(n-|\mathcal{B}|)}{n|\mathcal{B}|}\leq 1$, a step size of the form $\eta_{s}=O^{*}(\sqrt{(n-|\mathcal{B}|)/n|\mathcal{B}|})$ will improve the dependence on $e^{s}$ and $\sigma^{2}$ compared to the dependence on $e^{s}$ in the pure SVRG method.

Appendix C Proximal and Regularized SVRG

In this section we consider objectives of the form

where $g(x)=\frac{1}{n}\sum_{i=1}^{n}g_{i}(x)$. We first consider the case where $h$ is non-smooth and consider a proximal-gradient variant of SVRG where there is an error in the calculation of the gradient (Algorithm 4). We then consider smooth functions $h$ where we use a modified SVRG iteration,

Similar to the work of , in this section we assume that $f,g$ and $h$ are $\mu$-, $\mu_{g}$-, $\mu_{h}$-strongly convex (respectively). As before, we assume each $g_{i}$ is convex and has an $L$-Lipschitz continuous gradient, but $h$ can potentially be non-smooth. The algorithm we propose here extends the algorithm of , but adding an error term. In the algorithm we use the proximal operator which is defined by

Below, we give a convergence rate for this algorithm with an error $e^{s}$.

To prove Proposition 5, we use Lemma 1,2 and 3 from , which are unchanged when we allow an error. Below we modify their Corollary 3 and then the proof of their main theorem.

Using Lemma 1 from and bounding the two expectations gives the result. ∎

Following the proof of Theroem 1 in , we have

Combining with the two previous inequalities we get

If we take the expectation with respect to $i_{t}$ we have

Now by using the Lemma 5 and $\|\bar{x}_{t}-x^{*}\|<Z$ we have

The rest of the proof follows the argument of , and is simlar to the previous proofs in this appendix. We take the expectation and sum up values, using convexity to give

By dividing both sides to $2\eta(1-4L\eta)m$, we get the result. ∎

C.2 Proof of Proposition 3

Now using that $\|h^{\prime}(x^{s})-h^{\prime}(x^{*})\|^{2}\geq 0$ and $\|f(x)-f(y)\|^{2}\leq 2L[f(x)-f(y)-\left<f^{\prime}(y),x-y\right>]$,

From this point, we follow the standard SVRG argument to obtain

Appendix D Mini-Batch

We first give an analysis of SVRG where mini-batches are selected by sampling propotional to the Lipschitz constants of the gradients. We then consider the mixed deterministic/random sampling scheme described in the main paper.

Here we consider using a ‘mini-batch’ of examples in the inner SVRG loop. We use $M$ to denote the batch size, and we assume that the elements of the mini-batch are sampled with a probability of $p_{i}=L_{i}/n\bar{L}$. This gives a search direction and inner iteration of:

It follows from the definition of $p_{i}$ that

D.2 Proof of Proposition 4

We now consider the case where we have $g(x)=(1/n)\sum_{i\notin[\mathcal{B}_{f}]}f_{i}(x)$ and $h(x)=(1/n)\sum_{i\in[\mathcal{B}_{f}]}f_{i}(x)$ for some batch $\mathcal{B}_{f}$. We assume that we sample $M_{r}$ elements of $g$ with probability of $p_{i}=\frac{L_{i}}{(n-M_{f})\bar{L}_{r}}$ and that we use:

where the inequality uses $\|a+b\|^{2}\leq 2\|a\|^{2}+2\|b\|^{2}$. Now we bound each of the above terms separately,

Now following the proof technique we used several times, we can show that:

where $\zeta=\frac{(n-M_{f})\bar{L}_{r}}{(M-M_{f})n}$ and $\kappa=\max\{\frac{L_{1}}{n},\zeta\}$.

Appendix E Learning efficiency

In this section we closely follow Bottou and Bousquet to discuss the performance of SVRG, and other linearly-convergent stochastic methods, as learning algorithms. In the typical supervised learning setting, we are giving $n$ independently drawn input-output pairs $(x_{i},y_{i})$ from some distribution $P(x,y)$ and we seek to minimize the empirical risk,

which tells us how well we do on test data from the same distribution. We use $f^{*}$ to denote the minimizer of the expected risk,

which is the best that a learner can hope to achieve.

Consider a family $\mathcal{F}$ of possible functions that we use to predict $y_{i}$ from $x_{i}$. We write the minimizer of the expected risk over this restricted set as $f_{\mathcal{F}}^{*}$,

while we denote the empirical risk minimizer within this family as $f_{n}$,

where the expectation is taken with respect to the output of the algorithm and with respect to the training examples that we sample. This decomposition shows how three intuitive terms affect the sub-optimality:

$\mathcal{E}_{\text{app}}$ is the approximation error: it measures the effect of restricting attention to the function class $\mathcal{F}$.

$\mathcal{E}_{\text{est}}$ is the estimation error: it measures the effect of only using a finite number of samples.

$\mathcal{E}_{\text{opt}}$ is the optimization error: it measures the effect of inexactly solving the optimization problem.

All other sections of this work focus on the case of finite datasets where we can afford to do several passes through the data (small-scale learning problems in the language of Bottou & Bousquet). In this setting, $\mathcal{E}_{\text{est}}$ is fixed so all we can do to minimize $\mathcal{E}$ is drive the optimization error $\rho$ as small as possible. In this section we consider the case where we do not have enough time to process all available examples, or we have an infinite number of possible examples (large-scale learning problems in the language of Bottou & Bousquet). In this setting, the time restriction means we need to make a trade-off between the optimization error and the estimation error: should we increase $n$ in order to decrease the estimation error $\mathcal{E}_{\text{est}}$ or should we revisit examples to try to more quickly decrease the optimization error $\mathcal{E}_{\text{opt}}$ while keeping the estimation error fixed?

Bottou & Bousquet discuss how under various assumptions we have the variance condition

To make the second and third terms comparable, we can take $\rho=\left(\frac{d}{n}\log\frac{n}{d}\right)^{\alpha}$. Then to achieve an accuracy of $O(\mathcal{E}_{\text{app}}+\epsilon)$ it is sufficient to take $n=O\left(\frac{d}{\epsilon^{1/\alpha}}\log(1/\epsilon)\right)$ samples:

The results presented in the main paper follow from noting that (i) the iteration cost of SVRG is $O(d)$ and (ii) that the number of iterations for SVRG to reach an accuracy of $\rho$ is $O((n+\kappa)\log(1/\rho))$.

Appendix F Additional Experimental Results

We list properties of the dataset considered in the experiments in Table 1. In Figures 1-4, we plot the performance on the various datasets in terms of both the training objective and test error, showing the maximum/mean/minimum performance across 10 random trials. In these plots, we see a clear advantage for the Grow strategy on the largest datasets (bottom row), but less of an advantage or no advantage on the smaller datasets. The advantage of using support vectors seemed less dependent on the data size, as it helped in some small datasets as well as some large datasets, while in some small/large datasets it did not make a big difference.

In Figures 5-6, we give the result of experiments comparing different mini-batch selection strategies. In particular, we consider mini-batch SVRG with a batch size of $16$ and compare the following methods: uniform sampling of the mini-batch (Uniform), sampling proportional to the Lipschitz constants (Lipschitz), and a third strategy based on Proposition 4 in the main paper (Lipschitz+). On each iteration, the Lipschitz+ strategy constructs the mini-batch using the $100$ examples with the largest Lipschitz constants (the ‘fixed’ set) in addition to $16$ examples sampled according to their Lipschitz constants from among the remaining examples. We assume that the fixed set is computed ‘for free’ by calculating these gradients on a GPU or FPGA. In these experiments, there was often no difference between the various methods because the rows of the data were normalized. For the two Lipschitz sampling strategies, we used a step size of $1/\bar{L}$. In some cases, the new sampling scheme may have given a small improvement, but in general the theoretical advantage of this method was not reflected in our experiments.

In Figure 7-8, we repeat the mini-batch experiment but include two additional method: sampling example $i$ proportional to $f_{i}(x^{s})$ (Function) and sampling $i$ proportional to $\|f_{i}^{\prime}(x^{s})\|$ (Gradient). For these strategies we used a step size of $1/\bar{L}$, and on eight of the nine datasets we were surprised that these strategies had similar performance to the Lipschitz sampling strategy (even though they do not have access to the $L_{i}$). However, both of these strategies had strange behaviour on one of the datasets. On the covertype dataset, the Function method seemed to diverge in terms of training objective and test error while the Gradient seemed to converge to a sub-optimal solution in terms of training objective but achieved close to the optimal test error.

References