In -estimation, the minimization of empirical risk functions (RFs) provides point estimates of the model parameters. Statistical inference then seeks to assess the quality of these estimates; e.g.
, by obtaining confidence intervals or solving hypothesis testing problems. Within this context, a classical result in statistics states that the asymptotic distribution of the empirical RF’s minimizer is normal, centered around the population RF’s minimizer[vdV00]
. Thus, given the mean and covariance of this normal distribution, we can infer a range of values, along with probabilities, that allows us to quantify the probability that this interval includes the true minimizer.
The Bootstrap [Efr82, ET94] is a classical tool for obtaining estimates of the mean and covariance of this distribution. The Bootstrap operates by generating samples from this distribution (usually, by re-sampling with or without replacement from the entire data set) and repeating the estimation procedure over these different re-samplings. As parameter dimensionality and data size grow, the Bootstrap becomes increasingly –even prohibitively– expensive.
In this context, we follow a different path: we show that inference can also be accomplished by directly using stochastic gradient descent (SGD), both for point estimates and inference, with a fixed step size over the data set. It is well-established that fixed step-size SGD is by and large the dominant method used for large scale data analysis. We prove, and also demonstrate empirically, that the average of SGD sequences, obtained by minimizing RFs, can also be used for statistical inference. Unlike the Bootstrap, our approach does not require creating many large-size subsamples from the data, neither re-running SGD from scratch for each of these subsamples. Our method only uses first order information from gradient computations, and does not require any second order information. Both of these are important for large scale problems, where re-sampling many times, or computing Hessians, may be computationally prohibitive.
Outline and main contributions:
This paper studies and analyzes a simple, fixed step size111Fixed step size means we use the same step size every iteration, but the step size is smaller with more total number of iterations. In contrast, constant step size means the step size is constant no matter how many iterations taken., SGD-based algorithm for inference in -estimation problems. Our algorithm produces samples, whose covariance converges to the covariance of the -estimate, without relying on bootstrap-based schemes, and also avoiding direct and costly computation of second order information. Much work has been done on the asymptotic normality of SGD, as well as on the Stochastic Gradient Langevin Dynamics (and variants) in the Bayesian setting. As we discuss in detail in Section 4, this is the first work to provide finite sample inference results, using fixed step size, and without imposing overly restrictive assumptions on the convergence of fixed step size SGD.
The remainder of the paper is organized as follows. In the next section, we define the inference problem for -estimation, and recall basic results of asymptotic normality and how these are used. Section 3 is the main body of the paper: we provide the algorithm for creating bootstrap-like samples, and also provide the main theorem of this work. As the details are involved, we provide an intuitive analysis of our algorithm and explanation of our main results, using an asymptotic Ornstein-Uhlenbeck process approximation for the SGD process [KH81, Pfl86, BPM90, KY03, MHB16]
, and we postpone the full proof until the appendix. We specialize our main theorem to the case of linear regression (see supplementary material), and also that of logistic regression. For logistic regression in particular, we require a somewhat different approach, as the logistic regression objective is not strongly convex. InSection 4, we present related work and elaborate how this work differs from existing research in the literature. Finally, in the experimental section, we provide parts of our numerical experiments that illustrate the behavior of our algorithm, and corroborate our theoretical findings. We do this using synthetic data for linear and logistic regression, and also by considering the Higgs detection [BSW14] and the LIBSVM Splice data sets. A considerably expanded set of empirical results is deferred to the appendix.
Supporting our theoretical results, our empirical findings suggest that the SGD inference procedure produces results similar to bootstrap while using far fewer operations. Thereby, we produce a more efficient inference procedure applicable in large scale settings, where other approaches fail.
2 Statistical inference for -estimators
Consider the problem of estimating a set of parameters using samples , drawn from some distribution on the sample space . In frequentist inference, we are interested in estimating the minimizer of the population risk:
where we assume that is real-valued and convex; further, we will use , unless otherwise stated. In practice, the distribution is unknown. We thus estimate by solving an empirical risk minimization (ERM) problem, where we use the estimate :
Statistical inference consists of techniques for obtaining information beyond point estimates , such as confidence intervals. These can be performed if there is an asymptotic limiting distribution associated with [Was13]. Indeed, under standard and well-understood regularity conditions, the solution to -estimation problems satisfies asymptotic normality. That is, the distribution converges weakly to a normal distribution:
see also Theorem 5.21 in [vdV00]. We can therefore use this result, as long as we have a good estimate of the covariance matrix: . The central goal of this paper is obtaining accurate estimates for .
A naive way to estimate is through the empirical estimator where:
Beyond calculating222In the case of maximum likelihood estimation, we have —which is called Fisher information. Thus, the covariance of interest is . This can be estimated either using or . and , this computation requires an inversion of and matrix-matrix multiplications in order to compute —a key computational bottleneck in high dimensions. Instead, our method uses SGD to directly estimate .
3 Statistical inference using SGD
Consider the optimization problem in (2). For instance, in maximum likelihood estimation (MLE), is a negative log-likelihood function. For simplicity of notation, we use and for and , respectively, for the rest of the paper.
The SGD algorithm with a fixed step size , is given by the iteration
is an unbiased estimator of the gradient,i.e., , where the expectation is w.r.t. the stochasticity in the calculation. A classical example of an unbiased estimator of the gradient is , where is a uniformly random index over the samples .
Our inference procedure uses the average of consecutive SGD iterations. In particular, the algorithm proceeds as follows: Given a sequence of SGD iterates, we use the first SGD iterates as a burn in period; we discard these iterates. Next, for each “segment” of iterates, we use the first iterates to compute and discard the last iterates, where indicates the -th segment. This procedure is illustrated in Figure 1. As the final empirical minimum , we use in practice [Bub15].
Some practical aspects of our scheme are discussed below.
Step size selection and length : Theorem 1 below is consistent only for SGD with fixed step size that depends on the number of samples taken. Our experiments, however, demonstrate that choosing a constant (large) gives equally accurate results with significantly reduced running time. We conjecture that a better understanding of ’s and
’s influence requires stronger bounds for SGD with constant step size. Heuristically, calibration methods for parameter tuning in subsampling methods ([PRW12], Ch. 9) could be used for hyper-parameter tuning in our SGD procedure. We leave the problem of finding maximal (provable) learning rates for future work.
Discarded length : Based on the analysis of mean estimation in the appendix, if we discard SGD iterates in every segment, the correlation between consecutive and is of the order of , where and are data dependent constants. This can be used as a rule of thumb to reduce correlation between samples from our SGD inference procedure.
Burn-in period : The purpose of the burn-in period , is to ensure that samples are generated when SGD iterates are sufficiently close to the optimum. This can be determined using heuristics for SGD convergence diagnostics. Another approach is to use other methods (e.g., SVRG [JZ13]) to find the optimum, and use a relatively small
for SGD to reach stationarity, similar to Markov Chain Monte Carlo burn-in.
Statistical inference using and : Similar to ensemble learning [OM99], we use estimators for statistical inference:
Here, is a scaling factor that depends on how the stochastic gradient is computed. We show examples offor statistical inference.
3.1 Theoretical guarantees
Next, we provide the main theorem of our paper. Essentially, this provides conditions under which our algorithm is guaranteed to succeed, and hence has inference capabilities.
For a differentiable convex function , with gradient , let be its minimizer, according to (2), and denote its Hessian at by . Assume that , satisfies:
Weak strong convexity: , for constant ,
Lipschitz gradient continuity: , for constant ,
Bounded Taylor remainder: , for constant ,
Bounded Hessian spectrum at : , .
Furthermore, let be a stochastic gradient of , satisfying:
where and, for positive, data dependent constants , for .
Assume that ; then for sufficiently small step size , the average SGD sequence, , satisfies:
We provide the full proof in the appendix, and also we give precise (data-dependent) formulas for the above constants. For ease of exposition, we leave them as constants in the expressions above. Further, in the next section, we relate a continuous approximation of SGD to Ornstein-Uhlenbeck process [RM51] to give an intuitive explanation of our results.
Discussion. For linear regression, assumptions (), (), (), and () are satisfied when the empirical risk function is not degenerate. In mini batch SGD using sampling with replacement, assumptions (), (), (), and () are satisfied. Linear regression’s result is presented in Corollary 2 in the appendix.
For logistic regression, assumption () is not satisfied because the empirical risk function in this case is strictly but not strongly convex. Thus, we cannot apply Theorem 1 directly. Instead, we consider the use of SGD on the square of the empirical risk function plus a constant; see eq. (3.3) below. When the empirical risk function is not degenerate, (3.3) satisfies assumptions (), (), (), and (). We cannot directly use vanilla SGD to minimize (3.3), instead we describe a modified SGD procedure for minimizing (3.3) in Section 3.3, which satisfies assumptions (), (), (), and (). We believe that this result is of interest by its own. We present the result specialized for logistic regression in Corollary 1.
Note that Theorem 1 proves consistency for SGD with fixed step size, requiring when . However, we empirically observe in our experiments that a sufficiently large constant gives better results. We conjecture that the average of consecutive iterates in SGD with larger constant step size converges to the optimum and we consider it for future work.
3.2 Intuitive interpretation via the Ornstein-Uhlenbeck process approximation
Here, we describe a continuous approximation of the discrete SGD process and relate it to the Ornstein-Uhlenbeck process [RM51], to give an intuitive explanation of our results. In particular, under regularity conditions, the stochastic process asymptotically converges to an Ornstein-Uhlenbeck process , [KH81, Pfl86, BPM90, KY03, MHB16] that satisfies:
where is a standard Brownian motion. Given (8), can be approximated as
3.3 Logistic regression
We next apply our method to logistic regression. We have samples where consists of features and is the label. We estimate
of a linear classifierby:
We cannot apply Theorem 1 directly because the empirical logistic risk is not strongly convex; it does not satisfy assumption (). Instead, we consider the convex function
|where (e.g., ).||(11)|
The gradient of is a product of two terms
Therefore, we can compute , using two independent random variables satisfying
using two independent random variables satisfyingand . For , we have , where are indices sampled from uniformly at random with replacement. For , we have , where are indices uniformly sampled from with or without replacement. Given the above, we have for some constant by the generalized self-concordance of logistic regression [Bac10, Bac14], and therefore the assumptions are now satisfied.
For convenience, we write where . Thus , , and .
Assume ; also , are bounded. Then, we have
where . Here, with depending on how indexes are sampled to compute :
with replacement: ,
no replacement: .
Quantities other than and are data dependent constants.
As with the results above, in the appendix we give data-dependent expressions for the constants. Simulations suggest that the term in our bound is an artifact of our analysis. Because in logistic regression the estimate’s covariance is , we set the scaling factor in (6) for statistical inference. Note that for sufficiently large .
4 Related work
First and second order iterative optimization algorithms –including SGD, gradient descent, and variants– naturally define a Markov chain. Based on this principle, most related to this work is the case of stochastic gradient Langevin dynamics (SGLD) for Bayesian inference – namely, for sampling from the posterior distributions – using a variant of SGD[WT11, BEL15, MHB16, MHB17]. We note that, here as well, the vast majority of the results rely on using a decreasing step size. Very recently, [MHB17] uses a heuristic approximation for Bayesian inference, and provides results for fixed step size.
Our problem is different in important ways from the Bayesian inference problem. In such parameter estimation problems, the covariance of the estimator only depends on the gradient of the likelihood function. This is not the case, however, in general frequentist -estimation problems (e.g., linear regression). In these cases, the covariance of the estimator depends both on the gradient and Hessian of the empirical risk function. For this reason, without second order information, SGLD methods are poorly suited for general -estimation problems in frequentist inference. In contrast, our method exploits properties of averaged SGD, and computes the estimator’s covariance without second order information.
Connection with Bootstrap methods: The classical approach for statistical inference is to use the bootstrap [ET94, ST12]. Bootstrap samples are generated by replicating the entire data set by resampling, and then solving the optimization problem on each generated set of the data. We identify our algorithm and its analysis as an alternative to bootstrap methods. Our analysis is also specific to SGD, and thus sheds light on the statistical properties of this very widely used algorithm.
Connection with stochastic approximation methods: It has been long observed in stochastic approximation that under certain conditions, SGD displays asymptotic normality for both the setting of decreasing step size, e.g., [LPW12, PJ92], and more recently, [TA14, CLTZ16]; and also for fixed step size, e.g., [BPM90], Chapter 4. All of these results, however, provide their guarantees with the requirement that the stochastic approximation iterate converges to the optimum. For decreasing step size, this is not an overly burdensome assumption, since with mild assumptions it can be shown directly. As far as we know, however, it is not clear if this holds in the fixed step size regime. To side-step this issue, [BPM90] provides results only when the (constant) step-size approaches 0 (see Section 4.4 and 4.6, and in particular Theorem 7 in [BPM90]). Similarly, while [KY03] has asymptotic results on the average of consecutive stochastic approximation iterates with constant step size, it assumes convergence of iterates (assumption A1.7 in Ch. 10) – an assumption we are unable to justify in even simple settings.
Beyond the critical difference in the assumptions, the majority of the “classical” subject matter seeks to prove asymptotic results about different flavors of SGD, but does not properly consider its use for inference. Key exceptions are the recent work in [TA14] and [CLTZ16], which follow up on [PJ92]. Both of these rely on decreasing step size, for reasons mentioned above. The work in [CLTZ16] uses SGD with decreasing step size for estimating an -estimate’s covariance. Work in [TA14] studies implicit SGD with decreasing step size and proves results similar to [PJ92], however it does not use SGD to compute confidence intervals.
Overall, to the best of our knowledge, there are no prior results establishing asymptotic normality for SGD with fixed step size for general M-estimation problems (that do not rely on overly restrictive assumptions, as discussed).
5.1 Synthetic data
The coverage probability is defined as where , and is the estimated confidence interval for the coordinate. The average confidence interval width is defined as where is the estimated confidence interval for the coordinate. In our experiments, coverage probability and average confidence interval width are estimated through simulation. We use the empirical quantile of our SGD inference procedure and bootstrap to compute the 95% confidence intervals for each coordinate of the parameter. For results given as a pair , it usually indicates (coverage probability, confidence interval length).
5.1.1 Univariate models
In Figure 2, we compare our SGD inference procedure with Bootstrap and normal approximation with inverse Fisher information in univariate models. We observe that our method and Bootstrap have similar statistical properties. Figure 5 in the appendix shows Q-Q plots of samples from our SGD inference procedure.
Normal distribution mean estimation: Figure (a)a compares 500 samples from SGD inference procedure and Bootstrap versus the distribution , using i.i.d. samples from . We used mini batch SGD described in Sec. A. For the parameters, we used , , , , and mini batch size of 2. Our SGD inference procedure gives (0.916 , 0.806), Bootstrap gives (0.926 , 0.841), and normal approximation gives (0.922 , 0.851).
Exponential distribution parameter estimation: Figure (b)b compares 500 samples from inference procedure and Bootstrap, using
samples from an exponential distribution with PDFwhere . We used SGD for MLE with mini batch sampled with replacement. For the parameters, we used , , , , and mini batch size of 5. Our SGD inference procedure gives (0.922, 0.364), Bootstrap gives (0.942 , 0.392), and normal approximation gives (0.922, 0.393).
Poisson distribution parameter estimation: Figure (c)c compares 500 samples from inference procedure and Bootstrap, using
samples from a Poisson distribution with PDFwhere . We used SGD for MLE with mini batch sampled with replacement. For the parameters, we used , , , , and mini batch size of 5. Our SGD inference procedure gives (0.942 , 0.364), Bootstrap gives (0.946 , 0.386), and normal approximation gives (0.960 , 0.393).
5.1.2 Multivariate models
In these experiments, we set , used mini-batch size of , and used 200 SGD samples. In all cases, we compared with Bootstrap using 200 replicates. We computed the coverage probabilities using 500 simulations. Also, we denote . Additional simulations comparing covariance matrix computed with different methods are given in Sec. D.1.1.
Linear regression: Experiment 1: Results for the case where , , , and with samples is given in Table (a)a. Bootstrap gives (0.941, 4.14), and confidence intervals computed using the error covariance and normal approximation gives (0.928, 3.87). Experiment 2: Results for the case where , , , , and with samples is given in Table (b)b. Bootstrap gives (0.938, 4.47), and confidence intervals computed using the error covariance and normal approximation gives (0.925, 4.18).
Logistic regression: Here we show results for logistic regression trained using vanilla SGD with mini batch sampled with replacement. Results for modified SGD (Sec. 3.3) are given in Sec. D.1.1. Experiment 1: Results for the case where , with samples is given in Table (a)a. Bootstrap gives (0.932, 0.245), and confidence intervals computed using inverse Fisher matrix as the error covariance and normal approximation gives (0.954, 0.256). Experiment 2: Results for the case where , , with samples is given in Table (b)b. Bootstrap gives (0.932, 0.253), and confidence intervals computed using inverse Fisher matrix as the error covariance and normal approximation gives (0.957, 0.264).
5.2 Real data
Here, we compare covariance matrices computed using our SGD inference procedure, bootstrap, and inverse Fisher information matrix on the LIBSVM Splice data set, and we observe that they have similar statistical properties.
5.2.1 Splice data set
The Splice data set 333https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/binary.html contains 60 distinct features with 1000 data samples. This is a classification problem between two classes of splice junctions in a DNA sequence. We use a logistic regression model trained using vanilla SGD.
In Figure 3, we compare the covariance matrix computed using our SGD inference procedure and bootstrap samples. We used 10000 samples from both bootstrap and our SGD inference procedure with , , , and mini batch size of .
Here, we train a binary logistic regression classifier to classify 0/1 using a noisy MNIST data set, and demonstrate that adversarial examples produced by gradient attack [GSS15]
(perturbing an image in the direction of loss function’s gradient with respect to data) can be detected using prediction intervals. We flatten each
image into a 784 dimensional vector, and train a linear classifier using pixel values as features. To add noise to each image, where each original pixel is either 0 or 1, we randomly changed 70% pixels to random numbers uniformly on. Next we train the classifier on the noisy MNIST data set, and generate adversarial examples using this noisy MNIST data set. Figure 4 shows each image’s logit value () and its 95% confidence interval (CI) computed using quantiles from our SGD inference procedure.
In our experiments, we observed that using a larger step size produces accurate results with significantly accelerated convergence time. This might imply that the term in Theorem 1’s bound is an artifact of our analysis. Indeed, although Theorem 1 only applies to SGD with fixed step size, where and imply that the step size should be smaller when the number of consecutive iterates used for the average is larger, our experiments suggest that we can use a (data dependent) constant step size and only require .
In the experiments, our SGD inference procedure uses operations to produce a sample, and Newton method uses operations to produce a sample. The experiments therefore suggest that our SGD inference procedure produces results similar to Bootstrap while using far fewer operations.
This work was partially supported by NSF Grants 1609279, 1704778 and 1764037, and also by the USDoT through the Data-Supported Transportation Operations and Planning (D-STOP) Tier 1 University Transportation Center. A.K. is supported by the IBM Goldstine fellowship. We thank Xi Chen, Philipp Krähenbühl, Matthijs Snel, and Tom Spangenberg for insightful discussions.
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Appendix A Exact analysis of mean estimation
In this section, we give an exact analysis of our method in the least squares, mean estimation problem. For i.i.d. samples , the mean is estimated by solving the following optimization problem
In the case of mini-batch SGD, we sample indexes uniformly randomly with replacement from ; denote that index set as . For convenience, we write , Then, in the mini batch SGD step, the update step is
which is the same as the exponential moving average. And we have
Assume that , then from Chebyshev’s inequality almost surely when
. By the central limit theorem,converges weakly to with . From (12), we have uniformly for all , where the constant is data dependent. Thus, for our SGD inference procedure, we have . Our SGD inference procedure does not generate samples that are independent conditioned on the data, whereas replicates are independent conditioned on the data in bootstrap, but this suggests that our SGD inference procedure can produce “almost independent” samples if we discard sufficient number of SGD iterates in each segment.
When estimating a mean using our SGD inference procedure where each mini batch is elements sampled with replacement, we set in (6).
Appendix B Linear Regression
In linear regression, the empirical risk function satisfies:
where denotes the observations of the linear model and are the regressors. To find an estimate to , one can use SGD with stochastic gradient give by:
where are indices uniformly sampled from with replacement.
Next, we state a special case of Theorem 1. Because the Taylor remainder , linear regression has a stronger result than general -estimation problems.
Assume that , we have
where and .
We assume that is bounded, and quantities other than and are data dependent constants.
As with our main theorem, in the appendix we provide explicit data-dependent expressions for the constants in the result. Because in linear regression the estimate’s covariance is , we set the scaling factor in (6) for statistical inference.
Appendix C Proofs
c.1 Proof of Theorem 1
We first assume that . For ease of notation, we denote
and, without loss of generality, we assume that . The stochastic gradient descent recursion satisfies:
where . Note that is a martingale difference sequence. We use
to denote the gradient component at index , and the Hessian component at index , at optimum , respectively. Note that and .
For each , its Taylor expansion around is
where is the remainder term. For convenience, we write .
For the proof, we require the following lemmata. The following lemma states that as and .
For data dependent, positive constants according to assumptions and in Theorem 1, and given assumption , we have
under the assumption .
As already stated, we assume without loss of generality that . This further implies that: , and
Given the above and assuming expectation w.r.t. the selection of a sample from , we have: