Variable Metric Stochastic Approximation Theory

TL;DR

This paper develops a convergence theory for variable metric stochastic approximation methods, including online BFGS and LBFGS algorithms, with implications for learning from expert advice.

Contribution

It introduces a comprehensive convergence framework for a broad class of online variable metric algorithms, extending existing methods like BFGS and LBFGS.

Findings

Convergence guarantees for online BFGS and LBFGS algorithms.

Applicability of the theory to learning from expert advice.

Enhanced understanding of variable metric stochastic approximation methods.

Abstract

We provide a variable metric stochastic approximation theory. In doing so, we provide a convergence theory for a large class of online variable metric methods including the recently introduced online versions of the BFGS algorithm and its limited-memory LBFGS variant. We also discuss the implications of our results for learning from expert advice.

Equations

$$w_{t+1}=w_t-a_t{\nabla }_w^{\kappa }C(w_t,z_t),a_t>0.$$

$$w_{t+1}=w_t-a_t{B}_t{\nabla }_{w}C(w_t,z_t),$$

$$\frac{c}{2}\Vert w{\Vert }^2+\frac{1}{n}\sum _{i=1}^{n}l(⟨w,x_i⟩,y_i)$$

$$\frac{c}{2}\Vert w{\Vert }^2+\frac{1}{b}\sum _{(x,y)\in Z_t}l(⟨w,x⟩,y),$$

$$w_{t+1}=\underset{w}{\mathrm{argmin}}{\Delta }_H(w,w_t)+a_t{L}_t(w)$$

$${\Delta }_H(p,q)=H(p)-H(q)+\nabla H(q)\cdot (p-q)$$

$$\nabla ({\Delta }_H(w,w_t)$$

$$h(w)-h(w_t)+a_t\nabla L_t(w)$$

$$w_{t+1}=h^{-1}(h(w_t)-a_t\nabla L_t(w_t)).$$

$$h(w_{t+1})=h(w_t)-a_t{\nabla }_w{L}_t(w_t).$$

$${\gamma }_{t+1}={\gamma }_t-a_t{\nabla }_{\gamma }{\tilde{L}}_t({\gamma }_t)$$

$$w_{t+1}=w_t-a_t{Y}_t,$$

$$\inf \{(x-w^{\ast }{)}^{T}f(x):\epsilon <\Vert x-w^{\ast }\Vert <{\epsilon }^{-1}\}>0.$$

$$E(U_{t+1}|{\mathcal{F}}_t)\le (1+{\beta }_t)U_t+{\xi }_t-{\zeta }_t,t=1,2,\dots$$

$$w_{t+1}=w_t-a_t{B}_t{Y}_t,$$

$$C(w_{t+1})=C(w_t-a_t{B}_t{Y}_t)\le$$

$$C(w_t)-a_t[{\nabla }_{w}C(w_t){]}^T{B}_t{Y}_t+K{M}^2{a}_t^2\Vert Y_t{\Vert }^2$$

$$E_{t}C(w_{t+1})\le C(w_t)+K{M}^2{a}_t^2[A+BC(w_t)]-$$

$$a_t[{\nabla }_{w}C(w_t){]}^T{B}_t{\nabla }_{w}C(w_t).$$

$$U_t(1+a_t^{2}BK{M}^2)+AK{M}^2{a}_t^2-m{a}_t\Vert {\nabla }_{w}C(w_t){\Vert }^2.$$

$$\sum _t{a}_t\Vert {\nabla }_{w}C(w_t){\Vert }^2<\infty .$$

$$\Vert ({\nabla }_{w}C(w_t)){\Vert }^2\to 0$$

$$\frac{C(w_t)-C(w^{\ast })}{\Vert \nabla C(w_t){\Vert }^2}\le D<\infty \forall t.$$

$$E_{t}C(w_{t+1})-C(w^{\ast })\le$$

$$[C(w_t)-C(w^{\ast })](1-a_{t}m/D)+AK{M}^2{a}_t^2.$$

$$EC(w_{t+1})-C(w^{\ast })\le$$

$$[EC(w_t)-C(w^{\ast })](1-a_{t}m/D)+AK{M}^2{a}_t^2.$$

$$w_{t+1}=h^{-1}(h(w_t)-a_t\nabla L_t(w_t)).$$

$$w_{t+1}=h^{-1}(h(w_t)-a_t{B}_t\nabla L_t(w_t)).$$

$${\gamma }_{t+1}={\gamma }_t-a_t{B}_t^{-1}{\nabla }_{\gamma }{\tilde{L}}_t({\gamma }_t).$$

Full text

1 Introduction

We begin by introducing online optimization and in particular stochastic gradient descent methods.

1.1 Online Optimization

There exists an abundance of applications that can lead us to online optimization problems where we are trying to find the minimum of a data-dependent function $C:\mathbb{R}^{n}\to\mathbb{R},w\mapsto C(w)=C(w,Z)$, where $Z$ represents data or a probability distribution that generates data. During an online optimization procedure, a sequence of parameters $w_{t}$, $t=1,2,\ldots$ is created by using an update rule for how to define $w_{t+1}$ given the earlier parameters and the new data, $z_{t}$, that has arrived at time $t$. Given a sequence of data samples $z_{1},...,z_{t}$ drawn from a fixed distribution, we will use the notation $C_{t}(\cdot)=\frac{1}{t}\sum_{i=1}^{t}C(\cdot,z_{i})$ for the empirical objective which is the average of online/stochastic/instantaneous objectives $C(\cdot,z)$. We will refer to $C(\cdot)=\operatorname{\mathbb{E}}_{z}C(\cdot,z)$ as the true objective.

1.2 Stochastic Gradient Descent

If one has defined a metric $\kappa$ on the parameter space by supplying a dot product, and if the online objectives are differentiable, then we can use the gradients $\nabla^{\kappa}$ with respect to that metric to define the update equation

[TABLE]

If one uses metrics defined by different dot products for different $t$, then one can let $\nabla$ denote the gradient with respect to the standard Euclidean dot product and instead let the updates take the form

[TABLE]

where the $B_{t}$ are positive definite and symmetric matrices. One example of considering variable metrics is the study of information geometry (Amari and Nagaoka, 1993), where the Fisher information matrix is used to define a metric tensor on a family of probability distributions. More specific examples will be provided in Section 2 below.

1.3 Outline and Summary

We investigate the theoretical foundations for using online updates that include scaling matrices, that is stochastic gradient descent where the gradients are taken with respect to time-varying metrics. Among other results, this provides a convergence proof for a large class of variable metric methods including the recent online (L)BFGS algorithm (Schraudolph et al., 2007). In Section 3 we employ the Robbins-Siegmund theorem to prove $O(1/t)$ convergence in function values $C(w_{t})$ for a class of functions that is useful in machine learning. This is the best possible rate (Bottou and LeCun, 2005; Amari, 1998; Murata, 1998), limited only by the rate at which information arrives. Under weaker assumptions we show almost sure convergence without rate for a larger class. Our results extend those in Bottou and LeCun (2005) by not demanding that the metrics converge to an asymptotic metric.

We first introduce motivating application areas in Section 2. Then in Section 3 we both provide a background in stochastic approximation theory and our new theorems. We apply our theorems to prove convergence for online BFGS in Section 4. Furthermore, we consider implications in the area of learning from expert advice in Section 5. The paper concludes with a discussion in Section 6.

2 Examples

In this section we provide different objective functions and describe the problem settings they appear in.

2.1 Online Risk Minimization

The goal of many machine learning algorithms is to minimize the risk $\operatorname{\mathbb{E}}_{z}l(\left\langle w,x\right\rangle,y)$, where the expectation is taken with respect to a fixed but unknown probability distribution $Z$ that generates instance-label pairs $z:=(x,y)$. The loss $l$ is a non-negative convex function of the parameters $w$ and measures the discrepancy between the labels $y$ and the predictions arising from $x$ and $w$ via their inner product $\left\langle w,x\right\rangle$ (often the Euclidean dot product if $x,w\in\mathbb{R}^{k}$).

In the absence of complete knowledge about the underlying distribution, an empirical sample $Z=\{(x_{i},y_{i}),i=1,\ldots,n\}$111With some abuse of notation we use $Z$ to represent either a data set or a distribution that generates data. is often used to minimize the regularized empirical risk

[TABLE]

where the $L_{2}$-regularization term $\frac{c}{2}\|w\|^{2}$ is introduced for well-posedness. With larger $n$ one can use smaller $c>0$. The true regularized risk that is estimated by (3) is $\frac{c}{2}\|w\|^{2}+\operatorname{\mathbb{E}}_{z}l(\left\langle w,x\right\rangle,y)$.

Batch optimization algorithms, including quasi-Newton and bundle-based methods, are available and widely used to minimize (3), but they are computationally expensive. Gradient-based batch methods may also fail to converge if the loss is non-smooth. Therefore, online optimization methods that work with small subsamples of training data have received considerable attention recently (Kivinen and Warmuth, 1997; Schraudolph, 2002; Azoury and Warmuth, 2001; Shalev-Shwartz et al., 2007; Schraudolph et al., 2007).

In the online setting, we replace the objective (3) with approximations based on subsets (“mini-batches”) of the samples:

[TABLE]

where $Z_{t}\subset Z$ with $|Z_{t}|=b\ll n$. Furthermore, in online learning we can consider using $c=0$ and thereby aiming directly at minimizing the true objective $\operatorname{\mathbb{E}}_{z}l(\left\langle w,x\right\rangle,y)$. During online optimization, a sequence of parameters $w_{t}$, $t=1,2,\ldots$ arises from an update rule that computes $w_{t+1}$ from the previous state and the new information at time $t$. In addition to alleviating the high computational cost of batch methods, the online setting also arises when the data itself is streaming, that is, we are receiving partial information about $C(\cdot)$ in a sequence of small packages.

Including second-order information in the online optimization procedure can accelerate the convergence (Schraudolph et al., 2007). This is particularly true in a setting where only one pass through a dataset is performed. Bottou and LeCun (2005) and Murata (1998) point out that minimizing the empirical objective is different from minimizing the true objective, and show that the result of an online second-order gradient descent procedure can be as close to the true optimum as the minimum of the empirical objective.

2.2 Filtering

The goal in filtering is to separate the signal from the noise in a stream of data. Kalman algorithms in particular use the Euclidean distance (sum-squared loss) and track the minimizer of the empirical objectives $C_{t}(w)=\sum_{i=1}^{t}(y_{t}-w\cdot x_{t})^{2}$. The inverse of the Hessian of $C_{t}$ is used as $B_{t}$ , $a_{t}=1$ and $w_{0}=0$ for the update in (2). $B_{t+1}$ is found from $B_{t}$ with an update whose cost is order $n^{2}$. The result is that $w_{t}=\operatorname*{\mathop{\mathrm{argmin}}}_{w}C_{t}(w)$. Therefore, if we have a fixed distribution the sequence will converge to the optimal parameters.

The same algorithm can be extended to a more general setting where the sum-squared loss is replaced by arbitrary convex functions. The resulting algorithm was called the online Newton-step algorithm by Hazan et al. (2007) and described as an approximate “follow the leader” algorithm, i.e., an algorithm that approximately follows the optimum that the Kalman filter tracks exactly for the sum-squared loss.

2.3 Learning from Expert Advice with Bregman Divergences

The general convex optimization framework by Zinkevich (2003) that Hazan et al. (2007) worked in is also related to the expert advice framework by Azoury and Warmuth (2001). In the expert advice framework one encounters a sequence of loss functions $L_{t}$ ($L_{t}(w)=C(w,z_{t})$ here) and one wants to perform the implicitly defined update

[TABLE]

where $\Delta_{H}$ is the Bregman divergence

[TABLE]

defined by the differentiable convex function $H$. The squared E uclidean distance is an example of a Bregman divergence, corresponding to $H(p)=\frac{1}{2}\|p\|_{2}^{2}$. The goal for Azoury and Warmuth (2001), as it was for Zinkevich (2003), was to have a small total accumulated loss $\sum_{j=1}^{t}L_{j}(w_{j})$ and in particular small regret, $\sum_{j=1}^{t}L_{j}(w_{j})-\min_{w}\sum_{j=1}^{t}L_{j}(w)$. To derive an explicit update, Azoury and Warmuth (2001) differentiate the expression to be minimized

[TABLE]

and by using the approximation $\nabla L_{t}(w)\approx\nabla L_{t}(w_{t})$, they arrive at the updates

[TABLE]

This reduces to online gradient descent, $w_{t+1}=w_{t}-\nabla L_{t}(w_{t})$, for the case $h(w)=w$, i.e., when $H(w)=\frac{1}{2}\|w\|_{2}^{2}$.

Azoury and Warmuth (2001) in particular consider loss functions of a special form, namely the case where $L_{t}(w)=\Delta_{G}(x_{t}\cdot w,g^{-1}(y_{t}))$ where $g=\nabla G$ is an increasing continuous function from $\mathbb{R}$ to itself and where $(x_{t},y_{t})=z_{t}$ is an example. This results in simpler updates since it implies that $\nabla L_{t}(w)=(g(w\cdot x_{t})-y_{t})x_{t}$. Note that, given a parametrization of $\mathbb{R}^{d}$, a one-dimensional transfer function $g$ can be used to define a $d$-dimensional continuous bijection by applying it coordinate-wise.

It is interesting to compare (8) to a reparametrization where one makes a coordinate change $\gamma=h(w)$. Then (8) can be written as

[TABLE]

The one obstacle to identifying (9) with

[TABLE]

where $\tilde{L}_{t}(\gamma)=L(h^{-1}(\gamma))$, is that the gradient of $L_{t}$ is taken with respect to $w$. We know that $\nabla_{\gamma}\tilde{L}_{t}(\gamma_{t})=B_{t}\nabla_{w}L_{t}(w_{t})$, where $B_{t}$ is the inverse Hessian of $H$ at $w_{t}$, in other words the inverse Jacobian of $h$.

A reason for doing a reparametrization is that the loss functions $L_{t}$ might not satisfy the conditions that are needed for convergence of stochastic gradient descent, but it can be possible to find a transfer function $h$ such that $\tilde{L}_{t}$ does. In that sense, the update (8) can be a way of trying to do this even if we only have the gradients $\nabla_{w}L_{t}$ and do not want to calculate the matrices $B_{t}$. Our main theorems will tell us when it is acceptable to omit $B_{t}$, and thereby use an SGD update with respect to varying metrics.

3 Stochastic Approximation Theory

Robbins and Monro (1951) proved a theorem that implies convergence for one-dimensional stochastic gradient descent; Blum (1954) generalized it to the multivariate case. Robbins and Siegmund (1971) achieved a stronger result of wider applicability in supermartingale theory. Here we extend the known convergence results (Bottou and LeCun, 2005) in two ways: a) We prove that updates that include scaling matrices with eigenvalues bounded by positive constants from above and below will converge almost surely; b) under slightly stronger assumptions we obtain a $O(1/t)$ rate of convergence in the function values.

3.1 The Multivariate Robbins-Monro Procedure

Suppose that $\nabla C=f:\mathbb{R}^{k}\to\mathbb{R}^{k}$ is an unknown continuous function that we want to find a root of. Furthermore, suppose that there is a unique root and denote it by $w^{*}$. Given an initial estimate $w_{1}$, the procedure constructs a sequence of estimates $w_{t}$ such that $w_{t}\to w^{*}$ as $t\to\infty$. For any random vector $X$ let $\operatorname{\mathbb{E}}_{t}(X)$ be the conditional expectation given $w_{1},\ldots,w_{t}$. Given $w_{1},\ldots,w_{t}$, we assume that we observe an unbiased estimate $Y_{t}$ of $f(w_{t})=\nabla C(w_{t})$, i.e., $\operatorname{\mathbb{E}}Y_{t}=f(w_{t})$. Given $Y_{t}$ and $w_{t}$ we define $w_{t+1}$ by

[TABLE]

where $a_{t}>0$ for all $t$, $\sum a_{t}=\infty$ and $\sum a_{t}^{2}<\infty$. The $Y_{t}$ are assumed to be drawn from a family $Y(x)$ of random vectors defined for all $x\in\mathbb{R}^{k}$, and $Y_{t}$ is distributed as $Y(w_{t})$. To ensure that $w_{t}$ converges to $w^{*}$ almost surely it is sufficient to assume that there are finite constants $A$ and $B$ such that $\operatorname{\mathbb{E}}\|Y(x)\|^{2}\leq A+B\|x-w^{*}\|^{2}$ for all $x$, and that for all $\varepsilon>0$

[TABLE]

For instance, strictly convex functions satisfy (12). This classical convergence result is implied by the following theorem on almost positive supermartingales, which we will also use to prove our results:

Theorem 3.1** (Robbins and Siegmund, 1971)**

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{F}_{1}\subseteq\mathcal{F}_{2}\subseteq\ldots$ a sequence of sub-$\sigma$-fields of $\mathcal{F}$. Let $U_{t},\beta_{t},\xi_{t}$ and $\zeta_{t}$, $t=1,2,\ldots$ be non-negative $\mathcal{F}_{t}$-measurable random variables such that

[TABLE]

Then on the set $\{\sum_{t}\beta_{t}<\infty,\sum_{t}\xi_{t}<\infty\}$, $U_{t}$ converges almost surely to a random variable, and $\sum_{t}\zeta_{t}<\infty$ almost surely.

3.2 Convergence for Updates with Scaling Matrices

Bottou and LeCun (2005) previously presented results on the rate of convergence in parameter space (using the Euclidean norm) of online updates with convergent scaling matrices, and remark that bounds on the eigenvalues of the scaling matrix will be essential to extending convergence guarantees beyond that. Since recent online quasi-Newton methods (Schraudolph et al., 2007) do not provide convergence of their scaling matrices, we now employ the Robbins-Siegmund theorem to prove that such updates are still guaranteed almost sure convergence, provided the spectrum of their (possibly non-convergent) scaling matrices is uniformly bounded from above by a finite constant and from below by a strictly positive constant.

Theorem 3.2

Let $C:\mathbb{R}^{n}\to\mathbb{R}$ be a twice differentiable cost function with unique minimum $w^{*}$, and let

[TABLE]

*where $B_{t}$ is symmetric and only depends on information available at time $t$.

Then $w_{t}$ converges to $w^{*}$ almost surely if the following conditions hold:*

C.1

$(\forall t)~{}\operatorname{\mathbb{E}}_{t}Y_{t}=\nabla_{w}C(w_{t})$;

C.2

$(\exists K)\,(\forall w)~{}\|\nabla_{w}^{2}C(w)\|\leq 2K$;

C.3

$(\forall\delta>0)~{}\inf_{C(w)-C(w^{*})>\delta}\|\nabla_{w}C(w)\|>0$;

C.4

$(\exists A,B)\,(\forall t)~{}\operatorname{\mathbb{E}}\|Y_{t}\|^{2}\leq A+BC(w_{t})$;

C.5

$(\exists~{}m,M\!:0<m<M<\infty)\,(\forall t)~{}mI\prec B_{t}\prec MI$,  where $I$ is the identity matrix;

C.6

$\sum_{t}a_{t}^{2}<\infty$*  and  $\sum_{t}a_{t}=\infty$.*

**Proof ** Since $C$ is twice differentiable and has bounded Hessian (C.2) we can use Taylor expansion and the upper eigenvalue bound (C.5) to prove that

[TABLE]

[TABLE]

which implies, using (C.1) and (C.4), that

[TABLE]

[TABLE]

If we let $U_{t}=C(w_{t})$ and merge the terms containing $U_{t}$ it follows that $\operatorname{\mathbb{E}}_{t}U_{t+1}\leq$

[TABLE]

Since $\sum_{t}a_{t}^{2}<\infty$ (C.6), the Robbins-Siegmund theorem can now be applied. We find that

[TABLE]

Since $\sum a_{t}=\infty$ (C.6) it follows from (C.3) that

[TABLE]

and that $C(w_{t})\to C(w^{*})$ as $t\to\infty$.

Remark 3.3

The assumption that $C$ is twice differentiable is only needed for the Taylor expansion we use to obtain (15). If we have such a property from elsewhere we do not need twice-differentiability.

3.3 Asymptotic Rates

Consider the situation described in Theorem 3.2. We now strengthen assumption (C.3), which demands that the function $C$ is not so flat around the minimum that we may never approach it, to instead assuming that

[TABLE]

Condition (20) is implied by strong convexity if we know that the $w_{t}$ tend to the optimum $w^{*}$. Since $\nabla C(w^{*})=0$ we can use first-order Taylor expansion of $\nabla C$ around $w^{*}$ to approximate $\|\nabla C(w)\|^{2}$ by $(w-w^{*})^{T}\nabla^{2}C(w^{*})(w-w^{*})$.

We will also modify assumption (C.4) by setting $B=0$. Theorem 3.2 guarantees (under the weaker conditions) that the procedure will almost surely generate a converging sequence which is therefore contained in some ball around $w^{*}$. This makes the new condition reasonable. Bottou (1998) contains a more elaborate discussion on what is there called ”Global Confinement”.

We need a result on what the expected improvement is, given the step size $a_{t}$, the uniform bound on the Hessian, and the uniform eigenvalue bounds. The key to achieving this is (15). This section’s counterpart to (16) under the new conditions is

[TABLE]

[TABLE]

We want to know the rate of the sequence $\operatorname{\mathbb{E}}C(w_{t})-\inf_{\tilde{w}}C(\tilde{w})$, and will use the fact that taking the unconditional expectation of (21) yields the result

[TABLE]

[TABLE]

We are now in a position to state our result:

Theorem 3.4

Let $C:\mathbb{R}^{n}\to\mathbb{R}$ be a twice differentiable cost function with unique minimum $w^{*}$. Assume that

Conditions C.1–C.6 from Theorem 3.2 hold with $B=0$ in C.4. 2. 2.

Equation (20) holds. 3. 3.

$a_{t}=\frac{\tau}{t}$* with $\tau>D/m$.*

Then $\operatorname{\mathbb{E}}C(w_{t})-\inf_{\tilde{w}}C(\tilde{w})$ is equivalent to $\frac{1}{t}$ as $t\to\infty$.

**Proof ** Bottou and LeCun (2005, A.4) state: if $u_{t}=[1-\frac{\alpha}{t}+o(1/t)]u_{t-1}+\frac{\beta}{t^{2}}+o(1/t^{2})$ with $\alpha>1$ and $\beta>0$, then $tu_{t}\to\frac{\beta}{\alpha-1}$. Theorem 3.4 follows from (22) by setting $u_{t}=\operatorname{\mathbb{E}}C(w_{t})-C(w^{*})$.

4 Quasi-Newton Methods

Quasi-Newton methods are optimization methods with updates of the form $w_{t+1}\leftarrow w_{t}-a_{t}\,B_{t}\nabla_{w}C(w_{t})$, where $a_{t}>0$ is a scalar gain (typically set by a line search) and $B_{t}$ a positive-semidefinite scaling matrix. If $B_{t}=I$, we have simple gradient descent; setting $B_{t}$ to the inverse Hessian of $C(w)$ we recover Newton’s method. Inverting the $k\times k$ Hessian is a computationally challenging task if $k$ is large; quasi-Newton methods reduce the computational cost by incrementally maintaining a symmetric positive-definite estimate $B_{t}$ of the inverse Hessian of the objective function.

4.1 (L)BFGS

The BFGS algorithm (Nocedal and Wright, 1999) was developed independently by Broyden, Fletcher, Goldfarb, and Shanno in 1970. It incrementally maintains its estimate $B_{t}$ of the inverse Hessian via a rank-two update that minimizes a weighted Frobenius norm $\|B_{t+1}-B_{t}\|_{W}$ subject to the secant equation $s_{t}=B_{t+1}y_{t}$, where $s_{t}:=w_{t+1}-w_{t}$ and $y_{t}:=\nabla_{w}C(w_{t+1})-\nabla_{w}C(w_{t})$ denote the most recent step along the optimization trajectory in parameter and gradient space, respectively. LBFGS is a limited-memory (matrix-free) version of BFGS.

4.2 Online (L)BFGS

Recently developed online variants of BFGS and LBFGS, called oBFGS resp. oLBFGS, are amenable to stochastic approximation of the gradient (Schraudolph et al., 2007). The key differences between the online and batch algorithms can be summarized as follows:

The gradient of the objective is estimated from small samples (mini-batches) of data. The difference $y_{t}$ of gradients is computed with gradients for the same data sample, i.e., for the same function $C(\cdot,z_{t})$. Line search is replaced by a gain sequence $a_{t}$. A trust region parameter $\lambda$ is introduced modifying the algorithm to estimate $(H_{t}+\lambda I)^{-1}$, where $H_{t}$ is the Hessian at iteration $t$; this prevents the largest eigenvalue of $B_{t}$ from exceeding $\lambda^{-1}$.

See Schraudolph et al. (2007) for a more detailed description of the oLBFGS and oBFGS algorithms, and for experimental results on quadratic bowl objectives and conditional random fields (CRFs), which are instances of risk minimization.

Remark 4.1

To guarantee a uniform lower eigenvalue bound for the updates of Schraudolph et al. (2007) we would have to use $B_{t}+\gamma I$ for some $\gamma>0$, effectively interpolating between o(L)BFGS as defined by Schraudolph et al. (2007) and simple online gradient descent. This lower bound is not needed for convergence per se but to prove that the convergence is to the minimum.

4.3 Filtering

Kailath et al. (2000, Chapter 14) present assumptions in control theory that imply either convergence of the matrices $B_{t}$ or upper and lower eigenvalue bounds. The relevant control theory is too extensive and complicated to be reviewed here and we therefore only point out the connection.

5 Expert Advice with Bregman Divergences

We now compare the updates in the expert advice framework by Azoury and Warmuth (2001) to the SGD updates that would result from a non-linear reparametrization.

As outlined in section 2.3 the difference between the update

[TABLE]

and performing stochastic gradient descent with respect to a new variable $\gamma=h(w)$ is that the latter would include matrices $B_{t}$ which are the inverse Jacobians of $h$ at $w_{t}$, i.e. the inverse Hessian of $H$ where $\nabla H=h$. The update $\gamma_{t+1}=\gamma_{t}-a_{t}\nabla_{\gamma}\tilde{L}_{t}(\gamma_{t})$ where $\tilde{L}_{t}(\gamma)=L_{t}(h^{-1}(\gamma))$, expressed with respect to $w$ looks as follows

[TABLE]

Therefore (23) can be written as

[TABLE]

The main point of the reparametrization would be to change variables so that SGD converges with an optimal rate for the new objectives. We would like the new objective to be approximately quadratic. Assuming that we have chosen the transfer function in such a manner that our main theorems apply to the new objective function, it remains to check the scaling matrix condition.

To satisfy the conditions for the scaling matrices we need the Jacobian of the transfer function to have upper and lower eigenvalue bounds. In the literature that these methods are studied in, an assumption of uniformly bounded gradients $\nabla L_{t}(w_{t})$ (Azoury and Warmuth, 2001) is often used, or the parameters are restricted to a compact set (Zinkevich, 2003). In that case the conditions on scaling matrices become easier to satisfy: in any compact set, popular functions like $e^{\theta}$, $(1+e^{\theta})^{-1}$, or other sigmoid functions (though not, e.g., $\theta^{3}$ which is flat at the origin) have derivatives that are bounded from above and below by a strictly positive constant.

6 Conclusion

We provide a variable metric stochastic approximation theory which implies convergence for stochastic gradient descent even when the gradients are calculated with respect to variable metrics. Metrics are sometimes changed for optimization purposes, as in the case of using quasi-Newton methods. Our main theorems imply convergence results for online versions of the BFGS and LBFGS optimization methods. Kalman filters are a class of well-known algorithms that can be viewed as an online Newton method for the special case of square losses, since the procedure at every step performs a gradient step where the gradient is defined using the Hessian of the loss for the examples seen so far. Finally we investigate the task of learning from expert advice where Bregman divergences are frequently used to achieve updates that are suitable for the task at hand. We interpret the resulting updates as stochastic gradient descent in a space that has undergone a non-linear reparametrization, and where we use different metrics depending on the point we are at.

Acknowledgements

The authors are very grateful to Leon Bottou at NEC Labs in Princeton, NJ for his help with the main theorem.

The first author is funded by NICTA which is funded through the Australian Government’s Backing Australia’s Ability initiative, in part through the Australian Research Council.