Stochastic subgradient method converges on tame functions

Introduction

In this work, we study the long term behavior of the stochastic subgradient method on nonsmooth and nonconvex functions. Setting the stage, consider the optimization problem

Here $\partial f(x)$ denotes the Clarke subdifferential . Informally, the set $\partial f(x)$ is the convex hull of limits of gradients at nearby differentiable points. In classical circumstances, the subdifferential reduces to more familiar objects. Namely, when $f$ is $C^{1}$-smooth at $x$, the subdifferential $\partial f(x)$ consists only of the gradient $\nabla f(x)$, while for convex functions, it reduces to the subdifferential in the sense of convex analysis. The positive sequence $\{\alpha_{k}\}_{k\geq 0}$ is user specified, and it controls the step-sizes of the algorithm. As is typical for stochastic subgradient methods, we will assume that this sequence is square summable but not summable, meaning $\sum_{k}\alpha_{k}=\infty$ and $\sum_{k}\alpha_{k}^{2}<\infty$. Finally, the stochasticity is modeled by the random (noise) sequence $\{\xi_{k}\}_{k\geq 1}$. We make the standard assumption that conditioned on the past, each random variable $\xi_{k}$ has mean zero and its second moment grows at a controlled rate.

Though variants of the stochastic subgradient method (1.1) date back to Robbins-Monro’s pioneering 1951 work , their convergence behavior is still largely not understood in nonsmooth and nonconvex settings. In particular, the following question remains open.

Does the (stochastic) subgradient method have any convergence guarantees on locally Lipschitz functions, which may be neither smooth nor convex?

That this question remains unanswered is somewhat concerning as the stochastic subgradient method forms a core numerical subroutine for several widely used solvers, including Google’s TensorFlow and the open source PyTorch library.

Though widely applicable, these previous results on the convergence of the stochastic subgradient method do not apply to even relatively simple non-pathological functions, such as $f(x,y)=(|x|-|y|)^{2}$ and $f(x)=(1-\max\{x,0\})^{2}$. It is not only toy examples, however, that lack convergence guarantees, but the entire class of deep neural networks with nonsmooth activation functions (e.g., ReLU). Since such networks are routinely trained in practice, it is worthwhile to understand if indeed the iterates $x_{k}$ tend to a meaningful limit.

In this paper, we provide a positive answer to this question for a wide class of locally Lipschitz functions; indeed, the function class we consider is virtually exhaustive in data scientific contexts (see Corollary 5.11 for consequences in deep learning). Aside from mild technical conditions, the only meaningful assumption we make is that $f$ strictly decreases along any trajectory $x(\cdot)$ of the differential inclusion $\dot{x}(t)\in-\partial f(x(t))$ emanating from a noncritical point. Under this assumption, a standard Lyapunov-type argument shows that every limit point of the stochastic subgradient method is critical for $f$, almost surely. Techniques of this type can be found for example in the monograph of Kushner-Yin [22, Theorem 5.2.1] and the landmark papers of Benaïm-Hofbauer-Sorin . Here, we provide a self-contained treatment, which facilitates direct extensions to “proximal” variants of the stochastic subgradient method.Concurrent to this work, the independent preprint also provides convergence guarantees for the stochastic projected subgradient method, under the assumption that the objective function is “subdifferentially regular” and the constraint set is convex. Subdifferential regularity rules out functions with downward kinks and cusps, such as deep networks with the Relu($\cdot$) activation functions. Besides subsuming the subdifferentially regular case, the results of the current paper apply to the broad class of Whitney stratifiable functions, which includes all popular deep network architectures. In particular, our analysis follows closely the recent work of Duchi-Ruan [17, Section 3.4.1] on convex composite minimization.

An elementary linear algebraic argument then shows that if $x$ satisfies $\dot{x}(t)\in-\partial f(x(t))$ a.e., then automatically $-\dot{x}(t)$ is the minimal norm element of $\partial f(x(t))$. Therefore, integrating (1.2) yields the desired descent guarantee

Evidently, exactly the same argument yields the chain rule (1.2) for subdifferentially regular functions. These are the functions $f$ such that each subgradient $v\in\partial f(x)$ defines a linear lower-estimator of $f$ up to first-order; see for example [10, Section 2.4] or [31, Definition 7.25]. Nonetheless, subdifferentially regular functions preclude “downwards cusps”, and therefore still do not capture such simple examples as $f(x)=(1-\max\{x,0\})^{2}$. It is worthwhile to mention that one can not expect (1.3) to always hold. Indeed, there are pathological locally Lipschitz functions $f$ that do not satisfy (1.3); one example is the univariate 1-Lipschitz function whose Clarke subdifferential is the unit interval at every point .

In this work, we isolate a different structural property on the function $f$, which guarantees the validity of (1.2) and therefore of the descent condition (1.3). We will assume that the graph of the function $f$ admits a partition into finitely many smooth manifolds, which fit together in a regular pattern. Formally, we require the graph of $f$ to admit a so-called Whitney stratification, and we will call such functions Whitney stratifiable. Whitney stratifications have already figured prominently in optimization, beginning with the seminal work . An important subclass of Whitney stratifiable functions consists of semi-algebraic functions – meaning those whose graphs can be written as a finite union of sets each defined by finitely many polynomial inequalities. Semialgebraicity is preserved under all the typical functional operations in optimization (e.g. sums, compositions, inf-projections) and therefore semi-algebraic functions are usually easy to recognize. More generally still, “semianalytic” functions and those that are “definable in an o-minimal structure” are Whitney stratifiable . The latter function class, in particular, shares all the robustness and analytic properties of semi-algebraic functions, while encompassing many more examples. Case in point, Wilkie famously showed that there is an o-minimal structure that contains both the exponential $x\mapsto e^{x}$ and all semi-algebraic functions.The term “tame” used in the title has a technical meaning. Tame sets are those whose intersection with any ball is definable in some o-minimal structure. The manuscript provides a nice exposition on the role of tame sets and functions in optimization.

The key observation for us, which originates in [16, Section 5.1], is that any locally Lipschitz Whitney stratifiable function necessarily satisfies the chain rule (1.2) along any absolutely continuous curve. Consequently, the descent guarantee (1.3) holds along any subgradient trajectory, and our convergence guarantees for the stochastic subgradient method become applicable. Since the composition of two definable functions is definable, it follows immediately from Wilkie’s o-minimal structure that nonsmooth deep neural networks built from definable pieces—such as quadratics $t^{2}$, hinge losses $\max\{0,t\}$, and log-exp $\log(1+e^{t})$ functions—are themselves definable. Hence, the results of this paper endow stochastic subgradient methods, applied to definable deep networks, with rigorous convergence guarantees.

Validity of the chain rule (1.2) for Whitney stratifiable functions is not new. It was already proved in [16, Section 5.1] for semi-algebraic functions, though identical arguments hold more broadly for Whitney stratifiable functions. These results, however, are somewhat hidden in the paper , which is possibly why they have thus far been underutilized. In this manuscript, we provide a self-contained review of the material from [16, Section 5.1], highlighting only the most essential ingredients and streamlining some of the arguments.

Though the discussion above is for unconstrained problems, the techniques we develop apply much more broadly to constrained problems of the form

Here $f$ and $g$ are locally-Lipschitz continuous functions and $\mathcal{X}$ is an arbitrary closed set. The popular proximal stochastic subgradient method simply iterates the steps

Combining our techniques with those in quickly yields subsequential convergence guarantees for this algorithm. Note that we impose no convexity assumptions on $f$, $g$, or $\mathcal{X}$.

The outline of this paper is as follows. In Section 2, we fix the notation for the rest of the manuscript. Section 3 provides a self-contained treatment of asymptotic consistency for discrete approximations of differential inclusions. In Section 4, we specialize the results of the previous section to the stochastic subgradient method. Finally, in Section 5, we verify the sufficient conditions for subsequential convergence for a broad class of locally Lipschitz functions, including those that are subdifferentially regular and Whitney stratifiable. In particular, we specialize our results to deep learning settings in Corollary 5.11. In the final Section 6, we extend the results of the previous sections to the proximal setting.

Preliminaries

2 Set-valued maps and the Clarke subdifferential

Differential inclusions and discrete approximations

In this section, we discuss the asymptotic behavior of discrete approximations of differential inclusions. All the elements of the analysis we present, in varying generality, can be found in the works of Benaïm-Hofbauer-Sorin , Borkar , and Duchi-Ruan . Out of these, we most closely follow the work of Duchi-Ruan .

Notice that the image of any arc $x$ is automatically contained in $\mathcal{X}$, since arcs are continuous and $\mathcal{X}$ is closed. In this work, we will primarily focus on iterative algorithms that aim to asymptotically track a trajectory of the differential inclusion (3.1) using a noisy discretization with vanishing step-sizes. Though our discussion allows for an arbitrary set-valued map $G$, the reader should keep in mind that the most important example for us will be $G=-\partial f$, where $f$ is a locally Lipschitz function.

Throughout, we will consider the following iteration sequence:

Here $\alpha_{k}>0$ is a sequence of step-sizes, $y_{k}$ should be thought of as an approximate evaluation of $G$ at some point near $x_{k}$, and $\xi_{k}$ is a sequence of “errors”.

Our immediate goal is to isolate reasonable conditions, under which the sequence $\{x_{k}\}$ asymptotically tracks a trajectory of the differential inclusion (3.1). Following the work of Duchi-Ruan on stochastic approximation, we stipulate the following assumptions.

All limit points of $\{x_{k}\}$ lie in $\mathcal{X}$.

The iterates are bounded, i.e., $\sup_{k\geq 1}\|x_{k}\|<\infty$ and $\sup_{k\geq 1}\|y_{k}\|<\infty$.

The sequence $\{\alpha_{k}\}$ is nonnegative, square summable, but not summable:

The weighted noise sequence is convergent: $\sum_{k=1}^{n}\alpha_{k}\xi_{k}\rightarrow v$ for some $v$ as $k\rightarrow\infty$.

Some comments are in order. Conditions 1, 2, and 3 are in some sense minimal, though the boundedness condition must be checked for each particular algorithm. Condition 4 guarantees that the noise sequence $\xi_{k}$ does not grow too quickly relative to the rate at which $\alpha_{k}$ decrease. The key Condition 5 summarizes the way in which the values $y_{k}$ are approximate evaluations of $G$, up to convexification.

To formalize the idea of asymptotic approximation, let us define the time points $t_{0}=0$ and $t_{m}=\sum_{k=1}^{m-1}\alpha_{k}$, for $m\geq 1$. Let $x(\cdot)$ now be the linear interpolation of the discrete path:

For each $\tau\geq 0$, define the time-shifted curve $x^{\tau}(\cdot)=x(\tau+\cdot)$.

2 Subsequential convergence to equilibrium points

A primary application of the discrete process (3.2) is to solve the inclusion

Indeed, one can consider the points satisfying (3.4) as equilibrium (constant) trajectories of the differential inclusion (3.1). Ideally, one would like to find conditions guaranteeing that every limit point $\bar{x}$ of the sequence $\{x_{k}\}$, produced by the recursion (3.2), satisfies the desired inclusion (3.4). Making such a leap rigorous typically relies on combining the asymptotic convergence guarantee of Theorem 3.1 with existence of a Lyapunov-like function $\varphi(\cdot)$ for the continuous dynamics; see e.g. . Let us therefore introduce the following assumption.

As we have alluded to above, the following theorem shows that under Assumptions A and B, every limit point $\bar{x}$ of $\{x_{k}\}$ indeed satisfies the inclusion $0\in G(\bar{x})$. We were unable to find this result stated and proved in this generality. Therefore, we record a complete proof in Section 3.3. The idea of the proof is of course not new, and can already be seen for example in . Upon first reading, the reader can safely skip to Section 4.

Suppose that Assumptions A and B hold. Then every limit point of $\{x_{k}\}_{k\geq 1}$ lies in $G^{-1}(0)$ and the function values $\{\varphi(x_{k})\}_{k\geq 1}$ converge.

3 Proof of Theorem 3.2

In this section, we will prove Theorem 3.2. The argument we present is rooted in the “non-escape argument” for ODEs, using $\varphi$ as a Lyapunov function for the continuous dynamics. In particular, the proof we present is in the same spirit as that in [22, Theorem 5.2.1] and [17, Section 3.4.1].

Henceforth, we will suppose that Assumptions A and B hold. We first collect two elementary lemmas.

The equality $\lim_{k\to\infty}\|x_{k+1}-x_{k}\|=0$ holds.

From the recurrence (3.2), we have $\|x_{k+1}-x_{k}\|\leq\alpha_{k}\|y_{k}\|+\alpha_{k}\|\xi_{k}\|.$ Assumption A guarantees $\alpha_{k}\to 0$ and $\{y_{k}\}$ are bounded, and therefore $\alpha_{k}\|y_{k}\|\to 0$. Moreover, since the sequence $\sum_{k=1}^{n}\alpha_{k}\xi_{k}$ is convergent, we deduce $\alpha_{k}\|\xi_{k}\|\to 0$. The result follows. ∎

Clearly, the inequalities $\leq$ and $\geq$ hold in (3.5), respectively. We will argue that the reverse inequalities are valid. To this end, let $\tau_{i}\to\infty$ be an arbitrary sequence with $x(\tau_{i})$ converging to some point $x^{*}$ as $i\to\infty$.

Lemma 3.3 implies that the right-hand-side tends to zero, and hence $x_{k_{i}}\to x^{*}$. Continuity of $\varphi$ then directly yields the guarantee $\varphi(x_{k_{i}})\rightarrow\varphi(x^{*})$.

In particular, we may take $\tau_{i}\to\infty$ to be a sequence realizing $\operatornamewithlimits{liminf}_{t\to\infty}\varphi(x(t))$. Since the curve $x(\cdot)$ is bounded, we may suppose that up to taking a subsequence, $x(\tau_{i})$ converges to some point $x^{*}$. We therefore deduce

thereby establishing the first equality in (3.5). The second equality follows analogously. ∎

The proof of Theorem 3.3 will follow quickly from the following proposition.

The values $\varphi(x(t))$ have a limit as $t\to\infty$.

Choose any $\epsilon>0$ satisfying $\epsilon\notin\varphi(G^{-1}(0))$. Note that by Assumption B, we can let $\epsilon>0$ be as small as we wish. By the first equality in (3.5), there are infinitely many indices $k$ such that $\varphi(x_{k})<\epsilon$. The following elementary observation shows that for all large $k$, if $x_{k}$ lies in $\mathcal{L}_{\epsilon}$ then the next iterate $x_{k+1}$ lies in $\mathcal{L}_{2\epsilon}$.

$x_{i_{1}+1}\in\mathcal{L}_{2\epsilon}\setminus\mathcal{L}_{\epsilon}$, and

Then let $i_{2}>i_{1}$ be the next smallest index satisfying the same property, and so on. See Figure 1 for an illustration. The following claim will be key.

This process must terminate, that is $\{x_{k}\}$ exits $\mathcal{L}_{2\epsilon}$ only finitely many times.

Before proving the claim, let us see how it immediately yields the validity of the theorem. To this end, observe that Claims 1 and 2 immediately imply $x_{k}\in\mathcal{L}_{2\epsilon}$ for all large $k$. Since $\epsilon>0$ can be made arbitrarily small, we deduce $\lim_{k\to\infty}\varphi(x_{k})=0$. Equation (3.5) then directly implies $\lim_{t\to\infty}\varphi(x(t))=0$, as claimed.

By construction, we have $\varphi(x_{i_{j}})\leq\epsilon$ and $\varphi(x_{i_{j}+1})>\epsilon$. We therefore deduce

Recall $x_{i_{j}}\to z(0)$ as $j\to\infty$. Lemma 3.3 in turn implies $\|x_{i_{j}}-x_{i_{j}+1}\|\to 0$ and therefore $x_{i_{j}+1}\to z(0)$ as well. Continuity of $\varphi$ then guarantees that the right-hand-side of (3.6) tends to $\epsilon$, and hence $\varphi(z(0))=\lim_{j\to\infty}\varphi(x_{i_{j}})=\epsilon$. In particular, $z(0)$ is not an equilibrium point of $G$. Hence, Assumption B yields a real $T>0$ such that

In particular, there exists a real $\delta>0$ satisfying $\varphi(z(T))\leq\epsilon-2\delta.$

Appealing to uniform convergence on $[0,T]$, we conclude

Hence, for all large $j$, all the curves $x^{\tau_{j}}$ map $[0,T]$ into $\mathcal{L}_{2\epsilon}$. We conclude that the exit time satisfies

We will show that the bound $\varphi(z(T))\leq\epsilon-2\delta$ yields the opposite inequality $t_{e_{j}}\leq\tau_{j}+T$, which will lead to a contradiction.

The proof of the lemma is now complete. ∎

We can now prove the main convergence theorem.

On the other hand, we successively deduce

where the last two equalities follow from Proposition 3.5 and continuity of $\varphi$. We have thus arrived at a contradiction, and the theorem is proved. ∎

Subgradient dynamical system

Assumptions A and B, taken together, provide a powerful framework for proving subsequential convergence of algorithms to a zero of the set-valued map $G$. Note that the two assumptions are qualitatively different. Assumption A is a property of both the algorithm (3.2) and the map $G$, while Assumption B is a property of $G$ alone.

For the rest of our discussion, we apply the differential inclusion approach outlined above to optimization problems. Setting the notation, consider the optimization task

and subsequentially converge to critical points of $f$. Discrete processes of the type (3.2) for the optimization problem (4.1) are often called stochastic approximation algorithms. Here we study two such prototypical methods: the stochastic subgradient method in this section and the stochastic proximal subgradient in Section 6. Each fits under the umbrella of Assumption A.

Setting the stage, the stochastic subgradient method simply iterates the steps:

where $\{\alpha_{k}\}_{k\geq 1}$ is a step-size sequence and $\{\xi_{k}\}_{k\geq 1}$ is now a sequence of random variables (the “noise”) on some probability space. Let us now isolate the following standard assumptions (e.g. ) for the method and see how they immediately imply Assumption A.

The sequence $\{\alpha_{k}\}$ is nonnegative, square summable, but not summable:

Almost surely, the stochastic subgradient iterates are bounded: $\sup_{k\geq 1}\|x_{k}\|<\infty$.

$\{\xi_{k}\}$ is a martingale difference sequence w.r.t the increasing $\sigma$-fields

Assumption C guarantees that almost surely Assumption A holds.

Suppose Assumption C holds. Clearly A.1 and A.3 hold vacuously, while A.2 follows immediately from C.2 and local Lipschitz continuity of $f$. Assumption A.5 follows quickly from the fact the $\partial f$ outer-semicontinuous and compact-convex valued; we leave the details to the reader. Thus we must only verify A.4, which follows quickly from standard martingale arguments. Indeed, notice from Assumption C, we have

Define the $L^{2}$ martingale $X_{k}=\sum_{i=1}^{k}\alpha_{i}\xi_{i}$. Thus the limit $\langle X\rangle_{\infty}$ of the predictable compensator

exists. Applying [15, Theorem 5.3.33(a)], we deduce that almost surely $X_{k}$ converges to a finite limit. ∎

Thus applying Theorem 3.1, we deduce that under Assumption C, almost surely, the stochastic subgradient path tracks a trajectory of the differential inclusion (4.2). As we saw in Section 3, proving subsequential convergence to critical points requires existence of a Lyapunov-type function $\varphi$ for the continuous dynamics. Henceforth, let us assume that the Lyapunov function $\varphi$ is $f$ itself. Section 5 is devoted entirely to justifying this assumption for two broad classes of functions that are virtually exhaustive in data scientific contexts.

Thus applying Theorem 3.2, we have arrived at the following guarantee for the stochastic subgradient method.

Suppose that Assumptions C and D hold. Then almost surely, every limit point of stochastic subgradient iterates $\{x_{k}\}_{k\geq 1}$ is critical for $f$ and the function values $\{f(x_{k})\}_{k\geq 1}$ converge.

Verifying the descent condition

In light of Theorems 3.2 and 4.2, it is important to isolate a class of functions that automatically satisfy Assumption D.2. In this section, we do exactly that, focusing on two problem classes: (1) subdifferentially regular functions and (2) those functions whose graphs are Whitney stratifiable. We will see that the latter problem class also satisfies D.1.

The material in this section is not new. In particular, the results of this section have appeared in [16, Section 5.1]. These results, however, are somewhat hidden in the paper and are difficult to parse. Moreover, at the time of writing [16, Section 5.1], there was no clear application of the techniques, in contrast to our current paper. Since we do not expect the readers to be experts in variational analysis and semialgebraic geometry, we provide here a self-contained treatment, highlighting only the most essential ingredients and streamlining some of the arguments.

Let us begin with the following definition, whose importance for verifying Property 2 in Assumption D will become clear shortly.

The importance of the chain rule becomes immediately clear with the following lemma.

Then equality $\|\dot{z}(t)\|={\rm dist}(0,\partial f(z(t)))$ holds for a.e. $t\geq 0$, and therefore

In particular, property 2 of Assumption D holds.

Fix a real $t\geq 0$ satisfying $(f\circ z)^{\prime}(t)=\langle\partial f(z(t)),\dot{z}(t)\rangle$ . Observe then the equality

To simplify the notation, set $S:=\partial f(z(t))$, $W:=\text{span}(S-S)$, and $y:=-\dot{z}(t)$. Appealing to (5.2), we conclude $y\in W^{\perp}$, and therefore trivially we have

Basic linear algebra implies $\|y\|={\rm dist}(0;y+W)$. Noting $\partial f(z(t))\subset y+W$, we deduce $\|\dot{z}(t)\|\leq{\rm dist}(0;\partial f(z(t)))$ as claimed. Since the reverse inequality trivially holds, we obtain the claimed equality, $\|\dot{z}(t)\|={\rm dist}(0;\partial f(z(t)))$.

Since $f$ admits a chain rule, we conclude for a.e. $\tau\geq 0$ the estimate

Since $f$ is locally Lipschitz, the composition $f\circ z$ is absolutely continuous. Hence integrating over the interval $[0,t]$ yields (5.1).

Suppose now that the point $z(0)$ is noncritical. Then by outer semi-continuity of $\partial f$, the exists $T>0$ such that $z(\tau)$ is noncritical for any $\tau\in[0,T]$. It follows immediately that the value $\int_{0}^{t}{\rm dist}^{2}\left(0;\partial f(z(\tau))\right)\,d\tau$ is strictly increasing in $t\in[0,T]$, and therefore by (5.1) that $f\circ z$ is strictly decreasing. Hence item 2 of Assumption D holds, as claimed. ∎

Thus property 2 of Assumption D is sure to hold as long as $f$ admits a chain rule. In the following two sections, we identify two different function classes that indeed admit the chain rule.

The first function class we consider consists of subdifferentially regular functions. Such functions play a prominent role in variational analysis due to their close connection with convex functions; we refer the reader to the monograph for details. In essence, subdifferential regularity forbids downward facing cusps in the graph of the function; e.g. $f(x)=-|x|$ is not subdifferentially regular. We now present the formal definition.

The following lemma shows that any locally Lipschitz function that is subdifferentially regular indeed admits a chain rule.

Any locally Lipschitz function that is subdifferentially regular admits a chain rule and therefore item 2 of Assumption D holds.

Instead, equating $(f\circ x)^{\prime}(t)$ with the left limit of the difference quotient yields the reverse inequality $(f\circ x)^{\prime}(t)\leq\langle v,\dot{x}(t)\rangle$. Thus $f$ admits a chain rule and item 2 of Assumption D holds by Lemma 5.2. ∎

Thus we have arrived at the following corollary. For ease of reference, we state subsequential convergence guarantees both for the general process (3.2) and for the specific stochastic subgradient method (4.3).

(Stochastic approximation) Consider the iterates $\{x_{k}\}_{k\geq 1}$ produced by (3.2) and suppose that Assumption A holds with $G=-\partial f$. Then every limit point of the iterates $\{x_{k}\}_{k\geq 1}$ is critical for $f$ and the function values $\{f(x_{k})\}_{k\geq 1}$ converge.

(Stochastic subgradient method) Consider the iterates $\{x_{k}\}_{k\geq 1}$ produced by the stochastic subgradient method (4.3) and suppose that Assumption C holds. Then almost surely, every limit point of the iterates $\{x_{k}\}_{k\geq 1}$ is critical for $f$ and the function values $\{f(x_{k})\}_{k\geq 1}$ converge.

Though subdifferentially regular functions are widespread in applications, they preclude “downwards cusps”, and therefore do not capture such simple examples as $f(x,y)=(|x|-|y|)^{2}$ and $f(x)=(1-\max\{x,0\})^{2}$. The following section concerns a different function class that does capture these two nonpathological examples.

2 Stratifiable functions

As we saw in the previous section, subdifferential regularity is a local property that implies the desired item 2 of Assumption D. In this section, we instead focus on a broad class of functions satisfying a global geometric property, which eliminates pathological examples from consideration.

Frontier condition: For any two strata $L$ and $M$, the implication

Whitney condition (a): For any sequence of points $z_{k}$ in a stratum $M$ converging to a point $\bar{z}$ in a stratum $L$, if the corresponding normal vectors $v_{k}\in N_{M}(z_{k})$ converge to a vector $v$, then the inclusion $v\in N_{L}(\bar{z})$ holds.

The following theorem, which first appeared in [4, Corollary 5], shows that Whitney stratifiable functions automatically satisfy the weak Sard property of Assumption D. We present a quick argument here for completeness. It is worthwhile to mention that such a Sard type result holds more generally for any stratifiable set-valued map; see the original work or the monograph [21, Section 8.4].

Next, we prove the chain rule for any Whitney stratifiable function.

To see this, fix a manifold $\mathcal{M}_{i}$ and let $\Omega_{i}$ be the set of all $t\geq 0$ such that $x(t)\in\mathcal{M}_{i}$, the derivative $\dot{x}(t)$ exists, and we have $\dot{x}(t)\notin T_{\mathcal{M}_{i}}(x(t))$. If we argue that $\Omega_{i}$ has zero measure, then so does the union $\cup_{i}\Omega_{i}$ and the claim is proved. Fix an arbitrary $t\in\Omega_{i}$. There exists a closed interval $I$ around $t$ such that $x$ restricted to $I$ intersects $\mathcal{M}_{i}$ only at $x(t)$, since otherwise we would deduce that $\dot{x}(t)$ lies in $T_{\mathcal{M}_{i}}(x(t))$ by definition of the tangent space. We may further shrink $I$ such that its endpoints are rational. It follows that $\Omega_{i}$ may be covered by disjoint closed intervals with rational endpoints. Hence $\Omega_{i}$ is countable and therefore zero measure, as claimed.

Notice $\tfrac{|f(x(t+r))-f(\gamma(r))|}{r}\leq L\left\|\tfrac{x(t+r)-x(t)+\gamma(0)-\gamma(r)}{r}\right\|\rightarrow L\|\dot{x}(t)-\dot{\gamma}(0)\|=0$ as $r\rightarrow 0.$ Thus

where the last equality follows from (5.4). ∎

Putting together Theorems 3.2, 4.2, 5.8 and Lemma 5.7, we arrive at the main result of our paper. Again for ease of reference, we state subsequential convergence guarantees both for the general process (3.2) and for the specific stochastic subgradient method (4.3).

(Stochastic approximation) Consider the iterates $\{x_{k}\}_{k\geq 1}$ produced by (3.2) and suppose that Assumption A holds with $G=-\partial f$. Then every limit point of the iterates $\{x_{k}\}_{k\geq 1}$ is critical for $f$ and the function values $\{f(x_{k})\}_{k\geq 1}$ converge.

(Stochastic subgradient method) Consider the iterates $\{x_{k}\}_{k\geq 1}$ produced by the stochastic subgradient method (4.3) and suppose that Assumption C holds. Then almost surely, every limit point of the iterates $\{x_{k}\}_{k\geq 1}$ is critical for $f$ and the function values $\{f(x_{k})\}_{k\geq 1}$ converge.

Verifying Whitney stratifiability is often an easy task. Indeed, there are a number of well-known and easy to recognize function classes, whose members are automatically Whitney stratifiable. We now briefly review such classes, beginning with the semianalytic setting.

A closed set $Q$ is called semianalytic if it can be written as a finite union of sets, each having the form

For example, every semialgebraic set is semianalytic, but in contrast to the semianalytic case, semi-algebraic sets are stable with respect to all boolean operations and projections onto subspaces. The latter property is a direct consequence of the celebrated Tarski-Seidenberg Theorem. Moreover, semialgebraic sets are typically easy to recognize using quantifier elimination; see [12, Chapter 2] for a detailed discussion. Importantly, compositions of semialgebraic functions are semialgebraic.

A far reaching axiomatic extension of semialgebraic sets, whose members are also Whitney stratifiable, is built from “o-minimal structures”. Loosely speaking, sets that are definable in an o-minimal structure share the same robustness properties and attractive analytic features as semialgebraic sets. For the sake of completeness, let us give a formal definition, following Coste and van den Dries-Miller.

the elements of $\mathcal{O}_{1}$ are exactly the finite unions of intervals (possibly infinite) and points.

As in the semialgebraic setting, any function definable in an o-minimal structure admits a Whitney $C^{p}$ stratification, for any $p\geq 1$ (see e.g. ). Beyond semialgebraicity, Wilkie showed that that there is an o-minimal structure that simultaneously contains both the graph of the exponential function $x\mapsto e^{x}$ and all semi-algebraic sets .

Since the composition of two definable functions is definable, we conclude that nonsmooth deep neural networks built from definable pieces—such as ReLU, quadratics $t^{2}$, hinge losses $\max\{0,t\}$, and SoftPlus $\log(1+e^{t})$ functions—are themselves definable. Hence, the results of this paper endow stochastic subgradient methods, applied to definable deep networks, with rigorous convergence guarantees. Due to the importance of subgradient methods in deep learning, we make this observation precise in the following corollary which provides a rigorous convergence guarantee for a wide class of deep learning loss functions that are recursively defined, including convolutional neural networks, recurrent neural networks, and feed-forward networks.

For each given data pair $(x_{j},y_{j})$ with $j=1,\ldots,n$, recursively define:

$V_{i}(\cdot)$ are linear maps into the space of matrices.

$\rho_{i}$ are definable activation functions applied coordinate wise, such as those whose domain can be decomposed into finitely many intervals on which it coincides with $\log t$, $\exp(t)$, $\max(0,t)$, or $\log(1+e^{t})$.

Let $\{w_{k}\}_{k\geq 1}$ be the iterates produced by the stochastic subgradient method on the deep neural network loss $f(w):=\sum_{j=1}^{n}f(w;x_{j},y_{j})$, and suppose that the standing assumption C holds.In the assumption, replace $x_{k}$ with $w_{k}$, since we now use $w_{k}$ to denote the stochastic subgradient iterates. Then almost surely, every limit point $w^{*}$ of the iterates $\{w_{k}\}_{k\geq 1}$ is critical for $f$, meaning $0\in\partial f(w^{*})$, and the function values $\{f(w_{k})\}_{k\geq 1}$ converge.

Proximal extensions

In this section, we extend most of the results in Sections 4 and 5 on unconstrained problems to a “proximal” setting and comment on sufficient conditions to ensure boundedness of the iterates. The arguments follow quickly by combining the techniques developed by Duchi-Ruan with those presented in Section 5. Consequently, all the proofs are in Appendix A.

Setting the stage, consider the composite optimization problem

Thus we will be interested in an extension of the stochastic subgradient method that tracks a trajectory of the differential inclusion

and subsequentially converges to a composite critical point of (6.1). Seeking to apply the techniques of Section 3, we can simply set $G=-\partial f-\partial g-N_{\mathcal{X}}$ in the notation therein. Note that $G$ thus defined is not necessarily a subdifferential of a single function because equality in the subdifferential sum rule [31, Corollary 10.9] can fail when the summands are not subdifferentially regular.

We now aim to describe the proximal stochastic subgradient method for the problem (6.1). There are two ingredients we must introduce: a stochastic subgradient oracle for $f$ and the proximity map of $g+\delta_{\mathcal{X}}$, where $\delta_{\mathcal{X}}$ is the indicator function of $\mathcal{X}$. We describe the two ingredients in turn.

Thus after sampling $\omega\sim P$, the vector $\zeta(x,\omega)$ can serve as a stochastic estimator for a true subgradient of $f$.

Standard deterministic proximal splitting methods utilize the proximal map of $g+\delta_{\mathcal{X}}$, namely:

We can now formally state the algorithm. Given an iterate $x_{k}\in\mathcal{X}$, the proximal stochastic subgradient method performs the update

$\mathcal{X}$ is closed, $f$ and $g$ are locally Lipschitz, and $g$ is bounded from below on $\mathcal{X}$.

The sequence $\{\alpha_{k}\}_{k\geq 1}$ is nonnegative, square summable, but not summable:

Almost surely, the iterates are bounded: $\sup_{k\geq 1}\|x_{k}\|<\infty$.

For every convergent sequence $\{z_{k}\}_{k\geq 1}$, we have

and $z(0)$ is not a composite critical point of (6.1), there exists a real $T>0$ satisfying

Let us make a few comments. Properties E.1, E.3, E.4, and E.5 are mild and completely expected in light of the results in the previous sections. Property E.6 is a technical condition ensuring that the expected maximal noise in the stochastic subgradient is bounded along any convergent sequence. Finally, property E.2 is a mild technical condition on function $g$ that we allow. In particular, it holds for any convex, globally Lipschitz, or coercive locally Lipschitz function. We record this observation in the following lemma.

Fix a point $x$ and a point $z$ satisfying $g(z)\leq g(x)$. Let us look at each case and upper bound the error $g(x)-g(z)$. Suppose first that $g$ is convex. Then for the vector $v\in\partial g(x)$ of minimal norm, we have

where $L(x):={\rm dist}(0,\partial g(x))$. If $f$ is globally Lipschitz, then clearly we have

where $L(x)$ is identically equal to the global Lipschitz constant of $g$. Finally, in the third case, suppose that $g$ is coercive and locally Lipschitz. We deduce

where $L(x)$ is the Lipschitz modulus of $g$ on the compact sublevel set $[g\leq g(x)]$. Since $g$ is locally Lipschitz continuous, in all three cases, the function $L(\cdot)$ is bounded on bounded sets. ∎

Under the two assumptions, E and F, we obtain the following subsequential convergence guarantee. The argument in the appendix is an application of Theorem 3.2. To this end, we show that Assumption E implies Assumption A almost surely, while Assumption F is clearly equivalent to Assumption B.

Suppose that Assumptions E and F hold. Then almost surely, every limit point of the iterates $\{x_{k}\}_{k\geq 1}$ produced by the proximal stochastic subgradient method (6.4) is composite critical for (6.1) and the function values $\{\varphi(x_{k})\}_{k\geq 1}$ converge.

Whenever $\mathcal{X}$ is Clarke regular or Whitney stratifiable, $\mathcal{X}$ automatically admits a chain rule. Indeed, the argument is identical to that of Lemma 5.4 and Theorem 5.8. As in the unconstrained case, Assumption F.2 is true as long as $f$, $g$, and $\mathcal{X}$ admit a chain rule.

Then equality $\|\dot{z}(t)\|={\rm dist}(0,G(z(t)))$ holds for a.e. $t\geq 0$, and therefore we have the estimate

In particular, property 2 of Assumption F holds.

We now arrive at the main result of the section.

Suppose that Assumption E holds and that $f$, $g$, and $\mathcal{X}$ are definable in an o-minimal structure. Let $\{x_{k}\}_{k\geq 1}$ be the iterates produced by the proximal stochastic subgradient method (6.4). Then almost surely, every limit point of the iterates $\{x_{k}\}_{k\geq 1}$ is composite critical for the problem (6.1) and the function values $\{\varphi(x_{k})\}_{k\geq 1}$ converge.

1 Comments on boundedness

Thus far, all of our results have assumed that the subgradient iterates $\{x_{k}\}$ satisfy $\sup_{k\geq 1}\|x_{k}\|<\infty$ almost surely. One may enforce this assumption in several ways, most easily by assuming the constraint set $\mathcal{X}$ is bounded. Beyond boundedness of $\mathcal{X}$, proper choice of regularizer $g$ may also ensure boundedness of $\{x_{k}\}$. Indeed, this observation was already made by Duchi-Ruan [17, Lemma 3.15]. Following their work, let us isolate the following assumption.

$g$ is convex and $\beta$-coercive, meaning $\lim_{k\rightarrow\infty}g(x)/\|x\|^{\beta}=\infty.$

There exists $\lambda\in(0,1]$ such that $g(x)\geq g(\lambda x)$ for $x$ with sufficiently large norm.

A natural regularizer satisfying this assumption is $\|x\|^{\beta+\epsilon}$ for any $\epsilon>0$. The following theorem, whose proof is identical to that of [17, Lemma 3.15], shows that with Assumption G in place, the stochastic proximal subgradient methods produces bounded iterates.

We note that in the special (deterministic) case that $\zeta(x,\omega)\in\partial f(x)$ for all $\omega$, the assumption on $\zeta(x,\omega)$ reduces to $\sup_{\zeta\in\partial f(x)}\|\zeta\|\leq L(1+\|x\|^{\nu})$, which stipulates that $g$ grows more quickly than $f$.

Appendix A Proofs for the proximal extension

Let us now formally define the normal cone constructions of variational analysis. For any point $x\in\mathcal{X}$, the proximal normal cone to $\mathcal{X}$ at $x$ is the set

In this subsection, we record a few auxiliary lemmas to be used in the sequel.

where we set $x_{+}:=T_{\alpha}(x-\alpha v)$.

Let $L(\cdot)$ be the function from property E.2. From the definition of the proximal map, we deduce

Dividing both sides by $\tfrac{1}{2}\|x_{+}-x\|$ yields the result. ∎

Notice that because $\{z_{k}\}_{k\geq 1}$ is bounded, it follows that $\{p(z_{k})\}$ is bounded. Now consider the random variable $X_{k}=\beta_{k}^{2}\|\zeta(z_{k},\cdot)\|^{2}$. Due to the estimate

standard results in measure theory (e.g., [33, Exercise 1.5.5]) imply that $X_{k}\rightarrow 0$ almost surely. ∎

Almost surely, we have $\alpha_{k}\|\zeta(x_{k},\omega_{k})\|\to 0$ as $k\rightarrow\infty$.

Therefore, the following infinite sum is a.s. finite:

A.2 Proof Theorem 6.2

For each index $k$, define the set-valued map

Note that $G_{k}$ is a deterministic map, with $k$ only signifying the dependence on the deterministic sequence $\alpha_{k}$. Define now the noise sequence

Let us now write the proximal stochastic subgradient method in the form (3.2).

Notice that for every index $k\geq 0$, we have

The following lemma shows that A.4 holds almost surely.

The limit $\displaystyle\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\alpha_{i}\xi_{i}$ exists almost surely.

We first prove that $\{\alpha_{k}\xi_{k}\}$ is an $L_{2}$ martingale difference sequence, meaning that for all $k$, we have

Notice that because $\{x_{k}\}$ is bounded a.s., it follows that $\{L(x_{k})\}$ and $\{p(x_{k})\}$ are bounded a.s. Therefore, because

Now, define the $L^{2}$ martingale $X_{k}=\sum_{i=1}^{k}\alpha_{i}\xi_{i}$. Thus, the limit $\langle X\rangle_{\infty}$ of the predictable compensator

exists. Applying [15, Theorem 5.3.33(a)], we deduce that almost surely $X_{k}$ converges to a finite limit, which completes the proof of the claim. ∎

Almost surely, the sequence $\{y_{k}\}$ is bounded.

Because the sequence $\{x_{k}\}$ is almost surely bounded and $g$ is locally Lipschitz, clearly we have

almost surely. Thus, we need only show that

almost surely. To this end, by the triangle inequality and Lemma A.1, we have for any fixed $\omega\in\Omega$ the bound

Therefore, by Jensen’s inequality, we have that

which is almost surely bounded for all $k$. Taking the supremum yields the result. ∎

It will be convenient to prove a more general statement, which is independent of the iterate sequence $\{x_{k}\}$, and instead only depends on the maps $G_{k}$. Namely, consider any sequence $\{z_{k}\}\subseteq\mathcal{X}$ converging to a point $z\in\mathcal{X}$ and an arbitrary sequence $w_{k}^{f}\in\partial f(z_{k})$. Let $\{n_{k}\}$ be an unbounded increasing sequence of indices. Observe that since $G(z)$ is convex and using Jensen’s inequality, we have

Our goal is to prove that the right-hand-side tends to zero almost surely, which directly implies validity of A.5

Our immediate goal is to apply the dominated convergence theorem to each term in the above finite sum to conclude that each term converges to zero. To that end, we must show two properties: for every fixed $\omega$, each term in the sum tends to zero, and that each term is bounded by an integrable function. We now prove both properties.

Almost surely in $\omega\sim P$, we have that

Optimality conditions [31, Exercise 10.10] of the proximal subproblem imply

for some $w^{g}_{k}(\omega)\in\partial g(T_{\alpha_{n_{k}}}(z_{k}-\alpha_{n_{k}}\zeta(z_{k},\omega)))$ and $w^{\mathcal{X}}_{k}(\omega)\in N_{\mathcal{X}}^{L}(T_{\alpha_{n_{k}}}(z_{k}-\alpha_{n_{k}}\zeta(z_{k},\omega)))$, and where $N_{\mathcal{X}}^{L}$ denotes the limiting normal cone. Observe that by continuity and the fact that $\sum_{k=1}^{\infty}\alpha_{n_{k}}^{2}<\infty$ and $\alpha_{n_{k}}\zeta(z_{k},\omega)\rightarrow 0$ as $k\rightarrow\infty$ a.e. (see Lemma A.2), it follows that

Indeed, setting $z_{k}^{+}=T_{\alpha_{n_{k}}}(z_{k}-\alpha_{n_{k}}\zeta(z_{k},\omega))$, we have that by Lemma A.1,

which implies that $\lim_{k\rightarrow\infty}z_{k}^{+}=\lim_{k\rightarrow\infty}z_{k}=z$.

We furthermore deduce that $w^{\mathcal{X}}_{k}(\omega)$ and $w_{k}^{g}(\omega)$ are bounded almost surely. Indeed, $w_{k}^{g}(\omega)$ is bounded since $g$ is locally Lipschitz and $z_{k}^{+}$ are bounded. Moreover, Lemma A.1 implies

Observe that the right hand-side is a.s. bounded by item 6 of Assumption E. Thus, since $w_{k}^{g}(\omega)+w_{k}^{\mathcal{X}}(\omega)$ and $w_{k}^{g}(\omega)$ are a.s. bounded, it follows that $w_{k}^{\mathcal{X}}(\omega)$ must also be a.s. bounded, as desired.

Appealing to outer semicontinuity of $\partial f,\partial g,$ and $N^{L}_{\mathcal{X}}$ (e.g. [31, Propostion 6.6]), the inclusion $N^{L}_{\mathcal{X}}\subset N_{\mathcal{X}}$, and the boundedness of $\{w^{f}_{k}\},\{w^{g}_{k}(\omega)\},$ and $\{w^{\mathcal{X}}_{k}(\omega)\}$, it follows that

as $k\rightarrow\infty$. Consequently, almost surely we have that

Let $L_{f}:=\sup_{k\geq 1}{\rm dist}(0,\partial f(z_{k}))$ and $L_{g}:=\sup_{k\geq 1}L(z_{k})$. Then for all $k\geq 0$, the functions

are uniformly dominated by an integrable function in $\omega$.

For each $k$, Lemma A.1 implies the bound

which is integrable by Item 6 of Assumption E. ∎

Applying the dominated convergence theorem, it follows that

as $k\rightarrow\infty$. Notice the simple fact that for any real sequence $b_{k}\to 0$, it must be that $\frac{1}{n}\sum_{k=1}^{n}b_{k}\rightarrow 0$ as $n\rightarrow\infty$. Consequently

as $n\rightarrow\infty$. This completes the proof. ∎

We have now verified all parts of Theorem 3.1. Therefore, the proof is complete.

A.3 Verifying Assumption F for composite problems

for a.e. $t\geq 0$. Adding the three equations yields

Suppose now that $z(\cdot)$ satisfies $\dot{z}(t)\in-G(z(t))$ for a.e. $t\geq 0$. Then the same linear algebraic argument as in Lemma 5.2 yields the equality $\|\dot{z}(t)\|={\rm dist}(0;G(z(t)))$ for a.e. $t\geq 0$ and consequently the equation (6.6).

To complete the proof, we must only show that property 2 of Assumption F holds. To this end, suppose that $z(0)$ is not composite critical and let $T>0$ be arbitrary. Appealing to (6.6), clearly $\sup_{t\in[0,T]}\varphi(z(t))\leq\varphi(z(0))$. Thus we must only argue $\varphi(z(T))<\varphi(z(0))$. According to (6.6), if this were not the case, then we would deduce ${\rm dist}(0;G(z(t)))=0$ for a.e. $t\in[0,T]$. Appealing to the equality $\|\dot{z}\|={\rm dist}(0;G(z(t)))$, we therefore conclude $\|\dot{z}\|=0$ for a.e. $t\in[0,T]$. Since $z(\cdot)$ is absolutely continuous, it must therefore be constant $z(\cdot)\equiv z(0)$, but this is a contradiction since $0\notin G(z(0))$. Thus property 2 of Assumption F holds, as claimed. ∎

Consider an arbitrary stratum $M\in\mathcal{A}$ intersecting $\mathcal{X}$ (and therefore contained in $\mathcal{X}$) and a point $x\in M$. Consider now the (unique) strata $M_{f}\in\Pi\mathcal{A}_{f}$, $M_{g}\in\Pi\mathcal{A}_{g}$, and $M_{\mathcal{X}}\in\mathcal{A}_{\mathcal{X}}$ containing $x$. Let $\widehat{f}$ and $\widehat{g}$ be $C^{d}$-smooth functions agreeing with $f$ and $g$ on a neighborhood of $x$ in $M_{f}$ and $M_{g}$, respectively. Appealing to (5.4), we conclude

The Whitney condition in turn directly implies $N_{\mathcal{X}}(x)\subset N_{M_{\mathcal{X}}}(x).$ Hence summing yields

where the last inclusion follows from the compatibility $M\subset M_{f}$ and $M\subset M_{g}$. Notice that $\widehat{f}+\widehat{g}$ agrees with $f+g$ on a neighborhood of $x$ in $M$. Hence if the inclusion, $0\in\partial f(x)+\partial g(x)+N_{\mathcal{X}}(x)$, holds it must be that $x$ is a critical point of the $C^{d}$-smooth function $f+g$ restricted to $M$, in the classical sense. Applying the standard Sard’s theorem to each manifold $M$, the result follows. ∎

References