Efficient Algorithms for Smooth Minimax Optimization

Introduction

In this paper we study smooth minimax problems of the form:

The problem has applications in several domains such as machine learning [Goo+14, Mad+17], optimization [Ber14], statistics [Ber13], mathematics [KS80], and game theory [Mye13]. Given the importance of these problems, there is an extensive body of work that studies various algorithms and their convergence properties. The vast majority of existing results for this problem focus on the convex-concave setting, where $g(\cdot,y)$ is convex for every $y$ and $g(x,\cdot)$ is concave for every $x$. The best known convergence rate in this setting is $O(1/k)$ for the primal-dual gap, achieved for example by Mirror-Prox [Nem04]. This rate is also known to be optimal for the class of smooth convex-concave problems [OX18]. A natural question is whether we can achieve a faster convergence if we have strong convexity (as opposed to just convexity) of $g(\cdot,y)$. We answer this in the affirmative, by introducing an algorithm that achieves a convergence rate of $\widetilde{O}\left(1/k^{2}\right)$ for the general smooth, strongly-convex–concave minimax problem. The algorithm we propose is a novel combination of Mirror-Prox and Nesterov’s accelerated gradient descent. This matches the known lower bound of $\Omega(1/k^{2})$ from [OX18], closing the gap up to a poly-logarithmic factor. The only known upper bounds that obtain a rate of $O(1/k^{2})$ in this context are for very special cases, where $x$ and $y$ are connected through a bi-linear term or $g(x,\cdot)$ is linear in $y$ [Nes05, JN11, Gol+14, CP16, HM16, Xu17, HA18, XS19].

While most theoretical results focus on the convex-concave setting, several real world problems fall outside this class. A slightly larger class, which captures several more applications, is the class of smooth nonconvex–concave minimax problems, where $g(x,\cdot)$ is concave for every $x$ but $g(\cdot,y)$ can be nonconvex. For example, finite minimax problems, i.e., $\min_{x}\max_{i=1}^{m}f_{i}(x)=\min_{x}\max_{0\preceq y\preceq 1,\sum_{i=1}^{m}y_{i}=1}\sum_{i}y_{i}\cdot f_{i}(x)\mathrel{\mathop{:}}=g(x,y)$ belong to this class, and so do nonconvex constrained optimization problems [Kom+18]. In addition, several machine learning problems with non-decomposable loss functions [KNJ15] also belong to this class.

In this general nonconvex concave setting however, we cannot hope to find global optimum efficiently as even the special case of nonconvex optimization is NP-hard. Similar to nonconvex optimization, we might hope to find an approximate stationary point [Nes98].

Our second contribution is a new algorithm and a faster rate for the general smooth nonconvex–concave minimax problem. Our algorithm is an inexact proximal point method for the nonconvex function $f(x)\mathrel{\mathop{:}}=\max_{y\in\mathcal{Y}}g(x,y)$. The key insight is that the proximal point problem in each iteration results in a strongly-convex concave minimax problem, for which we use our improved algorithm to obtain the overall computation/iteration complexity of $\widetilde{O}\left(1/k^{1/3}\right)$ thus improving over the previous best known rate of $O(1/k^{1/5})$ [JNJ19]While [JNJ19] gives a rate of $O\left(1/k^{1/4}\right)$ with an approximate maximization oracle for $\max_{y\in\mathcal{Y}}g(x,y)$, taking into account the cost of implementing such a maximization oracle gives a rate of $O\left(1/k^{1/5}\right)$..

Finally, we specialize our result to finite minimax problems, i.e., $\min_{x}\max_{1\leq i\leq m}f_{i}(x)$ where $f_{i}(x)$ can be nonconvex function but each $f_{i}$ is a smooth function; nonconvex constrained optimization problems can be reduced to such finite minimax problems. For these, we obtain a rate of $\widetilde{O}\left(m(\log m)^{3/2}/k^{1/3}\right)$ total gradient computations which improves upon the state-of-the-art rate ($O(m\sqrt{\log m}/k^{1/5})$) in this setting as well.

Summary of contributions: See also Table 1. 1. $\widetilde{O}\left(1/k^{2}\right)$ convergence rate for smooth, strongly-convex – concave problems, improving upon the previous best known rate of $O\left(1/k\right)$ and, 2. $\widetilde{O}\left(1/k^{1/3}\right)$ convergence rate for smooth, nonconvex – concave problems, improving upon the previous best known rate of $O\left(1/k^{1/5}\right)$.

Related works: For strongly-convex-concave minimax problems with special structures, several algorithms have been proposed. In an increasing order of generality, [Gol+14, Xu17, XZ18] study optimizing a strongly convex function with linear constraints, which can be posed as a special case of minimax optimization, [Nes05] studies a minimax problem where $x$ and $y$ are connected only through a bi-linear term, and [HA18] and [JN11] study a case where $g(x,\cdot)$ is linear in $y$. In all these cases, it is shown that $O(1/k^{2})$ convergence rate is achievable if $g(\cdot,y)$ is strongly-convex $\forall\;y$. Recently, [Zha19] provides a unified approach, that achieves $O(1/k)$ convergence rate for general convex-concave case and $O(1/k^{2})$ for a special case with strongly-convex $g(\cdot,y)$ and linear $g(x,\cdot)$. However, it has remained an open question if the fast rate of $O(1/k^{2})$ can be achieved for general strongly-convex-concave minimax problems.

For nonconvex-concave minimax problems, [Raf+18] considers both deterministic and stochastic settings, and proposes inexact proximal point methods for solving smooth nonconvex–concave problems. In the deterministic setting, their result guarantees an error of $O(1/k^{1/6})$. We note that there have also been other notions of stationarity proposed in literature for nonconvex-concave minimax problems [Lu+19, Nou+19]. These notions however are weaker than the one considered in this paper, in the sense that, our notion of stationarity implies these other notions (without loss in parameters). For one such weaker notion, [Nou+19] proposes an algorithm with a convergence rate of $O\left(1/k^{3.5}\right)$. Since the notion they consider is weaker, it does not imply the same convergence rate in our setting.

We would also like to highlight the work on variational inequalities that are a generalization of minimax optimization problems. In particular, monotone variational inequalities generalizes the convex-concave minimax problems and have applications in solving differential equations [KS80]. There have also been a large number of works designing efficient algorithms for finding solutions to monotone variational inequalities [BJ77, Nem81, Nem04].

Paper organization: In Section 2, we present preliminaries and all relevant background. In Section 3, we present our results for strongly-convex–concave setting and in section 4, results for nonconvex–concave setting. In Section 5, we present empirical evaluation of our algorithm for nonconvex-concave setting and compare it to a state-of-the-art algorithm. We conclude in Section 6. Several technical details are presented in the appendix.

Preliminaries and background material

In this section, we will present some preliminaries, describing the setup and reviewing some background material that will be useful in the sequel.

We are interested in the minimax problems of the form (1) where $g(x,y)$ is a smooth function.

A function $g(x,y)$ is said to be $L$-smooth if:

Throughout, we assume that $g(x,.)$ is concave for every $x\in\mathcal{X}$. For $g(\cdot,y)$ behavior in terms of $x$, there are broadly two settings:

In this setting, $g(\cdot,y)$ is convex $\forall\;y\in\mathcal{Y}$. Given any $g$ and $\forall({\widehat{x}},{\widehat{y}})$, the following holds trivially:

which then implies that $\max_{y\in\mathcal{Y}}\min_{x\in\mathcal{X}}g(x,y)\leq\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}}g(x,y)$. The celebrated minimax theorem for the convex-concave setting [Sio58] says that if $\mathcal{Y}$ is a compact set then the above inequality is in fact an equality, i.e., $\max_{y\in\mathcal{Y}}\min_{x\in\mathcal{X}}g(x,y)=\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}}g(x,y)$. Furthermore, any point $(x^{*},y^{*})$ is an optimal solution to (1) if and only if:

Hence, our goal is to find $\varepsilon$-primal-dual pair $({\widehat{x}},{\widehat{y}})$ with small primal-dual gap: $\max_{y\in\mathcal{Y}}g({\widehat{x}},y)-\min_{x\in\mathcal{X}}g(x,{\widehat{y}})$.

1.2 Nonconvex-concave setting

In this setting the function $g(\cdot,y)$ need not be convex. One cannot hope to solve such problems in general, since the special case of nonconvex optimization is already NP-hard [NLR18]. Furthermore, the minimax theorem no longer holds, i.e., $\max_{y\in\mathcal{Y}}\min_{x\in\mathcal{X}}g(x,y)$ can be strictly smaller than $\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}}g(x,y)$. Oftentimes the order of $min$ and $max$ might be important for a given application i.e., we might be interested only in minimax but not maximin (or vice versa). So, the primal-dual gap may not be a meaningful quantity to measure convergence. One approach, inspired by nonconvex optimization, to measure convergence is to consider the function $f(x)=\max_{y\in\mathcal{Y}}g(x,y)$ and consider the convergence rate to approximate first order stationary points (i.e., $\nabla f(x)$ is small)[Raf+18, JNJ19]. But as $f(x)$ could be non-smooth, $\nabla f(x)$ might not even be defined. It turns out that whenever $g(x,y)$ is smooth, $f(x)$ is weakly convex (Definition 4) for which first order stationarity notions are well-studied and are discussed below.

Approximate first-order stationary point for weakly convex functions: We first need to generalize the notion of gradient for a non-smooth function.

The Fréchet sub-differential of a function $f(\cdot)$ at $x$ is defined as the set, $\partial f(x)=\{u\,|\,\ \underset{x^{\prime}\to x}{\lim\inf}{f(x^{\prime})-f(x)-\left\langle u,x^{\prime}-x\right\rangle}/{\|x^{\prime}-x\|}\geq 0\}$.

In order to define approximate stationary points, we also need the notion of weakly convex function and Moreau envelope.

for all Fréchet subgradients $u_{x}\in\partial f(x)$.

The following lemma provides some useful properties of the Moreau envelope for weakly convex functions. The proof can be found in Appendix B.2.

The minimizer $\hat{x}_{\lambda}(x)=\arg\min_{x^{\prime}\in\mathcal{X}}f(x^{\prime})+\frac{1}{2\lambda}\|x-x^{\prime}\|^{2}$ is unique and $f(\hat{x}_{\lambda}(x))\leq f_{\lambda}(x)\leq f(x)$. Furthermore, $\arg\min_{x}f(x)=\arg\min_{x}f_{\lambda}(x)$.

$f_{\lambda}$ is \big{(}\frac{1}{\lambda}+\frac{1}{\lambda(1-\lambda L)}\big{)}-smooth and thus differentiable, and

$\min_{u\in\partial f(\hat{x}_{\lambda}(x))}\|u\|\leq(1/\lambda)\|\hat{x}_{\lambda}(x)-x\|=\|\nabla f_{\lambda}(x)\|$.

Now, first order stationary point of a non-smooth nonconvex function is well-defined, i.e., $x^{*}$ is a first order stationary point (FOSP) of a function $f(x)$ if, $0\in\partial f(x^{*})$ (see Definition 3). However, unlike smooth functions, it is nontrivial to define an approximate FOSP. For example, if we define an $\varepsilon$-FOSP as the point $x$ with $\min_{u\in\partial f(x)}\|u\|\leq\varepsilon$, there may never exist such a point for sufficiently small $\varepsilon$, unless $x$ is exactly a FOSP. In contrast, by using above properties of the Moreau envelope of a weakly convex function, it’s approximate FOSP can be defined as [DD18]:

Given an $L$-weakly convex function $f$, we say that $x^{*}$ is an $\varepsilon$-first order stationary point ($\varepsilon$-FOSP) if, $\|\nabla f_{\frac{1}{2L}}(x^{*})\|\leq\varepsilon$, where $f_{\frac{1}{2L}}$ is the Moreau envelope with parameter $1/2L$.

Using Lemma 1, we can show that for any $\varepsilon$-FOSP $x^{*}$, there exists $\hat{x}$ such that $\|\hat{x}-x^{*}\|\leq\varepsilon/2L$ and $\min_{u\in\partial f(\hat{x})}\|u\|\leq\varepsilon$. In other words, an $\varepsilon$-FOSP is $O(\varepsilon)$ close to a point $\hat{x}$ which has a subgradient smaller than $\varepsilon$. We note that other notions of FOSP have also been proposed recently such as in [Nou+19]. However, it can be shown that an $\varepsilon$-FOSP according to the above definition is also an $\epsilon$-FOSP with [Nou+19]’s definition as well, but the reverse is not necessarily true.

2 Mirror-Prox

Mirror-Prox [Nem04] is a popular algorithm proposed for solving convex-concave minimax problems (1). It achieves a convergence rate of $O\left(1/k\right)$ for the primal dual gap. The original Mirror-Prox paper [Nem04] motivates the algorithm through a conceptual Mirror-Prox (CMP) method, which brings out the main idea behind its convergence rate of $O\left(1/k\right)$. CMP does the following update:

The main difference between CMP and standard gradient descent ascent (GDA) is that in the $k^{\textrm{th}}$ step, while GDA uses gradients at $(x_{k},y_{k})$, CMP uses gradients at $(x_{k+1},y_{k+1})$. The key observation of [Nem04] is that if $g(\cdot,\cdot)$ is smooth, it can be implemented efficiently. CMP is analyzed as follows: Implementability of CMP: Let $(x_{k}^{(0)},y_{k}^{(0)})=(x_{k},y_{k})$. For $\beta<\frac{1}{L}$, the iteration

can be shown to be $\frac{1}{\sqrt{2}}$-contraction (when $g(\cdot,\cdot)$ is smooth) and that its fixed point is $\left(x_{k+1},y_{k+1}\right)$. So, in $\log\frac{1}{\epsilon}$ iterations of (6), we can obtain an accurate version of the update required by CMP. In fact, [Nem04] showed that just two iterations of (6) suffice. Convergence rate of CMP: Using CMP update with simple manipulations leads to the following:

$O\left(1/k\right)$ convergence rate follows easily using the above result.

Finally, our method and analysis also requires Nesterov’s accelerated gradient descent method (see Algorithm 4 in Appendix A)and it’s per-step analysis by [BG17] (Lemma 4 in Appendix A).

Strongly-convex concave saddle point problem

We first study the minimax problem of the form:

Our objective here is to find an $\epsilon$-primal-dual pair $({\widehat{x}},{\widehat{y}})$ (see Definition 2). Now the fact that $f(\hat{x})-f^{*}\leq\max_{y\in\mathcal{Y}}g({\widehat{x}},y)-\min_{x\in\mathcal{X}}g(x,{\widehat{y}})$ implies that if $(\hat{x},\hat{y})$ is an $\varepsilon$-primal-dual-pair, then $\hat{x}$ is also an $\varepsilon$-approximate minima of $f$. Furthermore, by Sion’s minimax theorem [Kom88], strong-convexity–concavity of $g(\cdot,\cdot)$ ensures that: $\min_{x}[f(x)\mathrel{\mathop{:}}=\max_{y}g(x,y)]=\max_{y}[h(y)\mathrel{\mathop{:}}=\min_{x}g(x,y)]$. Hence, one approach to efficiently solving the problem is by optimizing the dual problem $\max_{y}h(y)$. By Lemma 2, $h(y)$ is an $L(1+L/\sigma)$-smooth function. So we can use AGD to ensure that $h(y_{k})-h(y^{*})=O(1/k^{2})$. Now, each step of AGD requires computing $\arg\min_{x}g(x,y_{k})$ which can be done efficiently (i.e., logarithmic number of steps) as $g(\cdot,y_{k})$ is strongly-convex and smooth. So, the overall first-order oracle complexity is $h(y_{k})-h(y^{*})=\widetilde{O}\left(1/k^{2}\right)$.

For a $\sigma$-strongly-convex–concave $L$-smooth function $g(\cdot,\cdot)$, $h(u)=\min_{x\in\mathcal{X}}g(x,u)$ is an L\big{(}1+\frac{L}{\sigma}\big{)}-smooth concave function.

The pseudocode for C-DIAG algorithm is presented in Algorithm 1. The main idea of the algorithm is in Step 4, where we simultaneously find $x_{k+1}$ and $y_{k+1}$ satisfying the following requirements:

$x_{k+1}$ is the minimizer of $g(\cdot,y_{k+1})$, and

$y_{k+1}$ corresponds to an AGD step (see Algorithm 4 in Appendix A) for $g(x_{k+1},\cdot)$

Implementability: The first question is whether it is easy enough to implement such a step? It turns out that it is indeed possible to quickly find points $x_{k+1}$ and $y_{k+1}$ that approximately satisfy the above requirements. The reason is that:

Since $g(\cdot,y)$ is smooth and strongly convex for every $y\in\mathcal{Y}$, we can find $\epsilon$-approximate minimizer for a given $y$ in $O\left(\log\frac{1}{\epsilon}\right)$ iterations.

Convergence rate: Since $y_{k+1}$ and $z_{k+1}$ correspond to an AGD update for $g(x_{k+1},\cdot)$, we can use the potential function decrease argument for AGD (Lemma 4 in Appendix A) to conclude that $\forall y\in\mathcal{Y}$,

Since $g(x_{k+1},y_{k+1})\leq g(x,y_{k+1})$ for every $x\in\mathcal{X}$, we have

where $\bar{x}_{k+1}\mathrel{\mathop{:}}=\frac{1}{(k+1)(k+2)}\sum_{i=1}^{k+1}(2i)\cdot x_{i}$. Since $x$ and $y$ are arbitrary above, this gives a $O\left(1/k^{2}\right)$ convergence rate for the primal dual gap.

2 Error analysis

The main issue with Algorithm 1 is that the update step is not exactly implementable. However, as we noted in the previous section, we can quickly find updates that almost satisfy the requirements. Algorithm 2 presents this inexact version. The following theorem states our formal result and a detailed proof is provided in Appendix B.4.

Remark 2: Unlike standard AGD for $h(y)$, which only updates $y_{k}$ in the outer-loop, DIAG’s outer-step updates both $x_{k}$ and $y_{k}$ thus allowing us to better track the primal-dual gap. However, DIAG’s dependence on the condition number ${L}/{\sigma}$ seems sub-optimal and can perhaps be improved if we do not compute Imp-STEP nearly optimally allowing for inexact updates; we leave further investigation into improved dependence on the condition number for future work.

Nonconvex concave saddle point problem

Let $g(\cdot,y)$ be continuous and $\mathcal{Y}$ be compact. Then $f(x)=\max_{y\in\mathcal{Y}}g(x,y)$ is $L$-weakly convex, if $g$ is $L$-weakly convex in $x$ (Definition 1), or if $g$ is $L$-smooth in $x$ .

See Appendix B.3 for the proof. The arguments of [JNJ19] easily extend to show that applying subgradient method on $f(x)$, [DD18] gives a convergence rate of $O\left(1/k^{1/5}\right)$. Instead, we exploit the smooth minimax form of $f(\cdot)$ to design a faster converging scheme. The main intuition comes from the proximal viewpoint that gradient descent can be viewed as iteratively forming and optimizing local quadratic upper bounds. As $f$ is weakly convex, adding enough quadratic regularization should ensure that the resulting sequence of problems are all strongly-convex–concave. We then exploit DIAG to efficiently solve such local quadratic problems to obtain improved convergence rates. Concretely, let

By $L$-weak-convexity of $f$, $\widehat{f}(x;x_{k})$ is strongly-convex–concave (Lemma 5) that can be solved using DIAG up to certain accuracy to obtain $x_{k+1}$. We refer to this algorithm as Prox-DIAG and provide a pseudo-code for the same in Algorithm 3.

The following theorem gives convergence guarantees for Prox-DIAG.

steps outputs an $\varepsilon$-FOSP. The total first-order oracle complexity to output $\varepsilon$-FOSP is: O\big{(}{\frac{L^{2}D_{\mathcal{Y}}(f(x_{0})-f^{*})}{\varepsilon^{3}}}\log^{2}\big{(}1/{\varepsilon}\big{)}\big{)}\,.

Note that Prox-DIAG solves the quadratic approximation problem to higher accuracy of $O(\epsilon^{2})$ which then helps bounding the gradient of the Moreau envelope. Also due to the modular structure of the argument, a faster inner loop for special settings, e.g., when $g(x,y)$ is a finite-sum, can ensure more efficient algorithm. While our algorithm is able to significantly improve upon existing state-of-the-art rate of $O(1/\varepsilon^{5})$ in general nonconvex-concave setting [JNJ19], it is unclear if the rate can be further improved. In fact, precise lower-bounds for this setting are mostly unexplored and we leave further investigation into lower-bounds as a topic of future research.

We first note that by Lemma 5 and $L$-weak convexity of $g(\cdot,y)$ and $2L$-strong convexity of $L\|x-x_{k}\|^{2}$, $\widehat{g}(x,y;x_{k}):=g(x,y)+L\|x-x_{k}\|^{2}$ is $L$-strongly-convex. Similarly, $\widehat{f}(\cdot;x_{k}):=\max_{y\in\mathcal{Y}}[\widehat{g}(x,y;x_{k})=g(x,y)+L\|x-x_{k}\|^{2}]$ is also $L$-strongly-convex.

We now divide the analysis of each iteration of our algorithm into two cases:

Define $x_{k}^{*}$ as the point satisfying $x^{*}_{k}=\arg\min_{x}\widehat{f}(x;x_{k})$. By $L$-strong convexity of $\widehat{f}(\cdot;x_{k})$ (9), we prove that $x_{k}$ is close to $x_{k}^{*}$:

Therefore the number of gradient computations required for each iteration of inner problem is O\Big{(}\frac{LD_{\mathcal{Y}}}{\epsilon}\log^{2}\Big{(}\frac{1}{\varepsilon}\Big{)}\Big{)} (Theorem 1), which along with the bound on the number of outer iterations establishes the Theorem’s upper bound on the number of first-order oracle calls.∎

As a special case of nonconvex–concave minimax problem, consider minimizing a weakly convex $f(x)$, with a special structure of finite max-type function:

If the functional components $f_{i}(x)$’s are $G$-Lipschitz and $L$-smooth, and the optimal solution is bounded below, i.e. $f(x)\geq f^{*}>-\infty$, then after: K=\bigg{\lceil}\frac{4^{4}L(f(x_{0})-f^{*})}{3\varepsilon^{2}}\bigg{\rceil} outer steps, Prox-FDIAG outputs an $\varepsilon$-FOSP. The total first-order oracle complexity to find $\varepsilon$-FOSP is: \bigg{\lceil}\frac{4^{4}L(f(x_{0})-f^{*})}{3\varepsilon^{2}}\bigg{\rceil}\cdot\bigg{\lceil}\frac{2^{4}G}{\varepsilon}(m\log^{3/2}m){{}{}}\bigg{\rceil}.

See Appendix B.6 for a proof. Current best rate for this problem is achieved by subgradient methods. As the subgradient of a finite minimax function $\nabla_{i^{*}}f(x)$ is easy to evaluate, where $i^{*}\in\arg\max_{i}f_{i}(x)$, a rate of $O(m/\varepsilon^{4})$ first-order oracle and function calls is achieved by the state-of-the-art subgradient method in [DD18]. We can obtain a similar result using Algorithm 2 but it requires extension to non-Euclidean settings with the framework of Bregman divergences. This is fairly standard and will be updated in the next version of the paper.

Experiments

We empirically verify the performance of Prox-FDIAG (Algorithm 5) on a synthetic finite max-type nonconvex minimization problem (P3). We consider the following problem.

where $f_{i}(x)=q_{(-1,\,(X^{(1)}_{i},X^{(2)}_{i}),\,C_{i})}(x)$ for all $1\leq i\leq 8$, where $q_{(a,b,c)}(x)=a\|x-b\|_{2}^{2}+c$, $X^{(1)}_{i}$ and $X^{(2)}_{i}$ are generated from the interval $[-3.0,3.0]$ uniformly at random, and $C_{i}$ is generated from the interval $[1.0,5.0]$ uniformly at random. We fix the last component $f_{9}(x)=q_{(0.5,\,(0,0),\,0)}(x)$. Each $f_{i}$ is smooth with parameter $L=1$, which implies that $f$ is $L$-weakly convex.

All the algorithms are initialized with the point $x_{0}=(4,4)$ and are given a Lipschitzness parameter of $G=2\,L\,\|x_{0}\|_{2}$. We run the algorithms ten times with randomly generated instances of the objective function $f(x)$. In Figure 1, we plot the norm of gradient of Moreau envelope $\|\nabla f_{\frac{1}{2L}}(x_{k})\|_{2}$ against the number of iterations $k$ in log-log scale. We compute the gradient of the Moreau envelope at any point $x$, by solving the corresponding convex-concave saddle point problem (23) using Mirror-Prox [Nem04] method with appropriate primal-dual gap based stopping criteria and then using Lemma 1(c). For Prox-FDIAG ( red circles), we show in a scatter plot the gradient norm $\|\nabla f_{\frac{1}{2L}}(x_{K(\varepsilon)})\|_{2}$ at the final output of Prox-FDIAG $x_{K(\varepsilon)}$ versus the total number of inner iterations (of excessive gap technique) taken, for $\varepsilon=10^{0},10^{-1},10^{-2},10^{-3}$ over the 10 functions. For Adaptive Prox-FDIAG ( black dots) in a scatter plot, we plot the gradient norm $\|\nabla f_{\frac{1}{2L}}(x^{\prime})\|_{2}$ at the output $x^{\prime}$ of each inner sub-problem (excessive gap technique) of each inner Prox-FDIAG step versus the total number of inner iterations (of excessive gap technique) taken to reach that point from the beginning, for $\varepsilon=10^{-7}$ over the 10 functions. For Prox-FDIAG and Adaptive Prox-FDIAG, using solid red and black (respectively) lines we also plot the best linear function (in log-scale) which fits the scatter points (using default parameters of scipy.stats.linregresshttps://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.linregress.html). For the subgradient method ( blue triangles), we plot the mean and standard error of gradient norm $\max_{0\leq k^{\prime}\leq k}\|\nabla f_{\frac{1}{2L}}(x_{\hat{k}(k^{\prime})})\|_{2}$ over the 10 instances at iterations $k=10^{0},10^{1},\ldots,10^{7}$. The estimate at each iteration is the best one so far in the function value, i.e. $\hat{k}(k)\in{\arg\min}_{0\leq k^{\prime}\leq k}f(x_{k^{\prime}})$. We see that, Prox-FDIAG and Adaptive Prox-FDIAG have a faster convergence rate than subgradient method. Further, in the same vein as analogous variants in convex non-smooth optimization, Adaptive Prox-FDIAG is faster than Prox-FDIAG almost always.

Subgradient method has a theoretical convergence rate of $O(\frac{1}{\sqrt{K}})$ for a fixed number of iterations $K$ and a constant step-size $\gamma/{\sqrt{K+1}}$ [DD18, Corollary 2.2]. However, similar to the case of convex non-smooth problems, we observe that fixed step-size results in a slow convergence. In our experiments, we achieve a faster convergence for the subgradient method by using a diminishing, non-summable but square-summable step-size, $\gamma/{\sqrt{k+1}}$, which varies with the iteration number $k$. This step-size has convergence rate of $O(\frac{\log(k)}{\sqrt{k}})$ [DD18, Theorem 2.1], but in practice we observe a faster convergence rate than the constant step-size. After a very simple parameter search, we set $\gamma$ as $0.1\times G\times L^{3/2}$. We ran subgradient method for a total of $K=10^{7}$ number of iterations. Since, subgradient method is not a descent method, at any iteration $k$, we keep track of the best point among all the points we have observed so far, $\{x_{0},\cdots,x_{k-1}\}$. Ideally, we should keep track of the point with the minimum norm for the gradient of the Moreau envelope, $\|\nabla f_{\frac{1}{2L}}(x_{k})\|_{2}$, but since the computation of the gradient of Moreau envelope is costly, we only keep track of the point with the minimum function value we have observed so far.

Conclusion

In this paper, we study smooth minimax problems, where the maximization is concave but the minimization is either strongly convex or nonconvex. In both of these settings, we present new algorithms improving state-of-the-art. The key ideas are i) a novel way to combine Mirror-Prox and Nesterov’s AGD for strongly convex case that can tightly bound primal-dual gap and ii) an inexact prox method with good convergence rate to stationary points for the nonconvex case. While we only present our results for the Euclidean setting, generalizing it to non-Euclidean settings with the framework of Bregman divergences should be straight forward. Finally, we showcase the empirical superiority of our nonconvex algorithm over state-of-the-art subgradient method for a case of finite max-type nonconvex minimization problems. Some of the more interesting questions would be to understand the optimality of the rates that we obtain and dependence on the strong convexity parameter. Further extensions of these results to the stochastic setting would also be quite interesting.

References

Appendix

Appendix A Nesterov’s accelerated gradient descent

Nesterov’s accelerated gradient descent [Nes83] is an optimal method for minimizing smooth convex functions (or equivalently maximizing smooth concave functions). In order to simplify the exposition in the sequel, we will consider the algorithm for maximizing concave functions. The pseudocode for this is presented in Algorithm 4. Fix any point $y\in\mathcal{Y}$. Consider the potential function

The following lemma (from [BG17]) is the key result that helps us obtain the convergence rate of Algorithm 4. Here $\mathcal{P}_{\mathcal{Y}}\left({\cdot}\right)$ denotes projection onto $\mathcal{Y}$.

[BG17] Suppose $h(\cdot)$ is an $L$-smooth concave function and the parameters of Algorithm 4 are chosen so that $\beta>L$, $\eta_{k}=\frac{k+1}{2\beta}$ and $\tau_{k}=\frac{2}{k+2}$. Then, we have

we bound the three terms appearing in separate lines above. Firstly, for the third term, $\|{z_{k+1}-y}\|^{2}\leq\|{z_{k}+\eta_{k}\nabla h(w_{k})-y}\|^{2}-\|{z_{k+1}-z_{k}-\eta_{k}\nabla h(w_{k})}\|^{2}$ due to Pythagoras theorem and so

where we used $w_{k}=(1-\tau_{k})y_{k}+\tau_{k}z_{k}$ in the last step. Substituting (21), (20) and (19) in (18) proves the lemma. ∎

Appendix B Proofs

Since $f$ is $L$-weakly convex and $f$ is $\sigma$-strongly convex we get that,

B.2 Proof of Lemma 1

We re-write $f_{\lambda}(x)$ as minimum value of a $(\frac{1}{\lambda}-L)$-strong convex function $\phi_{\lambda,x}$, as $f$ is $L$-weakly convex (Definition 3) and $\frac{1}{2\lambda}\|x-x^{\prime}\|^{2}$ is differentiable and $\frac{1}{\lambda}$-strongly convex (Lemma 5),

Then first part of (a) follows trivially by the strong convexity. For the second part notice the following,

Thus $\arg\min_{x}f_{\lambda}(x)=\arg\min_{x}f(x)$. For $(b)$ we can re-write the Moreau envelope $f_{\lambda}$ as,

where $(\cdot)^{*}$ is the Fenchel conjugation operator. Since $L<1/\lambda$, using $L$-weak convexity of $f$, it is easy to see that $\lambda f(x^{\prime})+\frac{\|x^{\prime}\|^{2}}{2}$ is $(1-\lambda L)$-strongly convex, therefore its Fenchel conjugate would be $\frac{1}{(1-\lambda L)}$-smooth [KSST09, Theorem 6]. This, along with $\frac{1}{\lambda}$-smoothness of first quadratic term implies that $f_{\lambda}(x)$ is \big{(}\frac{1}{\lambda}+\frac{1}{\lambda(1-\lambda L)}\big{)}-smooth, and thus differentiable.

For $(c)$ we again use the reformulation of $f_{\lambda}(x)$ as $\min_{x^{\prime}\in\mathcal{X}}\phi_{\lambda,x}(x^{\prime})$ (23). Then by first-order necessary condition for optimality of $\hat{x}_{\lambda}(x)$, we have that $x-\hat{x}_{\lambda}(x)\in\lambda\partial f(x)$. Further, from proof of part (a) we have that $\phi_{\lambda,x}(x^{\prime})$ $(1-\lambda L)$-strongly-convex in $x^{\prime}$ and it is quadratic (and thus convex) in $x$. Then we can use Danskin’s theorem [Ber09, Section 6.11] to prove that, $\nabla f_{\lambda}(x)=(x-\hat{x}_{\lambda}(x))/\lambda\in\partial f(x)$.

B.3 Proof of Lemma 3

It is easy to see that $g(\cdot,y)$ is $L$-weakly convex if it is $L$-smooth: $g(x^{\prime},y)\geq g(x,y)+\left\langle\nabla_{x}g(x,y),x^{\prime}-x\right\rangle-\frac{L}{2}\|x^{\prime}-x\|^{2}$. Thus we only need to prove the case of $L$-weakly convex $g(\cdot,y)$. Since $g(\cdot,y)$ is $L$-weakly convex we get that,

where $(a)$ uses $L$-weak convexity of $g(\cdot,y)$, and $(b)$ uses (25) and $v_{x}\in\partial f(x)$.

B.4 Proof of Theorem 1

A cursory glance of the DIAG (Algorithm 2) reveals that it is a modified version of projected accelerated gradient ascent (Algorithm 4) on some function of $y$ with a modified step given by Imp-STEP, which is inspired from the conceptual Mirror-Prox method of [Nem04]. In the following lemma we analyze the Imp-STEP sub-routine, which is the most non-trivial step of the algorithm.

If $\beta=2{\frac{L^{2}}{\sigma}}$, the sub-routine Imp-STEP($g$, $L$, $\sigma$, $w$, $\beta$, $\varepsilon_{\rm step}$) of Algorithm 2, returns a pair of points $(\hat{x}_{R},y_{R+1})\in\mathcal{X}\times\mathcal{Y}$, such that,

in R=\lceil\log_{2}\big{(}({5LD_{\mathcal{Y}}}/{\sigma})\sqrt{{L}/{2\varepsilon_{\rm step}}}\big{)}\rceil iterations with O\Big{(}\sqrt{{L}/\sigma}\log\Big{(}1/{\varepsilon_{\rm step}}\Big{)}\Big{)} gradient computations per iterations.

A proof for this lemma is provided in Appendix B.4.1. The above lemma guarantees that the Imp-STEP sub-routine converges fast (linear time), in $O(\log(1/\varepsilon_{\rm step}))$ steps with $O(\sqrt{{L}/\sigma}\log^{2}(1/\varepsilon_{\rm step}))$ number of gradient computations.

In the rest of the proof we will utilize the recently proposed potential-function based proof for accelerated gradient decent (AGD) [BG17, Section 5.2]. Analyzing AGD using potential-function has an advantage over the standard analysis because, even though AGD does not decrease the function value monotonically the former constructs a potential-function which monotonically decreases over the iterations. Given the guarantees (Lemma 6) for the Imp-STEP sub-routine we can re-write an iteration of the DIAG algorithm by the following steps:

Since $g(x,\cdot)$ is $L$-smooth, it is also $\frac{2L^{3}}{\sigma}$-smooth ($\sigma\leq L$). Then, using Lemma 4 , we see that for a step-size of $\frac{1}{\beta}=\frac{\sigma}{2L^{2}}$, the potential function $\Phi^{h_{k}}(k)$ decrease at step of $k$ of the algorithm: $\Phi^{h_{k+1}}({k+1})\leq\Phi^{h_{k+1}}({k})$. Thus,

Where $(a)$ follows from Lemma 6 and $g(x_{k},y_{k})-g(x_{k+1},y_{k})\leq g(x_{k},y_{k})-\min_{x}g(x,y_{k})\leq\varepsilon^{(k)}_{\rm step}$, $(b)$ is obtained summing (32) over $k=\{0,\ldots,K-1\}$. Rearranging the terms of (33) we get,

Further, using Lemma 6 and $\varepsilon^{(k)}_{\rm step}=\frac{L^{2}D^{2}_{\mathcal{Y}}}{\sigma k^{3}(k+1)}$, we get that the total number of gradient computations at iteration $k$ is at most O\big{(}\sqrt{\frac{L}{\sigma}}\log^{2}(k)\big{)}:

Note that in updating $y_{k+1}$ in Eq. (29) and $x_{k+1}$ in Imp-STEP sub-routine, we were applying the principle of conceptual Mirror-Prox, where the update needs to satisfy some fixed point equation. This is critical in proving the above fast convergence rate.

For brevity, we define the following operations,

$x^{*}(y)$ is unique since $g(\cdot,y)$ is strongly convex. We first prove that, $x^{*}(y)$ is ${\frac{L}{\sigma}}$-Lipschitz continuous as follows.

where $(a)$ uses $\sigma$-strong convexity of $g(\cdot,y)$, $(b)$ and $(c)$ use the necessary first order optimality conditions for $x^{*}(y_{1})$ and $x^{*}(y_{2})$: $\left\langle\nabla_{x}g(x^{*}(y),y),x-x^{*}(y)\right\rangle\geq 0$, and $(d)$ uses Cauchy-Schwarz inequality and $L$-smoothness of $g$ (Definition 1). Next we prove that the operation $(\cdot)^{+}$ is a contraction as follows,

where $(a)$ uses triangle inequality, and $(b)$ uses (40) and 42, and $(c)$ uses (44) and the fact that $R=\lceil\log_{2}\frac{2D_{\mathcal{Y}}}{\varepsilon_{\rm mp}}\rceil$. Finally, we prove that $(x_{R},y_{R+1})$ satisfies (26).

where $(a)$ uses $L$-smoothness of $g(\cdot,y)$, $(b)$ uses necessary first order optimality condition: ${\left\langle\nabla_{x}g(x^{*}(y),y),x-x^{*}(y)\right\rangle=0}$ and (45), and $(c)$ uses $\varepsilon_{\rm mp}=\frac{2\sigma}{5L}\sqrt{\frac{2\varepsilon_{\rm step}}{L}}$.

Let the number of gradient computations done per iteration of Imp-STEP (a run of accelerated gradient ascent) be $T_{r}$ and $\kappa=\sqrt{L/\sigma}$. Then, from guarantee on AGD ([BG17, Eqn. (5.68)]), we get that,

where $\min_{y^{\prime}\in D_{\mathcal{Y}}}h(y^{\prime})$ is well-defined since $\mathcal{Y}$ is compact and $h$ is smooth (Lemma 2). This means that if we want $g(\hat{x}_{r},y_{r})-g(x^{*}(y_{r}),y_{r})\leq\varepsilon_{\rm agd}$, then required number of steps $T_{r}$ is at most,

B.5 Proof of Lemma 2

We know that $h(y)=\min_{x\in\mathcal{X}}g(x,y)$, where $g(\cdot,y)$ is $\sigma$-strongly convex, $g(x,\cdot)$ is concave, $g$ is $L$-smooth (Definition 1). Since $g(\cdot,y)$ is strongly convex, the minimizer $x^{*}(y)=\arg\min_{x\in\mathcal{X}}g(x,y)$ unique. Then by Danskin’s theorem [Ber09, Section 6.11], $h$ is differentiable and $\nabla h(y)=\nabla_{y}g(x^{*}(y),y)$. Then $h$ can be show to be smooth as follows,

where $(a)$ uses $L$-smoothness of $g$ and $(b)$ uses (40).

B.6 Proof of Corollary 1

be a quadratic approximation of the finite max-type function $f(x)$ at $x_{k}$. Then, $\widehat{f}(\cdot;x_{k})$ is $L$-strongly convex, since it is a maximum of convex functions and the quadratic term in (51) is independent of $i$.

Proof is similar to that of Theorem 2. We divide the analysis of each iteration of our algorithm into two cases.

Define $x_{k}^{*}$ as the point satisfying $x^{*}_{k}=\arg\min_{x}\widehat{f}(x;x_{k})$. By $L$-strong convexity of $\widehat{f}(\cdot,x_{k})$ (51), we prove that $x_{k}$ is close to $x_{k}^{*}$:

Putting these together we see that the total number of inner steps to reach $\varepsilon$-FOSP is,

B.7 Adaptive Prox-FDIAG algorithm

In this section, we provide the Adaptive Prox-FDIAG (Algorithm 6) to find an $\varepsilon$-FOSP of the finite max-type nonconvex minimax problem P3 with $L$-smooth components. Adaptive Prox-FDIAG is a variation of the Prox-FDIAG (Algorithm 5). Adaptive Prox-FDIAG uses Prox-FDIAG as a sub-routine and successively finds $\varepsilon^{\prime}$-FOSPs, for geometrically decreasing values of $\varepsilon^{\prime}$ starting from $\varepsilon_{0}$ ($\geq\varepsilon$) until $\varepsilon^{\prime}$ becomes equal to $\varepsilon$. It uses the $\varepsilon^{\prime}$-FOSP as the starting point to find an $\varepsilon^{\prime}/2$-FOSP. In the following corollary, we show that Adaptive Prox-FDIAG has the same the first-order oracle complexity (up to a $O(\log(\frac{1}{\varepsilon}))$ factor) as the Prox-FDIAG.

If the functional components $f_{i}(x)$’s are $G$-Lipschitz and $L$-smooth, and the optimal solution is bounded below, i.e. $f(x)\geq f^{*}>-\infty$, then after: K=\bigg{\lceil}\log_{2}\frac{\varepsilon_{0}}{\varepsilon}\bigg{\rceil} outer steps, Adaptive Prox-FDIAG outputs an $\varepsilon$-FOSP. The total first-order oracle complexity to find $\varepsilon$-FOSP is: \bigg{\lceil}\log_{2}\frac{\varepsilon_{0}}{\varepsilon}\bigg{\rceil}\bigg{\lceil}\frac{4^{4}L(f(x_{0})-f^{*})}{3\varepsilon^{2}}\bigg{\rceil}\cdot\bigg{\lceil}\frac{2^{4}G}{\varepsilon}(m\log^{3/2}m){{}{}}\bigg{\rceil}.

Notice that, each iteration of Adaptive Prox-FDIAG for finding an $\varepsilon^{\prime}$-FOSP, is a run of Prox-FDIAG (Algorithm 5), which has a maximum first-order oracle complexity of \bigg{\lceil}\frac{4^{4}L(f(x_{0})-f^{*})}{3\varepsilon^{2}}\bigg{\rceil}\cdot\bigg{\lceil}\frac{2^{4}G}{\varepsilon}(m\log^{3/2}m){{}{}}\bigg{\rceil} for finding an $\varepsilon^{\prime}$-FOSP (Corollary 1), as $\varepsilon\leq\varepsilon^{\prime}$. Further, since $\varepsilon^{\prime}$ starts at $\varepsilon_{0}$ and halves after each iteration until $\varepsilon^{\prime}$ becomes less than or equal to $\varepsilon$, the total number of outer iterations is K=\bigg{\lceil}\log_{2}\frac{\varepsilon_{0}}{\varepsilon}\bigg{\rceil}. ∎

Therefore, Adaptive Prox-FDIAG has the same first-order oracle complexity as Prox-FDIAG, up to a $O(\log(\frac{1}{\varepsilon}))$ factor. However, we observe that Adaptive Prox-FDIAG converges faster than Prox-FDIAG in our experiments.